openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Analysis /AnalyticDeterminantCollision.lean
| import OAI.NumberTheory.PiExponent.Analysis.AnalyticRowSeries | |
| import OAI.NumberTheory.PiExponent.Analysis.CollisionExponential | |
| import OAI.NumberTheory.PiExponent.Analysis.CollisionLimit | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace PiExponent.AnalyticCollision | |
| theorem translated_row_determinant_bound | |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] | |
| (group : ι → κ) (f : κ → ι → ℂ → ℂ) | |
| (center : ι → ℂ) (ell : ι → ℕ) | |
| (R : NNReal) (hR : 0 < R) (D s : ℝ) (hD : 0 ≤ D) | |
| (hs0 : 0 ≤ s) (hs1 : s < 1) (hss : s * s = 1 / 2) | |
| (hf : ∀ a j, Differentiable ℂ (f a j)) | |
| (hfD : ∀ a j z, z ∈ Metric.sphere (0 : ℂ) (R : ℝ) → ‖f a j z‖ ≤ D) | |
| (hc : ∀ i, ‖center i‖ + 3 / 4 ≤ (R : ℝ) / 2) : | |
| ‖Matrix.det (fun i j => rowTest (ell i) | |
| (fun t => f (group i) j (center i + Complex.log (1 + t))))‖ ≤ | |
| ((Fintype.card ι).factorial * D ^ Fintype.card ι * (∏ i, (2 : ℝ) ^ ell i)) * | |
| s ^ (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) * | |
| ((1 - s)⁻¹) ^ Fintype.card ι := by | |
| let B : ι → ℕ → ℂ := fun i d => rowTest (ell i) | |
| (fun t => ((center i + Complex.log (1 + t)) / (R : ℂ)) ^ d) | |
| let C : κ → ℕ → ι → ℂ := fun a d j => normalizedTaylorCoeff (f a j) R d | |
| apply Collision.infinite_collision_bound group B C D s (fun i => (2 : ℝ) ^ ell i) | |
| hD hs0 hs1 (fun _ => by positivity) | |
| · intro a d j | |
| exact norm_normalizedTaylorCoeff_le (by exact_mod_cast hR) (hfD a j) d | |
| · intro i d | |
| dsimp [B] | |
| rw [hss] | |
| exact norm_rowTest_shifted_power_le (by exact_mod_cast hR) (hc i) (ell i) d | |
| · intro i j | |
| exact hasSum_translated_row (hf (group i) j) hR hD (hfD (group i) j) (hc i) (ell i) | |
| theorem translated_row_determinant_bound_sum_order | |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] | |
| (group : ι → κ) (f : κ → ι → ℂ → ℂ) | |
| (center : ι → ℂ) (ell : ι → ℕ) | |
| (R : NNReal) (hR : 0 < R) (D s : ℝ) (hD : 0 ≤ D) | |
| (hs0 : 0 ≤ s) (hs1 : s < 1) (hss : s * s = 1 / 2) | |
| (hf : ∀ a j, Differentiable ℂ (f a j)) | |
| (hfD : ∀ a j z, z ∈ Metric.sphere (0 : ℂ) (R : ℝ) → ‖f a j z‖ ≤ D) | |
| (hc : ∀ i, ‖center i‖ + 3 / 4 ≤ (R : ℝ) / 2) : | |
| ‖Matrix.det (fun i j => rowTest (ell i) | |
| (fun t => f (group i) j (center i + Complex.log (1 + t))))‖ ≤ | |
| ((Fintype.card ι).factorial * D ^ Fintype.card ι * (2 : ℝ) ^ (∑ i, ell i)) * | |
| s ^ (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) * | |
| ((1 - s)⁻¹) ^ Fintype.card ι := by | |
| simpa only [Finset.prod_pow_eq_pow_sum] using | |
| translated_row_determinant_bound group f center ell R hR D s hD hs0 hs1 hss hf hfD hc | |
| theorem translated_row_determinant_exp_bound | |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] | |
| (group : ι → κ) (f : κ → ι → ℂ → ℂ) | |
| (center : ι → ℂ) (ell : ι → ℕ) | |
| (R : NNReal) (hR : 0 < R) {H A v : ℝ} (hH : 0 < H) | |
| (hf : ∀ a j, Differentiable ℂ (f a j)) | |
| (hfD : ∀ a j z, z ∈ Metric.sphere (0 : ℂ) (R : ℝ) → | |
| ‖f a j z‖ ≤ Real.exp (H * A)) | |
| (hc : ∀ i, ‖center i‖ + 3 / 4 ≤ (R : ℝ) / 2) | |
| (hell : ((∑ i, ell i : ℕ) : ℝ) ≤ (Fintype.card ι : ℝ) * H / v) : | |
| ‖Matrix.det (fun i j => rowTest (ell i) | |
| (fun t => f (group i) j (center i + Complex.log (1 + t))))‖ ≤ | |
| Real.exp (-(Real.log 2 / 4) * | |
| (∑ a, (Collision.multiplicity Finset.univ group a : ℝ) ^ 2) + | |
| (Fintype.card ι : ℝ) * H * | |
| (A + Real.log 2 / v + Collision.collisionRemainder (Fintype.card ι) H)) := by | |
| have hfinite := translated_row_determinant_bound_sum_order group f center ell R hR | |
| (Real.exp (H * A)) Collision.collisionRatio (Real.exp_pos _).le | |
| Collision.collisionRatio_pos.le Collision.collisionRatio_lt_one | |
| Collision.collisionRatio_mul_self hf hfD hc | |
| exact hfinite.trans (Collision.collision_factor_le_exp_normalized (Fintype.card ι) | |
| (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) (∑ i, ell i) | |
| hH hell (Collision.twice_collision_count_eq group)) | |
| end PiExponent.AnalyticCollision | |
| end OAI | |