| 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 | |