import Mathlib import OAI.LinearAlgebra.MatrixFields.Construction.JointLossCounts import OAI.LinearAlgebra.MatrixFields.Entropy.JointExtractionRates namespace OAI namespace MatrixAllFields open scoped BigOperators Topology Polynomial section namespace MatrixMultiplication.JointHashing open scoped BigOperators attribute [local instance] Classical.propDecidable variable {P R : Type*} [Fintype P] [CommRing R] def dot (v w : P → R) : R := ∑ i, v i * w i @[simp] theorem dot_add_right (v w u : P → R) : dot v (w + u) = dot v w + dot v u := by simp [dot, mul_add, Finset.sum_add_distrib] @[simp] theorem dot_sub_right (v w u : P → R) : dot v (w - u) = dot v w - dot v u := by simp [dot, mul_sub, Finset.sum_sub_distrib] @[simp] theorem dot_add_left (v w u : P → R) : dot (v + w) u = dot v u + dot w u := by simp [dot, add_mul, Finset.sum_add_distrib] @[simp] theorem dot_sub_left (v w u : P → R) : dot (v - w) u = dot v u - dot w u := by simp [dot, sub_mul, Finset.sum_sub_distrib] abbrev Sample (P R : Type*) := (P → R) × R × R def xHash (s : Sample P R) (I : P → R) := s.2.1 + dot s.1 I def yHash (s : Sample P R) (J : P → R) := s.2.2 + dot s.1 J def zHash (two : Rˣ) (s : Sample P R) (S K : P → R) := (↑(two⁻¹) : R) * (s.2.1 + s.2.2 + dot s.1 (S - K)) def Survives (U : Set R) (I J : P → R) (s : Sample P R) : Prop := xHash s I = yHash s J ∧ xHash s I ∈ U def survivalEquiv (U : Set R) (I J : P → R) : {s : Sample P R // Survives U I J s} ≃ (P → R) × U where toFun s := (s.val.1, ⟨xHash s.val I, s.property.2⟩) invFun vt := ⟨(vt.1, vt.2.val - dot vt.1 I, vt.2.val - dot vt.1 J), by simp [Survives, xHash, yHash, vt.2.property]⟩ left_inv s := by apply Subtype.ext rcases s with ⟨⟨v, a, b⟩, hs⟩ dsimp [Survives, xHash, yHash] at hs ⊢ congr 1 apply Prod.ext · ring · linear_combination hs.1 right_inv vt := by rcases vt with ⟨v, t, ht⟩ simp [xHash] theorem hash_relation (two : Rˣ) (htwo : (two : R) = 2) (s : Sample P R) (I J K S : P → R) (hs : I + J + K = S) : xHash s I + yHash s J = 2 * zHash two s S K := by have hSK : S - K = I + J := by rw [← hs]; abel rw [zHash, ← htwo, ← mul_assoc] rw [Units.mul_inv, one_mul, hSK, dot_add_right] simp only [xHash, yHash] ring theorem full_survival_iff (two : Rˣ) (htwo : (two : R) = 2) (U : Set R) (hAP : ∀ x ∈ U, ∀ y ∈ U, ∀ z ∈ U, x + y = 2 * z → x = z ∧ y = z) (s : Sample P R) (I J K S : P → R) (hs : I + J + K = S) : (xHash s I ∈ U ∧ yHash s J ∈ U ∧ zHash two s S K ∈ U) ↔ Survives U I J s := by have hr := hash_relation two htwo s I J K S hs constructor · rintro ⟨hx, hy, hz⟩ obtain ⟨h1, h2⟩ := hAP _ hx _ hy _ hz hr exact ⟨h1.trans h2.symm, hx⟩ · rintro ⟨hxy, hx⟩ have hz : zHash two s S K = xHash s I := by have hh : (two : R) * zHash two s S K = (two : R) * xHash s I := by rw [htwo, ← hr, hxy]; ring exact two.isUnit.mul_left_cancel hh exact ⟨hx, hxy ▸ hx, hz ▸ hx⟩ theorem survival_card [Fintype R] (U : Finset R) (I J : P → R) : Fintype.card {s : Sample P R // Survives (U : Set R) I J s} = Fintype.card R ^ Fintype.card P * U.card := by classical rw [Fintype.card_congr (survivalEquiv (U : Set R) I J)] simp omit [CommRing R] in @[simp] theorem sample_card [Fintype R] : Fintype.card (Sample P R) = Fintype.card R ^ Fintype.card P * Fintype.card R ^ 2 := by simp [Sample, pow_two] theorem shared_x_collision_iff (U : Set R) (I J J' : P → R) (s : Sample P R) (hs : Survives U I J s) : Survives U I J' s ↔ dot s.1 (J' - J) = 0 := by rcases hs with ⟨heq, hU⟩ simp only [Survives, dot_sub_right] constructor · intro h dsimp [xHash, yHash] at heq h linear_combination heq - h.1 · intro h refine ⟨?_, hU⟩ dsimp [xHash, yHash] at heq ⊢ linear_combination heq - h theorem shared_y_collision_iff (U : Set R) (I I' J : P → R) (s : Sample P R) (hs : Survives U I J s) : Survives U I' J s ↔ dot s.1 (I' - I) = 0 := by rcases hs with ⟨heq, hU⟩ simp only [Survives, dot_sub_right] constructor · intro h dsimp [xHash, yHash] at heq h linear_combination h.1 - heq · intro h have hx : xHash s I' = xHash s I := by dsimp [xHash] linear_combination h exact ⟨hx.trans heq, hx ▸ hU⟩ def pivot [DecidableEq P] (i : P) (u : Rˣ) (t : R) : P → R := Pi.single i ((↑(u⁻¹) : R) * t) @[simp] theorem dot_pivot [DecidableEq P] (d : P → R) (i : P) (u : Rˣ) (hu : d i = u) (t : R) : dot (pivot i u t) d = t := by simp [dot, pivot, Pi.single_apply, hu, mul_comm, mul_left_comm] def linearSplitEquiv [DecidableEq P] (d : P → R) (i : P) (u : Rˣ) (hu : d i = u) : (P → R) ≃ R × {v : P → R // dot v d = 0} where toFun v := (dot v d, ⟨v - pivot i u (dot v d), by simp [hu]⟩) invFun t := t.2.val + pivot i u t.1 left_inv v := by simp right_inv t := by rcases t with ⟨t, v, hv⟩ apply Prod.ext · simp [hv, hu] · apply Subtype.ext simp [hv, hu] theorem linear_zero_fiber_card [Fintype R] (d : P → R) (i : P) (u : Rˣ) (hu : d i = u) : Fintype.card R * Fintype.card {v : P → R // dot v d = 0} = Fintype.card R ^ Fintype.card P := by classical have h := Fintype.card_congr (linearSplitEquiv d i u hu) simpa only [Fintype.card_prod, Fintype.card_fun] using h.symm def collisionEquiv (U : Set R) (I J d : P → R) : {s : Sample P R // Survives U I J s ∧ dot s.1 d = 0} ≃ {v : P → R // dot v d = 0} × U where toFun s := (⟨s.val.1, s.property.2⟩, ⟨xHash s.val I, s.property.1.2⟩) invFun vt := ⟨(vt.1.val, vt.2.val - dot vt.1.val I, vt.2.val - dot vt.1.val J), by exact ⟨by simp [Survives, xHash, yHash, vt.2.property], vt.1.property⟩⟩ left_inv s := by apply Subtype.ext exact congrArg (fun t : {s : Sample P R // Survives U I J s} => t.val) ((survivalEquiv U I J).left_inv ⟨s.val, s.property.1⟩) right_inv vt := by rcases vt with ⟨⟨v, hv⟩, ⟨t, ht⟩⟩ simp [xHash] theorem collision_card_mul [Fintype R] (U : Finset R) (I J d : P → R) (i : P) (u : Rˣ) (hu : d i = u) : Fintype.card R * Fintype.card {s : Sample P R // Survives (U : Set R) I J s ∧ dot s.1 d = 0} = Fintype.card {s : Sample P R // Survives (U : Set R) I J s} := by classical have hcard : Fintype.card {s : Sample P R // Survives (U : Set R) I J s ∧ dot s.1 d = 0} = Fintype.card {v : P → R // dot v d = 0} * U.card := by simpa using Fintype.card_congr (collisionEquiv (U : Set R) I J d) calc _ = Fintype.card R * (Fintype.card {v : P → R // dot v d = 0} * U.card) := congrArg (fun n : ℕ => Fintype.card R * n) hcard _ = (Fintype.card R * Fintype.card {v : P → R // dot v d = 0}) * U.card := (mul_assoc _ _ _).symm _ = (Fintype.card R ^ Fintype.card P) * U.card := congrArg (fun n : ℕ => n * U.card) (linear_zero_fiber_card d i u hu) _ = _ := (survival_card U I J).symm theorem shared_x_collision_card_mul [Fintype R] (U : Finset R) (I J J' : P → R) (i : P) (u : Rˣ) (hu : J' i - J i = u) : Fintype.card R * Fintype.card {s : Sample P R // Survives (U : Set R) I J s ∧ Survives (U : Set R) I J' s} = Fintype.card {s : Sample P R // Survives (U : Set R) I J s} := by classical let e : {s : Sample P R // Survives (U : Set R) I J s ∧ Survives (U : Set R) I J' s} ≃ {s : Sample P R // Survives (U : Set R) I J s ∧ dot s.1 (J' - J) = 0} := Equiv.subtypeEquivRight (fun s => and_congr_right (fun hs => shared_x_collision_iff (U : Set R) I J J' s hs)) rw [Fintype.card_congr e] exact collision_card_mul U I J (J' - J) i u hu theorem shared_y_collision_card_mul [Fintype R] (U : Finset R) (I I' J : P → R) (i : P) (u : Rˣ) (hu : I' i - I i = u) : Fintype.card R * Fintype.card {s : Sample P R // Survives (U : Set R) I J s ∧ Survives (U : Set R) I' J s} = Fintype.card {s : Sample P R // Survives (U : Set R) I J s} := by classical let e : {s : Sample P R // Survives (U : Set R) I J s ∧ Survives (U : Set R) I' J s} ≃ {s : Sample P R // Survives (U : Set R) I J s ∧ dot s.1 (I' - I) = 0} := Equiv.subtypeEquivRight (fun s => and_congr_right (fun hs => shared_y_collision_iff (U : Set R) I I' J s hs)) rw [Fintype.card_congr e] exact collision_card_mul U I J (I' - I) i u hu theorem survival_fraction [Fintype R] (U : Finset R) (I J : P → R) : (Fintype.card {s : Sample P R // Survives (U : Set R) I J s} : ℝ) / Fintype.card (Sample P R) = (U.card : ℝ) / (Fintype.card R : ℝ) ^ 2 := by have hR : (Fintype.card R : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero rw [survival_card, sample_card] push_cast field_simp theorem collision_fraction [Fintype R] (U : Finset R) (hU : U.Nonempty) (I J d : P → R) (i : P) (u : Rˣ) (hu : d i = u) : (Fintype.card {s : Sample P R // Survives (U : Set R) I J s ∧ dot s.1 d = 0} : ℝ) / Fintype.card {s : Sample P R // Survives (U : Set R) I J s} = 1 / (Fintype.card R : ℝ) := by have hR : (Fintype.card R : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero have hS : (Fintype.card {s : Sample P R // Survives (U : Set R) I J s} : ℝ) ≠ 0 := by obtain ⟨t, ht⟩ := hU let : Nonempty {s : Sample P R // Survives (U : Set R) I J s} := ⟨(survivalEquiv (U : Set R) I J).symm (0, ⟨t, ht⟩)⟩ exact Nat.cast_ne_zero.mpr Fintype.card_ne_zero apply (div_eq_div_iff hS hR).mpr simpa only [Nat.cast_mul, one_mul, mul_one, mul_comm] using (congrArg (fun n : ℕ => (n : ℝ)) (collision_card_mul U I J d i u hu)) theorem small_nat_isUnit (k n : ℕ) (hn : 0 < n) (hn16 : n ≤ 16) : IsUnit (n : ZMod (37 ^ k)) := by apply (ZMod.isUnit_iff_coprime _ _).mpr apply Nat.Coprime.pow_right apply Nat.Coprime.symm apply (Nat.Prime.coprime_iff_not_dvd (by decide : Nat.Prime 37)).mpr intro h have := Nat.le_of_dvd hn h omega theorem coarse_difference_isUnit (k : ℕ) (a b : Fin 17) (hab : a ≠ b) : IsUnit ((a.val : ZMod (37 ^ k)) - (b.val : ZMod (37 ^ k))) := by have hne : a.val ≠ b.val := fun h => hab (Fin.ext h) rcases lt_or_gt_of_ne hne with h | h · have hu := small_nat_isUnit k (b.val - a.val) (Nat.sub_pos_of_lt h) (by omega) rw [Nat.cast_sub h.le] at hu simpa only [neg_sub] using hu.neg · have hu := small_nat_isUnit k (a.val - b.val) (Nat.sub_pos_of_lt h) (by omega) rwa [Nat.cast_sub h.le] at hu theorem coarse_words_unit_coordinate (k : ℕ) (I J : P → Fin 17) (hIJ : I ≠ J) : ∃ i : P, IsUnit (((I i : ℕ) : ZMod (37 ^ k)) - ((J i : ℕ) : ZMod (37 ^ k))) := by classical have hi : ∃ i, I i ≠ J i := by by_contra h push Not at h exact hIJ (funext h) obtain ⟨i, hi⟩ := hi exact ⟨i, coarse_difference_isUnit k (I i) (J i) hi⟩ theorem shared_z_collision_iff (two : Rˣ) (htwo : (two : R) = 2) (U : Set R) (I I' J J' : P → R) (hIJ : I' + J' = I + J) (s : Sample P R) (hs : Survives U I J s) : Survives U I' J' s ↔ dot s.1 (I' - I) = 0 := by have hsum : dot s.1 I' + dot s.1 J' = dot s.1 I + dot s.1 J := by simpa using congrArg (dot s.1) hIJ rcases hs with ⟨heq, hU⟩ rw [dot_sub_right] constructor · rintro ⟨heq', _⟩ dsimp [xHash, yHash] at heq heq' have hz : (two : R) * (dot s.1 I' - dot s.1 I) = (two : R) * 0 := by rw [htwo] linear_combination heq' - heq + hsum exact two.isUnit.mul_left_cancel hz · intro hd have hx : xHash s I' = xHash s I := by dsimp [xHash] linear_combination hd have hy : yHash s J' = yHash s J := by dsimp [yHash] linear_combination hsum - hd exact ⟨hx.trans (heq.trans hy.symm), hx ▸ hU⟩ theorem shared_z_collision_card_mul [Fintype R] (two : Rˣ) (htwo : (two : R) = 2) (U : Finset R) (I I' J J' : P → R) (hIJ : I' + J' = I + J) (i : P) (u : Rˣ) (hu : I' i - I i = u) : Fintype.card R * Fintype.card {s : Sample P R // Survives (U : Set R) I J s ∧ Survives (U : Set R) I' J' s} = Fintype.card {s : Sample P R // Survives (U : Set R) I J s} := by classical let e : {s : Sample P R // Survives (U : Set R) I J s ∧ Survives (U : Set R) I' J' s} ≃ {s : Sample P R // Survives (U : Set R) I J s ∧ dot s.1 (I' - I) = 0} := Equiv.subtypeEquivRight (fun s => and_congr_right (fun hs => shared_z_collision_iff two htwo (U : Set R) I I' J J' hIJ s hs)) rw [Fintype.card_congr e] exact collision_card_mul U I J (I' - I) i u hu end MatrixMultiplication.JointHashing noncomputable section namespace MatrixMultiplication.JointCoarseHashing attribute [local instance] Classical.propDecidable abbrev Word (P : Type*) := P → Fin 17 abbrev Triple (P : Type*) := Word P × Word P × Word P variable {P : Type*} def HasSupportSum (S : P → ℕ) (e : Triple P) : Prop := ∀ p, (e.1 p).val + (e.2.1 p).val + (e.2.2 p).val = S p theorem eq_of_shared_xy (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hx : e.1 = f.1) (hy : e.2.1 = f.2.1) : e = f := by have hz : e.2.2 = f.2.2 := by funext p apply Fin.ext have he' := he p have hf' := hf p rw [hx, hy] at he' omega exact Prod.ext hx (Prod.ext hy hz) theorem eq_of_shared_xz (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hx : e.1 = f.1) (hz : e.2.2 = f.2.2) : e = f := by have hy : e.2.1 = f.2.1 := by funext p apply Fin.ext have he' := he p have hf' := hf p rw [hx, hz] at he' omega exact Prod.ext hx (Prod.ext hy hz) theorem eq_of_shared_yz (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hy : e.2.1 = f.2.1) (hz : e.2.2 = f.2.2) : e = f := by have hx : e.1 = f.1 := by funext p apply Fin.ext have he' := he p have hf' := hf p rw [hy, hz] at he' omega exact Prod.ext hx (Prod.ext hy hz) def castWord (k : ℕ) (w : Word P) : P → ZMod (37 ^ k) := fun p => (w p).val def twoUnit (k : ℕ) : (ZMod (37 ^ k))ˣ := (JointHashing.small_nat_isUnit k 2 (by decide) (by decide)).unit theorem twoUnit_coe (k : ℕ) : (twoUnit k : ZMod (37 ^ k)) = 2 := by exact (JointHashing.small_nat_isUnit k 2 (by decide) (by decide)).unit_spec theorem supportSum_cast (k : ℕ) (S : P → ℕ) (e : Triple P) (he : HasSupportSum S e) : castWord k e.1 + castWord k e.2.1 + castWord k e.2.2 = fun p => (S p : ZMod (37 ^ k)) := by funext p have h := congrArg (fun n : ℕ => (n : ZMod (37 ^ k))) (he p) simpa only [Nat.cast_add, Pi.add_apply, castWord] using h theorem shared_z_cast_sum (k : ℕ) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hz : f.2.2 = e.2.2) : castWord k f.1 + castWord k f.2.1 = castWord k e.1 + castWord k e.2.1 := by have h := (supportSum_cast k S f hf).trans (supportSum_cast k S e he).symm rw [hz] at h exact add_right_cancel h section FiniteHashing variable [Fintype P] abbrev Sample (P : Type*) (k : ℕ) := JointHashing.Sample P (ZMod (37 ^ k)) def Survives (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : Triple P) (s : Sample P k) : Prop := JointHashing.Survives (U : Set (ZMod (37 ^ k))) (castWord k e.1) (castWord k e.2.1) s theorem full_survival_iff (k : ℕ) (U : Finset (ZMod (37 ^ k))) (hAP : ∀ x ∈ U, ∀ y ∈ U, ∀ z ∈ U, x + y = 2 * z → x = z ∧ y = z) (S : P → ℕ) (e : Triple P) (he : HasSupportSum S e) (s : Sample P k) : (JointHashing.xHash s (castWord k e.1) ∈ U ∧ JointHashing.yHash s (castWord k e.2.1) ∈ U ∧ JointHashing.zHash (twoUnit k) s (fun p => (S p : ZMod (37 ^ k))) (castWord k e.2.2) ∈ U) ↔ Survives k U e s := by exact JointHashing.full_survival_iff (twoUnit k) (twoUnit_coe k) (U : Set (ZMod (37 ^ k))) hAP s (castWord k e.1) (castWord k e.2.1) (castWord k e.2.2) (fun p => (S p : ZMod (37 ^ k))) (supportSum_cast k S e he) theorem survival_card (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : Triple P) : Fintype.card {s : Sample P k // Survives k U e s} = (37 ^ k) ^ Fintype.card P * U.card := by have hcard := JointHashing.survival_card U (castWord k e.1) (castWord k e.2.1) simp only [ZMod.card] at hcard simp only [Fintype.card_eq_nat_card] at hcard ⊢ exact hcard theorem shared_x_card_mul (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hne : f ≠ e) (hx : f.1 = e.1) : (37 ^ k) * Fintype.card {s : Sample P k // Survives k U e s ∧ Survives k U f s} = Fintype.card {s : Sample P k // Survives k U e s} := by classical have hy : f.2.1 ≠ e.2.1 := fun hy => hne (eq_of_shared_xy S f e hf he hx hy) obtain ⟨i, hi⟩ := JointHashing.coarse_words_unit_coordinate k f.2.1 e.2.1 hy have h := JointHashing.shared_x_collision_card_mul U (castWord k e.1) (castWord k e.2.1) (castWord k f.2.1) i hi.unit hi.unit_spec.symm simp only [ZMod.card] at h simp only [Fintype.card_eq_nat_card] at h ⊢ simpa only [Survives, hx] using h theorem shared_y_card_mul (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hne : f ≠ e) (hy : f.2.1 = e.2.1) : (37 ^ k) * Fintype.card {s : Sample P k // Survives k U e s ∧ Survives k U f s} = Fintype.card {s : Sample P k // Survives k U e s} := by classical have hx : f.1 ≠ e.1 := fun hx => hne (eq_of_shared_xy S f e hf he hx hy) obtain ⟨i, hi⟩ := JointHashing.coarse_words_unit_coordinate k f.1 e.1 hx have h := JointHashing.shared_y_collision_card_mul U (castWord k e.1) (castWord k f.1) (castWord k e.2.1) i hi.unit hi.unit_spec.symm simp only [ZMod.card] at h simp only [Fintype.card_eq_nat_card] at h ⊢ simpa only [Survives, hy] using h theorem shared_z_card_mul (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hne : f ≠ e) (hz : f.2.2 = e.2.2) : (37 ^ k) * Fintype.card {s : Sample P k // Survives k U e s ∧ Survives k U f s} = Fintype.card {s : Sample P k // Survives k U e s} := by classical have hx : f.1 ≠ e.1 := fun hx => hne (eq_of_shared_xz S f e hf he hx hz) obtain ⟨i, hi⟩ := JointHashing.coarse_words_unit_coordinate k f.1 e.1 hx have h := JointHashing.shared_z_collision_card_mul (twoUnit k) (twoUnit_coe k) U (castWord k e.1) (castWord k f.1) (castWord k e.2.1) (castWord k f.2.1) (shared_z_cast_sum k S e f he hf hz) i hi.unit hi.unit_spec.symm simp only [ZMod.card] at h simp only [Fintype.card_eq_nat_card] at h ⊢ exact h def SharesSide (e f : Triple P) : Prop := f.1 = e.1 ∨ f.2.1 = e.2.1 ∨ f.2.2 = e.2.2 theorem pair_survival_card_mul (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hne : f ≠ e) (hshare : SharesSide e f) : (37 ^ k) * Fintype.card {s : Sample P k // Survives k U e s ∧ Survives k U f s} = Fintype.card {s : Sample P k // Survives k U e s} := by rcases hshare with hx | hy | hz · exact shared_x_card_mul k U S e f he hf hne hx · exact shared_y_card_mul k U S e f he hf hne hy · exact shared_z_card_mul k U S e f he hf hne hz def ownEvent (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : Triple P) : Finset (Sample P k) := Finset.univ.filter (Survives k U e) theorem ownEvent_card (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : Triple P) : (ownEvent k U e).card = (37 ^ k) ^ Fintype.card P * U.card := by simpa only [ownEvent, Fintype.card_subtype] using survival_card k U e theorem pair_event_card_mul (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hne : f ≠ e) (hshare : SharesSide e f) : (37 ^ k) * ((ownEvent k U e).filter (Survives k U f)).card = (ownEvent k U e).card := by simpa only [ownEvent, Finset.filter_filter, Fintype.card_subtype] using pair_survival_card_mul k U S e f he hf hne hshare theorem competitor_union_card_mul_le (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e : Triple P) (he : HasSupportSum S e) (competitors : Finset (Triple P)) (hf : ∀ f ∈ competitors, HasSupportSum S f) (hne : ∀ f ∈ competitors, f ≠ e) (hshare : ∀ f ∈ competitors, SharesSide e f) : (37 ^ k) * ((ownEvent k U e).filter (fun s => ∃ f ∈ competitors, Survives k U f s)).card ≤ competitors.card * (ownEvent k U e).card := by apply JointLossCounts.competitor_union_card_mul_le intro f hf' exact (pair_event_card_mul k U S e f he (hf f hf') (hne f hf') (hshare f hf')).le theorem competitor_union_card_le (k : ℕ) (U : Finset (ZMod (37 ^ k))) (S : P → ℕ) (e : Triple P) (he : HasSupportSum S e) (competitors : Finset (Triple P)) (hf : ∀ f ∈ competitors, HasSupportSum S f) (hne : ∀ f ∈ competitors, f ≠ e) (hshare : ∀ f ∈ competitors, SharesSide e f) : (((ownEvent k U e).filter (fun s => ∃ f ∈ competitors, Survives k U f s)).card : ℝ) ≤ ((competitors.card : ℝ) / ((37 ^ k : ℕ) : ℝ)) * ((ownEvent k U e).card : ℝ) := by have hM : (0 : ℝ) < ((37 ^ k : ℕ) : ℝ) := by exact_mod_cast (pow_pos (by decide : 0 < (37 : ℕ)) k) have h := competitor_union_card_mul_le k U S e he competitors hf hne hshare have hR : ((37 ^ k : ℕ) : ℝ) * (((ownEvent k U e).filter (fun s => ∃ f ∈ competitors, Survives k U f s)).card : ℝ) ≤ (competitors.card : ℝ) * ((ownEvent k U e).card : ℝ) := by exact_mod_cast h rw [div_mul_eq_mul_div] exact (le_div_iff₀ hM).2 (by simpa only [mul_comm] using hR) theorem pair_survival_fraction (k : ℕ) (U : Finset (ZMod (37 ^ k))) (hU : U.Nonempty) (S : P → ℕ) (e f : Triple P) (he : HasSupportSum S e) (hf : HasSupportSum S f) (hne : f ≠ e) (hshare : SharesSide e f) : (((ownEvent k U e).filter (Survives k U f)).card : ℝ) / ((ownEvent k U e).card : ℝ) = 1 / ((37 ^ k : ℕ) : ℝ) := by have hM : (0 : ℕ) < 37 ^ k := pow_pos (by decide) k have hS : 0 < (ownEvent k U e).card := by rw [ownEvent_card] exact Nat.mul_pos (pow_pos hM _) (Finset.card_pos.mpr hU) apply (div_eq_div_iff (by exact_mod_cast Nat.ne_of_gt hS) (by exact_mod_cast Nat.ne_of_gt hM)).2 have h := congrArg (fun n : ℕ => (n : ℝ)) (pair_event_card_mul k U S e f he hf hne hshare) simpa only [Nat.cast_mul, one_mul, mul_comm] using h end FiniteHashing end MatrixMultiplication.JointCoarseHashing namespace MatrixMultiplication.JointOrdinarySelection open JointCoarseHashing JointExtractionRates Filter open scoped BigOperators Topology attribute [local instance] Classical.propDecidable variable {P E : Type*} def neighbors (ambient : Finset (Triple P)) (e : Triple P) : Finset (Triple P) := ambient.filter (fun f => f ≠ e ∧ SharesSide e f) @[simp] theorem mem_neighbors (ambient : Finset (Triple P)) (e f : Triple P) : f ∈ neighbors ambient e ↔ f ∈ ambient ∧ f ≠ e ∧ SharesSide e f := by simp only [neighbors, Finset.mem_filter] section Finite variable [Fintype P] def Bad (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (e : Triple P) (s : Sample P k) : Prop := ∃ f ∈ neighbors ambient e, Survives k U f s def Good (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (e : Triple P) (s : Sample P k) : Prop := Survives k U e s ∧ ¬ Bad k U ambient e s def goodEvent (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (e : Triple P) : Finset (Sample P k) := (ownEvent k U e).filter (fun s => ¬ Bad k U ambient e s) @[simp] theorem mem_goodEvent (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (e : Triple P) (s : Sample P k) : s ∈ goodEvent k U ambient e ↔ Good k U ambient e s := by simp only [goodEvent, ownEvent, Finset.mem_filter, Finset.mem_univ, true_and, Good] theorem good_excludes_neighbor (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (e f : Triple P) (s : Sample P k) (hg : Good k U ambient e s) (hf : f ∈ ambient) (hne : f ≠ e) (hshare : SharesSide e f) : ¬ Survives k U f s := by intro hs exact hg.2 ⟨f, (mem_neighbors ambient e f).2 ⟨hf, hne, hshare⟩, hs⟩ theorem good_no_shared_side (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (e f : Triple P) (s : Sample P k) (he : Good k U ambient e s) (hf : Good k U ambient f s) (hmem : f ∈ ambient) (hne : f ≠ e) : ¬ SharesSide e f := by intro hshare exact good_excludes_neighbor k U ambient e f s he hmem hne hshare hf.1 theorem ownEvent_card_real (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : Triple P) : ((ownEvent k U e).card : ℝ) = ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2) * (Fintype.card (Sample P k) : ℝ) := by rw [ownEvent_card, JointHashing.sample_card, ZMod.card] push_cast field_simp theorem bad_card_le (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (S : P → ℕ) (hsupport : ∀ f ∈ ambient, HasSupportSum S f) (e : Triple P) (he : e ∈ ambient) : (((ownEvent k U e).filter (Bad k U ambient e)).card : ℝ) ≤ ((neighbors ambient e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) * ((ownEvent k U e).card : ℝ) := by have hset : (ownEvent k U e).filter (Bad k U ambient e) = (ownEvent k U e).filter (fun s => ∃ f ∈ neighbors ambient e, Survives k U f s) := by ext s simp only [Finset.mem_filter, Bad] rw [hset] apply competitor_union_card_le k U S e (hsupport e he) (neighbors ambient e) · intro f hf exact hsupport f ((mem_neighbors ambient e f).mp hf).1 · intro f hf exact ((mem_neighbors ambient e f).mp hf).2.1 · intro f hf exact ((mem_neighbors ambient e f).mp hf).2.2 theorem goodEvent_card_lower (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (S : P → ℕ) (hsupport : ∀ f ∈ ambient, HasSupportSum S f) (e : Triple P) (he : e ∈ ambient) (D : ℕ) (hdegree : (neighbors ambient e).card ≤ D) (hsmall : (D : ℝ) / ((37 ^ k : ℕ) : ℝ) ≤ 1 / 2) : ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2) * (Fintype.card (Sample P k) : ℝ) ≤ (goodEvent k U ambient e).card := by have hM : (0 : ℝ) ≤ ((37 ^ k : ℕ) : ℝ) := Nat.cast_nonneg _ have hdeg : ((neighbors ambient e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) ≤ 1 / 2 := (div_le_div_of_nonneg_right (by exact_mod_cast hdegree) hM).trans hsmall have hb : (((ownEvent k U e).filter (Bad k U ambient e)).card : ℝ) ≤ (1 / 2) * ((ownEvent k U e).card : ℝ) := (bad_card_le k U ambient S hsupport e he).trans (mul_le_mul_of_nonneg_right hdeg (Nat.cast_nonneg _)) have hg := JointLossCounts.good_card_lower (ownEvent k U e) (Bad k U ambient e) (1 / 2) hb rw [ownEvent_card_real] at hg change _ ≤ ((goodEvent k U ambient e).card : ℝ) at hg convert hg using 1 ring theorem goodIncidence_eq (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (coarse : E → Triple P) (e : E) : goodIncidence (fun s e => Good k U ambient (coarse e) s) e = (goodEvent k U ambient (coarse e)).card := by unfold goodIncidence congr 1 ext s simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_goodEvent] theorem exists_selection_of_degree_bound [Fintype E] (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (S : P → ℕ) (coarse : E → Triple P) (hinj : Function.Injective coarse) (hsupport : ∀ f ∈ ambient, HasSupportSum S f) (htarget : ∀ e, coarse e ∈ ambient) (D : ℕ) (hdegree : ∀ e, (neighbors ambient (coarse e)).card ≤ D) (hsmall : (D : ℝ) / ((37 ^ k : ℕ) : ℝ) ≤ 1 / 2) (K : ℕ) (hK : (K : ℝ) ≤ (Fintype.card E : ℝ) * ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2)) : ∃ (s : Sample P k) (G : Finset E), G.card = K ∧ (∀ e ∈ G, Good k U ambient (coarse e) s) ∧ (G : Set E).Pairwise (fun e f => ¬ SharesSide (coarse e) (coarse f)) := by classical have hi (e : E) : ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2) * (Fintype.card (Sample P k) : ℝ) ≤ (goodIncidence (fun s e => Good k U ambient (coarse e) s) e : ℝ) := by rw [goodIncidence_eq] exact goodEvent_card_lower k U ambient S hsupport (coarse e) (htarget e) D (hdegree e) hsmall obtain ⟨s, hs⟩ := exists_goodCount_ge (fun s e => Good k U ambient (coarse e) s) _ hi have hk : K ≤ goodCount (fun s e => Good k U ambient (coarse e) s) s := by exact_mod_cast hK.trans hs obtain ⟨G, hcard, hgood⟩ := exists_good_subfamily (fun s e => Good k U ambient (coarse e) s) s K hk refine ⟨s, G, hcard, hgood, ?_⟩ intro e he f hf hne exact good_no_shared_side k U ambient (coarse e) (coarse f) s (hgood e he) (hgood f hf) (htarget f) (fun h => hne (hinj h).symm) theorem exists_selection_floor [Fintype E] (k : ℕ) (U : Finset (ZMod (37 ^ k))) (ambient : Finset (Triple P)) (S : P → ℕ) (coarse : E → Triple P) (hinj : Function.Injective coarse) (hsupport : ∀ f ∈ ambient, HasSupportSum S f) (htarget : ∀ e, coarse e ∈ ambient) (D : ℕ) (hdegree : ∀ e, (neighbors ambient (coarse e)).card ≤ D) (hsmall : (D : ℝ) / ((37 ^ k : ℕ) : ℝ) ≤ 1 / 2) : ∃ (s : Sample P k) (G : Finset E), G.card = ⌊(Fintype.card E : ℝ) * ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2)⌋₊ ∧ (∀ e ∈ G, Good k U ambient (coarse e) s) ∧ (G : Set E).Pairwise (fun e f => ¬ SharesSide (coarse e) (coarse f)) := by apply exists_selection_of_degree_bound k U ambient S coarse hinj hsupport htarget D hdegree hsmall _ (Nat.floor_le ?_) positivity end Finite def hashLevel (H : ℝ) (N : ℕ) : ℕ := ⌊(N : ℝ) * H / Real.log 37⌋₊ + 1 def hashSet (H : ℝ) (N : ℕ) : Finset (ZMod (37 ^ hashLevel H N)) := JointAPFree.modularSet (hashModulus H N) theorem hashSet_AP (H : ℝ) (N : ℕ) : ∀ x ∈ hashSet H N, ∀ y ∈ hashSet H N, ∀ z ∈ hashSet H N, x + y = 2 * z → x = z ∧ y = z := JointAPFree.modularSet_hashAP (hashModulus H N) theorem eventually_exists_selection {P E : ℕ → Type*} [∀ N, Fintype (P N)] [∀ N, Fintype (E N)] (ambient : ∀ N, Finset (Triple (P N))) (S : ∀ N, P N → ℕ) (coarse : ∀ N, E N → Triple (P N)) (hinj : ∀ N, Function.Injective (coarse N)) (hsupport : ∀ N, ∀ f ∈ ambient N, HasSupportSum (S N) f) (htarget : ∀ N e, coarse N e ∈ ambient N) (degree : ℕ → ℕ) (hdegree : ∀ N e, (neighbors (ambient N) (coarse N e)).card ≤ degree N) {A D H : ℝ} (hH : 0 ≤ H) (hgap : D < H) (hdegreeRate : ∀ η : ℝ, 0 < η → ∀ᶠ N in atTop, Real.log (degree N : ℝ) / (N : ℝ) ≤ D + η) (hpos : ∀ᶠ N in atTop, 0 < Fintype.card (E N)) (hE : Tendsto (fun N => Real.log (Fintype.card (E N) : ℝ) / (N : ℝ)) atTop (𝓝 A)) {ε : ℝ} (hε : 0 < ε) : ∀ᶠ N in atTop, ∃ (s : Sample (P N) (hashLevel H N)) (G : Finset (E N)), G.card = ⌊Real.exp ((N : ℝ) * (A - H - ε))⌋₊ ∧ (∀ e ∈ G, Good (hashLevel H N) (hashSet H N) (ambient N) (coarse N e) s) ∧ (G : Set (E N)).Pairwise (fun e f => ¬ SharesSide (coarse N e) (coarse N f)) := by classical let c : ℝ := (H - D) / 2 have hc : 0 < c := by dsimp [c]; linarith have hclt : c < H - D := by dsimp [c]; linarith have hratio := eventually_degree_div_hashModulus_le_exp hH hdegreeRate hclt have hsmall : ∀ᶠ N in atTop, (degree N : ℝ) / (hashModulus H N : ℝ) ≤ 1 / 2 := by apply eventually_error_le_half 1 0 hc simpa only [pow_zero, one_mul] using hratio let good := fun N (s : Sample (P N) (hashLevel H N)) (e : E N) => Good (hashLevel H N) (hashSet H N) (ambient N) (coarse N e) s have hinc : ∀ᶠ N in atTop, ∀ e, (((JointAPFree.modularSet (hashModulus H N)).card : ℝ) / (hashModulus H N : ℝ) ^ 2 / 2) * (Fintype.card (Sample (P N) (hashLevel H N)) : ℝ) ≤ (goodIncidence (good N) e : ℝ) := by filter_upwards [hsmall] with N hN e rw [show goodIncidence (good N) e = (goodEvent (hashLevel H N) (hashSet H N) (ambient N) (coarse N e)).card from goodIncidence_eq _ _ _ _ _] exact goodEvent_card_lower (hashLevel H N) (hashSet H N) (ambient N) (S N) (hsupport N) (coarse N e) (htarget N e) (degree N) (hdegree N e) hN have hyield := eventually_exists_many_good_explicit good hH hpos hinc hE hε have hselection := eventually_exists_good_subfamily good (fun N => ⌊Real.exp ((N : ℝ) * (A - H - ε))⌋₊) hyield filter_upwards [hselection] with N hN obtain ⟨s, G, hcard, hgood⟩ := hN refine ⟨s, G, hcard, hgood, ?_⟩ intro e he f hf hne exact good_no_shared_side (hashLevel H N) (hashSet H N) (ambient N) (coarse N e) (coarse N f) s (hgood e he) (hgood f hf) (htarget N f) (fun h => hne ((hinj N) h).symm) end MatrixMultiplication.JointOrdinarySelection namespace MatrixMultiplication.JointMaskedSelection open JointCoarseHashing JointExtractionRates attribute [local instance] Classical.propDecidable variable {P E O V : Type*} structure OrbitData (P E O V : Type*) where ambient : Finset (Triple P) support : P → ℕ coarse : E → Triple P support_ambient : ∀ f ∈ ambient, HasSupportSum support f target_mem : ∀ e, coarse e ∈ ambient orbits : E → Finset O full : E → O → Finset V passing : E → O → Finset V full_pos : ∀ e, ∀ o ∈ orbits e, 0 < (full e o).card passing_subset : ∀ e, ∀ o ∈ orbits e, passing e o ⊆ full e o competitors : E → O → V → Finset (Triple P) competitor_mem : ∀ e, ∀ o ∈ orbits e, ∀ v ∈ passing e o, ∀ f ∈ competitors e o v, f ∈ ambient competitor_ne : ∀ e, ∀ o ∈ orbits e, ∀ v ∈ passing e o, ∀ f ∈ competitors e o v, f ≠ coarse e competitor_shares : ∀ e, ∀ o ∈ orbits e, ∀ v ∈ passing e o, ∀ f ∈ competitors e o v, SharesSide (coarse e) f namespace OrbitData def eligibilityCompetitors (d : OrbitData P E O V) (e : E) : Finset (Triple P) := d.ambient.filter (fun f => f ≠ d.coarse e ∧ f.1 = (d.coarse e).1) @[simp] theorem mem_eligibilityCompetitors (d : OrbitData P E O V) (e : E) (f : Triple P) : f ∈ d.eligibilityCompetitors e ↔ f ∈ d.ambient ∧ f ≠ d.coarse e ∧ f.1 = (d.coarse e).1 := by simp only [eligibilityCompetitors, Finset.mem_filter] section Finite variable [Fintype P] def variableBad (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : E) (o : O) (v : V) (s : Sample P k) : Prop := ∃ f ∈ d.competitors e o v, Survives k U f s def eligibilityGood (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : E) (s : Sample P k) : Prop := ¬ ∃ f ∈ d.eligibilityCompetitors e, Survives k U f s def Failure (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) (s : Sample P k) : Prop := JointLossCounts.targetBad (d.orbits e) (d.full e) (d.passing e) (d.variableBad k U e) (d.eligibilityGood k U e) N s def Good (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) (s : Sample P k) : Prop := Survives k U (d.coarse e) s ∧ ¬ d.Failure k U N e s def goodEvent (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) : Finset (Sample P k) := (ownEvent k U (d.coarse e)).filter (fun s => ¬ d.Failure k U N e s) @[simp] theorem mem_goodEvent (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) (s : Sample P k) : s ∈ d.goodEvent k U N e ↔ d.Good k U N e s := by simp only [goodEvent, ownEvent, Finset.mem_filter, Finset.mem_univ, true_and, Good] theorem variableBad_card_le (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : E) (o : O) (ho : o ∈ d.orbits e) (v : V) (hv : v ∈ d.passing e o) (D : ℕ) (hdegree : (d.competitors e o v).card ≤ D) : (((ownEvent k U (d.coarse e)).filter (d.variableBad k U e o v)).card : ℝ) ≤ ((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) * ((ownEvent k U (d.coarse e)).card : ℝ) := by have h := competitor_union_card_le k U d.support (d.coarse e) (d.support_ambient _ (d.target_mem e)) (d.competitors e o v) (fun f hf => d.support_ambient f (d.competitor_mem e o ho v hv f hf)) (d.competitor_ne e o ho v hv) (d.competitor_shares e o ho v hv) have hdegreeR : ((d.competitors e o v).card : ℝ) ≤ (D : ℝ) := by exact_mod_cast hdegree have hfilter : (ownEvent k U (d.coarse e)).filter (d.variableBad k U e o v) = (ownEvent k U (d.coarse e)).filter (fun s => ∃ f ∈ d.competitors e o v, Survives k U f s) := by ext s simp only [Finset.mem_filter, variableBad] rw [hfilter] exact h.trans (mul_le_mul_of_nonneg_right (div_le_div_of_nonneg_right hdegreeR (Nat.cast_nonneg _)) (Nat.cast_nonneg _)) theorem eligibilityFailure_card_le (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : E) : (((ownEvent k U (d.coarse e)).filter (fun s => ¬ d.eligibilityGood k U e s)).card : ℝ) ≤ (((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ)) * ((ownEvent k U (d.coarse e)).card : ℝ) := by have h := competitor_union_card_le k U d.support (d.coarse e) (d.support_ambient _ (d.target_mem e)) (d.eligibilityCompetitors e) (fun f hf => d.support_ambient f ((d.mem_eligibilityCompetitors e f).1 hf).1) (fun f hf => ((d.mem_eligibilityCompetitors e f).1 hf).2.1) (fun f hf => Or.inl ((d.mem_eligibilityCompetitors e f).1 hf).2.2) simpa only [eligibilityGood, not_not] using h theorem failure_card_le (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (hN : 0 < N) (e : E) (D : ℕ) (hdegree : ∀ o ∈ d.orbits e, ∀ v ∈ d.passing e o, (d.competitors e o v).card ≤ D) : (((ownEvent k U (d.coarse e)).filter (d.Failure k U N e)).card : ℝ) ≤ (((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) + ((d.orbits e).card : ℝ) * N * ((D : ℝ) / ((37 ^ k : ℕ) : ℝ))) * ((ownEvent k U (d.coarse e)).card : ℝ) := by exact JointLossCounts.targetBad_card_le (ownEvent k U (d.coarse e)) (d.orbits e) (d.full e) (d.passing e) (d.variableBad k U e) (d.eligibilityGood k U e) ((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) (((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ)) N (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) hN (d.full_pos e) (d.passing_subset e) (fun o ho v hv => d.variableBad_card_le k U e o ho v hv D (hdegree o ho v hv)) (d.eligibilityFailure_card_le k U e) theorem ownEvent_card_real (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (e : E) : ((ownEvent k U (d.coarse e)).card : ℝ) = ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2) * (Fintype.card (Sample P k) : ℝ) := by exact JointOrdinarySelection.ownEvent_card_real k U (d.coarse e) theorem goodEvent_card_lower (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (hN : 0 < N) (e : E) (D : ℕ) (hdegree : ∀ o ∈ d.orbits e, ∀ v ∈ d.passing e o, (d.competitors e o v).card ≤ D) (hsmall : ((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) + ((d.orbits e).card : ℝ) * N * ((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) ≤ 1 / 2) : ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2) * (Fintype.card (Sample P k) : ℝ) ≤ (d.goodEvent k U N e).card := by have hb : (((ownEvent k U (d.coarse e)).filter (d.Failure k U N e)).card : ℝ) ≤ (1 / 2) * ((ownEvent k U (d.coarse e)).card : ℝ) := (d.failure_card_le k U N hN e D hdegree).trans (mul_le_mul_of_nonneg_right hsmall (Nat.cast_nonneg _)) have hg := JointLossCounts.good_card_lower (ownEvent k U (d.coarse e)) (d.Failure k U N e) (1 / 2) hb rw [d.ownEvent_card_real] at hg change _ ≤ ((d.goodEvent k U N e).card : ℝ) at hg convert hg using 1 ring theorem goodIncidence_eq (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) : goodIncidence (fun s e => d.Good k U N e s) e = (d.goodEvent k U N e).card := by unfold goodIncidence congr 1 ext s simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_goodEvent] theorem good_eligibility (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) (s : Sample P k) (hg : d.Good k U N e s) : d.eligibilityGood k U e s := by by_contra h exact hg.2 (Or.inl h) theorem good_no_shared_x (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (N : ℝ) (e f : E) (s : Sample P k) (he : d.Good k U N e s) (hf : d.Good k U N f s) (hne : d.coarse f ≠ d.coarse e) : (d.coarse f).1 ≠ (d.coarse e).1 := by intro hx exact d.good_eligibility k U N e s he ⟨d.coarse f, (d.mem_eligibilityCompetitors e _).2 ⟨d.target_mem f, hne, hx⟩, hf.1⟩ theorem good_missingFraction_le (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (C₀ N : ℝ) (e : E) (s : Sample P k) (hg : d.Good k U N e s) (hdet : ∀ o ∈ d.orbits e, JointLossCounts.deterministicLossFraction (d.full e o) (d.passing e o) ≤ C₀ / N) (o : O) (ho : o ∈ d.orbits e) : JointLossCounts.missingFraction (d.full e o) (d.passing e o) (d.variableBad k U e o) s ≤ (C₀ + 1) / N := JointLossCounts.good_missingFraction_le (d.orbits e) (d.full e) (d.passing e) (d.variableBad k U e) (d.eligibilityGood k U e) C₀ N s hg.2 (d.passing_subset e) hdet o ho theorem exists_selection [Fintype E] (d : OrbitData P E O V) (hinj : Function.Injective d.coarse) (k : ℕ) (U : Finset (ZMod (37 ^ k))) (C₀ N : ℝ) (hN : 0 < N) (D K : ℕ) (hdegree : ∀ e, ∀ o ∈ d.orbits e, ∀ v ∈ d.passing e o, (d.competitors e o v).card ≤ D) (hsmall : ∀ e, ((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) + ((d.orbits e).card : ℝ) * N * ((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) ≤ 1 / 2) (hdet : ∀ e, ∀ o ∈ d.orbits e, JointLossCounts.deterministicLossFraction (d.full e o) (d.passing e o) ≤ C₀ / N) (hK : (K : ℝ) ≤ (Fintype.card E : ℝ) * ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2)) : ∃ (s : Sample P k) (G : Finset E), G.card = K ∧ (∀ e ∈ G, d.Good k U N e s) ∧ (∀ e ∈ G, ∀ f ∈ G, e ≠ f → (d.coarse e).1 ≠ (d.coarse f).1) ∧ ∀ e ∈ G, ∀ o ∈ d.orbits e, JointLossCounts.missingFraction (d.full e o) (d.passing e o) (d.variableBad k U e o) s ≤ (C₀ + 1) / N := by classical have hi (e : E) : ((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2) * (Fintype.card (Sample P k) : ℝ) ≤ (goodIncidence (fun s e => d.Good k U N e s) e : ℝ) := by rw [d.goodIncidence_eq] exact d.goodEvent_card_lower k U N hN e D (hdegree e) (hsmall e) obtain ⟨s, hs⟩ := exists_goodCount_ge (fun s e => d.Good k U N e s) _ hi have hk : K ≤ goodCount (fun s e => d.Good k U N e s) s := by exact_mod_cast hK.trans hs obtain ⟨G, hcard, hgood⟩ := exists_good_subfamily (fun s e => d.Good k U N e s) s K hk refine ⟨s, G, hcard, hgood, ?_, ?_⟩ · intro e he f hf hne exact d.good_no_shared_x k U N f e s (hgood f hf) (hgood e he) (fun h => hne (hinj h)) · intro e he o ho exact d.good_missingFraction_le k U C₀ N e s (hgood e he) (hdet e) o ho end Finite end OrbitData end MatrixMultiplication.JointMaskedSelection end end end MatrixAllFields end OAI