| import OAI.NumberTheory.PiExponent.Analysis.CollisionCombinatorics | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace PiExponent.Collision | |
| theorem sum_pow_fin_le_inv_one_sub {s : ℝ} (hs0 : 0 ≤ s) (hs1 : s < 1) | |
| (N : ℕ) : (∑ k : Fin N, s ^ (k : ℕ)) ≤ (1 - s)⁻¹ := by | |
| rw [Fin.sum_univ_eq_sum_range (fun k : ℕ => s ^ k), ← one_div] | |
| apply (le_div_iff₀ (sub_pos.mpr hs1)).2 | |
| rw [geom_sum_mul_neg] | |
| exact sub_le_self 1 (pow_nonneg hs0 N) | |
| theorem sum_pow_degree_assignments_eq {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (s : ℝ) (N : ℕ) : | |
| (∑ f : ι → Fin N, s ^ (∑ i, (f i : ℕ))) = | |
| (∑ k : Fin N, s ^ (k : ℕ)) ^ Fintype.card ι := by | |
| classical | |
| simp_rw [← Finset.prod_pow_eq_pow_sum] | |
| rw [← Fintype.prod_sum (fun (_ : ι) (k : Fin N) => s ^ (k : ℕ))] | |
| simp | |
| theorem sum_pow_degree_assignments_le {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| {s : ℝ} (hs0 : 0 ≤ s) (hs1 : s < 1) (N : ℕ) : | |
| (∑ f : ι → Fin N, s ^ (∑ i, (f i : ℕ))) ≤ | |
| ((1 - s)⁻¹) ^ Fintype.card ι := by | |
| rw [sum_pow_degree_assignments_eq] | |
| exact pow_le_pow_left₀ (Finset.sum_nonneg fun k _ => pow_nonneg hs0 _) | |
| (sum_pow_fin_le_inv_one_sub hs0 hs1 N) _ | |
| theorem collision_geometric_sum_le | |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] | |
| (group : ι → κ) {s : ℝ} (hs0 : 0 ≤ s) (hs1 : s < 1) (N : ℕ) : | |
| (∑ f ∈ (Finset.univ : Finset (ι → Fin N)).filter | |
| (fun f => Function.Injective (fun i => (group i, (f i : ℕ)))), | |
| (s * s) ^ (∑ i, (f i : ℕ))) ≤ | |
| s ^ (∑ a, (multiplicity Finset.univ group a).choose 2) * | |
| ((1 - s)⁻¹) ^ Fintype.card ι := by | |
| classical | |
| let good := (Finset.univ : Finset (ι → Fin N)).filter | |
| (fun f => Function.Injective (fun i => (group i, (f i : ℕ)))) | |
| let C := ∑ a, (multiplicity Finset.univ group a).choose 2 | |
| have hpoint : ∀ f ∈ good, | |
| (s * s) ^ (∑ i, (f i : ℕ)) ≤ s ^ C * s ^ (∑ i, (f i : ℕ)) := by | |
| intro f hf | |
| have hinj := (Finset.mem_filter.mp hf).2 | |
| have hdeg : C ≤ ∑ i, (f i : ℕ) := | |
| sum_choose_multiplicity_le_sum_degree_univ group (fun i => (f i : ℕ)) hinj | |
| rw [mul_pow] | |
| exact mul_le_mul_of_nonneg_right (pow_le_pow_of_le_one hs0 hs1.le hdeg) | |
| (pow_nonneg hs0 _) | |
| calc | |
| _ ≤ ∑ f ∈ good, s ^ C * s ^ (∑ i, (f i : ℕ)) := | |
| Finset.sum_le_sum hpoint | |
| _ = s ^ C * ∑ f ∈ good, s ^ (∑ i, (f i : ℕ)) := by rw [Finset.mul_sum] | |
| _ ≤ s ^ C * ∑ f : ι → Fin N, s ^ (∑ i, (f i : ℕ)) := by | |
| apply mul_le_mul_of_nonneg_left _ (pow_nonneg hs0 C) | |
| apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) | |
| intro f _ _ | |
| exact pow_nonneg hs0 _ | |
| _ ≤ _ := mul_le_mul_of_nonneg_left (sum_pow_degree_assignments_le hs0 hs1 N) | |
| (pow_nonneg hs0 C) | |
| end PiExponent.Collision | |
| end OAI | |