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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 | import Mathlib
namespace OAI
namespace PiExponent.Collision
open scoped BigOperators
variable {G : Type*}
theorem sum_sq_le_card_mul_sum_sq (S : Finset G) (n : G → ℝ) :
(∑ a ∈ S, n a) ^ 2 ≤ (S.card : ℝ) * ∑ a ∈ S, n a ^ 2 := by
simpa using Finset.sum_mul_sq_le_sq_mul_sq S (fun _ : G => (1 : ℝ)) n
theorem sum_sq_le_card_mul_total_sq (S U : Finset G) (hSU : S ⊆ U)
(n : G → ℝ) :
(∑ a ∈ S, n a) ^ 2 ≤ (S.card : ℝ) * ∑ a ∈ U, n a ^ 2 := by
apply (sum_sq_le_card_mul_sum_sq S n).trans
apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
exact Finset.sum_le_sum_of_subset_of_nonneg hSU (fun a _ _ => sq_nonneg (n a))
theorem mass_sq_div_card_le_total_sq (S U : Finset G) (hSU : S ⊆ U)
(n : G → ℝ) {mass : ℝ} (hmass : 0 ≤ mass)
(hlow : mass ≤ ∑ a ∈ S, n a) (hcard : 0 < S.card) :
mass ^ 2 / (S.card : ℝ) ≤ ∑ a ∈ U, n a ^ 2 := by
have hsq : mass ^ 2 ≤ (∑ a ∈ S, n a) ^ 2 := by
exact pow_le_pow_left₀ hmass hlow 2
have hc : (0 : ℝ) < S.card := by exact_mod_cast hcard
apply (div_le_iff₀ hc).2
calc
mass ^ 2 ≤ (∑ a ∈ S, n a) ^ 2 := hsq
_ ≤ (S.card : ℝ) * ∑ a ∈ U, n a ^ 2 :=
sum_sq_le_card_mul_total_sq S U hSU n
_ = (∑ a ∈ U, n a ^ 2) * (S.card : ℝ) := mul_comm _ _
theorem fraction_sq_mul_mass_sq_div_card_le_total_sq
(S U : Finset G) (hSU : S ⊆ U) (n : G → ℝ)
{η M : ℝ} (hη : 0 ≤ η) (hM : 0 ≤ M)
(hlow : η * M ≤ ∑ a ∈ S, n a) (hcard : 0 < S.card) :
η ^ 2 * M ^ 2 / (S.card : ℝ) ≤ ∑ a ∈ U, n a ^ 2 := by
simpa only [mul_pow] using
mass_sq_div_card_le_total_sq S U hSU n (mul_nonneg hη hM) hlow hcard
variable [Fintype G]
theorem sum_sq_le_card_mul_univ_sq (S : Finset G) (n : G → ℝ) :
(∑ a ∈ S, n a) ^ 2 ≤ (S.card : ℝ) * ∑ a, n a ^ 2 := by
exact sum_sq_le_card_mul_total_sq S Finset.univ (Finset.subset_univ S) n
theorem nat_occupancy_square_lower_bound (S : Finset G) (n : G → ℕ)
{η : ℝ} {M : ℕ} (hη : 0 ≤ η)
(hlow : η * (M : ℝ) ≤ ∑ a ∈ S, (n a : ℝ)) (hcard : 0 < S.card) :
η ^ 2 * (M : ℝ) ^ 2 / (S.card : ℝ) ≤ ∑ a, (n a : ℝ) ^ 2 := by
exact fraction_sq_mul_mass_sq_div_card_le_total_sq S Finset.univ
(Finset.subset_univ S) (fun a => (n a : ℝ)) hη (Nat.cast_nonneg M) hlow hcard
end PiExponent.Collision
end OAI
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