import OAI.NumberTheory.PiExponent.Analysis.Collision import OAI.NumberTheory.PiExponent.Analysis.CollisionOccupancy namespace OAI open scoped BigOperators namespace PiExponent.DeterminantAnalyticBound theorem two_alternative_exponent {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] (group : ι → κ) (weight : κ → ℝ) {H A eta nu c N bbar E : ℝ} (hH : 0 < H) (hA : 0 ≤ A) (heta : 0 ≤ eta) (hnu : 0 ≤ nu) (hc : 0 ≤ c) (hN : 0 < N) (hw : ∀ a, 0 ≤ weight a) (hbeta : (Fintype.card ι : ℝ) * H * bbar ≤ ∑ i, weight (group i)) (hcap : (((Finset.univ : Finset κ).filter (fun a => weight a ≤ A * H)).card : ℝ) ≤ N) : -nu * ((∑ i, weight (group i)) - (Fintype.card ι : ℝ) * H * bbar) - c * (∑ a, (Collision.multiplicity Finset.univ group a : ℝ) ^ 2) + (Fintype.card ι : ℝ) * H * E ≤ (Fintype.card ι : ℝ) * H * (E + max (-c * eta ^ 2 * (Fintype.card ι : ℝ) / (H * N)) (-nu * (A * (1 - eta) - bbar))) := by classical let S := (Finset.univ : Finset κ).filter (fun a => weight a ≤ A * H) let M : ℝ := Fintype.card ι let Z : ℝ := ∑ a, (Collision.multiplicity Finset.univ group a : ℝ) ^ 2 have hM : 0 ≤ M := Nat.cast_nonneg _ have hZ : 0 ≤ Z := Finset.sum_nonneg fun _ _ => sq_nonneg _ have hmass : (∑ a ∈ S, (Collision.multiplicity Finset.univ group a : ℝ)) = (((Finset.univ : Finset ι).filter (fun i => weight (group i) ≤ A * H)).card : ℝ) := by have hn := Collision.sum_multiplicity_eq_card_filter (Finset.univ : Finset ι) S group have hn' : (∑ a ∈ S, Collision.multiplicity Finset.univ group a) = ((Finset.univ : Finset ι).filter (fun i => weight (group i) ≤ A * H)).card := by simpa only [S, Finset.mem_filter, Finset.mem_univ, true_and] using hn exact_mod_cast hn' by_cases hlow : eta * M ≤ ∑ a ∈ S, (Collision.multiplicity Finset.univ group a : ℝ) · have hsq : eta ^ 2 * M ^ 2 ≤ N * Z := by calc _ = (eta * M) ^ 2 := by ring _ ≤ (∑ a ∈ S, (Collision.multiplicity Finset.univ group a : ℝ)) ^ 2 := pow_le_pow_left₀ (mul_nonneg heta hM) hlow 2 _ ≤ (S.card : ℝ) * Z := Collision.sum_sq_le_card_mul_univ_sq S _ _ ≤ N * Z := mul_le_mul_of_nonneg_right hcap hZ have hcollision : -c * Z ≤ M * H * (-c * eta ^ 2 * M / (H * N)) := by have hz : eta ^ 2 * M ^ 2 / N ≤ Z := (div_le_iff₀ hN).mpr (by nlinarith [hsq]) have hh := mul_le_mul_of_nonneg_left hz hc have he : M * H * (-c * eta ^ 2 * M / (H * N)) = -c * (eta ^ 2 * M ^ 2 / N) := by field_simp rw [he] nlinarith [hh] have hscalar : -nu * ((∑ i, weight (group i)) - M * H * bbar) ≤ 0 := by have hp := mul_nonneg hnu (sub_nonneg.mpr hbeta) nlinarith [hp] have hmax := mul_le_mul_of_nonneg_left (le_max_left (-c * eta ^ 2 * M / (H * N)) (-nu * (A * (1 - eta) - bbar))) (mul_nonneg hM hH.le) change -nu * ((∑ i, weight (group i)) - M * H * bbar) - c * Z + M * H * E ≤ _ change _ ≤ M * H * (E + max (-c * eta ^ 2 * M / (H * N)) (-nu * (A * (1 - eta) - bbar))) nlinarith · have hl : (((Finset.univ : Finset ι).filter (fun i => weight (group i) ≤ A * H)).card : ℝ) ≤ eta * (Finset.univ : Finset ι).card := by rw [← hmass] exact (le_of_not_ge hlow) have hwgt := Collision.sum_weight_ge_of_low_count_le (Finset.univ : Finset ι) (fun i => weight (group i)) (mul_nonneg hA hH.le) (fun i _ => hw _) hl have hwgt' : A * H * (1 - eta) * M ≤ ∑ i, weight (group i) := by simpa only [Finset.card_univ, M] using hwgt have hmul := mul_le_mul_of_nonneg_left hwgt' hnu have hscalar : -nu * ((∑ i, weight (group i)) - M * H * bbar) ≤ M * H * (-nu * (A * (1 - eta) - bbar)) := by nlinarith [hmul] have hcollision : -c * Z ≤ 0 := by nlinarith [mul_nonneg hc hZ] have hmax := mul_le_mul_of_nonneg_left (le_max_right (-c * eta ^ 2 * M / (H * N)) (-nu * (A * (1 - eta) - bbar))) (mul_nonneg hM hH.le) change -nu * ((∑ i, weight (group i)) - M * H * bbar) - c * Z + M * H * E ≤ _ change _ ≤ M * H * (E + max (-c * eta ^ 2 * M / (H * N)) (-nu * (A * (1 - eta) - bbar))) nlinarith theorem translated_summand_bound {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] (group : ι → κ) (weight : κ → ℝ) (scalar determinant : ℂ) {H A eta nu c N bbar Etr Ehol rho : ℝ} (hH : 0 < H) (hA : 0 ≤ A) (heta : 0 ≤ eta) (hnu : 0 ≤ nu) (hc : 0 ≤ c) (hN : 0 < N) (hw : ∀ a, 0 ≤ weight a) (hbeta : (Fintype.card ι : ℝ) * H * bbar ≤ ∑ i, weight (group i)) (hcap : (((Finset.univ : Finset κ).filter (fun a => weight a ≤ A * H)).card : ℝ) ≤ N) (hscalar : ‖scalar‖ ≤ Real.exp (-nu * ((∑ i, weight (group i)) - (Fintype.card ι : ℝ) * H * bbar) + (Fintype.card ι : ℝ) * H * Etr)) (hdet : ‖determinant‖ ≤ Real.exp (-c * (∑ a, (Collision.multiplicity Finset.univ group a : ℝ) ^ 2) + (Fintype.card ι : ℝ) * H * (Ehol + rho))) : ‖scalar * determinant‖ ≤ Real.exp ((Fintype.card ι : ℝ) * H * (Etr + Ehol + rho + max (-c * eta ^ 2 * (Fintype.card ι : ℝ) / (H * N)) (-nu * (A * (1 - eta) - bbar)))) := by rw [norm_mul] apply (mul_le_mul hscalar hdet (norm_nonneg _) (Real.exp_pos _).le).trans rw [← Real.exp_add] apply Real.exp_le_exp.mpr have h := two_alternative_exponent group weight hH hA heta hnu hc hN hw hbeta hcap (E := Etr + Ehol + rho) nlinarith [h] theorem log_norm_sum_le {Ω : Type*} [Fintype Ω] (term : Ω → ℂ) (delta : ℂ) {M : ℕ} {H Q E : ℝ} (hM : 0 < M) (hH : 0 < H) (hQ : 0 < Q) (hcard : (Fintype.card Ω : ℝ) ≤ Q ^ M) (hexpansion : delta = ∑ x, term x) (hne : delta ≠ 0) (hterm : ∀ x, ‖term x‖ ≤ Real.exp ((M : ℝ) * H * E)) : Real.log ‖delta‖ / ((M : ℝ) * H) ≤ E + Real.log Q / H := by have hbound : ‖delta‖ ≤ Real.exp ((M : ℝ) * H * (E + Real.log Q / H)) := by calc _ = ‖∑ x, term x‖ := by rw [hexpansion] _ ≤ ∑ x, ‖term x‖ := norm_sum_le _ _ _ ≤ ∑ _x : Ω, Real.exp ((M : ℝ) * H * E) := Finset.sum_le_sum fun x _ => hterm x _ = (Fintype.card Ω : ℝ) * Real.exp ((M : ℝ) * H * E) := by simp _ ≤ Q ^ M * Real.exp ((M : ℝ) * H * E) := mul_le_mul_of_nonneg_right hcard (Real.exp_pos _).le _ = _ := by have he : Q ^ M = Real.exp ((M : ℝ) * Real.log Q) := by rw [Real.exp_nat_mul, Real.exp_log hQ] rw [he, ← Real.exp_add] congr 1 field_simp ring have hlog := Real.log_le_log (norm_pos_iff.mpr hne) hbound rw [Real.log_exp] at hlog apply (div_le_iff₀ (mul_pos (by exact_mod_cast hM) hH)).mpr nlinarith [hlog] end PiExponent.DeterminantAnalyticBound end OAI