import OAI.Combinatorics.ListHadwiger.Thinning namespace OAI namespace LinearListHadwiger.Sampling open Finset variable {κ ι Ω V : Type} [Fintype κ] [Fintype ι] [Fintype Ω] [Fintype V] [DecidableEq V] [DecidableEq κ] [DecidableEq ι] lemma probability_product_curry (μ : κ → ι → Ω → ℝ) (P : (κ → ι → Ω) → Prop) : probability (productMass (fun k => productMass (μ k))) P = probability (productMass (fun j : κ × ι => μ j.1 j.2)) (fun x => P (fun k i => x (k,i))) := by classical let e : (κ × ι → Ω) ≃ (κ → ι → Ω) := Equiv.curry κ ι Ω have hm (x : κ × ι → Ω) : productMass (fun k => productMass (μ k)) (e x) = productMass (fun j : κ × ι => μ j.1 j.2) x := by simp [productMass,e,Fintype.prod_prod_type] unfold probability rw [←e.sum_comp] apply sum_congr rfl intro x _ change (if P (fun k i => x (k,i)) then _ else 0) = _ rw [hm] lemma unionRemainders_survival {π : Ω → ℝ} (hπ : ∀ ω, 0 ≤ π ω) (hs : ∑ ω, π ω=1) (E : ι → Finset V) (I : Ω → Finset V) (v : V) : probability (productMass (fun _ : κ × ι => π)) (fun x => v∈unionRemainders E I x) ≤ (Fintype.card κ : ℝ)*probability (productMass (fun _ : ι => π)) (fun x => v∈remainder E I x) := by classical let μ := productMass (fun _ : ι => π) have hμ : ∀ ω, 0 ≤ μ ω := productMass_nonneg (fun _ => hπ) have hsum : ∑ ω, μ ω=1 := productMass_sum (fun _ => hs) have he : probability (productMass (fun _ : κ × ι => π)) (fun x => v∈unionRemainders E I x) = probability (productMass (fun _ : κ => μ)) (fun x => ∃ k, v∈remainder E I (x k)) := by rw [probability_product_curry (fun _ _ => π)] congr 1 funext x simp only [unionRemainders,mem_biUnion,mem_univ,true_and] rw [he] calc _ ≤ ∑ k, probability (productMass (fun _ : κ => μ)) (fun x => v∈remainder E I (x k)) := probability_union_le (productMass_nonneg (fun _ => hμ)) _ _ = ∑ _ : κ, probability μ (fun x => v∈remainder E I x) := by apply sum_congr rfl intro k _ exact probability_coordinate (fun _ => hsum) k (fun x => v∈remainder E I x) _ = _ := by simp [μ] theorem bounded_sampler (G : SimpleGraph V) {n r : ℝ} (hn : 1 ≤ n) (hr : 1 ≤ r) (horder : (Fintype.card V : ℝ) ≤ n) (hf : Fractional.number G ≤ r) : ∃ (π : Finset V → ℝ) (I : Finset V → Finset V), (∀ s, 0 ≤ π s) ∧ (∑ s, π s=1) ∧ (∀ s, Fractional.Stable G (I s) ∧ (I s).card ≤ ⌈n/r⌉₊) ∧ ∀ v, probability π (fun s => v∈I s)=1/(2*r) := by classical obtain ⟨π,hπ,hs,hstable,hmarg⟩ := Fractional.bounded_sampling G hn hr horder hf let I (s : Finset V) : Finset V := if Fractional.Stable G s ∧ s.card ≤ ⌈n/r⌉₊ then s else ∅ refine ⟨π,I,hπ,hs,?_,?_⟩ · intro s dsimp [I] split_ifs with h · exact h · exact ⟨Fractional.stable_empty G,by simp⟩ · intro v rw [←hmarg v] unfold probability Fractional.marginal apply sum_congr rfl intro s _ by_cases hπs : π s=0 · simp [hπs] · simp only [I,ite_eq_left (hstable s hπs)] end LinearListHadwiger.Sampling namespace LinearListHadwiger.Sampling open Filter Finset Fractional theorem source_private_fractional : ∃ T : ℝ, ∀ t n r : ℝ, T ≤ t → 1 ≤ n → n ≤ t^(11/10:ℝ) → t/Real.log t ≤ r → r ≤ 4*t → let y := max 1 (t/r) let N := ⌈192*r*Real.log (2*y)⌉₊ let M := ⌈n/r⌉₊ let q := ⌈t/(⌊r/200⌋₊:ℝ)⌉₊^2 ∀ (V Ω κ : Type) [Fintype V] [DecidableEq V] [Fintype Ω] [Fintype κ] [DecidableEq κ], Fintype.card κ=q → ∀ G : SimpleGraph V, ∀ π : Ω → ℝ, (∀ ω, 0 ≤ π ω) → (∑ ω, π ω=1) → ∀ (I : Ω → Finset V) (E : Fin N → Finset V) (H : Finset V), (∀ ω, (I ω).card ≤ M) → (∀ v, probability π (fun ω => v∈I ω)=1/(2*r)) → (∀ v, (N:ℝ)/2 ≤ (univ.filter (fun j => v∈E j)).card) → r/4 < expectation (productMass (fun _ : Fin N => π)) (fun x => number (G.induce (↑(remainder E I x ∩ H) : Set V))) → probability (productMass (fun _ : κ => productMass (fun _ : Fin N => π))) (fun x => ∃ i, number (G.induce (↑(privateCopy (fun z => remainder E I z ∩ H) i x) : Set V)) ≤ r/8) < 1/50 := by classical obtain ⟨T₀,hT₀⟩ := eventually_atTop.mp parameter_range_cutoff obtain ⟨T₁,hT₁⟩ := source_uniform_concentration (by norm_num : (1:ℕ)≤192) (by norm_num : (0:ℝ)<1/20) obtain ⟨T₂,hT₂⟩ := eventually_atTop.mp private_tail_cutoff refine ⟨max T₀ (max T₁ T₂),?_⟩ intro t n r ht hn hnt hrl hru dsimp only intro V Ω κ _ _ _ _ _ hκ G π hπ hs I E H hI hmarg hE hmean have hp := hT₀ t ((le_max_left _ _).trans ht) obtain ⟨hr,hy,hyt,hry,hry'⟩ := hp.2.2 r hrl hru have htpos : 0 < t := by linarith [hp.1] obtain ⟨hN,hN',hM,hM'⟩ := sample_size_parameters hp.1 hn hnt hr hy hry (by norm_num : (1:ℕ)≤192) have hq := source_q_pos htpos hr let N := ⌈192*r*Real.log (2*max 1 (t/r))⌉₊ let q := ⌈t/(⌊r/200⌋₊:ℝ)⌉₊^2 let p := Real.exp (-((N:ℝ)/(4*r))) let μ := productMass (fun _ : Fin N => π) let W := fun z => remainder E I z ∩ H have hμ : ∀ x, 0 ≤ μ x := productMass_nonneg (fun _ => hπ) have hμs : ∑ x, μ x=1 := productMass_sum (fun _ => hs) have hcard : Fintype.card (Fin N) ≤ q*N := by rw [Fintype.card_fin] simpa only [one_mul] using Nat.mul_le_mul_right N hq have hNcard : 1 ≤ Fintype.card (Fin N) := by simpa [N] using hN have hc := hT₁ t n r ((le_max_left _ _).trans ((le_max_right _ _).trans ht)) hn hnt hrl hru (Fin N) Ω hNcard hcard (fun _ => π) (fun _ => hπ) (fun _ => hs) have hsurv (v : V) : probability μ (fun x => v∈W x) ≤ p := by have hh := remainder_survival_bound (fun _ : Fin N => hs) E I (by linarith : 1 ≤ r) (fun _ => hmarg) (by simpa [N] using hE) v exact (probability_mono hμ (fun _ h => (mem_inter.mp h).1)).trans (by simpa only [μ,p,Fintype.card_fin,N,neg_div] using hh) have hcomm (z : Fin N → Ω) : number (G.induce (↑(H ∩ remainder E I z) : Set V)) = number (G.induce (↑(W z) : Set V)) := congrArg (fun S : Finset V => number (G.induce (↑S : Set V))) (inter_comm _ _) have htail := hc (fun z => number (G.induce (↑(H ∩ remainder E I z) : Set V))) (remainder_fractional_bounded_changes G E I hI H) have hhigh : probability μ (fun z => number (G.induce (↑(W z) : Set V)) ≤ r/5) ≤ 2*Real.exp (-t^(7/10:ℝ)) := by have hh := probability_mono hμ (A := fun z => number (G.induce (↑(W z) : Set V)) ≤ r/5) (B := fun z => r/20 ≤ |number (G.induce (↑(W z) : Set V)) - expectation μ (fun x => number (G.induce (↑(W x) : Set V)))|) (by intro z hz; have ha := neg_le_abs (number (G.induce (↑(W z) : Set V)) - expectation μ (fun x => number (G.induce (↑(W x) : Set V)))); change r/4 < expectation μ (fun x => number (G.induce (↑(W x) : Set V))) at hmean linarith) exact hh.trans (by simpa only [hcomm,show (1/20:ℝ)*r=r/20 by ring] using htail) have hfail := private_fractional_failure (ι := κ) G hμ hμs W (by linarith : 0 < r) (Real.exp_pos _).le hsurv hhigh have hsparse := (source_sparseness hp.1.le hr hru).2.1 have hqt := hT₂ t ((le_max_right _ _).trans ((le_max_right _ _).trans ht)) r hrl hru have hqm : ((q-1:ℕ):ℝ) ≤ q := by exact_mod_cast Nat.sub_le q 1 have hcollision : 4*(q:ℝ)*((q-1:ℕ):ℝ)*p ≤ 4/10000 := by calc _ ≤ 4*(q:ℝ)*(q:ℝ)*p := by gcongr _ = 4*((q:ℝ)^2*p) := by ring _ ≤ _ := by dsimp [p,N,q]; linarith change probability (productMass (fun _ : κ => μ)) (fun x => ∃ i, number (G.induce (↑(privateCopy W i x) : Set V)) ≤ r/8) < 1/50 rw [hκ] at hfail dsimp only [q] at hcollision linarith end LinearListHadwiger.Sampling namespace LinearListHadwiger.Knitting open Finset variable {V W : Type} [Fintype V] [DecidableEq V] [Fintype W] [DecidableEq W] lemma kConnected_image {V W : Type} [Fintype V] [DecidableEq V] [Fintype W] [DecidableEq W] (G : SimpleGraph V) (f : W → V) (hf : Function.Injective f) {T : Finset W} {k : ℕ} (hc : KConnectedOn (G.comap f) T k) : KConnectedOn G (T.image f) k := by classical refine ⟨by simpa only [card_image_of_injective _ hf] using hc.1,?_⟩ intro X hX hx let Y := T.filter (fun w => f w∈X) have hYT : Y ⊆ T := filter_subset _ _ have hYX : Y.image f=X := by ext v constructor · rintro hv obtain ⟨w,hw,rfl⟩ := mem_image.mp hv exact (mem_filter.mp hw).2 · intro hv obtain ⟨w,hw,rfl⟩ := mem_image.mp (hX hv) exact mem_image.mpr ⟨w,mem_filter.mpr ⟨hw,hv⟩,rfl⟩ have hYc : Y.card=X.card := by rw [←hYX,card_image_of_injective _ hf] have hcon := hc.2 Y hYT (hYc ▸ hx) let g : (G.comap f).induce (↑(T\Y) : Set W) →g G.induce (↑(T.image f\X) : Set V) := { toFun := fun w => ⟨f w.val,mem_sdiff.mpr ⟨mem_image.mpr ⟨w.val,(mem_sdiff.mp w.property).1,rfl⟩,by intro h exact (mem_sdiff.mp w.property).2 (mem_filter.mpr ⟨(mem_sdiff.mp w.property).1,h⟩)⟩⟩ map_rel' := by intro u v h; exact h } apply hcon.map g intro v obtain ⟨w,hw,he⟩ := mem_image.mp (mem_sdiff.mp v.property).1 have hwY : w∉Y := by intro hh exact (mem_sdiff.mp v.property).2 (he ▸ (mem_filter.mp hh).2) exact ⟨⟨w,mem_sdiff.mpr ⟨hw,hwY⟩⟩,Subtype.ext he⟩ end LinearListHadwiger.Knitting namespace LinearListHadwiger.Sampling open Finset variable {V Ω : Type} [Fintype V] [DecidableEq V] [Fintype Ω] theorem complement_failure_on (G : SimpleGraph V) [DecidableRel G.Adj] {n r ε : ℝ} {k : ℕ} (hk : 64 ≤ k) (hr : 0 < r) (hrk : r ≤ k) (H : Finset V) (hord : (H.card : ℝ) ≤ n) (hconn : KConnectedOn G H k) {μ : Ω → ℝ} (hμ : ∀ ω, 0 ≤ μ ω) (U : Ω → Finset V) (hsurv : ∀ v, probability μ (fun ω => v∈U ω) ≤ 1/10000) (hε : 0 ≤ ε) (htail : ∀ A : Finset V, probability μ (fun ω => r/100 ≤ |((U ω ∩ A).card : ℝ)-expectation μ (fun z => ((U z ∩ A).card : ℝ))|) ≤ ε) : probability μ (fun ω => ¬((∀ v∈H, k ≤ 2*((H\U ω).filter (G.Adj v)).card) ∧ KConnectedOn G (H\U ω) (k/32))) ≤ (n+(n+1)^(⌈10*(n/k)*Real.log (2*n)⌉₊+1))*ε := by classical let U' (ω : Ω) : Finset ↥H := univ.filter (fun v => v.val∈U ω) have hsurv' (v : ↥H) : probability μ (fun ω => v∈U' ω) ≤ 1/10000 := by simpa only [U',mem_filter,mem_univ,true_and] using hsurv v.val have hcard (A : Finset ↥H) (ω : Ω) : (U' ω ∩ A).card=(U ω ∩ A.image Subtype.val).card := by rw [←card_image_of_injective (U' ω ∩ A) Subtype.val_injective] congr 1 ext v constructor · intro hv obtain ⟨w,hw,rfl⟩ := mem_image.mp hv exact mem_inter.mpr ⟨(mem_filter.mp (mem_inter.mp hw).1).2, mem_image.mpr ⟨w,(mem_inter.mp hw).2,rfl⟩⟩ · intro hv obtain ⟨w,hw,rfl⟩ := mem_image.mp (mem_inter.mp hv).2 exact mem_image.mpr ⟨w,mem_inter.mpr ⟨mem_filter.mpr ⟨mem_univ _,(mem_inter.mp hv).1⟩,hw⟩,rfl⟩ have ht' (A : Finset ↥H) := htail (A.image Subtype.val) have hc := complement_failure (G.induce (↑H : Set V)) hk hr hrk (by simpa using hord) (Knitting.induce_connected G H hconn) hμ U' hsurv' hε (by intro A; simpa only [hcard] using ht' A) have himage (ω : Ω) : (univ\U' ω).image Subtype.val=H\U ω := by ext v simp only [mem_image,mem_sdiff,mem_univ,true_and,U',mem_filter,mem_univ,true_and] constructor · rintro ⟨w,hw,rfl⟩ exact ⟨w.property,hw⟩ · rintro ⟨hv,hu⟩ exact ⟨⟨v,hv⟩,hu,rfl⟩ have hncard (ω : Ω) (v : ↥H) : (((univ\U' ω).filter ((G.induce (↑H : Set V)).Adj v)).card) = ((H\U ω).filter (G.Adj v.val)).card := by rw [←card_image_of_injective _ Subtype.val_injective,←himage ω] congr 1 ext w simp only [mem_image,mem_filter] constructor · rintro ⟨z,⟨hz,hzv⟩,rfl⟩ exact ⟨⟨z,hz,rfl⟩,hzv⟩ · rintro ⟨⟨z,hz,rfl⟩,hzv⟩ exact ⟨z,⟨hz,hzv⟩,rfl⟩ have hg (ω : Ω) (hh : (∀ v : ↥H, k ≤ 2*((univ\U' ω).filter ((G.induce (↑H : Set V)).Adj v)).card) ∧ KConnectedOn (G.induce (↑H : Set V)) (univ\U' ω) (k/32)) : (∀ v∈H, k ≤ 2*((H\U ω).filter (G.Adj v)).card) ∧ KConnectedOn G (H\U ω) (k/32) := by constructor · intro v hv simpa only [hncard] using hh.1 ⟨v,hv⟩ · have hi := Knitting.kConnected_image G Subtype.val Subtype.val_injective hh.2 exact (congrArg (fun T : Finset V => KConnectedOn G T (k/32)) (himage ω)).mp hi exact (probability_mono hμ (fun ω hbad hgood => hbad (hg ω hgood))).trans hc end LinearListHadwiger.Sampling namespace LinearListHadwiger.Sampling open Filter Finset theorem source_complement_on {B : ℕ} (hB : 1 ≤ B) : ∃ T : ℝ, ∀ t n r : ℝ, T ≤ t → 1 ≤ n → n ≤ t^(11/10:ℝ) → t/Real.log t ≤ r → r ≤ 4*t → let y := max 1 (t/r) let N := ⌈B*r*Real.log (2*y)⌉₊ let M := ⌈n/r⌉₊ let q := ⌈t/(⌊r/200⌋₊ : ℝ)⌉₊^2 ∀ (V ι Ω : Type) [Fintype V] [DecidableEq V] [Fintype ι] [Fintype Ω] [DecidableEq ι], ∀ (G : SimpleGraph V) [DecidableRel G.Adj] (k : ℕ) (H : Finset V), 64 ≤ k → r ≤ k → t ≤ k → (H.card : ℝ) ≤ n → KConnectedOn G H k → 1 ≤ Fintype.card ι → Fintype.card ι ≤ q*N → ∀ (μ : ι → Ω → ℝ), (∀ i ω, 0 ≤ μ i ω) → (∀ i, ∑ ω, μ i ω=1) → ∀ U : (ι → Ω) → Finset V, (∀ v, probability (productMass μ) (fun x => v∈U x) ≤ 1/10000) → (∀ (A : Finset V) i x x', (∀ j, j ≠ i → x j=x' j) → |((A ∩ U x).card : ℝ)-(A ∩ U x').card| ≤ 2*M) → probability (productMass μ) (fun x => ¬((∀ v∈H, k ≤ 2*((H \ U x).filter (G.Adj v)).card) ∧ KConnectedOn G (H \ U x) (k/32))) ≤ 1/100 := by classical obtain ⟨T₀,hT₀⟩ := eventually_atTop.mp parameter_range_cutoff obtain ⟨T₁,hT₁⟩ := source_uniform_concentration hB (show (0:ℝ)<1/100 by norm_num) obtain ⟨T₂,hT₂⟩ := eventually_atTop.mp complement_numeric_cutoff refine ⟨max T₀ (max T₁ T₂),?_⟩ intro t n r ht hn hnt hrl hru dsimp only intro V ι Ω _ _ _ _ _ G _ k H hk hrk htk hord hconn hi hiq μ hμ hs U hsurv hchange have hr := (hT₀ t ((le_max_left _ _).trans ht)).2.2 r hrl hru have hconc := hT₁ t n r ((le_max_left _ _).trans ((le_max_right _ _).trans ht)) hn hnt hrl hru ι Ω hi hiq μ hμ hs have htail (A : Finset V) : probability (productMass μ) (fun x => r/100 ≤ |((U x ∩ A).card : ℝ)-expectation (productMass μ) (fun z => ((U z ∩ A).card : ℝ))|) ≤ 2*Real.exp (-t^(7/10:ℝ)) := by have hh := hconc (fun z => ((A ∩ U z).card : ℝ)) (hchange A) simpa only [inter_comm,show (1/100:ℝ)*r=r/100 by ring] using hh have hh := complement_failure_on G hk (by linarith [hr.1]) hrk H hord hconn (productMass_nonneg hμ) U hsurv (by positivity) htail exact hh.trans (hT₂ t ((le_max_right _ _).trans ((le_max_right _ _).trans ht)) n k hn hnt htk) end LinearListHadwiger.Sampling end OAI