File size: 3,386 Bytes
7d9acbd | 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 | import OAI.Geometry.HeilbronnTriangle.ConditionalMatrix
import OAI.Geometry.HeilbronnTriangle.MinorInvariant
import OAI.Geometry.HeilbronnTriangle.RepeatedLabels
namespace OAI
namespace Problem355.ConditionalMatrix
open scoped BigOperators
theorem arrayMatrix_map {X : Type*} {B k j : ℕ} (hjk : j ≤ k)
(digit : Fin 3 → Fin 3 → Fin k → X → Fin B)
(f : Fin 3 → Fin 3 → Fin k → X) :
(arrayMatrix digit f k).map (ZMod.castHom (Nat.pow_dvd_pow B hjk) (ZMod (B ^ j))) =
arrayMatrix digit f j := by
ext a c
simp only [Matrix.map_apply, arrayMatrix, map_natCast]
theorem array_moment_le_two_of_diagonal {X : Type*} [Fintype X] [Nonempty X]
{B k : ℕ} (hB : B.Prime) (hk : 0 < k)
(digit : Fin 3 → Fin 3 → Fin k → X → Fin B)
(hinj : ∀ a c v, Function.Injective (digit a c v))
(hpos : ∀ a c v x, 0 < (digit a c v x).val)
(b e : (Fin 3 → Fin 3 → Fin k → X) → ℕ)
(hbe : ∀ f, b f ≤ e f) (hek : ∀ f, e f ≤ k)
(P Q : (Fin 3 → Fin 3 → Fin k → X) →
Matrix.GeneralLinearGroup (Fin 3) (ZMod (B ^ k)))
(hdiag : ∀ f, arrayMatrix digit f k =
(P f : Matrix (Fin 3) (Fin 3) (ZMod (B ^ k))) *
Matrix.diagonal ![1, ((B ^ b f : ℕ) : ZMod (B ^ k)),
((B ^ e f : ℕ) : ZMod (B ^ k))] *
(Q f : Matrix (Fin 3) (Fin 3) (ZMod (B ^ k))))
(hsmall : (B : ℝ) * (1 / (Fintype.card X : ℝ)) ^ 4 ≤ 1 / 2) :
(∑ f : Fin 3 → Fin 3 → Fin k → X,
(1 / (Fintype.card (Fin 3 → Fin 3 → Fin k → X) : ℝ)) * (B : ℝ) ^ b f) ≤ 2 := by
apply array_moment_le_two hB hk digit hinj hpos b
(fun f => (hbe f).trans (hek f)) _ hsmall
intro j hjpos hj f hbf
have h := (UnitPivot.minorsVanish_reduce_prime_power_iff hB (hbe f) hj
(P f) (Q f) (arrayMatrix digit f k) (hdiag f)).mpr hbf
simpa only [arrayMatrix_map hj] using h
theorem collision_moment_le_six_div {Λ X : Type*}
[Fintype Λ] [DecidableEq Λ] [Nonempty Λ] [Fintype X] [Nonempty X]
{B k : ℕ} (hB : B.Prime) (hk : 0 < k)
(digit : Λ → Λ → Λ → Fin 3 → Fin 3 → Fin k → X → Fin B)
(hinj : ∀ a b c i l v, Function.Injective (digit a b c i l v))
(hpos : ∀ a b c i l v x, 0 < (digit a b c i l v x).val)
(val : Λ → Λ → Λ → (Fin 3 → Fin 3 → Fin k → X) → ℕ)
(hval : ∀ a b c f, val a b c f ≤ k)
(hminor : ∀ a b c j, 1 ≤ j → j ≤ k → ∀ f, j ≤ val a b c f →
UnitPivot.MinorsVanish (arrayMatrix (digit a b c) f j))
(hsmall : (B : ℝ) * (1 / (Fintype.card X : ℝ)) ^ 4 ≤ 1 / 2) :
(∑ a : Λ, ∑ b : Λ, ∑ c : Λ, ∑ f : Fin 3 → Fin 3 → Fin k → X,
(1 / (Fintype.card (Fin 3 → Fin 3 → Fin k → X) : ℝ)) *
(if a = b ∨ a = c ∨ b = c then (B : ℝ) ^ val a b c f else 0)) /
(Fintype.card Λ : ℝ) ^ 3 ≤ 6 / (Fintype.card Λ : ℝ) := by
have h := Sampling.conditional_collision_moment_le
(α := Λ) (Ω := Fin 3 → Fin 3 → Fin k → X)
(fun _ _ _ _ => 1 / (Fintype.card (Fin 3 → Fin 3 → Fin k → X) : ℝ))
(fun a b c f => (B : ℝ) ^ val a b c f) 2 (by norm_num)
(fun a b c => array_moment_le_two hB hk (digit a b c) (hinj a b c)
(hpos a b c) (val a b c) (hval a b c) (hminor a b c) hsmall)
norm_num only [show (3 : ℝ) * 2 = 6 by norm_num] at h
exact h
end Problem355.ConditionalMatrix
end OAI
|