File size: 2,896 Bytes
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 | 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
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