formal
stringlengths 41
427k
| informal
stringclasses 1
value |
|---|---|
List.measurable_prod' ** M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l : List (α → M) hl : ∀ (f : α → M), f ∈ l → Measurable f ⊢ Measurable (prod l) ** induction' l with f l ihl ** case cons M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l✝ : List (α → M) hl✝ : ∀ (f : α → M), f ∈ l✝ → Measurable f f : α → M l : List (α → M) ihl : (∀ (f : α → M), f ∈ l → Measurable f) → Measurable (prod l) hl : ∀ (f_1 : α → M), f_1 ∈ f :: l → Measurable f_1 ⊢ Measurable (prod (f :: l)) ** rw [List.forall_mem_cons] at hl ** case cons M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l✝ : List (α → M) hl✝ : ∀ (f : α → M), f ∈ l✝ → Measurable f f : α → M l : List (α → M) ihl : (∀ (f : α → M), f ∈ l → Measurable f) → Measurable (prod l) hl : Measurable f ∧ ∀ (x : α → M), x ∈ l → Measurable x ⊢ Measurable (prod (f :: l)) ** rw [List.prod_cons] ** case cons M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l✝ : List (α → M) hl✝ : ∀ (f : α → M), f ∈ l✝ → Measurable f f : α → M l : List (α → M) ihl : (∀ (f : α → M), f ∈ l → Measurable f) → Measurable (prod l) hl : Measurable f ∧ ∀ (x : α → M), x ∈ l → Measurable x ⊢ Measurable (f * prod l) ** exact hl.1.mul (ihl hl.2) ** case nil M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l : List (α → M) hl✝ : ∀ (f : α → M), f ∈ l → Measurable f hl : ∀ (f : α → M), f ∈ [] → Measurable f ⊢ Measurable (prod []) ** exact measurable_one ** Qed
| |
List.measurable_prod ** M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l : List (α → M) hl : ∀ (f : α → M), f ∈ l → Measurable f ⊢ Measurable fun x => prod (map (fun f => f x) l) ** simpa only [← Pi.list_prod_apply] using l.measurable_prod' hl ** Qed
| |
List.aemeasurable_prod ** M : Type u_1 α : Type u_2 inst✝² : Monoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α l : List (α → M) hl : ∀ (f : α → M), f ∈ l → AEMeasurable f ⊢ AEMeasurable fun x => prod (map (fun f => f x) l) ** simpa only [← Pi.list_prod_apply] using l.aemeasurable_prod' hl ** Qed
| |
Multiset.measurable_prod' ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M l : Multiset (α → M) hl : ∀ (f : α → M), f ∈ l → Measurable f ⊢ Measurable (prod l) ** rcases l with ⟨l⟩ ** case mk M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M l✝ : Multiset (α → M) l : List (α → M) hl : ∀ (f : α → M), f ∈ Quot.mk Setoid.r l → Measurable f ⊢ Measurable (prod (Quot.mk Setoid.r l)) ** simpa using l.measurable_prod' (by simpa using hl) ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M l✝ : Multiset (α → M) l : List (α → M) hl : ∀ (f : α → M), f ∈ Quot.mk Setoid.r l → Measurable f ⊢ ∀ (f : α → M), f ∈ l → Measurable f ** simpa using hl ** Qed
| |
Multiset.aemeasurable_prod' ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M l : Multiset (α → M) hl : ∀ (f : α → M), f ∈ l → AEMeasurable f ⊢ AEMeasurable (prod l) ** rcases l with ⟨l⟩ ** case mk M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M l✝ : Multiset (α → M) l : List (α → M) hl : ∀ (f : α → M), f ∈ Quot.mk Setoid.r l → AEMeasurable f ⊢ AEMeasurable (prod (Quot.mk Setoid.r l)) ** simpa using l.aemeasurable_prod' (by simpa using hl) ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M l✝ : Multiset (α → M) l : List (α → M) hl : ∀ (f : α → M), f ∈ Quot.mk Setoid.r l → AEMeasurable f ⊢ ∀ (f : α → M), f ∈ l → AEMeasurable f ** simpa using hl ** Qed
| |
Multiset.measurable_prod ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M s : Multiset (α → M) hs : ∀ (f : α → M), f ∈ s → Measurable f ⊢ Measurable fun x => prod (map (fun f => f x) s) ** simpa only [← Pi.multiset_prod_apply] using s.measurable_prod' hs ** Qed
| |
Multiset.aemeasurable_prod ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M s : Multiset (α → M) hs : ∀ (f : α → M), f ∈ s → AEMeasurable f ⊢ AEMeasurable fun x => prod (map (fun f => f x) s) ** simpa only [← Pi.multiset_prod_apply] using s.aemeasurable_prod' hs ** Qed
| |
Finset.measurable_prod ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M s : Finset ι hf : ∀ (i : ι), i ∈ s → Measurable (f i) ⊢ Measurable fun a => ∏ i in s, f i a ** simpa only [← Finset.prod_apply] using s.measurable_prod' hf ** Qed
| |
Finset.aemeasurable_prod ** M : Type u_1 ι : Type u_2 α : Type u_3 inst✝² : CommMonoid M inst✝¹ : MeasurableSpace M inst✝ : MeasurableMul₂ M m : MeasurableSpace α μ : Measure α f : ι → α → M s : Finset ι hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) ⊢ AEMeasurable fun a => ∏ i in s, f i a ** simpa only [← Finset.prod_apply] using s.aemeasurable_prod' hf ** Qed
| |
VitaliFamily.FineSubfamilyOn.measure_le_tsum_of_absolutelyContinuous ** α : Type u_1 inst✝¹ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ f : α → Set (Set α) s : Set α h : FineSubfamilyOn v f s inst✝ : SecondCountableTopology α ρ : Measure α hρ : ρ ≪ μ ⊢ s ⊆ (s \ ⋃ p ∈ FineSubfamilyOn.index h, FineSubfamilyOn.covering h p) ∪ ⋃ p ∈ FineSubfamilyOn.index h, FineSubfamilyOn.covering h p ** simp only [subset_union_left, diff_union_self] ** α : Type u_1 inst✝¹ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ f : α → Set (Set α) s : Set α h : FineSubfamilyOn v f s inst✝ : SecondCountableTopology α ρ : Measure α hρ : ρ ≪ μ ⊢ ↑↑ρ (s \ ⋃ p ∈ FineSubfamilyOn.index h, FineSubfamilyOn.covering h p) + ↑↑ρ (⋃ p ∈ FineSubfamilyOn.index h, FineSubfamilyOn.covering h p) = ∑' (p : ↑(FineSubfamilyOn.index h)), ↑↑ρ (FineSubfamilyOn.covering h ↑p) ** rw [hρ h.measure_diff_biUnion, zero_add,
measure_biUnion h.index_countable h.covering_disjoint fun x hx => h.measurableSet_u hx] ** Qed
| |
VitaliFamily.mem_filterAt_iff ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α s : Set (Set α) ⊢ s ∈ filterAt v x ↔ ∃ ε, ε > 0 ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s ** simp only [filterAt, exists_prop, gt_iff_lt] ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α s : Set (Set α) ⊢ s ∈ ⨅ ε ∈ Ioi 0, 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x ε} ↔ ∃ ε, 0 < ε ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s ** rw [mem_biInf_of_directed] ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α s : Set (Set α) ⊢ (∃ i, i ∈ Ioi 0 ∧ s ∈ 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x i}) ↔ ∃ ε, 0 < ε ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s ** simp only [subset_def, and_imp, exists_prop, mem_sep_iff, mem_Ioi, mem_principal] ** case h α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α s : Set (Set α) ⊢ DirectedOn ((fun ε => 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x ε}) ⁻¹'o fun x x_1 => x ≥ x_1) (Ioi 0) ** simp only [DirectedOn, exists_prop, ge_iff_le, le_principal_iff, mem_Ioi, Order.Preimage,
mem_principal] ** case h α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α s : Set (Set α) ⊢ ∀ (x_1 : ℝ), 0 < x_1 → ∀ (y : ℝ), 0 < y → ∃ z, 0 < z ∧ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x z} ⊆ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x x_1} ∧ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x z} ⊆ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x y} ** intro x hx y hy ** case h α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x✝ : α s : Set (Set α) x : ℝ hx : 0 < x y : ℝ hy : 0 < y ⊢ ∃ z, 0 < z ∧ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ z} ⊆ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ x} ∧ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ z} ⊆ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ y} ** refine' ⟨min x y, lt_min hx hy,
fun a ha => ⟨ha.1, ha.2.trans (closedBall_subset_closedBall (min_le_left _ _))⟩,
fun a ha => ⟨ha.1, ha.2.trans (closedBall_subset_closedBall (min_le_right _ _))⟩⟩ ** case ne α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α s : Set (Set α) ⊢ Set.Nonempty (Ioi 0) ** exact ⟨(1 : ℝ), mem_Ioi.2 zero_lt_one⟩ ** Qed
| |
VitaliFamily.eventually_filterAt_mem_sets ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α ⊢ ∀ᶠ (a : Set α) in filterAt v x, a ∈ setsAt v x ** simp (config := { contextual := true }) only [eventually_filterAt_iff, exists_prop, and_true_iff,
gt_iff_lt, imp_true_iff] ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α ⊢ ∃ ε, 0 < ε ** exact ⟨1, zero_lt_one⟩ ** Qed
| |
VitaliFamily.eventually_filterAt_subset_closedBall ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α ε : ℝ hε : 0 < ε ⊢ ∀ᶠ (a : Set α) in filterAt v x, a ⊆ closedBall x ε ** simp only [v.eventually_filterAt_iff] ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α ε : ℝ hε : 0 < ε ⊢ ∃ ε_1, ε_1 > 0 ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε_1 → a ⊆ closedBall x ε ** exact ⟨ε, hε, fun a _ ha' => ha'⟩ ** Qed
| |
VitaliFamily.tendsto_filterAt_iff ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ ι : Type u_2 l : Filter ι f : ι → Set α x : α ⊢ Tendsto f l (filterAt v x) ↔ (∀ᶠ (i : ι) in l, f i ∈ setsAt v x) ∧ ∀ (ε : ℝ), ε > 0 → ∀ᶠ (i : ι) in l, f i ⊆ closedBall x ε ** refine' ⟨fun H => ⟨H.eventually <| v.eventually_filterAt_mem_sets x,
fun ε hε => H.eventually <| v.eventually_filterAt_subset_closedBall x hε⟩,
fun H s hs => (_ : ∀ᶠ i in l, f i ∈ s)⟩ ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ ι : Type u_2 l : Filter ι f : ι → Set α x : α H : (∀ᶠ (i : ι) in l, f i ∈ setsAt v x) ∧ ∀ (ε : ℝ), ε > 0 → ∀ᶠ (i : ι) in l, f i ⊆ closedBall x ε s : Set (Set α) hs : s ∈ filterAt v x ⊢ ∀ᶠ (i : ι) in l, f i ∈ s ** obtain ⟨ε, εpos, hε⟩ := v.mem_filterAt_iff.mp hs ** case intro.intro α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ ι : Type u_2 l : Filter ι f : ι → Set α x : α H : (∀ᶠ (i : ι) in l, f i ∈ setsAt v x) ∧ ∀ (ε : ℝ), ε > 0 → ∀ᶠ (i : ι) in l, f i ⊆ closedBall x ε s : Set (Set α) hs : s ∈ filterAt v x ε : ℝ εpos : ε > 0 hε : ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s ⊢ ∀ᶠ (i : ι) in l, f i ∈ s ** filter_upwards [H.1, H.2 ε εpos] with i hi hiε using hε _ hi hiε ** Qed
| |
VitaliFamily.eventually_filterAt_measurableSet ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α ⊢ ∀ᶠ (a : Set α) in filterAt v x, MeasurableSet a ** filter_upwards [v.eventually_filterAt_mem_sets x] with _ ha using v.MeasurableSet' _ _ ha ** Qed
| |
VitaliFamily.frequently_filterAt_iff ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α P : Set α → Prop ⊢ (∃ᶠ (a : Set α) in filterAt v x, P a) ↔ ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ setsAt v x ∧ a ⊆ closedBall x ε ∧ P a ** simp only [Filter.Frequently, eventually_filterAt_iff, not_exists, exists_prop, not_and,
Classical.not_not, not_forall] ** Qed
| |
VitaliFamily.eventually_filterAt_subset_of_nhds ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α o : Set α hx : o ∈ 𝓝 x ⊢ ∀ᶠ (a : Set α) in filterAt v x, a ⊆ o ** rw [eventually_filterAt_iff] ** α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α o : Set α hx : o ∈ 𝓝 x ⊢ ∃ ε, ε > 0 ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ⊆ o ** rcases Metric.mem_nhds_iff.1 hx with ⟨ε, εpos, hε⟩ ** case intro.intro α : Type u_1 inst✝ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ x : α o : Set α hx : o ∈ 𝓝 x ε : ℝ εpos : ε > 0 hε : ball x ε ⊆ o ⊢ ∃ ε, ε > 0 ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ⊆ o ** exact ⟨ε / 2, half_pos εpos,
fun a _ ha => ha.trans ((closedBall_subset_ball (half_lt_self εpos)).trans hε)⟩ ** Qed
| |
MeasureTheory.JordanDecomposition.coe_smul ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ≥0 ⊢ ↑r • j = r • j ** rw [real_smul_def, if_pos (NNReal.coe_nonneg r), Real.toNNReal_coe] ** Qed
| |
MeasureTheory.JordanDecomposition.real_smul_posPart_nonneg ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ hr : 0 ≤ r ⊢ (r • j).posPart = Real.toNNReal r • j.posPart ** rw [real_smul_def, ← smul_posPart, if_pos hr] ** Qed
| |
MeasureTheory.JordanDecomposition.real_smul_negPart_nonneg ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ hr : 0 ≤ r ⊢ (r • j).negPart = Real.toNNReal r • j.negPart ** rw [real_smul_def, ← smul_negPart, if_pos hr] ** Qed
| |
MeasureTheory.JordanDecomposition.real_smul_posPart_neg ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ hr : r < 0 ⊢ (r • j).posPart = Real.toNNReal (-r) • j.negPart ** rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart] ** Qed
| |
MeasureTheory.JordanDecomposition.real_smul_negPart_neg ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ hr : r < 0 ⊢ (r • j).negPart = Real.toNNReal (-r) • j.posPart ** rw [real_smul_def, ← smul_posPart, if_neg (not_le.2 hr), neg_negPart] ** Qed
| |
MeasureTheory.JordanDecomposition.toSignedMeasure_neg ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α ⊢ toSignedMeasure (-j) = -toSignedMeasure j ** ext1 i hi ** case h α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α i : Set α hi : MeasurableSet i ⊢ ↑(toSignedMeasure (-j)) i = ↑(-toSignedMeasure j) i ** rw [neg_apply, toSignedMeasure, toSignedMeasure, toSignedMeasure_sub_apply hi,
toSignedMeasure_sub_apply hi, neg_sub, neg_posPart, neg_negPart] ** Qed
| |
MeasureTheory.JordanDecomposition.toSignedMeasure_smul ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ≥0 ⊢ toSignedMeasure (r • j) = r • toSignedMeasure j ** ext1 i hi ** case h α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α r : ℝ≥0 i : Set α hi : MeasurableSet i ⊢ ↑(toSignedMeasure (r • j)) i = ↑(r • toSignedMeasure j) i ** rw [VectorMeasure.smul_apply, toSignedMeasure, toSignedMeasure,
toSignedMeasure_sub_apply hi, toSignedMeasure_sub_apply hi, smul_sub, smul_posPart,
smul_negPart, ← ENNReal.toReal_smul, ← ENNReal.toReal_smul, smul_toOuterMeasure,
OuterMeasure.coe_smul, Pi.smul_apply, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply] ** Qed
| |
MeasureTheory.JordanDecomposition.exists_compl_positive_negative ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α ⊢ ∃ S, MeasurableSet S ∧ VectorMeasure.restrict (toSignedMeasure j) S ≤ VectorMeasure.restrict 0 S ∧ VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict (toSignedMeasure j) Sᶜ ∧ ↑↑j.posPart S = 0 ∧ ↑↑j.negPart Sᶜ = 0 ** obtain ⟨S, hS₁, hS₂, hS₃⟩ := j.mutuallySingular ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 ⊢ ∃ S, MeasurableSet S ∧ VectorMeasure.restrict (toSignedMeasure j) S ≤ VectorMeasure.restrict 0 S ∧ VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict (toSignedMeasure j) Sᶜ ∧ ↑↑j.posPart S = 0 ∧ ↑↑j.negPart Sᶜ = 0 ** refine' ⟨S, hS₁, _, _, hS₂, hS₃⟩ ** case intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 ⊢ VectorMeasure.restrict (toSignedMeasure j) S ≤ VectorMeasure.restrict 0 S ** refine' restrict_le_restrict_of_subset_le _ _ fun A hA hA₁ => _ ** case intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 A : Set α hA : MeasurableSet A hA₁ : A ⊆ S ⊢ ↑(toSignedMeasure j) A ≤ ↑0 A ** rw [toSignedMeasure, toSignedMeasure_sub_apply hA,
show j.posPart A = 0 from nonpos_iff_eq_zero.1 (hS₂ ▸ measure_mono hA₁), ENNReal.zero_toReal,
zero_sub, neg_le, zero_apply, neg_zero] ** case intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 A : Set α hA : MeasurableSet A hA₁ : A ⊆ S ⊢ 0 ≤ ENNReal.toReal (↑↑j.negPart A) ** exact ENNReal.toReal_nonneg ** case intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 ⊢ VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict (toSignedMeasure j) Sᶜ ** refine' restrict_le_restrict_of_subset_le _ _ fun A hA hA₁ => _ ** case intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 A : Set α hA : MeasurableSet A hA₁ : A ⊆ Sᶜ ⊢ ↑0 A ≤ ↑(toSignedMeasure j) A ** rw [toSignedMeasure, toSignedMeasure_sub_apply hA,
show j.negPart A = 0 from nonpos_iff_eq_zero.1 (hS₃ ▸ measure_mono hA₁), ENNReal.zero_toReal,
sub_zero] ** case intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑j.posPart S = 0 hS₃ : ↑↑j.negPart Sᶜ = 0 A : Set α hA : MeasurableSet A hA₁ : A ⊆ Sᶜ ⊢ ↑0 A ≤ ENNReal.toReal (↑↑j.posPart A) ** exact ENNReal.toReal_nonneg ** Qed
| |
MeasureTheory.SignedMeasure.toJordanDecomposition_spec ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α ⊢ ∃ i hi₁ hi₂ hi₃, (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ ∧ (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ** set i := choose s.exists_compl_positive_negative ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α := choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ) ⊢ ∃ i hi₁ hi₂ hi₃, (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ ∧ (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ** obtain ⟨hi₁, hi₂, hi₃⟩ := choose_spec s.exists_compl_positive_negative ** case intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α := choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ) hi₁ : MeasurableSet (choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ)) hi₂ : VectorMeasure.restrict 0 (choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ)) ≤ VectorMeasure.restrict s (choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ)) hi₃ : VectorMeasure.restrict s (choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ))ᶜ ≤ VectorMeasure.restrict 0 (choose (_ : ∃ i, MeasurableSet i ∧ VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i ∧ VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ))ᶜ ⊢ ∃ i hi₁ hi₂ hi₃, (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ ∧ (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ** exact ⟨i, hi₁, hi₂, hi₃, rfl, rfl⟩ ** Qed
| |
MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α ⊢ JordanDecomposition.toSignedMeasure (toJordanDecomposition s) = s ** obtain ⟨i, hi₁, hi₂, hi₃, hμ, hν⟩ := s.toJordanDecomposition_spec ** case intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hμ : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hν : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ JordanDecomposition.toSignedMeasure (toJordanDecomposition s) = s ** simp only [JordanDecomposition.toSignedMeasure, hμ, hν] ** case intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hμ : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hν : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ Measure.toSignedMeasure (toMeasureOfZeroLE s i hi₁ hi₂) - Measure.toSignedMeasure (toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃) = s ** ext k hk ** case intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hμ : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hν : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ k : Set α hk : MeasurableSet k ⊢ ↑(Measure.toSignedMeasure (toMeasureOfZeroLE s i hi₁ hi₂) - Measure.toSignedMeasure (toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃)) k = ↑s k ** rw [toSignedMeasure_sub_apply hk, toMeasureOfZeroLE_apply _ hi₂ hi₁ hk,
toMeasureOfLEZero_apply _ hi₃ hi₁.compl hk] ** case intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hμ : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hν : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ k : Set α hk : MeasurableSet k ⊢ ENNReal.toReal ↑{ val := ↑s (i ∩ k), property := (_ : 0 ≤ ↑s (i ∩ k)) } - ENNReal.toReal ↑{ val := -↑s (iᶜ ∩ k), property := (_ : 0 ≤ -↑s (iᶜ ∩ k)) } = ↑s k ** simp only [ENNReal.coe_toReal, NNReal.coe_mk, ENNReal.some_eq_coe, sub_neg_eq_add] ** case intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hμ : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hν : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ k : Set α hk : MeasurableSet k ⊢ ↑s (i ∩ k) + ↑s (iᶜ ∩ k) = ↑s k ** rw [← of_union _ (MeasurableSet.inter hi₁ hk) (MeasurableSet.inter hi₁.compl hk),
Set.inter_comm i, Set.inter_comm iᶜ, Set.inter_union_compl _ _] ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s✝ : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν s : SignedMeasure α i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hμ : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hν : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ k : Set α hk : MeasurableSet k ⊢ Disjoint (i ∩ k) (iᶜ ∩ k) ** exact (disjoint_compl_right.inf_left _).inf_right _ ** Qed
| |
MeasureTheory.SignedMeasure.subset_positive_null_set ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w ⊢ ↑s v = 0 ** have : s v + s (w \ v) = 0 := by
rw [← hw₁, ← of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,
Set.union_eq_self_of_subset_left hwt] ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w this : ↑s v + ↑s (w \ v) = 0 ⊢ ↑s v = 0 ** have h₁ := nonneg_of_zero_le_restrict _ (restrict_le_restrict_subset _ _ hu hsu (hwt.trans hw₂)) ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w this : ↑s v + ↑s (w \ v) = 0 h₁ : 0 ≤ ↑s v ⊢ ↑s v = 0 ** have h₂ :=
nonneg_of_zero_le_restrict _
(restrict_le_restrict_subset _ _ hu hsu ((w.diff_subset v).trans hw₂)) ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w this : ↑s v + ↑s (w \ v) = 0 h₁ : 0 ≤ ↑s v h₂ : 0 ≤ ↑s (w \ v) ⊢ ↑s v = 0 ** linarith ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w ⊢ ↑s v + ↑s (w \ v) = 0 ** rw [← hw₁, ← of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,
Set.union_eq_self_of_subset_left hwt] ** Qed
| |
MeasureTheory.SignedMeasure.subset_negative_null_set ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict s u ≤ VectorMeasure.restrict 0 u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w ⊢ ↑s v = 0 ** rw [← s.neg_le_neg_iff _ hu, neg_zero] at hsu ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w ⊢ ↑s v = 0 ** have := subset_positive_null_set hu hv hw hsu ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w this : ↑(-s) w = 0 → w ⊆ u → v ⊆ w → ↑(-s) v = 0 ⊢ ↑s v = 0 ** simp only [Pi.neg_apply, neg_eq_zero, coe_neg] at this ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hw₁ : ↑s w = 0 hw₂ : w ⊆ u hwt : v ⊆ w this : ↑s w = 0 → w ⊆ u → v ⊆ w → ↑s v = 0 ⊢ ↑s v = 0 ** exact this hw₁ hw₂ hwt ** Qed
| |
MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ** rw [restrict_le_restrict_iff] at hsu hsv ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u ∆ v) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet v case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** have a := hsu (hu.diff hv) (u.diff_subset v) ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u ∆ v) = 0 a : ↑0 (u \ v) ≤ ↑s (u \ v) ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet v case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** have b := hsv (hv.diff hu) (v.diff_subset u) ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u ∆ v) = 0 a : ↑0 (u \ v) ≤ ↑s (u \ v) b : ↑0 (v \ u) ≤ ↑s (v \ u) ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet v case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** erw [of_union (Set.disjoint_of_subset_left (u.diff_subset v) disjoint_sdiff_self_right)
(hu.diff hv) (hv.diff hu)] at hs ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u \ v) + ↑s (v \ u) = 0 a : ↑0 (u \ v) ≤ ↑s (u \ v) b : ↑0 (v \ u) ≤ ↑s (v \ u) ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet v case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** rw [zero_apply] at a b ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u \ v) + ↑s (v \ u) = 0 a : 0 ≤ ↑s (u \ v) b : 0 ≤ ↑s (v \ u) ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet v case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** constructor ** case left α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u \ v) + ↑s (v \ u) = 0 a : 0 ≤ ↑s (u \ v) b : 0 ≤ ↑s (v \ u) ⊢ ↑s (u \ v) = 0 case right α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u \ v) + ↑s (v \ u) = 0 a : 0 ≤ ↑s (u \ v) b : 0 ≤ ↑s (v \ u) ⊢ ↑s (v \ u) = 0 case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet v case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** all_goals first | linarith | infer_instance | assumption ** case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** first | linarith | infer_instance | assumption ** case right α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j hs : ↑s (u \ v) + ↑s (v \ u) = 0 a : 0 ≤ ↑s (u \ v) b : 0 ≤ ↑s (v \ u) ⊢ ↑s (v \ u) = 0 ** linarith ** case hi α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v hs : ↑s (u ∆ v) = 0 ⊢ MeasurableSet u ** assumption ** Qed
| |
MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_negative ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict s u ≤ VectorMeasure.restrict 0 u hsv : VectorMeasure.restrict s v ≤ VectorMeasure.restrict 0 v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ** rw [← s.neg_le_neg_iff _ hu, neg_zero] at hsu ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict s v ≤ VectorMeasure.restrict 0 v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ** rw [← s.neg_le_neg_iff _ hv, neg_zero] at hsv ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict (-s) v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ** have := of_diff_eq_zero_of_symmDiff_eq_zero_positive hu hv hsu hsv ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict (-s) v hs : ↑s (u ∆ v) = 0 this : ↑(-s) (u ∆ v) = 0 → ↑(-s) (u \ v) = 0 ∧ ↑(-s) (v \ u) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ** simp only [Pi.neg_apply, neg_eq_zero, coe_neg] at this ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict (-s) v hs : ↑s (u ∆ v) = 0 this : ↑s (u ∆ v) = 0 → ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 ** exact this hs ** Qed
| |
MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_negative ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict s u ≤ VectorMeasure.restrict 0 u hsv : VectorMeasure.restrict s v ≤ VectorMeasure.restrict 0 v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (w ∩ u) = ↑s (w ∩ v) ** rw [← s.neg_le_neg_iff _ hu, neg_zero] at hsu ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict s v ≤ VectorMeasure.restrict 0 v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (w ∩ u) = ↑s (w ∩ v) ** rw [← s.neg_le_neg_iff _ hv, neg_zero] at hsv ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict (-s) v hs : ↑s (u ∆ v) = 0 ⊢ ↑s (w ∩ u) = ↑s (w ∩ v) ** have := of_inter_eq_of_symmDiff_eq_zero_positive hu hv hw hsu hsv ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict (-s) v hs : ↑s (u ∆ v) = 0 this : ↑(-s) (u ∆ v) = 0 → ↑(-s) (w ∩ u) = ↑(-s) (w ∩ v) ⊢ ↑s (w ∩ u) = ↑s (w ∩ v) ** simp only [Pi.neg_apply, neg_inj, neg_eq_zero, coe_neg] at this ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α s : SignedMeasure α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν u v w : Set α hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (-s) u hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict (-s) v hs : ↑s (u ∆ v) = 0 this : ↑s (u ∆ v) = 0 → ↑s (w ∩ u) = ↑s (w ∩ v) ⊢ ↑s (w ∩ u) = ↑s (w ∩ v) ** exact this hs ** Qed
| |
MeasureTheory.JordanDecomposition.toJordanDecomposition_toSignedMeasure ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α j : JordanDecomposition α ⊢ toSignedMeasure j = toSignedMeasure (toJordanDecomposition (toSignedMeasure j)) ** simp ** Qed
| |
MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ ⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s ** by_cases hr : 0 ≤ r ** case pos α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : 0 ≤ r ⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s ** exact toJordanDecomposition_smul_real_nonneg s r hr ** case neg α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s ** ext1 ** case neg.posPart α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ (toJordanDecomposition (r • s)).posPart = (r • toJordanDecomposition s).posPart ** rw [real_smul_posPart_neg _ _ (not_le.1 hr),
show r • s = -(-r • s) by rw [neg_smul, neg_neg], toJordanDecomposition_neg, neg_posPart,
toJordanDecomposition_smul_real_nonneg, ← smul_negPart, real_smul_nonneg] ** case neg.posPart.hr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ 0 ≤ -r case neg.posPart.hr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ 0 ≤ -r ** all_goals exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr)) ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ r • s = -(-r • s) ** rw [neg_smul, neg_neg] ** case neg.posPart.hr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ 0 ≤ -r ** exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr)) ** case neg.negPart α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ (toJordanDecomposition (r • s)).negPart = (r • toJordanDecomposition s).negPart ** rw [real_smul_negPart_neg _ _ (not_le.1 hr),
show r • s = -(-r • s) by rw [neg_smul, neg_neg], toJordanDecomposition_neg, neg_negPart,
toJordanDecomposition_smul_real_nonneg, ← smul_posPart, real_smul_nonneg] ** case neg.negPart.hr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ 0 ≤ -r case neg.negPart.hr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ 0 ≤ -r ** all_goals exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr)) ** case neg.negPart.hr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α r : ℝ hr : ¬0 ≤ r ⊢ 0 ≤ -r ** exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr)) ** Qed
| |
MeasureTheory.SignedMeasure.totalVariation_neg ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α ⊢ totalVariation (-s) = totalVariation s ** simp [totalVariation, toJordanDecomposition_neg, add_comm] ** Qed
| |
MeasureTheory.SignedMeasure.totalVariation_absolutelyContinuous_iff ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α ⊢ totalVariation s ≪ μ ↔ (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ ** constructor <;> intro h ** case mp α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ ⊢ (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ ** constructor ** case mp.left α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ ⊢ (toJordanDecomposition s).posPart ≪ μ case mp.right α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ ⊢ (toJordanDecomposition s).negPart ≪ μ ** all_goals
refine' Measure.AbsolutelyContinuous.mk fun S _ hS₂ => _
have := h hS₂
rw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this ** case mp.left α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ S : Set α x✝ : MeasurableSet S hS₂ : ↑↑μ S = 0 this : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0 ⊢ ↑↑(toJordanDecomposition s).posPart S = 0 case mp.right α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ S : Set α x✝ : MeasurableSet S hS₂ : ↑↑μ S = 0 this : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0 ⊢ ↑↑(toJordanDecomposition s).negPart S = 0 ** exacts [this.1, this.2] ** case mp.right α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ ⊢ (toJordanDecomposition s).negPart ≪ μ ** refine' Measure.AbsolutelyContinuous.mk fun S _ hS₂ => _ ** case mp.right α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ S : Set α x✝ : MeasurableSet S hS₂ : ↑↑μ S = 0 ⊢ ↑↑(toJordanDecomposition s).negPart S = 0 ** have := h hS₂ ** case mp.right α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : totalVariation s ≪ μ S : Set α x✝ : MeasurableSet S hS₂ : ↑↑μ S = 0 this : ↑↑(totalVariation s) S = 0 ⊢ ↑↑(toJordanDecomposition s).negPart S = 0 ** rw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this ** case mpr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ ⊢ totalVariation s ≪ μ ** refine' Measure.AbsolutelyContinuous.mk fun S _ hS₂ => _ ** case mpr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : Measure α h : (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ S : Set α x✝ : MeasurableSet S hS₂ : ↑↑μ S = 0 ⊢ ↑↑(totalVariation s) S = 0 ** rw [totalVariation, Measure.add_apply, h.1 hS₂, h.2 hS₂, add_zero] ** Qed
| |
MeasureTheory.SignedMeasure.mutuallySingular_iff ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α ⊢ s ⟂ᵥ t ↔ totalVariation s ⟂ₘ totalVariation t ** constructor ** case mp α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α ⊢ s ⟂ᵥ t → totalVariation s ⟂ₘ totalVariation t ** rintro ⟨u, hmeas, hu₁, hu₂⟩ ** case mp.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 ⊢ totalVariation s ⟂ₘ totalVariation t ** obtain ⟨i, hi₁, hi₂, hi₃, hipos, hineg⟩ := s.toJordanDecomposition_spec ** case mp.intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ totalVariation s ⟂ₘ totalVariation t ** obtain ⟨j, hj₁, hj₂, hj₃, hjpos, hjneg⟩ := t.toJordanDecomposition_spec ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ j : Set α hj₁ : MeasurableSet j hj₂ : VectorMeasure.restrict 0 j ≤ VectorMeasure.restrict t j hj₃ : VectorMeasure.restrict t jᶜ ≤ VectorMeasure.restrict 0 jᶜ hjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj₁ hj₂ hjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t jᶜ (_ : MeasurableSet jᶜ) hj₃ ⊢ totalVariation s ⟂ₘ totalVariation t ** refine' ⟨u, hmeas, _, _⟩ ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ j : Set α hj₁ : MeasurableSet j hj₂ : VectorMeasure.restrict 0 j ≤ VectorMeasure.restrict t j hj₃ : VectorMeasure.restrict t jᶜ ≤ VectorMeasure.restrict 0 jᶜ hjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj₁ hj₂ hjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t jᶜ (_ : MeasurableSet jᶜ) hj₃ ⊢ ↑↑(totalVariation s) u = 0 ** rw [totalVariation, Measure.add_apply, hipos, hineg, toMeasureOfZeroLE_apply _ _ _ hmeas,
toMeasureOfLEZero_apply _ _ _ hmeas] ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ j : Set α hj₁ : MeasurableSet j hj₂ : VectorMeasure.restrict 0 j ≤ VectorMeasure.restrict t j hj₃ : VectorMeasure.restrict t jᶜ ≤ VectorMeasure.restrict 0 jᶜ hjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj₁ hj₂ hjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t jᶜ (_ : MeasurableSet jᶜ) hj₃ ⊢ ↑{ val := ↑s (i ∩ u), property := (_ : 0 ≤ ↑s (i ∩ u)) } + ↑{ val := -↑s (iᶜ ∩ u), property := (_ : 0 ≤ -↑s (iᶜ ∩ u)) } = 0 ** simp [hu₁ _ (Set.inter_subset_right _ _), ← NNReal.eq_iff] ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ j : Set α hj₁ : MeasurableSet j hj₂ : VectorMeasure.restrict 0 j ≤ VectorMeasure.restrict t j hj₃ : VectorMeasure.restrict t jᶜ ≤ VectorMeasure.restrict 0 jᶜ hjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj₁ hj₂ hjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t jᶜ (_ : MeasurableSet jᶜ) hj₃ ⊢ ↑↑(totalVariation t) uᶜ = 0 ** rw [totalVariation, Measure.add_apply, hjpos, hjneg,
toMeasureOfZeroLE_apply _ _ _ hmeas.compl,
toMeasureOfLEZero_apply _ _ _ hmeas.compl] ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t_1 : Set α), t_1 ⊆ uᶜ → ↑t t_1 = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ j : Set α hj₁ : MeasurableSet j hj₂ : VectorMeasure.restrict 0 j ≤ VectorMeasure.restrict t j hj₃ : VectorMeasure.restrict t jᶜ ≤ VectorMeasure.restrict 0 jᶜ hjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj₁ hj₂ hjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t jᶜ (_ : MeasurableSet jᶜ) hj₃ ⊢ ↑{ val := ↑t (j ∩ uᶜ), property := (_ : 0 ≤ ↑t (j ∩ uᶜ)) } + ↑{ val := -↑t (jᶜ ∩ uᶜ), property := (_ : 0 ≤ -↑t (jᶜ ∩ uᶜ)) } = 0 ** simp [hu₂ _ (Set.inter_subset_right _ _), ← NNReal.eq_iff] ** case mpr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α ⊢ totalVariation s ⟂ₘ totalVariation t → s ⟂ᵥ t ** rintro ⟨u, hmeas, hu₁, hu₂⟩ ** case mpr.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s t : SignedMeasure α u : Set α hmeas : MeasurableSet u hu₁ : ↑↑(totalVariation s) u = 0 hu₂ : ↑↑(totalVariation t) uᶜ = 0 ⊢ s ⟂ᵥ t ** exact
⟨u, hmeas, fun t htu => null_of_totalVariation_zero _ (measure_mono_null htu hu₁),
fun t htv => null_of_totalVariation_zero _ (measure_mono_null htv hu₂)⟩ ** Qed
| |
MeasureTheory.SignedMeasure.mutuallySingular_ennreal_iff ** α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ ⊢ s ⟂ᵥ μ ↔ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ ** constructor ** case mp α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ ⊢ s ⟂ᵥ μ → totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ ** rintro ⟨u, hmeas, hu₁, hu₂⟩ ** case mp.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0 ⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ ** obtain ⟨i, hi₁, hi₂, hi₃, hpos, hneg⟩ := s.toJordanDecomposition_spec ** case mp.intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ ** refine' ⟨u, hmeas, _, _⟩ ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ ↑↑(totalVariation s) u = 0 ** rw [totalVariation, Measure.add_apply, hpos, hneg, toMeasureOfZeroLE_apply _ _ _ hmeas,
toMeasureOfLEZero_apply _ _ _ hmeas] ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ ↑{ val := ↑s (i ∩ u), property := (_ : 0 ≤ ↑s (i ∩ u)) } + ↑{ val := -↑s (iᶜ ∩ u), property := (_ : 0 ≤ -↑s (iᶜ ∩ u)) } = 0 ** simp [hu₁ _ (Set.inter_subset_right _ _), ← NNReal.eq_iff] ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ ↑↑(VectorMeasure.ennrealToMeasure μ) uᶜ = 0 ** rw [VectorMeasure.ennrealToMeasure_apply hmeas.compl] ** case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0 hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0 i : Set α hi₁ : MeasurableSet i hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂ hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s iᶜ (_ : MeasurableSet iᶜ) hi₃ ⊢ ↑μ uᶜ = 0 ** exact hu₂ _ (Set.Subset.refl _) ** case mpr α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ ⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ → s ⟂ᵥ μ ** rintro ⟨u, hmeas, hu₁, hu₂⟩ ** case mpr.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ↑↑(totalVariation s) u = 0 hu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) uᶜ = 0 ⊢ s ⟂ᵥ μ ** refine'
VectorMeasure.MutuallySingular.mk u hmeas
(fun t htu _ => null_of_totalVariation_zero _ (measure_mono_null htu hu₁)) fun t htv hmt =>
_ ** case mpr.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ↑↑(totalVariation s) u = 0 hu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) uᶜ = 0 t : Set α htv : t ⊆ uᶜ hmt : MeasurableSet t ⊢ ↑μ t = 0 ** rw [← VectorMeasure.ennrealToMeasure_apply hmt] ** case mpr.intro.intro.intro α : Type u_1 β : Type u_2 inst✝ : MeasurableSpace α s : SignedMeasure α μ : VectorMeasure α ℝ≥0∞ u : Set α hmeas : MeasurableSet u hu₁ : ↑↑(totalVariation s) u = 0 hu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) uᶜ = 0 t : Set α htv : t ⊆ uᶜ hmt : MeasurableSet t ⊢ ↑↑(VectorMeasure.ennrealToMeasure μ) t = 0 ** exact measure_mono_null htv hu₂ ** Qed
| |
MeasureTheory.IsFundamentalDomain.mk' ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁶ : Group G inst✝⁵ : Group H inst✝⁴ : MulAction G α inst✝³ : MeasurableSpace α inst✝² : MulAction H β inst✝¹ : MeasurableSpace β inst✝ : NormedAddCommGroup E s t : Set α μ : Measure α h_meas : NullMeasurableSet s h_exists : ∀ (x : α), ∃! g, g • x ∈ s a b : G hab : a ≠ b x : α hxa : x ∈ (fun g => g • s) a hxb : x ∈ (fun g => g • s) b ⊢ False ** rw [mem_smul_set_iff_inv_smul_mem] at hxa hxb ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁶ : Group G inst✝⁵ : Group H inst✝⁴ : MulAction G α inst✝³ : MeasurableSpace α inst✝² : MulAction H β inst✝¹ : MeasurableSpace β inst✝ : NormedAddCommGroup E s t : Set α μ : Measure α h_meas : NullMeasurableSet s h_exists : ∀ (x : α), ∃! g, g • x ∈ s a b : G hab : a ≠ b x : α hxa : a⁻¹ • x ∈ s hxb : b⁻¹ • x ∈ s ⊢ False ** exact hab (inv_injective <| (h_exists x).unique hxa hxb) ** Qed
| |
MeasureTheory.IsFundamentalDomain.mk_of_measure_univ_le ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_meas : NullMeasurableSet s h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) ⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s ** replace h_meas : ∀ g : G, NullMeasurableSet (g • s) μ := fun g => by
rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹) ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) ⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s ** have h_meas' : NullMeasurableSet {a | ∃ g : G, g • a ∈ s} μ := by
rw [← iUnion_smul_eq_setOf_exists]; exact .iUnion h_meas ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) h_meas' : NullMeasurableSet {a | ∃ g, g • a ∈ s} ⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s ** rw [ae_iff_measure_eq h_meas', ← iUnion_smul_eq_setOf_exists] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) h_meas' : NullMeasurableSet {a | ∃ g, g • a ∈ s} ⊢ ↑↑μ (⋃ g, g • s) = ↑↑μ univ ** refine' le_antisymm (measure_mono <| subset_univ _) _ ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) h_meas' : NullMeasurableSet {a | ∃ g, g • a ∈ s} ⊢ ↑↑μ univ ≤ ↑↑μ (⋃ g, g • s) ** rw [measure_iUnion₀ aedisjoint h_meas] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) h_meas' : NullMeasurableSet {a | ∃ g, g • a ∈ s} ⊢ ↑↑μ univ ≤ ∑' (i : G), ↑↑μ (i • s) ** exact h_measure_univ_le ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_meas : NullMeasurableSet s h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) g : G ⊢ NullMeasurableSet (g • s) ** rw [← inv_inv g, ← preimage_smul] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_meas : NullMeasurableSet s h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) g : G ⊢ NullMeasurableSet ((fun x => g⁻¹ • x) ⁻¹' s) ** exact h_meas.preimage (h_qmp g⁻¹) ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) ⊢ NullMeasurableSet {a | ∃ g, g • a ∈ s} ** rw [← iUnion_smul_eq_setOf_exists] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁸ : Group G inst✝⁷ : Group H inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace α inst✝⁴ : MulAction H β inst✝³ : MeasurableSpace β inst✝² : NormedAddCommGroup E s t : Set α μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : Countable G h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s h_qmp : ∀ (g : G), QuasiMeasurePreserving ((fun x x_1 => x • x_1) g) h_measure_univ_le : ↑↑μ univ ≤ ∑' (g : G), ↑↑μ (g • s) aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s) h_meas : ∀ (g : G), NullMeasurableSet (g • s) ⊢ NullMeasurableSet (⋃ g, g • s) ** exact .iUnion h_meas ** Qed
| |
MeasureTheory.IsFundamentalDomain.image_of_equiv ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁶ : Group G inst✝⁵ : Group H inst✝⁴ : MulAction G α inst✝³ : MeasurableSpace α inst✝² : MulAction H β inst✝¹ : MeasurableSpace β inst✝ : NormedAddCommGroup E s t : Set α μ : Measure α ν : Measure β h : IsFundamentalDomain G s f : α ≃ β hf : QuasiMeasurePreserving ↑f.symm e : H ≃ G hef : ∀ (g : H), Semiconj (↑f) (fun x => ↑e g • x) fun x => g • x ⊢ IsFundamentalDomain H (↑f '' s) ** rw [f.image_eq_preimage] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁶ : Group G inst✝⁵ : Group H inst✝⁴ : MulAction G α inst✝³ : MeasurableSpace α inst✝² : MulAction H β inst✝¹ : MeasurableSpace β inst✝ : NormedAddCommGroup E s t : Set α μ : Measure α ν : Measure β h : IsFundamentalDomain G s f : α ≃ β hf : QuasiMeasurePreserving ↑f.symm e : H ≃ G hef : ∀ (g : H), Semiconj (↑f) (fun x => ↑e g • x) fun x => g • x ⊢ IsFundamentalDomain H (↑f.symm ⁻¹' s) ** refine' h.preimage_of_equiv hf e.symm.bijective fun g x => _ ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁶ : Group G inst✝⁵ : Group H inst✝⁴ : MulAction G α inst✝³ : MeasurableSpace α inst✝² : MulAction H β inst✝¹ : MeasurableSpace β inst✝ : NormedAddCommGroup E s t : Set α μ : Measure α ν : Measure β h : IsFundamentalDomain G s f : α ≃ β hf : QuasiMeasurePreserving ↑f.symm e : H ≃ G hef : ∀ (g : H), Semiconj (↑f) (fun x => ↑e g • x) fun x => g • x g : G x : β ⊢ ↑f.symm ((fun x => ↑e.symm g • x) x) = (fun x => g • x) (↑f.symm x) ** rcases f.surjective x with ⟨x, rfl⟩ ** case intro G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁶ : Group G inst✝⁵ : Group H inst✝⁴ : MulAction G α inst✝³ : MeasurableSpace α inst✝² : MulAction H β inst✝¹ : MeasurableSpace β inst✝ : NormedAddCommGroup E s t : Set α μ : Measure α ν : Measure β h : IsFundamentalDomain G s f : α ≃ β hf : QuasiMeasurePreserving ↑f.symm e : H ≃ G hef : ∀ (g : H), Semiconj (↑f) (fun x => ↑e g • x) fun x => g • x g : G x : α ⊢ ↑f.symm ((fun x => ↑e.symm g • x) (↑f x)) = (fun x => g • x) (↑f.symm (↑f x)) ** rw [← hef _ _, f.symm_apply_apply, f.symm_apply_apply, e.apply_symm_apply] ** Qed
| |
MeasureTheory.IsFundamentalDomain.smul ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁹ : Group G inst✝⁸ : Group H inst✝⁷ : MulAction G α inst✝⁶ : MeasurableSpace α inst✝⁵ : MulAction H β inst✝⁴ : MeasurableSpace β inst✝³ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α inst✝ : SMulInvariantMeasure G α μ h : IsFundamentalDomain G s g g' : G ⊢ (fun g' => g * g' * g⁻¹) ((fun g' => g⁻¹ * g' * g) g') = g' ** simp [mul_assoc] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁹ : Group G inst✝⁸ : Group H inst✝⁷ : MulAction G α inst✝⁶ : MeasurableSpace α inst✝⁵ : MulAction H β inst✝⁴ : MeasurableSpace β inst✝³ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α inst✝ : SMulInvariantMeasure G α μ h : IsFundamentalDomain G s g g' : G ⊢ (fun g' => g⁻¹ * g' * g) ((fun g' => g * g' * g⁻¹) g') = g' ** simp [mul_assoc] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁹ : Group G inst✝⁸ : Group H inst✝⁷ : MulAction G α inst✝⁶ : MeasurableSpace α inst✝⁵ : MulAction H β inst✝⁴ : MeasurableSpace β inst✝³ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α inst✝ : SMulInvariantMeasure G α μ h : IsFundamentalDomain G s g g' : G x : α ⊢ ↑(MulAction.toPerm g) ((fun x => ↑{ toFun := fun g' => g⁻¹ * g' * g, invFun := fun g' => g * g' * g⁻¹, left_inv := (_ : ∀ (g' : G), g * (g⁻¹ * g' * g) * g⁻¹ = g'), right_inv := (_ : ∀ (g' : G), g⁻¹ * (g * g' * g⁻¹) * g = g') } g' • x) x) = (fun x => g' • x) (↑(MulAction.toPerm g) x) ** simp [smul_smul, mul_assoc] ** Qed
| |
MeasureTheory.IsFundamentalDomain.sum_restrict_of_ac ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s hν : ν ≪ μ ⊢ (sum fun g => Measure.restrict ν (g • s)) = ν ** rw [← restrict_iUnion_ae (h.aedisjoint.mono fun i j h => hν h) fun g =>
(h.nullMeasurableSet_smul g).mono_ac hν,
restrict_congr_set (hν h.iUnion_smul_ae_eq), restrict_univ] ** Qed
| |
MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum_of_ac ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s hν : ν ≪ μ f : α → ℝ≥0∞ ⊢ ∫⁻ (x : α), f x ∂ν = ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂ν ** rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν] ** Qed
| |
MeasureTheory.IsFundamentalDomain.set_lintegral_eq_tsum ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s f : α → ℝ≥0∞ t : Set α ⊢ ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂Measure.restrict μ t = ∑' (g : G), ∫⁻ (x : α) in t ∩ g • s, f x ∂μ ** simp only [h.restrict_restrict, inter_comm] ** Qed
| |
MeasureTheory.IsFundamentalDomain.set_lintegral_eq_tsum' ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s f : α → ℝ≥0∞ t : Set α ⊢ ∑' (g : G), ∫⁻ (x : α) in t ∩ g⁻¹ • s, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g⁻¹ • (g • t ∩ s), f x ∂μ ** simp only [smul_set_inter, inv_smul_smul] ** Qed
| |
MeasureTheory.IsFundamentalDomain.measure_eq_tsum_of_ac ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s hν : ν ≪ μ t : Set α ⊢ ↑↑ν t = ∑' (g : G), ↑↑ν (t ∩ g • s) ** have H : ν.restrict t ≪ μ := Measure.restrict_le_self.absolutelyContinuous.trans hν ** G : Type u_1 H✝ : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H✝ inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H✝ β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s hν : ν ≪ μ t : Set α H : Measure.restrict ν t ≪ μ ⊢ ↑↑ν t = ∑' (g : G), ↑↑ν (t ∩ g • s) ** simpa only [set_lintegral_one, Pi.one_def,
Measure.restrict_apply₀ ((h.nullMeasurableSet_smul _).mono_ac H), inter_comm] using
h.lintegral_eq_tsum_of_ac H 1 ** Qed
| |
MeasureTheory.IsFundamentalDomain.measure_eq_tsum ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α h : IsFundamentalDomain G s t : Set α ⊢ ↑↑μ t = ∑' (g : G), ↑↑μ (g • t ∩ s) ** simpa only [set_lintegral_one] using h.set_lintegral_eq_tsum' (fun _ => 1) t ** Qed
| |
MeasureTheory.IsFundamentalDomain.measure_eq_card_smul_of_smul_ae_eq_self ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹¹ : Group G inst✝¹⁰ : Group H inst✝⁹ : MulAction G α inst✝⁸ : MeasurableSpace α inst✝⁷ : MulAction H β inst✝⁶ : MeasurableSpace β inst✝⁵ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSMul G α inst✝² : SMulInvariantMeasure G α μ inst✝¹ : Countable G ν : Measure α inst✝ : Finite G h : IsFundamentalDomain G s t : Set α ht : ∀ (g : G), g • t =ᶠ[ae μ] t ⊢ ↑↑μ t = Nat.card G • ↑↑μ (t ∩ s) ** haveI : Fintype G := Fintype.ofFinite G ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹¹ : Group G inst✝¹⁰ : Group H inst✝⁹ : MulAction G α inst✝⁸ : MeasurableSpace α inst✝⁷ : MulAction H β inst✝⁶ : MeasurableSpace β inst✝⁵ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSMul G α inst✝² : SMulInvariantMeasure G α μ inst✝¹ : Countable G ν : Measure α inst✝ : Finite G h : IsFundamentalDomain G s t : Set α ht : ∀ (g : G), g • t =ᶠ[ae μ] t this : Fintype G ⊢ ↑↑μ t = Nat.card G • ↑↑μ (t ∩ s) ** rw [h.measure_eq_tsum] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹¹ : Group G inst✝¹⁰ : Group H inst✝⁹ : MulAction G α inst✝⁸ : MeasurableSpace α inst✝⁷ : MulAction H β inst✝⁶ : MeasurableSpace β inst✝⁵ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSMul G α inst✝² : SMulInvariantMeasure G α μ inst✝¹ : Countable G ν : Measure α inst✝ : Finite G h : IsFundamentalDomain G s t : Set α ht : ∀ (g : G), g • t =ᶠ[ae μ] t this : Fintype G ⊢ ∑' (g : G), ↑↑μ (g • t ∩ s) = Nat.card G • ↑↑μ (t ∩ s) ** replace ht : ∀ g : G, (g • t ∩ s : Set α) =ᵐ[μ] (t ∩ s : Set α) := fun g =>
ae_eq_set_inter (ht g) (ae_eq_refl s) ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹¹ : Group G inst✝¹⁰ : Group H inst✝⁹ : MulAction G α inst✝⁸ : MeasurableSpace α inst✝⁷ : MulAction H β inst✝⁶ : MeasurableSpace β inst✝⁵ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSMul G α inst✝² : SMulInvariantMeasure G α μ inst✝¹ : Countable G ν : Measure α inst✝ : Finite G h : IsFundamentalDomain G s t : Set α this : Fintype G ht : ∀ (g : G), g • t ∩ s =ᶠ[ae μ] t ∩ s ⊢ ∑' (g : G), ↑↑μ (g • t ∩ s) = Nat.card G • ↑↑μ (t ∩ s) ** simp_rw [measure_congr (ht _), tsum_fintype, Finset.sum_const, Nat.card_eq_fintype_card,
Finset.card_univ] ** Qed
| |
MeasureTheory.IsFundamentalDomain.set_lintegral_eq ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → ℝ≥0∞ hf : ∀ (g : G) (x : α), f (g • x) = f x ⊢ ∑' (g : G), ∫⁻ (x : α) in s ∩ g • t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ ** simp only [hf, inter_comm] ** Qed
| |
MeasureTheory.IsFundamentalDomain.measure_set_eq ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t A : Set α hA₀ : MeasurableSet A hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A this : ∫⁻ (x : α) in s, indicator A 1 x ∂μ = ∫⁻ (x : α) in t, indicator A 1 x ∂μ ⊢ ↑↑μ (A ∩ s) = ↑↑μ (A ∩ t) ** simpa [Measure.restrict_apply hA₀, lintegral_indicator _ hA₀] using this ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t A : Set α hA₀ : MeasurableSet A hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A ⊢ ∫⁻ (x : α) in s, indicator A 1 x ∂μ = ∫⁻ (x : α) in t, indicator A 1 x ∂μ ** refine hs.set_lintegral_eq ht (Set.indicator A fun _ => 1) fun g x ↦ ?_ ** case h.e'_3.h.e'_4 G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t A : Set α hA₀ : MeasurableSet A hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A g : G x : α ⊢ A = (fun x => g • x) ⁻¹' A ** rw [hA g] ** Qed
| |
MeasureTheory.IsFundamentalDomain.measure_eq ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t ⊢ ↑↑μ s = ↑↑μ t ** simpa only [set_lintegral_one] using hs.set_lintegral_eq ht (fun _ => 1) fun _ _ => rfl ** Qed
| |
MeasureTheory.IsFundamentalDomain.hasFiniteIntegral_on_iff ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x ⊢ HasFiniteIntegral f ↔ HasFiniteIntegral f ** dsimp only [HasFiniteIntegral] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x ⊢ ∫⁻ (a : α) in s, ↑‖f a‖₊ ∂μ < ⊤ ↔ ∫⁻ (a : α) in t, ↑‖f a‖₊ ∂μ < ⊤ ** rw [hs.set_lintegral_eq ht] ** case hf G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x ⊢ ∀ (g : G) (x : α), ↑‖f (g • x)‖₊ = ↑‖f x‖₊ ** intro g x ** case hf G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹⁰ : Group G inst✝⁹ : Group H inst✝⁸ : MulAction G α inst✝⁷ : MeasurableSpace α inst✝⁶ : MulAction H β inst✝⁵ : MeasurableSpace β inst✝⁴ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝³ : MeasurableSpace G inst✝² : MeasurableSMul G α inst✝¹ : SMulInvariantMeasure G α μ inst✝ : Countable G ν : Measure α hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x g : G x : α ⊢ ↑‖f (g • x)‖₊ = ↑‖f x‖₊ ** rw [hf] ** Qed
| |
MeasureTheory.IsFundamentalDomain.integral_eq_tsum_of_ac ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E h : IsFundamentalDomain G s hν : ν ≪ μ f : α → E hf : Integrable f ⊢ ∫ (x : α), f x ∂ν = ∑' (g : G), ∫ (x : α) in g • s, f x ∂ν ** rw [← MeasureTheory.integral_sum_measure, h.sum_restrict_of_ac hν] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E h : IsFundamentalDomain G s hν : ν ≪ μ f : α → E hf : Integrable f ⊢ Integrable fun x => f x ** exact hf ** Qed
| |
MeasureTheory.IsFundamentalDomain.set_integral_eq_tsum ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E h : IsFundamentalDomain G s f : α → E t : Set α hf : IntegrableOn f t ⊢ ∑' (g : G), ∫ (x : α) in g • s, f x ∂Measure.restrict μ t = ∑' (g : G), ∫ (x : α) in t ∩ g • s, f x ∂μ ** simp only [h.restrict_restrict, measure_smul, inter_comm] ** Qed
| |
MeasureTheory.IsFundamentalDomain.set_integral_eq_tsum' ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t✝ : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E h : IsFundamentalDomain G s f : α → E t : Set α hf : IntegrableOn f t ⊢ ∑' (g : G), ∫ (x : α) in t ∩ g⁻¹ • s, f x ∂μ = ∑' (g : G), ∫ (x : α) in g⁻¹ • (g • t ∩ s), f x ∂μ ** simp only [smul_set_inter, inv_smul_smul] ** Qed
| |
MeasureTheory.IsFundamentalDomain.set_integral_eq ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x ⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in t, f x ∂μ ** by_cases hfs : IntegrableOn f s μ ** case pos G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x hfs : IntegrableOn f s ⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in t, f x ∂μ ** have hft : IntegrableOn f t μ := by rwa [ht.integrableOn_iff hs hf] ** case pos G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x hfs : IntegrableOn f s hft : IntegrableOn f t ⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in t, f x ∂μ ** calc
∫ x in s, f x ∂μ = ∑' g : G, ∫ x in s ∩ g • t, f x ∂μ := ht.set_integral_eq_tsum hfs
_ = ∑' g : G, ∫ x in g • t ∩ s, f (g⁻¹ • x) ∂μ := by simp only [hf, inter_comm]
_ = ∫ x in t, f x ∂μ := (hs.set_integral_eq_tsum' hft).symm ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x hfs : IntegrableOn f s ⊢ IntegrableOn f t ** rwa [ht.integrableOn_iff hs hf] ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x hfs : IntegrableOn f s hft : IntegrableOn f t ⊢ ∑' (g : G), ∫ (x : α) in s ∩ g • t, f x ∂μ = ∑' (g : G), ∫ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ ** simp only [hf, inter_comm] ** case neg G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x hfs : ¬IntegrableOn f s ⊢ ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in t, f x ∂μ ** rw [integral_undef hfs, integral_undef] ** case neg G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹² : Group G inst✝¹¹ : Group H inst✝¹⁰ : MulAction G α inst✝⁹ : MeasurableSpace α inst✝⁸ : MulAction H β inst✝⁷ : MeasurableSpace β inst✝⁶ : NormedAddCommGroup E s t : Set α μ : Measure α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α inst✝³ : SMulInvariantMeasure G α μ inst✝² : Countable G ν : Measure α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : α → E hf : ∀ (g : G) (x : α), f (g • x) = f x hfs : ¬IntegrableOn f s ⊢ ¬Integrable fun x => f x ** rwa [hs.integrableOn_iff ht hf] at hfs ** Qed
| |
MeasureTheory.mem_fundamentalFrontier ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹ : Group G inst✝ : MulAction G α s : Set α x : α ⊢ x ∈ fundamentalFrontier G s ↔ x ∈ s ∧ ∃ g, g ≠ 1 ∧ x ∈ g • s ** simp [fundamentalFrontier] ** Qed
| |
MeasureTheory.mem_fundamentalInterior ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝¹ : Group G inst✝ : MulAction G α s : Set α x : α ⊢ x ∈ fundamentalInterior G s ↔ x ∈ s ∧ ∀ (g : G), g ≠ 1 → ¬x ∈ g • s ** simp [fundamentalInterior] ** Qed
| |
MeasureTheory.fundamentalFrontier_smul ** G : Type u_1 H : Type u_2 α : Type u_3 β : Type u_4 E : Type u_5 inst✝⁴ : Group G inst✝³ : MulAction G α s : Set α x : α inst✝² : Group H inst✝¹ : MulAction H α inst✝ : SMulCommClass H G α g : H ⊢ fundamentalFrontier G (g • s) = g • fundamentalFrontier G s ** simp_rw [fundamentalFrontier, smul_set_inter, smul_set_iUnion, smul_comm g (_ : G) (_ : Set α)] ** Qed
| |
Besicovitch.SatelliteConfig.centerAndRescale_center ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ ⊢ c (centerAndRescale a) (last N) = 0 ** simp [SatelliteConfig.centerAndRescale] ** Qed
| |
Besicovitch.SatelliteConfig.centerAndRescale_radius ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N✝ : ℕ τ✝ : ℝ a✝ : SatelliteConfig E N✝ τ✝ N : ℕ τ : ℝ a : SatelliteConfig E N τ ⊢ r (centerAndRescale a) (last N) = 1 ** simp [SatelliteConfig.centerAndRescale, inv_mul_cancel (a.rpos _).ne'] ** Qed
| |
Besicovitch.card_le_of_separated ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** borelize E ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** let μ : Measure E := Measure.addHaar ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** let δ : ℝ := (1 : ℝ) / 2 ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** let ρ : ℝ := (5 : ℝ) / 2 ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** have ρpos : 0 < ρ := by norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** set A := ⋃ c ∈ s, ball (c : E) δ with hA ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** have D : Set.Pairwise (s : Set E) (Disjoint on fun c => ball (c : E) δ) := by
rintro c hc d hd hcd
apply ball_disjoint_ball
rw [dist_eq_norm]
convert h c hc d hd hcd
norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** have A_subset : A ⊆ ball (0 : E) ρ := by
refine' iUnion₂_subset fun x hx => _
apply ball_subset_ball'
calc
δ + dist x 0 ≤ δ + 2 := by rw [dist_zero_right]; exact add_le_add le_rfl (hs x hx)
_ = 5 / 2 := by norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** have I :
(s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) ≤
ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) :=
calc
(s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) = μ A := by
rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
have I : 0 < δ := by norm_num
simp only [div_pow, μ.addHaar_ball_of_pos _ I]
simp only [one_div, one_pow, Finset.sum_const, nsmul_eq_mul, mul_assoc]
_ ≤ μ (ball (0 : E) ρ) := (measure_mono A_subset)
_ = ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) := by
simp only [μ.addHaar_ball_of_pos _ ρpos] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) :=
(ENNReal.mul_le_mul_right (measure_ball_pos _ _ zero_lt_one).ne' measure_ball_lt_top.ne).1 I ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** have K : (s.card : ℝ) ≤ (5 : ℝ) ^ finrank ℝ E := by
have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
simp [ENNReal.toReal_mul] at this
simpa [div_eq_mul_inv, zero_le_two] using this ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) K : ↑(Finset.card s) ≤ 5 ^ finrank ℝ E ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** exact_mod_cast K ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ⊢ 0 < ρ ** norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ ⊢ Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) ** rintro c hc d hd hcd ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ c : E hc : c ∈ ↑s d : E hd : d ∈ ↑s hcd : c ≠ d ⊢ (Disjoint on fun c => ball c δ) c d ** apply ball_disjoint_ball ** case h E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ c : E hc : c ∈ ↑s d : E hd : d ∈ ↑s hcd : c ≠ d ⊢ δ + δ ≤ dist c d ** rw [dist_eq_norm] ** case h E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ c : E hc : c ∈ ↑s d : E hd : d ∈ ↑s hcd : c ≠ d ⊢ δ + δ ≤ ‖c - d‖ ** convert h c hc d hd hcd ** case h.e'_3 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ c : E hc : c ∈ ↑s d : E hd : d ∈ ↑s hcd : c ≠ d ⊢ δ + δ = 1 ** norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) ⊢ A ⊆ ball 0 ρ ** refine' iUnion₂_subset fun x hx => _ ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) x : E hx : x ∈ s ⊢ ball x δ ⊆ ball 0 ρ ** apply ball_subset_ball' ** case h E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) x : E hx : x ∈ s ⊢ δ + dist x 0 ≤ ρ ** calc
δ + dist x 0 ≤ δ + 2 := by rw [dist_zero_right]; exact add_le_add le_rfl (hs x hx)
_ = 5 / 2 := by norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) x : E hx : x ∈ s ⊢ δ + dist x 0 ≤ δ + 2 ** rw [dist_zero_right] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) x : E hx : x ∈ s ⊢ δ + ‖x‖ ≤ δ + 2 ** exact add_le_add le_rfl (hs x hx) ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) x : E hx : x ∈ s ⊢ δ + 2 = 5 / 2 ** norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ ⊢ ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) = ↑↑μ A ** rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ ⊢ ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) = Finset.sum s fun p => ↑↑μ (ball p δ) ** have I : 0 < δ := by norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : 0 < δ ⊢ ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) = Finset.sum s fun p => ↑↑μ (ball p δ) ** simp only [div_pow, μ.addHaar_ball_of_pos _ I] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : 0 < δ ⊢ ↑(Finset.card s) * ENNReal.ofReal (1 ^ finrank ℝ E / 2 ^ finrank ℝ E) * ↑↑Measure.addHaar (ball 0 1) = Finset.sum s fun x => ENNReal.ofReal (1 ^ finrank ℝ E / 2 ^ finrank ℝ E) * ↑↑Measure.addHaar (ball 0 1) ** simp only [one_div, one_pow, Finset.sum_const, nsmul_eq_mul, mul_assoc] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ ⊢ 0 < δ ** norm_num ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ ⊢ ↑↑μ (ball 0 ρ) = ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ** simp only [μ.addHaar_ball_of_pos _ ρpos] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) ⊢ ↑(Finset.card s) ≤ 5 ^ finrank ℝ E ** have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) this : ENNReal.toReal (↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E)) ≤ ρ ^ finrank ℝ E ⊢ ↑(Finset.card s) ≤ 5 ^ finrank ℝ E ** simp [ENNReal.toReal_mul] at this ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E μ : Measure E := Measure.addHaar δ : ℝ := 1 / 2 ρ : ℝ := 5 / 2 ρpos : 0 < ρ A : Set E := ⋃ c ∈ s, ball c δ hA : A = ⋃ c ∈ s, ball c δ D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ) A_subset : A ⊆ ball 0 ρ I : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1) J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) this : ↑(Finset.card s) * ENNReal.toReal (ENNReal.ofReal (2 ^ finrank ℝ E)⁻¹) ≤ 5 ^ finrank ℝ E / 2 ^ finrank ℝ E ⊢ ↑(Finset.card s) ≤ 5 ^ finrank ℝ E ** simpa [div_eq_mul_inv, zero_le_two] using this ** Qed
| |
Besicovitch.multiplicity_le ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E ⊢ multiplicity E ≤ 5 ^ finrank ℝ E ** apply csSup_le ** case h₁ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E ⊢ Set.Nonempty {N | ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} ** refine' ⟨0, ⟨∅, by simp⟩⟩ ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E ⊢ Finset.card ∅ = 0 ∧ (∀ (c : E), c ∈ ∅ → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ ∅ → ∀ (d : E), d ∈ ∅ → c ≠ d → 1 ≤ ‖c - d‖ ** simp ** case h₂ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E ⊢ ∀ (b : ℕ), b ∈ {N | ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} → b ≤ 5 ^ finrank ℝ E ** rintro _ ⟨s, ⟨rfl, h⟩⟩ ** case h₂.intro.intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E h : (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** exact Besicovitch.card_le_of_separated s h.1 h.2 ** Qed
| |
Besicovitch.card_le_multiplicity ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h's : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ Finset.card s ≤ multiplicity E ** apply le_csSup ** case h₁ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h's : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ BddAbove {N | ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} ** refine' ⟨5 ^ finrank ℝ E, _⟩ ** case h₁ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h's : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ 5 ^ finrank ℝ E ∈ upperBounds {N | ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} ** rintro _ ⟨s, ⟨rfl, h⟩⟩ ** case h₁.intro.intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s✝ : Finset E hs : ∀ (c : E), c ∈ s✝ → ‖c‖ ≤ 2 h's : ∀ (c : E), c ∈ s✝ → ∀ (d : E), d ∈ s✝ → c ≠ d → 1 ≤ ‖c - d‖ s : Finset E h : (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ Finset.card s ≤ 5 ^ finrank ℝ E ** exact Besicovitch.card_le_of_separated s h.1 h.2 ** case h₂ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h's : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ Finset.card s ∈ {N | ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} ** simp only [mem_setOf_eq, Ne.def] ** case h₂ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Finset E hs : ∀ (c : E), c ∈ s → ‖c‖ ≤ 2 h's : ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖ ⊢ ∃ s_1, Finset.card s_1 = Finset.card s ∧ (∀ (c : E), c ∈ s_1 → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s_1 → ∀ (d : E), d ∈ s_1 → ¬c = d → 1 ≤ ‖c - d‖ ** exact ⟨s, rfl, hs, h's⟩ ** Qed
| |
Besicovitch.one_lt_goodτ ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E ⊢ 1 < goodτ E ** dsimp [goodτ, goodδ] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E ⊢ 1 < 1 + Exists.choose (_ : ∃ δ, 0 < δ ∧ δ < 1 ∧ ∀ (s : Finset E), (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) → (∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 - δ ≤ ‖c - d‖) → Finset.card s ≤ multiplicity E) / 4 ** linarith [(exists_goodδ E).choose_spec.1] ** Qed
| |
Besicovitch.SatelliteConfig.exists_normalized_aux1 ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have ah :
∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
simpa only [dist_eq_norm] using a.h ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have I : (1 - δ / 4) * τ ≤ 1 :=
calc
(1 - δ / 4) * τ ≤ (1 - δ / 4) * (1 + δ / 4) := mul_le_mul_of_nonneg_left hδ1 D
_ = (1 : ℝ) - δ ^ 2 / 16 := by ring
_ ≤ 1 := by linarith only [sq_nonneg δ] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have J : 1 - δ ≤ 1 - δ / 4 := by linarith only [δnonneg] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** have K : 1 - δ / 4 ≤ τ⁻¹ := by rw [inv_eq_one_div, le_div_iff τpos]; exact I ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** suffices L : τ⁻¹ ≤ ‖a.c i - a.c j‖ ** case L E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ ⊢ τ⁻¹ ≤ ‖c a i - c a j‖ ** have hτ' : ∀ k, τ⁻¹ ≤ a.r k := by
intro k
rw [inv_eq_one_div, div_le_iff τpos, ← lastr, mul_comm]
exact a.hlast' k hτ ** case L E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ hτ' : ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k ⊢ τ⁻¹ ≤ ‖c a i - c a j‖ ** rcases ah i j inej with (H | H) ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ⊢ ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ** simpa only [dist_eq_norm] using a.h ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ 0 ≤ δ ** linarith only [hτ, hδ1] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ ⊢ 0 ≤ 1 - δ / 4 ** linarith only [hδ2] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ ⊢ (1 - δ / 4) * (1 + δ / 4) = 1 - δ ^ 2 / 16 ** ring ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ ⊢ 1 - δ ^ 2 / 16 ≤ 1 ** linarith only [sq_nonneg δ] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 ⊢ 1 - δ ≤ 1 - δ / 4 ** linarith only [δnonneg] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 ⊢ 1 - δ / 4 ≤ τ⁻¹ ** rw [inv_eq_one_div, le_div_iff τpos] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 ⊢ (1 - δ / 4) * τ ≤ 1 ** exact I ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ L : τ⁻¹ ≤ ‖c a i - c a j‖ ⊢ 1 - δ ≤ ‖c a i - c a j‖ ** linarith only [J, K, L] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ ⊢ ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k ** intro k ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ k : Fin (Nat.succ N) ⊢ τ⁻¹ ≤ r a k ** rw [inv_eq_one_div, div_le_iff τpos, ← lastr, mul_comm] ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ k : Fin (Nat.succ N) ⊢ r a (last N) ≤ τ * r a k ** exact a.hlast' k hτ ** case L.inl E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ hτ' : ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ⊢ τ⁻¹ ≤ ‖c a i - c a j‖ ** apply le_trans _ H.1 ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ hτ' : ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ⊢ τ⁻¹ ≤ r a i ** exact hτ' i ** case L.inr E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ hτ' : ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k H : r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ τ⁻¹ ≤ ‖c a i - c a j‖ ** rw [norm_sub_rev] ** case L.inr E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ hτ' : ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k H : r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ τ⁻¹ ≤ ‖c a j - c a i‖ ** apply le_trans _ H.1 ** E : Type u_1 inst✝ : NormedAddCommGroup E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ I : (1 - δ / 4) * τ ≤ 1 J : 1 - δ ≤ 1 - δ / 4 K : 1 - δ / 4 ≤ τ⁻¹ hτ' : ∀ (k : Fin (Nat.succ N)), τ⁻¹ ≤ r a k H : r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ τ⁻¹ ≤ r a j ** exact hτ' j ** Qed
| |
Besicovitch.SatelliteConfig.exists_normalized_aux2 ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** have ah :
∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
simpa only [dist_eq_norm] using a.h ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** have J : (1 - δ / 4) * τ ≤ 1 :=
calc
(1 - δ / 4) * τ ≤ (1 - δ / 4) * (1 + δ / 4) := mul_le_mul_of_nonneg_left hδ1 D
_ = (1 : ℝ) - δ ^ 2 / 16 := by ring
_ ≤ 1 := by linarith only [sq_nonneg δ] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ ⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖ ** set d := (2 / ‖a.c j‖) • a.c j with hd ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j ⊢ 1 - δ ≤ ‖c a i - d‖ ** have : a.r j - δ ≤ ‖a.c i - d‖ + (a.r j - 1) :=
calc
a.r j - δ ≤ ‖a.c i - a.c j‖ := A
_ ≤ ‖a.c i - d‖ + ‖d - a.c j‖ := by simp only [← dist_eq_norm, dist_triangle]
_ ≤ ‖a.c i - d‖ + (a.r j - 1) := by
apply add_le_add_left
have A : 0 ≤ 1 - 2 / ‖a.c j‖ := by simpa [div_le_iff (zero_le_two.trans_lt hj)] using hj.le
rw [← one_smul ℝ (a.c j), hd, ← sub_smul, norm_smul, norm_sub_rev, Real.norm_eq_abs,
abs_of_nonneg A, sub_mul]
field_simp [(zero_le_two.trans_lt hj).ne']
linarith only [hcrj] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j this : r a j - δ ≤ ‖c a i - d‖ + (r a j - 1) ⊢ 1 - δ ≤ ‖c a i - d‖ ** linarith only [this] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ⊢ ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ** simpa only [dist_eq_norm] using a.h ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ 0 ≤ δ ** linarith only [hτ, hδ1] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ ⊢ 0 ≤ 1 - δ / 4 ** linarith only [hδ2] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ ⊢ ‖c a j‖ ≤ r a j + 1 ** simpa only [lastc, lastr, dist_zero_right] using a.inter' j ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 ⊢ r a i ≤ 2 ** rcases lt_or_le i (last N) with (H | H) ** case inl E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 H : i < last N ⊢ r a i ≤ 2 ** apply (a.hlast i H).1.trans ** case inl E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 H : i < last N ⊢ dist (c a i) (c a (last N)) ≤ 2 ** simpa only [dist_eq_norm, lastc, sub_zero] using hi ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 H : last N ≤ i ⊢ r a i ≤ 2 ** have : i = last N := top_le_iff.1 H ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 H : last N ≤ i this : i = last N ⊢ r a i ≤ 2 ** rw [this, lastr] ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 H : last N ≤ i this : i = last N ⊢ 1 ≤ 2 ** exact one_le_two ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 ⊢ (1 - δ / 4) * (1 + δ / 4) = 1 - δ ^ 2 / 16 ** ring ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 ⊢ 1 - δ ^ 2 / 16 ≤ 1 ** linarith only [sq_nonneg δ] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 ⊢ r a j - δ ≤ ‖c a i - c a j‖ ** rcases ah j i inej.symm with (H | H) ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ⊢ r a j - δ ≤ ‖c a i - c a j‖ ** have C : a.r j ≤ 4 :=
calc
a.r j ≤ τ * a.r i := H.2
_ ≤ τ * 2 := (mul_le_mul_of_nonneg_left I τpos.le)
_ ≤ 5 / 4 * 2 := (mul_le_mul_of_nonneg_right (by linarith only [hδ1, hδ2]) zero_le_two)
_ ≤ 4 := by norm_num ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ r a j - δ ≤ ‖c a i - c a j‖ ** calc
a.r j - δ ≤ a.r j - a.r j / 4 * δ := by
refine' sub_le_sub le_rfl _
refine' mul_le_of_le_one_left δnonneg _
linarith only [C]
_ = (1 - δ / 4) * a.r j := by ring
_ ≤ (1 - δ / 4) * (τ * a.r i) := (mul_le_mul_of_nonneg_left H.2 D)
_ ≤ 1 * a.r i := by rw [← mul_assoc]; apply mul_le_mul_of_nonneg_right J (a.rpos _).le
_ ≤ ‖a.c i - a.c j‖ := by rw [one_mul]; exact H.1 ** case inl E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ r a j - δ ≤ ‖c a i - c a j‖ ** rw [norm_sub_rev] ** case inl E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ r a j - δ ≤ ‖c a j - c a i‖ ** linarith [H.1] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ⊢ τ ≤ 5 / 4 ** linarith only [hδ1, hδ2] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ⊢ 5 / 4 * 2 ≤ 4 ** norm_num ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ r a j - δ ≤ r a j - r a j / 4 * δ ** refine' sub_le_sub le_rfl _ ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ r a j / 4 * δ ≤ δ ** refine' mul_le_of_le_one_left δnonneg _ ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ r a j / 4 ≤ 1 ** linarith only [C] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ r a j - r a j / 4 * δ = (1 - δ / 4) * r a j ** ring ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ (1 - δ / 4) * (τ * r a i) ≤ 1 * r a i ** rw [← mul_assoc] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ (1 - δ / 4) * τ * r a i ≤ 1 * r a i ** apply mul_le_mul_of_nonneg_right J (a.rpos _).le ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ 1 * r a i ≤ ‖c a i - c a j‖ ** rw [one_mul] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i C : r a j ≤ 4 ⊢ r a i ≤ ‖c a i - c a j‖ ** exact H.1 ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j ⊢ ‖c a i - c a j‖ ≤ ‖c a i - d‖ + ‖d - c a j‖ ** simp only [← dist_eq_norm, dist_triangle] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j ⊢ ‖c a i - d‖ + ‖d - c a j‖ ≤ ‖c a i - d‖ + (r a j - 1) ** apply add_le_add_left ** case bc E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j ⊢ ‖d - c a j‖ ≤ r a j - 1 ** have A : 0 ≤ 1 - 2 / ‖a.c j‖ := by simpa [div_le_iff (zero_le_two.trans_lt hj)] using hj.le ** case bc E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A✝ : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j A : 0 ≤ 1 - 2 / ‖c a j‖ ⊢ ‖d - c a j‖ ≤ r a j - 1 ** rw [← one_smul ℝ (a.c j), hd, ← sub_smul, norm_smul, norm_sub_rev, Real.norm_eq_abs,
abs_of_nonneg A, sub_mul] ** case bc E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A✝ : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j A : 0 ≤ 1 - 2 / ‖c a j‖ ⊢ 1 * ‖c a j‖ - 2 / ‖c a j‖ * ‖c a j‖ ≤ r a j - 1 ** field_simp [(zero_le_two.trans_lt hj).ne'] ** case bc E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A✝ : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j A : 0 ≤ 1 - 2 / ‖c a j‖ ⊢ ‖c a j‖ ≤ r a j - 1 + 2 ** linarith only [hcrj] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 hδ2 : δ ≤ 1 i j : Fin (Nat.succ N) inej : i ≠ j hi : ‖c a i‖ ≤ 2 hj : 2 < ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ D : 0 ≤ 1 - δ / 4 τpos : 0 < τ hcrj : ‖c a j‖ ≤ r a j + 1 I : r a i ≤ 2 J : (1 - δ / 4) * τ ≤ 1 A : r a j - δ ≤ ‖c a i - c a j‖ d : E := (2 / ‖c a j‖) • c a j hd : d = (2 / ‖c a j‖) • c a j ⊢ 0 ≤ 1 - 2 / ‖c a j‖ ** simpa [div_le_iff (zero_le_two.trans_lt hj)] using hj.le ** Qed
| |
Besicovitch.SatelliteConfig.exists_normalized_aux3 ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖ ** have ah :
∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
simpa only [dist_eq_norm] using a.h ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖ ** have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ ⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖ ** have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 ⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖ ** have A : a.r i ≤ ‖a.c i‖ := by
have : i < last N := by
apply lt_top_iff_ne_top.2
intro iN
change i = last N at iN
rw [iN, lastc, norm_zero] at hi
exact lt_irrefl _ (zero_le_two.trans_lt hi)
convert (a.hlast i this).1 using 1
rw [dist_eq_norm, lastc, sub_zero] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 A : r a i ≤ ‖c a i‖ ⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖ ** have hj : 2 < ‖a.c j‖ := hi.trans_le hij ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 A : r a i ≤ ‖c a i‖ hj : 2 < ‖c a j‖ ⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖ ** set s := ‖a.c i‖ ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s ⊢ 1 - δ ≤ ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** have spos : 0 < s := zero_lt_two.trans hi ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s ⊢ 1 - δ ≤ ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** set d := (s / ‖a.c j‖) • a.c j with hd ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j ⊢ 1 - δ ≤ ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** have I : ‖a.c j - a.c i‖ ≤ ‖a.c j‖ - s + ‖d - a.c i‖ :=
calc
‖a.c j - a.c i‖ ≤ ‖a.c j - d‖ + ‖d - a.c i‖ := by simp [← dist_eq_norm, dist_triangle]
_ = ‖a.c j‖ - ‖a.c i‖ + ‖d - a.c i‖ := by
nth_rw 1 [← one_smul ℝ (a.c j)]
rw [add_left_inj, hd, ← sub_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg, sub_mul,
one_mul, div_mul_cancel _ (zero_le_two.trans_lt hj).ne']
rwa [sub_nonneg, div_le_iff (zero_lt_two.trans hj), one_mul] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ ⊢ 1 - δ ≤ ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** have invs_nonneg : 0 ≤ 2 / s := div_nonneg zero_le_two (zero_le_two.trans hi.le) ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ 1 - δ ≤ ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** calc
1 - δ = 2 / s * (s / 2 - s / 2 * δ) := by field_simp [spos.ne']; ring
_ ≤ 2 / s * ‖d - a.c i‖ :=
(mul_le_mul_of_nonneg_left (by linarith only [hcrj, I, J, hi]) invs_nonneg)
_ = ‖(2 / s) • a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
conv_lhs => rw [norm_sub_rev, ← abs_of_nonneg invs_nonneg]
rw [← Real.norm_eq_abs, ← norm_smul, smul_sub, hd, smul_smul]
congr 3
field_simp [spos.ne'] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ⊢ ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ** simpa only [dist_eq_norm] using a.h ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ 0 ≤ δ ** linarith only [hτ, hδ1] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ ⊢ ‖c a j‖ ≤ r a j + 1 ** simpa only [lastc, lastr, dist_zero_right] using a.inter' j ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 ⊢ r a i ≤ ‖c a i‖ ** have : i < last N := by
apply lt_top_iff_ne_top.2
intro iN
change i = last N at iN
rw [iN, lastc, norm_zero] at hi
exact lt_irrefl _ (zero_le_two.trans_lt hi) ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 this : i < last N ⊢ r a i ≤ ‖c a i‖ ** convert (a.hlast i this).1 using 1 ** case h.e'_4 E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 this : i < last N ⊢ ‖c a i‖ = dist (c a i) (c a (last N)) ** rw [dist_eq_norm, lastc, sub_zero] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 ⊢ i < last N ** apply lt_top_iff_ne_top.2 ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 ⊢ i ≠ ⊤ ** intro iN ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 iN : i = ⊤ ⊢ False ** change i = last N at iN ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < ‖c a i‖ hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 iN : i = last N ⊢ False ** rw [iN, lastc, norm_zero] at hi ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j hi : 2 < 0 hij : ‖c a i‖ ≤ ‖c a j‖ ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 iN : i = last N ⊢ False ** exact lt_irrefl _ (zero_le_two.trans_lt hi) ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j ⊢ ‖c a j - c a i‖ ≤ ‖c a j - d‖ + ‖d - c a i‖ ** simp [← dist_eq_norm, dist_triangle] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j ⊢ ‖c a j - d‖ + ‖d - c a i‖ = ‖c a j‖ - ‖c a i‖ + ‖d - c a i‖ ** nth_rw 1 [← one_smul ℝ (a.c j)] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j ⊢ ‖1 • c a j - d‖ + ‖d - c a i‖ = ‖c a j‖ - ‖c a i‖ + ‖d - c a i‖ ** rw [add_left_inj, hd, ← sub_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg, sub_mul,
one_mul, div_mul_cancel _ (zero_le_two.trans_lt hj).ne'] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j ⊢ 0 ≤ 1 - s / ‖c a j‖ ** rwa [sub_nonneg, div_le_iff (zero_lt_two.trans hj), one_mul] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ ⊢ r a j - ‖c a j - c a i‖ ≤ s * (τ - 1) ** rcases ah j i inej.symm with (H | H) ** case inl E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ H : r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j ⊢ r a j - ‖c a j - c a i‖ ≤ s * (τ - 1) ** calc
a.r j - ‖a.c j - a.c i‖ ≤ 0 := sub_nonpos.2 H.1
_ ≤ s * (τ - 1) := mul_nonneg spos.le (sub_nonneg.2 hτ) ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ H : r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ⊢ r a j - ‖c a j - c a i‖ ≤ s * (τ - 1) ** rw [norm_sub_rev] at H ** case inr E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ H : r a i ≤ ‖c a j - c a i‖ ∧ r a j ≤ τ * r a i ⊢ r a j - ‖c a j - c a i‖ ≤ s * (τ - 1) ** calc
a.r j - ‖a.c j - a.c i‖ ≤ τ * a.r i - a.r i := sub_le_sub H.2 H.1
_ = a.r i * (τ - 1) := by ring
_ ≤ s * (τ - 1) := mul_le_mul_of_nonneg_right A (sub_nonneg.2 hτ) ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ H : r a i ≤ ‖c a j - c a i‖ ∧ r a j ≤ τ * r a i ⊢ τ * r a i - r a i = r a i * (τ - 1) ** ring ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ ⊢ τ - 1 ≤ δ / 2 ** linarith only [δnonneg, hδ1] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ ⊢ s * (δ / 2) = s / 2 * δ ** ring ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ 1 - δ = 2 / s * (s / 2 - s / 2 * δ) ** field_simp [spos.ne'] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ (1 - δ) * (‖c a i‖ * 2) = 2 * (‖c a i‖ - ‖c a i‖ * δ) ** ring ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ s / 2 - s / 2 * δ ≤ ‖d - c a i‖ ** linarith only [hcrj, I, J, hi] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ 2 / s * ‖d - c a i‖ = ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** conv_lhs => rw [norm_sub_rev, ← abs_of_nonneg invs_nonneg] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ |2 / s| * ‖c a i - d‖ = ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** rw [← Real.norm_eq_abs, ← norm_smul, smul_sub, hd, smul_smul] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ ‖(2 / s) • c a i - (2 / s * (s / ‖c a j‖)) • c a j‖ = ‖(2 / s) • c a i - (2 / ‖c a j‖) • c a j‖ ** congr 3 ** case e_a.e_a.e_a E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E N : ℕ τ : ℝ a : SatelliteConfig E N τ lastc : c a (last N) = 0 lastr : r a (last N) = 1 hτ : 1 ≤ τ δ : ℝ hδ1 : τ ≤ 1 + δ / 4 i j : Fin (Nat.succ N) inej : i ≠ j ah : ∀ (i j : Fin (Nat.succ N)), i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j δnonneg : 0 ≤ δ hcrj : ‖c a j‖ ≤ r a j + 1 hj : 2 < ‖c a j‖ s : ℝ := ‖c a i‖ hi : 2 < s hij : s ≤ ‖c a j‖ A : r a i ≤ s spos : 0 < s d : E := (s / ‖c a j‖) • c a j hd : d = (s / ‖c a j‖) • c a j I : ‖c a j - c a i‖ ≤ ‖c a j‖ - s + ‖d - c a i‖ J : r a j - ‖c a j - c a i‖ ≤ s / 2 * δ invs_nonneg : 0 ≤ 2 / s ⊢ 2 / s * (s / ‖c a j‖) = 2 / ‖c a j‖ ** field_simp [spos.ne'] ** Qed
| |
Nat.Primrec.id ** n : ℕ ⊢ Nat.pair (unpair n).1 (unpair n).2 = id n ** simp ** Qed
| |
Nat.Primrec.prec1 ** f : ℕ → ℕ m : ℕ hf : Nat.Primrec f n : ℕ ⊢ unpaired (fun z n => Nat.rec m (fun y IH => f (unpair (Nat.pair z (Nat.pair y IH))).2) n) (Nat.pair 0 (id n)) = Nat.rec m (fun y IH => f (Nat.pair y IH)) n ** simp ** Qed
| |
Nat.Primrec.casesOn' ** f g : ℕ → ℕ hf : Nat.Primrec f hg : Nat.Primrec g n : ℕ ⊢ unpaired (fun z n => Nat.rec (f z) (fun y IH => g (Nat.pair (unpair (Nat.pair z (Nat.pair y IH))).1 (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).1)) n) n = unpaired (fun z n => Nat.casesOn n (f z) fun y => g (Nat.pair z y)) n ** simp ** Qed
| |
Nat.Primrec.swap ** n : ℕ ⊢ Nat.pair (unpair n).2 (unpair n).1 = unpaired (swap Nat.pair) n ** simp ** Qed
| |
Nat.Primrec.swap' ** f : ℕ → ℕ → ℕ hf : Nat.Primrec (unpaired f) n : ℕ ⊢ unpaired f (unpaired (swap Nat.pair) n) = unpaired (swap f) n ** simp ** Qed
| |
Nat.Primrec.pred ** n : ℕ ⊢ Nat.casesOn n 0 id = Nat.pred n ** cases n <;> simp [*] ** Qed
| |
Nat.Primrec.add ** p : ℕ ⊢ unpaired (fun z n => Nat.rec (id z) (fun y IH => succ (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).2) n) p = unpaired (fun x x_1 => x + x_1) p ** simp ** p : ℕ ⊢ Nat.rec (unpair p).1 (fun y IH => succ IH) (unpair p).2 = (unpair p).1 + (unpair p).2 ** induction p.unpair.2 <;> simp [*, add_succ] ** Qed
| |
Nat.Primrec.sub ** p : ℕ ⊢ unpaired (fun z n => Nat.rec (id z) (fun y IH => Nat.pred (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).2) n) p = unpaired (fun x x_1 => x - x_1) p ** simp ** p : ℕ ⊢ Nat.rec (unpair p).1 (fun y IH => Nat.pred IH) (unpair p).2 = (unpair p).1 - (unpair p).2 ** induction p.unpair.2 <;> simp [*, sub_succ] ** Qed
| |
Nat.Primrec.mul ** p : ℕ ⊢ unpaired (fun z n => Nat.rec 0 (fun y IH => unpaired (fun x x_1 => x + x_1) (Nat.pair (unpair (Nat.pair z (Nat.pair y IH))).1 (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).2)) n) p = unpaired (fun x x_1 => x * x_1) p ** simp ** p : ℕ ⊢ Nat.rec 0 (fun y IH => (unpair p).1 + IH) (unpair p).2 = (unpair p).1 * (unpair p).2 ** induction p.unpair.2 <;> simp [*, mul_succ, add_comm _ (unpair p).fst] ** Qed
| |
Nat.Primrec.pow ** p : ℕ ⊢ unpaired (fun z n => Nat.rec 1 (fun y IH => unpaired (fun x x_1 => x * x_1) (Nat.pair (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).2 (unpair (Nat.pair z (Nat.pair y IH))).1)) n) p = unpaired (fun x x_1 => x ^ x_1) p ** simp ** p : ℕ ⊢ Nat.rec 1 (fun y IH => IH * (unpair p).1) (unpair p).2 = (unpair p).1 ^ (unpair p).2 ** induction p.unpair.2 <;> simp [*, pow_succ] ** Qed
| |
Primrec.encode ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ n : ℕ ⊢ encode (decode n) = encode (Option.map encode (decode n)) ** cases @decode α _ n <;> rfl ** Qed
| |
Primrec.dom_denumerable ** α✝ : Type u_1 β✝ : Type u_2 σ : Type u_3 inst✝⁴ : Primcodable α✝ inst✝³ : Primcodable β✝ inst✝² : Primcodable σ α : Type u_4 β : Type u_5 inst✝¹ : Denumerable α inst✝ : Primcodable β f : α → β h : Primrec f n : ℕ ⊢ Nat.pred (encode (Option.map f (decode n))) = encode (f (ofNat α n)) ** simp ** α✝ : Type u_1 β✝ : Type u_2 σ : Type u_3 inst✝⁴ : Primcodable α✝ inst✝³ : Primcodable β✝ inst✝² : Primcodable σ α : Type u_4 β : Type u_5 inst✝¹ : Denumerable α inst✝ : Primcodable β f : α → β h : Nat.Primrec fun n => encode (f (ofNat α n)) n : ℕ ⊢ Nat.succ (encode (f (ofNat α n))) = encode (Option.map f (decode n)) ** simp ** Qed
| |
Primrec.option_some ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ n : ℕ ⊢ (Nat.casesOn (encode (decode n)) 0 fun n => Nat.succ (Nat.succ n)) = encode (Option.map some (decode n)) ** cases @decode α _ n <;> simp ** Qed
| |
Primrec.const ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ x : σ n : ℕ ⊢ (Nat.casesOn (encode (decode n)) 0 fun x_1 => Nat.succ (encode x)) = encode (Option.map (fun x_1 => x) (decode n)) ** cases @decode α _ n <;> rfl ** Qed
| |
Primrec.id ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ ⊢ ∀ (n : ℕ), encode (decode n) = encode (Option.map id (decode n)) ** simp ** Qed
| |
Primrec.comp ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : β → σ g : α → β hf : Primrec f hg : Primrec g n : ℕ ⊢ (Nat.casesOn (encode (decode n)) 0 fun n => encode (Option.map f (decode (Nat.pred (encode (Option.map g (decode n))))))) = encode (Option.map (fun a => f (g a)) (decode n)) ** cases @decode α _ n <;> simp [encodek] ** Qed
| |
Primrec.encode_iff ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : α → σ h : Primrec fun a => encode (f a) n : ℕ ⊢ encode (Option.map (fun a => encode (f a)) (decode n)) = encode (Option.map f (decode n)) ** cases @decode α _ n <;> rfl ** Qed
| |
Primrec.of_equiv_symm ** α : Type u_1 β✝ : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β✝ inst✝ : Primcodable σ β : Type u_4 e : β ≃ α this : Primcodable β := Primcodable.ofEquiv α e ⊢ Primrec fun a => encode (↑e (↑e.symm a)) ** simp [Primrec.encode] ** Qed
| |
Primrec.of_equiv_iff ** α : Type u_1 β✝ : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β✝ inst✝ : Primcodable σ β : Type u_4 e : β ≃ α f : σ → β this : Primcodable β := Primcodable.ofEquiv α e h : Primrec fun a => ↑e (f a) a : σ ⊢ ↑e.symm (↑e (f a)) = f a ** simp ** Qed
| |
Primrec.of_equiv_symm_iff ** α : Type u_1 β✝ : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β✝ inst✝ : Primcodable σ β : Type u_4 e : β ≃ α f : σ → α this : Primcodable β := Primcodable.ofEquiv α e h : Primrec fun a => ↑e.symm (f a) a : σ ⊢ ↑e (↑e.symm (f a)) = f a ** simp ** Qed
| |
Primrec.fst ** α✝ : Type u_1 σ : Type u_2 inst✝³ : Primcodable α✝ inst✝² : Primcodable σ α : Type u_3 β : Type u_4 inst✝¹ : Primcodable α inst✝ : Primcodable β n : ℕ ⊢ Nat.unpaired (fun z n => Nat.casesOn n 0 fun y => Nat.unpaired (fun z n => Nat.casesOn n 0 fun y => Nat.succ (Nat.unpair (Nat.pair z y)).1) (Nat.pair (Nat.unpair (Nat.pair z y)).2 (encode (decode (Nat.unpair (Nat.pair z y)).1)))) (Nat.pair (Nat.unpair n).2 (encode (decode (Nat.unpair n).1))) = encode (Option.map Prod.fst (decode n)) ** simp only [Nat.unpaired, Nat.unpair_pair, decode_prod_val] ** α✝ : Type u_1 σ : Type u_2 inst✝³ : Primcodable α✝ inst✝² : Primcodable σ α : Type u_3 β : Type u_4 inst✝¹ : Primcodable α inst✝ : Primcodable β n : ℕ ⊢ Nat.rec 0 (fun n_1 n_ih => Nat.rec 0 (fun n n_ih => Nat.succ n_1) (encode (decode (Nat.unpair n).2))) (encode (decode (Nat.unpair n).1)) = encode (Option.map Prod.fst (Option.bind (decode (Nat.unpair n).1) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).2))) ** cases @decode α _ n.unpair.1 <;> simp ** case some α✝ : Type u_1 σ : Type u_2 inst✝³ : Primcodable α✝ inst✝² : Primcodable σ α : Type u_3 β : Type u_4 inst✝¹ : Primcodable α inst✝ : Primcodable β n : ℕ val✝ : α ⊢ Nat.rec 0 (fun n n_ih => Nat.succ (encode val✝)) (encode (decode (Nat.unpair n).2)) = encode (Option.map (Function.const β val✝) (decode (Nat.unpair n).2)) ** cases @decode β _ n.unpair.2 <;> simp ** Qed
| |
Primrec.snd ** α✝ : Type u_1 σ : Type u_2 inst✝³ : Primcodable α✝ inst✝² : Primcodable σ α : Type u_3 β : Type u_4 inst✝¹ : Primcodable α inst✝ : Primcodable β n : ℕ ⊢ Nat.unpaired (fun z n => Nat.casesOn n 0 fun y => Nat.unpaired (fun z n => Nat.casesOn n 0 fun y => Nat.succ (Nat.unpair (Nat.pair z y)).2) (Nat.pair (Nat.unpair (Nat.pair z y)).2 (encode (decode (Nat.unpair (Nat.pair z y)).1)))) (Nat.pair (Nat.unpair n).2 (encode (decode (Nat.unpair n).1))) = encode (Option.map Prod.snd (decode n)) ** simp only [Nat.unpaired, Nat.unpair_pair, decode_prod_val] ** α✝ : Type u_1 σ : Type u_2 inst✝³ : Primcodable α✝ inst✝² : Primcodable σ α : Type u_3 β : Type u_4 inst✝¹ : Primcodable α inst✝ : Primcodable β n : ℕ ⊢ Nat.rec 0 (fun n_1 n_ih => Nat.rec 0 (fun n n_ih => Nat.succ n) (encode (decode (Nat.unpair n).2))) (encode (decode (Nat.unpair n).1)) = encode (Option.map Prod.snd (Option.bind (decode (Nat.unpair n).1) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).2))) ** cases @decode α _ n.unpair.1 <;> simp ** case some α✝ : Type u_1 σ : Type u_2 inst✝³ : Primcodable α✝ inst✝² : Primcodable σ α : Type u_3 β : Type u_4 inst✝¹ : Primcodable α inst✝ : Primcodable β n : ℕ val✝ : α ⊢ Nat.rec 0 (fun n n_ih => Nat.succ n) (encode (decode (Nat.unpair n).2)) = encode (decode (Nat.unpair n).2) ** cases @decode β _ n.unpair.2 <;> simp ** Qed
| |
Primrec.pair ** α✝ : Type u_1 σ : Type u_2 inst✝⁴ : Primcodable α✝ inst✝³ : Primcodable σ α : Type u_3 β : Type u_4 γ : Type u_5 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ f : α → β g : α → γ hf : Primrec f hg : Primrec g n : ℕ ⊢ (Nat.casesOn (encode (decode n)) 0 fun n => Nat.succ (Nat.pair (Nat.pred (encode (Option.map f (decode n)))) (Nat.pred (encode (Option.map g (decode n)))))) = encode (Option.map (fun a => (f a, g a)) (decode n)) ** cases @decode α _ n <;> simp [encodek] ** Qed
| |
Primrec.unpair ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ n : ℕ ⊢ ((Nat.unpair n).1, (Nat.unpair n).2) = Nat.unpair n ** simp ** Qed
| |
Primrec.list_get?₁ ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ a : α l : List α n : ℕ ⊢ (Nat.casesOn n (Nat.succ (encode a)) fun n => encode (List.get? l (ofNat ℕ n))) = encode (List.get? (a :: l) (ofNat ℕ n)) ** cases n <;> simp ** Qed
| |
Primrec₂.of_eq ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f g : α → β → σ hg : Primrec₂ f H : ∀ (a : α) (b : β), f a b = g a b ⊢ f = g ** funext a b ** case h.h α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f g : α → β → σ hg : Primrec₂ f H : ∀ (a : α) (b : β), f a b = g a b a : α b : β ⊢ f a b = g a b ** apply H ** Qed
| |
Primrec₂.natPair ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ ⊢ Primrec₂ Nat.pair ** simp [Primrec₂, Primrec] ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ ⊢ Nat.Primrec fun n => Nat.succ n ** constructor ** Qed
| |
Primrec₂.ofNat_iff ** α✝ : Type u_1 β✝ : Type u_2 σ✝ : Type u_3 inst✝⁵ : Primcodable α✝ inst✝⁴ : Primcodable β✝ inst✝³ : Primcodable σ✝ α : Type u_4 β : Type u_5 σ : Type u_6 inst✝² : Denumerable α inst✝¹ : Denumerable β inst✝ : Primcodable σ f : α → β → σ ⊢ (Primrec fun n => f (ofNat (α × β) n).1 (ofNat (α × β) n).2) ↔ Primrec (Nat.unpaired fun m n => f (ofNat α m) (ofNat β n)) ** simp ** Qed
| |
Primrec₂.uncurry ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : α → β → σ ⊢ Primrec (Function.uncurry f) ↔ Primrec₂ f ** rw [show Function.uncurry f = fun p : α × β => f p.1 p.2 from funext fun ⟨a, b⟩ => rfl] ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : α → β → σ ⊢ (Primrec fun p => f p.1 p.2) ↔ Primrec₂ f ** rfl ** Qed
| |
Primrec₂.nat_iff ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : α → β → σ ⊢ Primrec₂ f ↔ Nat.Primrec (Nat.unpaired fun m n => encode (Option.bind (decode m) fun a => Option.map (f a) (decode n))) ** have :
∀ (a : Option α) (b : Option β),
Option.map (fun p : α × β => f p.1 p.2)
(Option.bind a fun a : α => Option.map (Prod.mk a) b) =
Option.bind a fun a => Option.map (f a) b := fun a b => by
cases a <;> cases b <;> rfl ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : α → β → σ this : ∀ (a : Option α) (b : Option β), Option.map (fun p => f p.1 p.2) (Option.bind a fun a => Option.map (Prod.mk a) b) = Option.bind a fun a => Option.map (f a) b ⊢ Primrec₂ f ↔ Nat.Primrec (Nat.unpaired fun m n => encode (Option.bind (decode m) fun a => Option.map (f a) (decode n))) ** simp [Primrec₂, Primrec, this] ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ f : α → β → σ a : Option α b : Option β ⊢ Option.map (fun p => f p.1 p.2) (Option.bind a fun a => Option.map (Prod.mk a) b) = Option.bind a fun a => Option.map (f a) b ** cases a <;> cases b <;> rfl ** Qed
| |
Primrec.nat_iterate ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 σ : Type u_5 inst✝⁴ : Primcodable α inst✝³ : Primcodable β inst✝² : Primcodable γ inst✝¹ : Primcodable δ inst✝ : Primcodable σ f : α → ℕ g : α → β h : α → β → β hf : Primrec f hg : Primrec g hh : Primrec₂ h a : α ⊢ Nat.rec (g a) (fun n IH => h a (n, IH).2) (f a) = (h a)^[f a] (g a) ** induction f a <;> simp [*, -Function.iterate_succ, Function.iterate_succ'] ** Qed
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.