import Mathlib namespace OAI section section section noncomputable section section section section section section section section section section section section section section open Set MeasureTheory open scoped BigOperators namespace CannonAnnular theorem weighted_open_interval_cover {ι : Type*} [Fintype ι] (L U w : ι → ℝ) (a b c : ℝ) (hab : a ≤ b) (hLU : ∀ i, L i ≤ U i) (hw : ∀ i, 0 ≤ w i) (hcover : ∀ t ∈ Ioo a b, c ≤ ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => w i) t) : c * (b - a) ≤ ∑ i, w i * (U i - L i) := by have hint (l u v : ℝ) : Integrable ((Icc l u).indicator (fun _ : ℝ => v)) := by apply (integrable_indicator_iff measurableSet_Icc).mpr exact integrableOn_const (by simp [Real.volume_Icc]) have hinta : Integrable ((Ioo a b).indicator (fun _ : ℝ => c)) := by apply (integrable_indicator_iff measurableSet_Ioo).mpr exact integrableOn_const (by simp [Real.volume_Ioo]) have hle : (Ioo a b).indicator (fun _ : ℝ => c) ≤ fun t => ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => w i) t := by intro t by_cases ht : t ∈ Ioo a b · simpa [Set.indicator_of_mem ht] using hcover t ht · rw [Set.indicator_of_notMem ht] exact Finset.sum_nonneg (fun i _ => Set.indicator_nonneg (fun _ _ => hw i) _) have hi := integral_mono hinta (integrable_finsetSum _ (fun i _ => hint (L i) (U i) (w i))) hle rw [integral_finsetSum _ (fun i _ => hint (L i) (U i) (w i))] at hi simpa only [integral_indicator_const _ measurableSet_Ioo, integral_indicator_const _ measurableSet_Icc, Measure.real, Real.volume_Ioo, Real.volume_Icc, ENNReal.toReal_ofReal (sub_nonneg.mpr hab), ENNReal.toReal_ofReal (sub_nonneg.mpr (hLU _)), smul_eq_mul, mul_comm] using hi theorem interval_energy_cover {ι : Type*} [Fintype ι] (L U w : ι → ℝ) (a b : ℝ) (hab : a ≤ b) (hLU : ∀ i, L i ≤ U i) (hw : ∀ i, 0 ≤ w i) (hcover : ∀ t ∈ Ioo a b, 1 ≤ ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => w i) t) : (b - a)^2 ≤ (∑ i, (w i)^2) * (∑ i, (U i - L i)^2) := by have h := weighted_open_interval_cover L U w a b 1 hab hLU hw hcover simp only [one_mul] at h exact (pow_le_pow_left₀ (sub_nonneg.mpr hab) h 2).trans (Finset.sum_mul_sq_le_sq_mul_sq Finset.univ w (fun i => U i - L i)) theorem overlap_energy {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (R : α → Finset β) (w : β → ℝ) (kr kc : ℕ) (hrow : ∀ i, (R i).card ≤ kr) (hcol : ∀ j, (Finset.univ.filter (fun i => j ∈ R i)).card ≤ kc) : (∑ i, (∑ j ∈ R i, w j)^2) ≤ (kr : ℝ) * kc * (∑ j, (w j)^2) := by classical have hrowE (i : α) : (∑ j ∈ R i, w j)^2 ≤ (kr : ℝ) * (∑ j ∈ R i, (w j)^2) := by have h := Finset.sum_mul_sq_le_sq_mul_sq (R i) (fun _ => (1:ℝ)) w simp only [one_mul, one_pow, Finset.sum_const, nsmul_eq_mul, mul_one] at h exact h.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast hrow i) (Finset.sum_nonneg (fun j _ => sq_nonneg (w j)))) have hswap : (∑ i, ∑ j ∈ R i, (w j)^2) = ∑ j, ((Finset.univ.filter (fun i => j ∈ R i)).card : ℝ) * (w j)^2 := by calc _ = ∑ i, ∑ j : β, if j ∈ R i then (w j)^2 else 0 := by apply Finset.sum_congr rfl intro i _ rw [← Finset.sum_filter] simp _ = ∑ j, ∑ i ∈ Finset.univ.filter (fun i => j ∈ R i), (w j)^2 := by simp only [Finset.sum_filter] rw [Finset.sum_comm] _ = _ := by simp calc _ ≤ ∑ i, (kr : ℝ) * (∑ j ∈ R i, (w j)^2) := Finset.sum_le_sum (fun i _ => hrowE i) _ = (kr : ℝ) * (∑ i, ∑ j ∈ R i, (w j)^2) := (Finset.mul_sum ..).symm _ = (kr : ℝ) * (∑ j, ((Finset.univ.filter (fun i => j ∈ R i)).card : ℝ) * (w j)^2) := by rw [hswap] _ ≤ (kr : ℝ) * (∑ j, (kc : ℝ) * (w j)^2) := by apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _) exact Finset.sum_le_sum (fun j _ => mul_le_mul_of_nonneg_right (by exact_mod_cast hcol j) (sq_nonneg (w j))) _ = _ := by rw [← Finset.mul_sum]; ring theorem annular_energy_transfer {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ] [DecidableEq β] [DecidableEq γ] (R : α → Finset β) (S : α → Finset γ) (ω : β → ℝ) (ρ : γ → ℝ) (L U : α → ℝ) (a b : ℝ) (hab : a ≤ b) (hω : ∀ j, 0 ≤ ω j) (_hρ : ∀ j, 0 ≤ ρ j) (hLU : ∀ i, L i ≤ U i) (hlocal : ∀ i, U i - L i ≤ ∑ j ∈ S i, ρ j) (kr kc lr lc : ℕ) (hRrow : ∀ i, (R i).card ≤ kr) (hRcol : ∀ j, (Finset.univ.filter (fun i => j ∈ R i)).card ≤ kc) (hSrow : ∀ i, (S i).card ≤ lr) (hScol : ∀ j, (Finset.univ.filter (fun i => j ∈ S i)).card ≤ lc) (hcover : ∀ t ∈ Ioo a b, 1 ≤ ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => ∑ j ∈ R i, ω j) t) : (b-a)^2 ≤ ((kr:ℝ)*kc*lr*lc) * (∑ j, (ω j)^2) * (∑ j, (ρ j)^2) := by have henergy := interval_energy_cover L U (fun i => ∑ j ∈ R i, ω j) a b hab hLU (fun i => Finset.sum_nonneg (fun j _ => hω j)) hcover have hωE := overlap_energy R ω kr kc hRrow hRcol have hρE : (∑ i, (U i-L i)^2) ≤ (lr:ℝ)*lc*(∑ j, (ρ j)^2) := by apply le_trans _ (overlap_energy S ρ lr lc hSrow hScol) exact Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ (sub_nonneg.mpr (hLU i)) (hlocal i) 2) have hmul := mul_le_mul hωE hρE (Finset.sum_nonneg (fun i _ => sq_nonneg (U i-L i))) (by positivity) calc _ ≤ ((kr:ℝ)*kc*(∑ j, (ω j)^2)) * ((lr:ℝ)*lc*(∑ j, (ρ j)^2)) := henergy.trans hmul _ = _ := by ring end CannonAnnular namespace CannonAnnular theorem finite_charge_cover {α β : Type*} [Fintype α] [Fintype β] (A : Finset α) (B : Finset β) (R : α → Finset β) (w : β → ℝ) (hw : ∀ j, 0 ≤ w j) (hcharge : ∀ j ∈ B, ∃ i ∈ A, j ∈ R i) : (∑ j ∈ B, w j) ≤ ∑ i ∈ A, ∑ j ∈ R i, w j := by classical calc _ ≤ ∑ j ∈ B, ∑ i ∈ A, if j ∈ R i then w j else 0 := by apply Finset.sum_le_sum intro j hj obtain ⟨i, hi, hij⟩ := hcharge j hj have hh := Finset.single_le_sum (s := A) (f := fun i => if j ∈ R i then w j else 0) (fun i _ => by split_ifs; exact hw j; exact le_rfl) hi simpa only [ite_eq_left hij] using hh _ = ∑ i ∈ A, ∑ j ∈ B, if j ∈ R i then w j else 0 := Finset.sum_comm _ ≤ _ := by apply Finset.sum_le_sum intro i _ rw [← Finset.sum_filter] apply Finset.sum_le_sum_of_subset_of_nonneg · intro j hj exact (Finset.mem_filter.mp hj).2 · exact fun j _ _ => hw j theorem interval_cover_of_charging {α β : Type*} [Fintype α] [Fintype β] (R : α → Finset β) (w : β → ℝ) (L U : α → ℝ) (hw : ∀ j, 0 ≤ w j) (t : ℝ) (B : Finset β) (hB : 1 ≤ ∑ j ∈ B, w j) (hcharge : ∀ j ∈ B, ∃ i, t ∈ Icc (L i) (U i) ∧ j ∈ R i) : 1 ≤ ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => ∑ j ∈ R i, w j) t := by classical let A := Finset.univ.filter (fun i => t ∈ Icc (L i) (U i)) have hh := finite_charge_cover A B R w hw (by intro j hj obtain ⟨i, hit, hij⟩ := hcharge j hj exact ⟨i, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hit⟩, hij⟩) apply hB.trans (hh.trans_eq _) simp only [A, Finset.sum_filter] apply Finset.sum_congr rfl intro i _ by_cases hi : t ∈ Icc (L i) (U i) <;> simp [hi] end CannonAnnular namespace CannonAnnular theorem annular_net_charge {X Y α : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y] (q : α → X) (h : X → Y) (f : X → ℝ) (L U : α → ℝ) (x y z : X) (ξ : Y) (R r C t : ℝ) (hR : 0 < R) (hr : 0 ≤ r) (hrR : r ≤ R / 2) (hC : 0 ≤ C) (hy : R ≤ dist y x ∧ dist y x ≤ 2*R) (hnet : ∃ i, dist (q i) y ≤ r) (hzoom : ∀ u v, dist u x ≤ 3*R → dist v x ≤ 3*R → dist (h u) (h v) ≤ C / R * dist u v) (hξ : dist ξ (h y) ≤ 2 * (r / R)) (hz : dist z y ≤ r / 4) (hf : f z = t) (hrange : ∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i)) : ∃ i, R/2 ≤ dist (q i) x ∧ dist (q i) x ≤ 3*R ∧ t ∈ Icc (L i) (U i) ∧ dist ξ (h (q i)) ≤ (C+2)*(r/R) := by obtain ⟨i, hi⟩ := hnet have hqu : dist (q i) x ≤ 3*R := (dist_triangle (q i) y x).trans (by linarith [hy.2]) have hql : R/2 ≤ dist (q i) x := by have hh := dist_triangle y (q i) x rw [dist_comm y (q i)] at hh linarith [hy.1] refine ⟨i, hql, hqu, ?_, ?_⟩ · rw [← hf] apply hrange have hh := dist_triangle z y (q i) rw [dist_comm y (q i)] at hh linarith · have hqy : dist (h (q i)) (h y) ≤ C/R*r := by exact (hzoom _ _ hqu (by linarith [hy.2])).trans (mul_le_mul_of_nonneg_left hi (div_nonneg hC hR.le)) have hh := dist_triangle ξ (h y) (h (q i)) rw [dist_comm (h y) (h (q i))] at hh calc _ ≤ 2*(r/R) + C/R*r := hh.trans (add_le_add hξ hqy) _ = _ := by ring theorem charged_center_in_annulus {X : Type*} [PseudoMetricSpace X] (x q p : X) (R d : ℝ) (hR : 0 ≤ R) (hq : R/2 ≤ dist q x ∧ dist q x ≤ 3*R) (hpq : dist p q ≤ d) (hd : d ≤ R/4) : R/4 ≤ dist p x ∧ dist p x ≤ 4*R := by constructor · have hh := dist_triangle q p x rw [dist_comm q p] at hh linarith [hq.1] · have hh := dist_triangle p q x linarith [hq.2] end CannonAnnular end namespace CannonCapacity open Finset Set attribute [local instance] Classical.propDecidable theorem bin_interval_detection {ι β : Type*} [Fintype ι] [Fintype β] (near : β → ι → Prop) (L U v t : ι → ℝ) (a b B : ℝ) (hab : a < b) (hLU : ∀ i, L i ≤ U i) (hv : ∀ i, 0 ≤ v i) (hlen : ∀ i, U i-L i = (b-a)*t i) (hmult : ∀ i, ((univ.filter (fun z : β => near z i)).card : ℝ) ≤ B) (hcover : ∀ z s, s ∈ Ioo a b → 1 ≤ ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => if near z i then v i else 0) s) : (Fintype.card β : ℝ) ≤ B * ∑ i, v i*t i := by classical have ht : ∀ i, 0 ≤ t i := by intro i have := sub_nonneg.mpr (hLU i) rw [hlen] at this exact nonneg_of_mul_nonneg_right this (sub_pos.mpr hab) have hb (z : β) := CannonAnnular.weighted_open_interval_cover L U (fun i => if near z i then v i else 0) a b 1 hab.le hLU (fun i => by split_ifs; exact hv i; exact le_rfl) (hcover z) have hh := Finset.sum_le_sum (s := univ) (fun z _ => hb z) simp only [one_mul, sum_const, card_univ, nsmul_eq_mul] at hh have hr : ∑ z : β, ∑ i : ι, (if near z i then v i else 0)*(U i-L i) ≤ (b-a)*(B * ∑ i, v i*t i) := by rw [sum_comm] calc _ = ∑ i : ι, ((univ.filter (fun z : β => near z i)).card : ℝ)*v i*((b-a)*t i) := by apply sum_congr rfl intro i _ rw [← sum_mul,hlen,← sum_filter] simp _ ≤ ∑ i : ι, B*v i*((b-a)*t i) := by apply sum_le_sum intro i _ exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (hmult i) (hv i)) (mul_nonneg (sub_pos.mpr hab).le (ht i)) _ = _ := by simp_rw [show ∀ i, B*v i*((b-a)*t i) = (b-a)*B*(v i*t i) by intro i; ring] rw [← mul_sum]; ring have hfinal := hh.trans hr rw [mul_comm (Fintype.card β : ℝ)] at hfinal exact (mul_le_mul_iff_right₀ (sub_pos.mpr hab)).mp hfinal theorem mixed_capacity_of_detection {ι : Type*} [Fintype ι] (t σ : ι → ℝ) (ht : ∀ i, 0 ≤ t i) (hσ : ∀ i, 0 ≤ σ i) {L Q C B E bins : ℝ} (hL : 0 < L) (hQ : 0 < Q) (_hC : 0 ≤ C) (hB : 0 < B) (hE : 0 ≤ E) (hb : 0 ≤ bins) (henergy : ∑ i, (σ i)^2 ≤ E) (hlarge : 2*B*C ≤ L*bins) (hsmall : 4*B^2*E*C < Q*bins^2) (hdetect : bins ≤ B * ∑ i, (if L ≤ Q*t i then 1 else σ i)*t i) : C ≤ ∑ i, min (L*t i) (Q*(t i)^2) := by classical let large := univ.filter (fun i => L ≤ Q*t i) let small := univ.filter (fun i => ¬ L ≤ Q*t i) have hnon : ∀ i, 0 ≤ min (L*t i) (Q*(t i)^2) := fun i => le_min (mul_nonneg hL.le (ht i)) (mul_nonneg hQ.le (sq_nonneg _)) by_contra hn have hcap : ∑ i, min (L*t i) (Q*(t i)^2) < C := lt_of_not_ge hn have hla : L * (∑ i ∈ large, t i) ≤ C := by calc _ = ∑ i ∈ large, min (L*t i) (Q*(t i)^2) := by rw [mul_sum] apply sum_congr rfl intro i hi have hh := (mem_filter.mp hi).2 rw [min_eq_left (by nlinarith [mul_le_mul_of_nonneg_right hh (ht i)])] _ ≤ ∑ i, min (L*t i) (Q*(t i)^2) := sum_le_sum_of_subset_of_nonneg (filter_subset _ _) (fun i _ _ => hnon i) _ ≤ C := hcap.le have hsm : Q * (∑ i ∈ small, (t i)^2) ≤ C := by calc _ = ∑ i ∈ small, min (L*t i) (Q*(t i)^2) := by rw [mul_sum] apply sum_congr rfl intro i hi have hh := le_of_not_ge (mem_filter.mp hi).2 rw [min_eq_right (by nlinarith [mul_le_mul_of_nonneg_right hh (ht i)])] _ ≤ ∑ i, min (L*t i) (Q*(t i)^2) := sum_le_sum_of_subset_of_nonneg (filter_subset _ _) (fun i _ _ => hnon i) _ ≤ C := hcap.le have hde : bins ≤ B*((∑ i ∈ large, t i) + ∑ i ∈ small, σ i*t i) := by convert hdetect using 1 congr 1 simp only [large,small,sum_filter,← sum_add_distrib] apply sum_congr rfl intro i _ split_ifs <;> simp_all have hmass : bins ≤ 2*B*(∑ i ∈ small, σ i*t i) := by nlinarith have hCS := sum_mul_sq_le_sq_mul_sq small σ t have hsumσ : ∑ i ∈ small, (σ i)^2 ≤ E := (sum_le_sum_of_subset_of_nonneg (filter_subset _ _) (fun i _ _ => sq_nonneg _)).trans henergy have hsmt : 0 ≤ ∑ i ∈ small, (t i)^2 := sum_nonneg (fun _ _ => sq_nonneg _) have hsms : 0 ≤ ∑ i ∈ small, σ i*t i := sum_nonneg (fun i _ => mul_nonneg (hσ i) (ht i)) have hsquare : bins^2 ≤ 4*B^2*(∑ i ∈ small, σ i*t i)^2 := by nlinarith [sq_le_sq₀ hb (mul_nonneg (by positivity) hsms) |>.mpr hmass] have hcs' : (∑ i ∈ small, σ i*t i)^2 ≤ E*(∑ i ∈ small, (t i)^2) := hCS.trans (mul_le_mul_of_nonneg_right hsumσ hsmt) have hbound : Q*bins^2 ≤ 4*B^2*E*C := by calc _ ≤ Q*(4*B^2*(∑ i ∈ small, σ i*t i)^2) := mul_le_mul_of_nonneg_left hsquare hQ.le _ ≤ Q*(4*B^2*(E*(∑ i ∈ small, (t i)^2))) := by gcongr _ = (4*B^2*E)*(Q*(∑ i ∈ small, (t i)^2)) := by ring _ ≤ _ := mul_le_mul_of_nonneg_left hsm (by positivity) exact (not_lt_of_ge hbound) hsmall end CannonCapacity end section open Finset namespace CannonCapacity attribute [local instance] Classical.propDecidable theorem weighted_separated_subset {ι : Type*} (S : Finset ι) (rel : ι → ι → Prop) (hself : ∀ i, rel i i) (hsymm : ∀ i j, rel i j → rel j i) (w : ι → ℝ) (hw : ∀ i ∈ S, 0 ≤ w i) (N : ℕ) (hN : ∀ i ∈ S, (S.filter (rel i)).card ≤ N) : ∃ T ⊆ S, (∀ i ∈ T, ∀ j ∈ T, i ≠ j → ¬ rel i j) ∧ (∑ i ∈ S, w i) ≤ (N : ℝ) * ∑ i ∈ T, w i := by classical induction S using Finset.strongInductionOn with | _ S ih => by_cases hS : S.Nonempty · obtain ⟨i,hi,hmax⟩ := S.exists_max_image w hS let U := S.filter (fun j => ¬ rel i j) have hUS : U ⊆ S := filter_subset _ _ have hiU : i ∉ U := by simp [U,hself] have hss : U ⊂ S := by refine ⟨hUS,?_⟩ intro hh exact hiU (hh hi) obtain ⟨T,hTU,hsep,hbound⟩ := ih U hss (fun j hj => hw j (hUS hj)) (by intro j hj exact (Finset.card_le_card (Finset.filter_subset_filter _ hUS)).trans (hN j (hUS hj))) have hiT : i ∉ T := fun h => hiU (hTU h) refine ⟨insert i T,Finset.insert_subset_iff.mpr ⟨hi,hTU.trans hUS⟩,?_,?_⟩ · intro j hj k hk hjk rcases mem_insert.mp hj with rfl | hjT · rcases mem_insert.mp hk with rfl | hk · exact fun _ => hjk rfl · exact (mem_filter.mp (hTU hk)).2 · rcases mem_insert.mp hk with rfl | hk · exact fun h => (mem_filter.mp (hTU hjT)).2 (hsymm _ _ h) · exact hsep j hjT k hk hjk · have hnear : ∑ j ∈ S.filter (rel i), w j ≤ (N : ℝ)*w i := by calc _ ≤ ∑ j ∈ S.filter (rel i), w i := by exact Finset.sum_le_sum (fun j hj => hmax j (mem_filter.mp hj).1) _ = ((S.filter (rel i)).card : ℝ)*w i := by simp _ ≤ _ := mul_le_mul_of_nonneg_right (by exact_mod_cast hN i hi) (hw i hi) have hsplit : (∑ j ∈ S, w j) = (∑ j ∈ S.filter (rel i), w j) + ∑ j ∈ U, w j := by symm exact Finset.sum_filter_add_sum_filter_not S (rel i) w rw [hsplit,Finset.sum_insert hiT] nlinarith · refine ⟨∅,empty_subset _,by simp,?_⟩ simp [Finset.not_nonempty_iff_eq_empty.mp hS] end CannonCapacity end namespace CannonCapacity open Set Metric Finset attribute [local instance] Classical.propDecidable def clip (a b x : ℝ) : ℝ := max a (min b x) lemma clip_mono (a b : ℝ) : Monotone (clip a b) := by intro x y hxy exact max_le_max le_rfl (min_le_min le_rfl hxy) lemma mem_clipped_iff {a b l u t : ℝ} (ht : t ∈ Ioo a b) : t ∈ Icc (clip a b l) (clip a b u) ↔ t ∈ Icc l u := by simp only [Set.mem_Icc,clip] constructor · rintro ⟨hl,hu⟩ constructor · by_contra hh have hh' : t < l := lt_of_not_ge hh have : t < min b l := lt_min ht.2 hh' exact (not_lt_of_ge ((le_max_right _ _).trans hl)) this · by_contra hh have hh' : u < t := lt_of_not_ge hh have : max a (min b u) < t := max_lt ht.1 (lt_of_le_of_lt (min_le_right _ _) hh') exact (not_lt_of_ge hu) this · rintro ⟨hl,hu⟩ exact ⟨max_le ht.1.le ((min_le_right _ _).trans hl), (le_min ht.2.le hu).trans (le_max_right _ _)⟩ lemma clip_difference {a b l u : ℝ} (hab : a ≤ b) (hlu : l ≤ u) : clip a b u-clip a b l = max 0 (min b u-max a l) := by unfold clip rcases le_total l a with hla | hal · rw [min_eq_right (hla.trans hab),max_eq_left hla] rw [← max_sub_sub_right] simp only [sub_self,max_comm] · by_cases hlb : l ≤ b · rw [min_eq_right hlb,max_eq_right hal] have hmin : l ≤ min b u := le_min hlb hlu rw [max_eq_right (hal.trans hmin),max_eq_right (sub_nonneg.mpr hmin)] · have hbl := le_of_not_ge hlb simp only [min_eq_left (hbl.trans hlu),min_eq_left hbl,max_eq_right hab,sub_self, max_eq_right hal] exact (max_eq_left (by linarith)).symm theorem clipped_bin_detection {ι β : Type*} [Fintype ι] [Fintype β] (lo hi t σ : ι → ℝ) (large : ι → Prop) (near : β → ι → Prop) (a b B : ℝ) (hab : a < b) (hlohi : ∀ i, lo i ≤ hi i) (hlen : ∀ i, clip a b (hi i)-clip a b (lo i) = (b-a)*t i) (hσ : ∀ i, 0 ≤ σ i) (hmult : ∀ i, ((univ.filter (fun z => near z i)).card : ℝ) ≤ B) (hunmarked : ∀ z s, s ∈ Ioo a b → (∀ i, large i → near z i → s ∈ Icc (lo i) (hi i) → False) → 1 ≤ ∑ i, (Icc (lo i) (hi i)).indicator (fun _ : ℝ => if near z i ∧ ¬large i then σ i else 0) s) : (Fintype.card β : ℝ) ≤ B * ∑ i, (if large i then 1 else σ i)*t i := by apply bin_interval_detection near (fun i => clip a b (lo i)) (fun i => clip a b (hi i)) (fun i => if large i then 1 else σ i) t a b B hab (fun i => clip_mono a b (hlohi i)) (fun i => by split_ifs; norm_num; exact hσ i) hlen hmult intro z s hs by_cases hm : ∃ i, large i ∧ near z i ∧ s ∈ Icc (lo i) (hi i) · obtain ⟨i,hiL,hiN,hit⟩ := hm calc 1 = (Icc (clip a b (lo i)) (clip a b (hi i))).indicator (fun _ : ℝ => if near z i then if large i then 1 else σ i else 0) s := by rw [indicator_of_mem ((mem_clipped_iff hs).mpr hit),ite_eq_left hiN,ite_eq_left hiL] _ ≤ ∑ j, (Icc (clip a b (lo j)) (clip a b (hi j))).indicator (fun _ : ℝ => if near z j then if large j then 1 else σ j else 0) s := by apply single_le_sum _ (mem_univ i) intro j _ apply Set.indicator_nonneg intro _ _ split_ifs; norm_num; exact hσ j; exact le_rfl · apply (hunmarked z s hs (by simpa only [not_exists,not_and] using hm)).trans apply sum_le_sum intro i _ by_cases hit : s ∈ Icc (lo i) (hi i) · rw [indicator_of_mem hit,indicator_of_mem ((mem_clipped_iff hs).mpr hit)] by_cases hn : near z i <;> by_cases hl : large i <;> simp [hn,hl] · rw [indicator_of_notMem hit,indicator_of_notMem (fun hh => hit ((mem_clipped_iff hs).mp hh))] theorem separated_capacity_of_detection {X : Type*} [MetricSpace X] {ι β : Type*} [Fintype ι] [Fintype β] (q : ι → X) (lo hi t σ : ι → ℝ) (near : β → ι → Prop) (a b B L Q E H r : ℝ) (N : ℕ) (hab : a < b) (hlohi : ∀ i, lo i ≤ hi i) (ht : ∀ i, 0 ≤ t i) (hlen : ∀ i, clip a b (hi i)-clip a b (lo i) = (b-a)*t i) (hσ : ∀ i, 0 ≤ σ i) (hL : 0 < L) (hQ : 0 < Q) (hB : 0 < B) (hE : 0 ≤ E) (hHr : 0 ≤ H*r) (hN : 0 < N) (hdegree : ∀ i, (univ.filter (fun j => dist (q i) (q j) ≤ H*r)).card ≤ N) (hmult : ∀ i, ((univ.filter (fun z => near z i)).card : ℝ) ≤ B) (henergy : ∑ i, (σ i)^2 ≤ E) (hlarge : 2*B*N ≤ L*Fintype.card β) (hsmall : 4*B^2*E*N < Q*(Fintype.card β : ℝ)^2) (hunmarked : ∀ z s, s ∈ Ioo a b → (∀ i, L ≤ Q*t i → near z i → s ∈ Icc (lo i) (hi i) → False) → 1 ≤ ∑ i, (Icc (lo i) (hi i)).indicator (fun _ : ℝ => if near z i ∧ ¬L ≤ Q*t i then σ i else 0) s) : ∃ S : Finset ι, (∀ i ∈ S, ∀ j ∈ S, i ≠ j → H*r < dist (q i) (q j)) ∧ 1 ≤ ∑ i ∈ S, min (L*t i) (Q*(t i)^2) := by have hdetect := clipped_bin_detection lo hi t σ (fun i => L ≤ Q*t i) near a b B hab hlohi hlen hσ hmult hunmarked have hcap := mixed_capacity_of_detection t σ ht hσ hL hQ (by positivity : (0:ℝ) ≤ N) hB hE (by positivity) henergy hlarge hsmall hdetect obtain ⟨S,hS,hsep,hbound⟩ := weighted_separated_subset univ (fun i j => dist (q i) (q j) ≤ H*r) (by simpa using fun _i : ι => hHr) (fun i j h => by simpa only [dist_comm] using h) (fun i => min (L*t i) (Q*(t i)^2)) (fun i _ => le_min (mul_nonneg hL.le (ht i)) (mul_nonneg hQ.le (sq_nonneg _))) N (fun i _ => hdegree i) refine ⟨S,fun i hi j hj hij => lt_of_not_ge (hsep i hi j hj hij),?_⟩ have hN' : (0:ℝ) < N := by exact_mod_cast hN nlinarith [hcap.trans hbound] end CannonCapacity end section section open Set MeasureTheory open scoped BigOperators open Set unitInterval namespace CannonAnnular theorem radial_subinterval (g : I → ℝ) (hg : Continuous g) (r s : ℝ) (hrs : r < s) (hzero : g 0 ≤ r) (hone : s ≤ g 1) : ∃ a b : I, a < b ∧ g a = r ∧ g b = s ∧ ∀ t ∈ Icc a b, g t ∈ Icc r s := by have hnon : (g ⁻¹' {s}).Nonempty := by obtain ⟨t, _, ht⟩ := intermediate_value_Icc (show (0:I) ≤ 1 from zero_le) hg.continuousOn ⟨hzero.trans hrs.le, hone⟩ exact ⟨t, ht⟩ have hc : IsCompact (g ⁻¹' {s}) := (isClosed_singleton.preimage hg).isCompact obtain ⟨b, hb, hbmin⟩ := hc.exists_isMinOn hnon continuous_id.continuousOn have hgb : g b = s := hb have hnonr : (Icc (0:I) b ∩ g ⁻¹' {r}).Nonempty := by obtain ⟨t, ht, htr⟩ := intermediate_value_Icc (show (0:I) ≤ b from zero_le) hg.continuousOn ⟨hzero, hgb ▸ hrs.le⟩ exact ⟨t, ht, htr⟩ have hcr : IsCompact (Icc (0:I) b ∩ g ⁻¹' {r}) := isCompact_Icc.inter_right (isClosed_singleton.preimage hg) obtain ⟨a, ha, hamax⟩ := hcr.exists_isMaxOn hnonr continuous_id.continuousOn have hga : g a = r := ha.2 have hab : a < b := lt_of_le_of_ne ha.1.2 (by intro he; rw [he, hgb] at hga; exact hrs.ne hga.symm) refine ⟨a, b, hab, hga, hgb, ?_⟩ intro t ht constructor · by_contra hn have hlt : g t < r := lt_of_not_ge hn obtain ⟨z, hz, hzr⟩ := intermediate_value_Icc ht.2 hg.continuousOn ⟨hlt.le, hgb ▸ hrs.le⟩ have hza : z ≤ a := hamax ⟨⟨zero_le, hz.2⟩, hzr⟩ have htz : t < z := lt_of_le_of_ne hz.1 (by intro he; rw [← he] at hzr; exact hlt.ne hzr) exact (not_lt_of_ge (hza.trans ht.1)) htz · by_contra hn have hlt : s < g t := lt_of_not_ge hn obtain ⟨z, hz, hzs⟩ := intermediate_value_Icc (show (0:I) ≤ t from zero_le) hg.continuousOn ⟨hzero.trans hrs.le, hlt.le⟩ have hbz : b ≤ z := hbmin hzs have hzt : z < t := lt_of_le_of_ne hz.2 (by intro he; rw [he] at hzs; exact hlt.ne' hzs) exact (not_lt_of_ge (ht.2.trans hbz)) hzt end CannonAnnular namespace CannonAnnular theorem radial_subpath {X : Type*} [PseudoMetricSpace X] (o x y : X) (p : Path x y) (r s : ℝ) (hrs : r < s) (hx : dist x o ≤ r) (hy : s ≤ dist y o) : ∃ u v : X, ∃ q : Path u v, Set.range q ⊆ Set.range p ∧ dist u o = r ∧ dist v o = s ∧ s-r ≤ dist u v ∧ ∀ t, dist (q t) o ∈ Icc r s := by obtain ⟨a, b, hab, ha, hb, hbetween⟩ := radial_subinterval (fun t => dist (p t) o) (p.continuous.dist continuous_const) r s hrs (by simpa using hx) (by simpa using hy) refine ⟨p a, p b, p.subpath a b, ?_, ha, hb, ?_, ?_⟩ · rw [Path.range_subpath_of_le _ _ _ hab.le] exact Set.image_subset_range _ _ · have hh := dist_triangle (p b) (p a) o rw [dist_comm (p b) (p a), ha, hb] at hh linarith · intro t have ht : (p.subpath a b) t ∈ Set.range (p.subpath a b) := Set.mem_range_self _ rw [Path.range_subpath_of_le _ _ _ hab.le] at ht obtain ⟨z, hz, hzt⟩ := ht rw [← hzt] exact hbetween z hz theorem continuum_neighborhood_path {X : Type*} [TopologicalSpace X] [LocallyPathConnectedSpace X] {K V : Set X} (hK : IsPreconnected K) (hV : IsOpen V) (hKV : K ⊆ V) {x y : X} (hx : x ∈ K) (hy : y ∈ K) : JoinedIn V x y := by let : PreconnectedSpace K := Subtype.preconnectedSpace hK let : LocallyPathConnectedSpace V := hV.locallyPathConnectedSpace let f : K → V := fun z => ⟨z.val, hKV z.property⟩ have hf : Continuous f := continuous_subtype_val.subtype_mk _ have hconn : IsPreconnected (Set.range f) := isPreconnected_range hf have hmem : f ⟨y,hy⟩ ∈ connectedComponent (f ⟨x,hx⟩) := hconn.subset_connectedComponent ⟨⟨x,hx⟩, rfl⟩ ⟨⟨y,hy⟩, rfl⟩ rw [← pathComponent_eq_connectedComponent] at hmem obtain ⟨p⟩ := hmem exact ⟨p.map continuous_subtype_val, fun t => (p t).property⟩ theorem annular_path_near_continuum {X : Type*} [PseudoMetricSpace X] [LocallyPathConnectedSpace X] (o : X) (r s : ℝ) (hrs : r < s) (K V : Set X) (hK : IsPreconnected K) (hV : IsOpen V) (hKV : K ⊆ V) {x y : X} (hx : x ∈ K) (hy : y ∈ K) (hxr : dist x o ≤ r) (hys : s ≤ dist y o) : ∃ u v : X, ∃ q : Path u v, Set.range q ⊆ V ∧ dist u o = r ∧ dist v o = s ∧ s-r ≤ dist u v ∧ ∀ t, dist (q t) o ∈ Icc r s := by obtain ⟨p, hp⟩ := continuum_neighborhood_path hK hV hKV hx hy obtain ⟨u, v, q, hqp, hu, hv, huv, hq⟩ := radial_subpath o x y p r s hrs hxr hys refine ⟨u, v, q, ?_, hu, hv, huv, hq⟩ rintro z hz obtain ⟨t, rfl⟩ := hqp hz exact hp t end CannonAnnular end end end end end end end end end end end end end end end end end OAI