| import OAI.NumberTheory.PiExponent.Analysis.Collision |
| import OAI.NumberTheory.PiExponent.Analysis.CollisionOccupancy |
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| namespace OAI |
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|
| open scoped BigOperators |
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| namespace PiExponent.DeterminantAnalyticBound |
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| 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 |
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|
| 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 |
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| end OAI |
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