openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Counting /PrimeHashCollision.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.PrimeHash | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.ResidueCoverage | |
| import Mathlib.Data.Int.ModEq | |
| namespace OAI | |
| /-! | |
| # Prime hashing supplies the finite collision interface | |
| This connects the actual finite uniform prime sampler to the previously proved | |
| collision-to-occupancy theorem. Integer differences may have polynomially many | |
| bits; the parameter `u` remains explicit. | |
| -/ | |
| namespace LowSpaceSubsetSum | |
| open scoped BigOperators | |
| noncomputable def primeProbability (ell : ℕ) (_p : ℕ) : ℝ := | |
| 1 / (primeRange ell).card | |
| theorem primeRange_nonempty {ell : ℕ} (hell : 4 ≤ ell) : | |
| (primeRange ell).Nonempty := by | |
| apply Finset.card_pos.mp | |
| have h := primeRange_card_lower hell | |
| have hp : 0 < 2 ^ ell := pow_pos (by decide) _ | |
| nlinarith | |
| theorem primeProbability_sum {ell : ℕ} (hell : 4 ≤ ell) : | |
| (∑ p ∈ primeRange ell, primeProbability ell p) = 1 := by | |
| have h : ((primeRange ell).card : ℝ) ≠ 0 := by | |
| exact_mod_cast (primeRange_nonempty hell).card_pos.ne' | |
| simp [primeProbability, h] | |
| theorem prime_pairwise_collision_bound (X : Finset ℤ) {ell u : ℕ} | |
| (hell : 4 ≤ ell) (hellu : ell ≤ u) | |
| (hbits : ∀ x ∈ X, ∀ y ∈ X, (x - y).natAbs < 2 ^ (2 * u)) : | |
| PairwiseCollisionBound (primeRange ell) (primeProbability ell) X | |
| (fun p x => x % (p : ℤ)) (8 * (u : ℝ) ^ 2 / (2 : ℝ) ^ ell) := by | |
| intro x hx y hy hxy | |
| have hd : 0 < (x - y).natAbs := Int.natAbs_pos.mpr (sub_ne_zero.mpr hxy) | |
| have hiff (p : ℕ) : x % (p : ℤ) = y % (p : ℤ) ↔ p ∣ (x - y).natAbs := by | |
| rw [Int.emod_eq_emod_iff_emod_sub_eq_zero, ← Int.dvd_iff_emod_eq_zero, | |
| Int.natCast_dvd] | |
| have heq : (∑ p ∈ primeRange ell, | |
| primeProbability ell p * if x % (p : ℤ) = y % (p : ℤ) then (1 : ℝ) else 0) = | |
| (((primeRange ell).filter fun p => p ∣ (x - y).natAbs).card : ℝ) / | |
| (primeRange ell).card := by | |
| simp_rw [hiff, mul_ite, mul_one, mul_zero] | |
| rw [← Finset.sum_filter] | |
| simp [primeProbability, div_eq_mul_inv, mul_comm] | |
| rw [heq] | |
| exact prime_divisor_probability_le hell hellu hd (hbits x hx y hy) | |
| /-- The abstract occupancy result now has a proved prime-hashing input. -/ | |
| theorem prime_occupied_probability (X : Finset ℤ) {ell u : ℕ} | |
| (hell : 4 ≤ ell) (hellu : ell ≤ u) | |
| (hbits : ∀ x ∈ X, ∀ y ∈ X, (x - y).natAbs < 2 ^ (2 * u)) | |
| {T : ℝ} (hT : 0 < T) : | |
| 1 - ((X.card : ℝ) + (8 * (u : ℝ) ^ 2 / (2 : ℝ) ^ ell) * (X.card : ℝ) ^ 2) / T ≤ | |
| ∑ p ∈ (primeRange ell).filter | |
| (fun p : ℕ => (X.card : ℝ) ^ 2 / T ≤ ((X.image (fun x => x % (p : ℤ))).card : ℝ)), | |
| primeProbability ell p := by | |
| apply occupied_probability_of_collision_bound | |
| · intro p _ | |
| exact div_nonneg zero_le_one (Nat.cast_nonneg _) | |
| · exact primeProbability_sum hell | |
| · positivity | |
| · exact hT | |
| · exact prime_pairwise_collision_bound X hell hellu hbits | |
| end LowSpaceSubsetSum | |
| end OAI | |