| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Extraction.JointMaskedSelection |
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| namespace OAI |
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| namespace MatrixAllFields |
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| open scoped BigOperators Topology Polynomial |
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| section |
| noncomputable section |
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| namespace MatrixMultiplication.JointMaskedRates |
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| open Filter JointCoarseHashing JointExtractionRates JointMaskedSelection |
| open JointOrdinarySelection (hashLevel hashSet) |
| open scoped Topology |
|
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| attribute [local instance] Classical.propDecidable |
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| def errorBound (degree : ℕ → ℕ) (B p : ℕ) (H : ℝ) (N : ℕ) : ℝ := |
| (degree N : ℝ) / (hashModulus H N : ℝ) * |
| (1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ)) |
|
|
| theorem polynomialFactor_le (B p N : ℕ) : |
| 1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ) ≤ |
| ((B : ℝ) + 1) * ((N : ℝ) + 1) ^ (p + 1) := by |
| have hN : (0 : ℝ) ≤ N := Nat.cast_nonneg _ |
| have hx : (0 : ℝ) ≤ (N : ℝ) + 1 := by linarith |
| have hpower : (1 : ℝ) ≤ ((N : ℝ) + 1) ^ (p + 1) := |
| one_le_pow₀ (by linarith : (1 : ℝ) ≤ (N : ℝ) + 1) |
| calc |
| _ ≤ 1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * ((N : ℝ) + 1) := |
| add_le_add le_rfl (mul_le_mul_of_nonneg_left (by linarith) |
| (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hx _))) |
| _ = 1 + (B : ℝ) * ((N : ℝ) + 1) ^ (p + 1) := by rw [pow_succ]; ring |
| _ ≤ ((N : ℝ) + 1) ^ (p + 1) + (B : ℝ) * ((N : ℝ) + 1) ^ (p + 1) := |
| add_le_add hpower le_rfl |
| _ = _ := by ring |
|
|
| theorem eventually_errorBound_le_half (degree : ℕ → ℕ) (B p : ℕ) |
| {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) : |
| ∀ᶠ N in atTop, errorBound degree B p H N ≤ 1 / 2 := by |
| 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 |
| apply eventually_error_le_half ((B : ℝ) + 1) (p + 1) hc |
| filter_upwards [hratio] with N hN |
| have hx : (0 : ℝ) ≤ (N : ℝ) + 1 := add_nonneg (Nat.cast_nonneg N) zero_le_one |
| have hfactor : 0 ≤ 1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ) := |
| add_nonneg zero_le_one |
| (mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hx _)) (Nat.cast_nonneg _)) |
| calc |
| errorBound degree B p H N ≤ Real.exp (-c * (N : ℝ)) * |
| (1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ)) := |
| mul_le_mul_of_nonneg_right hN hfactor |
| _ ≤ Real.exp (-c * (N : ℝ)) * (((B : ℝ) + 1) * ((N : ℝ) + 1) ^ (p + 1)) := |
| mul_le_mul_of_nonneg_left (polynomialFactor_le B p N) (Real.exp_pos _).le |
| _ = ((B : ℝ) + 1) * (((N : ℝ) + 1) ^ (p + 1) * Real.exp (-c * (N : ℝ))) := by ring |
|
|
| theorem count_error_le_envelope {P E O V : Type*} (d : OrbitData P E O V) |
| (e : E) (degree B p N M : ℕ) |
| (heligibility : (d.eligibilityCompetitors e).card ≤ degree) |
| (horbits : (d.orbits e).card ≤ B * (N + 1) ^ p) : |
| ((d.eligibilityCompetitors e).card : ℝ) / (M : ℝ) + |
| ((d.orbits e).card : ℝ) * (N : ℝ) * ((degree : ℝ) / (M : ℝ)) ≤ |
| (degree : ℝ) / (M : ℝ) * (1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ)) := by |
| have hdegreeR : ((d.eligibilityCompetitors e).card : ℝ) ≤ (degree : ℝ) := by |
| exact_mod_cast heligibility |
| have horbitsR : ((d.orbits e).card : ℝ) ≤ (B : ℝ) * ((N : ℝ) + 1) ^ p := by |
| exact_mod_cast horbits |
| have hratio : (0 : ℝ) ≤ (degree : ℝ) / (M : ℝ) := |
| div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _) |
| calc |
| _ ≤ (degree : ℝ) / (M : ℝ) + |
| ((B : ℝ) * ((N : ℝ) + 1) ^ p) * (N : ℝ) * ((degree : ℝ) / (M : ℝ)) := |
| add_le_add (div_le_div_of_nonneg_right hdegreeR (Nat.cast_nonneg _)) |
| (mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right horbitsR (Nat.cast_nonneg _)) hratio) |
| _ = _ := by ring |
|
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| theorem eventually_exists_selection |
| {P E O V : ℕ → Type*} [∀ N, Fintype (P N)] [∀ N, Fintype (E N)] |
| (d : ∀ N, OrbitData (P N) (E N) (O N) (V N)) |
| (hinj : ∀ N, Function.Injective (d N).coarse) |
| (degree : ℕ → ℕ) (B p : ℕ) |
| (heligibility : ∀ N e, ((d N).eligibilityCompetitors e).card ≤ degree N) |
| (hdegree : ∀ N e, ∀ o ∈ (d N).orbits e, ∀ v ∈ (d N).passing e o, |
| ((d N).competitors e o v).card ≤ degree N) |
| (horbits : ∀ N e, ((d N).orbits e).card ≤ B * (N + 1) ^ p) |
| {A D H C₀ : 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) |
| (hdet : ∀ᶠ N in atTop, ∀ e, ∀ o ∈ (d N).orbits e, |
| JointLossCounts.deterministicLossFraction ((d N).full e o) ((d N).passing e o) ≤ |
| C₀ / (N : ℝ)) |
| (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, (d N).Good (hashLevel H N) (hashSet H N) (N : ℝ) e s) ∧ |
| (∀ e ∈ G, ∀ f ∈ G, e ≠ f → ((d N).coarse e).1 ≠ ((d N).coarse f).1) ∧ |
| ∀ e ∈ G, ∀ o ∈ (d N).orbits e, |
| JointLossCounts.missingFraction ((d N).full e o) ((d N).passing e o) |
| ((d N).variableBad (hashLevel H N) (hashSet H N) e o) s ≤ (C₀ + 1) / (N : ℝ) := by |
| classical |
| have huniform := eventually_errorBound_le_half degree B p hH hgap hdegreeRate |
| have hsmall : ∀ᶠ N in atTop, ∀ e, |
| (((d N).eligibilityCompetitors e).card : ℝ) / (hashModulus H N : ℝ) + |
| (((d N).orbits e).card : ℝ) * (N : ℝ) * |
| ((degree N : ℝ) / (hashModulus H N : ℝ)) ≤ 1 / 2 := by |
| filter_upwards [huniform] with N hN e |
| exact (count_error_le_envelope (d N) e (degree N) B p N (hashModulus H N) |
| (heligibility N e) (horbits N e)).trans hN |
| have hpositive : ∀ᶠ N : ℕ in atTop, (0 : ℝ) < N := by |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with N hN |
| exact_mod_cast hN |
| let good := fun N (s : Sample (P N) (hashLevel H N)) (e : E N) => |
| (d N).Good (hashLevel H N) (hashSet H N) (N : ℝ) 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, hpositive] with N hN hNpos e |
| rw [show goodIncidence (good N) e = |
| ((d N).goodEvent (hashLevel H N) (hashSet H N) (N : ℝ) e).card from |
| (d N).goodIncidence_eq _ _ _ _] |
| exact (d N).goodEvent_card_lower (hashLevel H N) (hashSet H N) (N : ℝ) hNpos e |
| (degree N) (hdegree N e) (hN e) |
| 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, hdet] with N hN hdetN |
| obtain ⟨s, G, hcard, hgood⟩ := hN |
| refine ⟨s, G, hcard, hgood, ?_, ?_⟩ |
| · intro e he f hf hne |
| exact (d N).good_no_shared_x (hashLevel H N) (hashSet H N) (N : ℝ) f e s |
| (hgood f hf) (hgood e he) (fun h => hne ((hinj N) h)) |
| · intro e he o ho |
| exact (d N).good_missingFraction_le (hashLevel H N) (hashSet H N) C₀ (N : ℝ) e s |
| (hgood e he) (hdetN e) o ho |
|
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| end MatrixMultiplication.JointMaskedRates |
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| end |
| end |
|
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| end MatrixAllFields |
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| end OAI |
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