| import OAI.NumberTheory.PiExponent.Analysis.CollisionCombinatorics | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace PiExponent.Collision | |
| noncomputable def collisionRatio : ℝ := Real.exp (-Real.log 2 / 2) | |
| theorem collisionRatio_pos : 0 < collisionRatio := Real.exp_pos _ | |
| theorem collisionRatio_lt_one : collisionRatio < 1 := by | |
| apply Real.exp_lt_one_iff.mpr | |
| have h : 0 < Real.log 2 := Real.log_pos (by norm_num) | |
| linarith | |
| theorem collisionRatio_mul_self : collisionRatio * collisionRatio = 1 / 2 := by | |
| unfold collisionRatio | |
| rw [← Real.exp_add] | |
| have heq : -Real.log 2 / 2 + -Real.log 2 / 2 = -Real.log 2 := by ring | |
| rw [heq, Real.exp_neg, Real.exp_log (by norm_num)] | |
| simp only [one_div] | |
| theorem factorial_le_exp_card_log (M : ℕ) : | |
| (M.factorial : ℝ) ≤ Real.exp ((M : ℝ) * Real.log M) := by | |
| by_cases hM : M = 0 | |
| · subst M | |
| norm_num | |
| · rw [Real.exp_nat_mul, Real.exp_log (by exact_mod_cast Nat.pos_of_ne_zero hM)] | |
| exact_mod_cast Nat.factorial_le_pow M | |
| theorem twice_collision_count_eq | |
| {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq κ] (group : ι → κ) : | |
| 2 * ((∑ a, (multiplicity Finset.univ group a).choose 2 : ℕ) : ℝ) = | |
| (∑ a, (multiplicity Finset.univ group a : ℝ) ^ 2) - Fintype.card ι := by | |
| have hsum : (∑ a, (multiplicity Finset.univ group a : ℝ)) = Fintype.card ι := by | |
| exact_mod_cast sum_multiplicity_univ group | |
| push_cast | |
| rw [Finset.mul_sum] | |
| calc | |
| _ = ∑ a, ((multiplicity Finset.univ group a : ℝ) ^ 2 - | |
| (multiplicity Finset.univ group a : ℝ)) := by | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| rw [Nat.cast_choose_two] | |
| ring | |
| _ = _ := by rw [Finset.sum_sub_distrib, hsum] | |
| theorem collision_factor_le_exp (M C ell : ℕ) {H A v S : ℝ} | |
| (hell : (ell : ℝ) ≤ (M : ℝ) * H / v) | |
| (hC : 2 * (C : ℝ) = S - M) : | |
| ((M.factorial : ℝ) * (Real.exp (H * A)) ^ M * (2 : ℝ) ^ ell) * | |
| collisionRatio ^ C * ((1 - collisionRatio)⁻¹) ^ M ≤ | |
| Real.exp (-(Real.log 2 / 4) * S + (M : ℝ) * H * (A + Real.log 2 / v) + | |
| (M : ℝ) * (Real.log M + Real.log 2 / 4 - Real.log (1 - collisionRatio))) := by | |
| have hlog2 : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num) | |
| have hr : 0 < 1 - collisionRatio := sub_pos.mpr collisionRatio_lt_one | |
| have ht : (2 : ℝ) ^ ell ≤ Real.exp ((M : ℝ) * H / v * Real.log 2) := by | |
| calc | |
| _ = Real.exp ((ell : ℝ) * Real.log 2) := by | |
| rw [Real.exp_nat_mul, Real.exp_log (by norm_num)] | |
| _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hell hlog2) | |
| calc | |
| _ ≤ (Real.exp ((M : ℝ) * Real.log M) * (Real.exp (H * A)) ^ M * | |
| Real.exp ((M : ℝ) * H / v * Real.log 2)) * | |
| collisionRatio ^ C * ((1 - collisionRatio)⁻¹) ^ M := by | |
| apply mul_le_mul_of_nonneg_right _ (pow_nonneg (inv_nonneg.mpr hr.le) M) | |
| apply mul_le_mul_of_nonneg_right _ (pow_nonneg collisionRatio_pos.le C) | |
| apply mul_le_mul _ ht (by positivity) (by positivity) | |
| exact mul_le_mul_of_nonneg_right (factorial_le_exp_card_log M) (by positivity) | |
| _ = _ := by | |
| rw [← Real.exp_nat_mul, collisionRatio, ← Real.exp_nat_mul] | |
| have hi : (1 - Real.exp (-Real.log 2 / 2))⁻¹ = | |
| Real.exp (-Real.log (1 - collisionRatio)) := by | |
| rw [Real.exp_neg, Real.exp_log hr] | |
| rfl | |
| rw [hi, ← Real.exp_nat_mul] | |
| simp only [← Real.exp_add] | |
| congr 1 | |
| have hc : (C : ℝ) = (S - M) / 2 := by linarith | |
| rw [hc] | |
| unfold collisionRatio | |
| ring | |
| noncomputable def collisionRemainder (M : ℕ) (H : ℝ) : ℝ := | |
| (Real.log M + Real.log 2 / 4 - Real.log (1 - collisionRatio)) / H | |
| theorem collision_factor_le_exp_normalized (M C ell : ℕ) {H A v S : ℝ} | |
| (hH : 0 < H) (hell : (ell : ℝ) ≤ (M : ℝ) * H / v) | |
| (hC : 2 * (C : ℝ) = S - M) : | |
| ((M.factorial : ℝ) * (Real.exp (H * A)) ^ M * (2 : ℝ) ^ ell) * | |
| collisionRatio ^ C * ((1 - collisionRatio)⁻¹) ^ M ≤ | |
| Real.exp (-(Real.log 2 / 4) * S + (M : ℝ) * H * | |
| (A + Real.log 2 / v + collisionRemainder M H)) := by | |
| convert collision_factor_le_exp M C ell hell hC using 1 | |
| congr 1 | |
| unfold collisionRemainder | |
| field_simp | |
| ring | |
| end PiExponent.Collision | |
| end OAI | |