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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
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