openAI-math-dataset / lean /OAI /NumberTheory /Ostmann /Arithmetic /AssignedPrimeCellReplacement.lean
| import OAI.NumberTheory.Ostmann.Arithmetic.LogCellPartitionReplacement | |
| import OAI.NumberTheory.Ostmann.Arithmetic.LogCellPartitionTuples | |
| import OAI.NumberTheory.Ostmann.Arithmetic.PrimeCellJointReplacementAlgebra | |
| namespace OAI | |
| open _root_.Erdos970 _root_.OAI.Erdos970 | |
| open Erdos970.Erdos970Dependency.SiegelWalfisz | |
| noncomputable section | |
| namespace Ostmann.Arithmetic.LogCellPartition | |
| open PrimeProgression PrimeCellReplacement | |
| open scoped BigOperators | |
| def assignedJointComplexTestSum {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (N : ι → ℕ) (M : ℕ) [NeZero M] (lo hi η Z : ι → ℝ) | |
| (j : GridBoxIndex lo hi η) (F : (ι → (ZMod M)ˣ) → ℂ) : ℂ := | |
| ∑ p : AssignedPrimeTuple N lo hi η j, | |
| (∏ i, (((Z i * ((p i).val : ℝ))⁻¹ : ℝ) : ℂ)) * | |
| jointUnitTest F (fun i => ((p i).val : ZMod M)) | |
| theorem assignedJointComplexTestSum_eq {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (N : ι → ℕ) (M : ℕ) [NeZero M] (lo hi η Z : ι → ℝ) | |
| (j : GridBoxIndex lo hi η) (F : (ι → (ZMod M)ˣ) → ℂ) : | |
| assignedJointComplexTestSum N M lo hi η Z j F = | |
| ∑ u : ι → (ZMod M)ˣ, | |
| (∏ i, (assignedResidueMass (N i) M (u i) (lo i) (hi i) (η i) (Z i) (j i) : ℂ)) * F u := by | |
| classical | |
| have hpart := finite_product_prior_partition | |
| (fun i (p : {p : ℕ // p ∈ assignedPrimeSupport (N i) (lo i) (hi i) (η i) (j i)}) => | |
| (p.val : ZMod M)) | |
| (fun i (p : {p : ℕ // p ∈ assignedPrimeSupport (N i) (lo i) (hi i) (η i) (j i)}) => | |
| (((Z i * (p.val : ℝ))⁻¹ : ℝ) : ℂ)) F | |
| change assignedJointComplexTestSum N M lo hi η Z j F = _ at hpart | |
| rw [hpart] | |
| apply Finset.sum_congr rfl | |
| intro u hu | |
| congr 1 | |
| apply Finset.prod_congr rfl | |
| intro i hi' | |
| rw [Finset.sum_coe_sort (assignedPrimeSupport (N i) (lo i) (hi i) (η i) (j i)) | |
| (fun p : ℕ => if (u i : ZMod M) = (p : ZMod M) then | |
| (((Z i * (p : ℝ))⁻¹ : ℝ) : ℂ) else 0)] | |
| simp only [assignedResidueMass, Complex.ofReal_sum, Finset.sum_filter] | |
| apply Finset.sum_congr rfl | |
| intro p hp | |
| by_cases he : (p : ZMod M) = (u i : ZMod M) | |
| · simp [he] | |
| · simp [he, Ne.symm he] | |
| theorem assignedJointComplexTestSum_error {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (N : ι → ℕ) (M : ℕ) [NeZero M] (lo hi η Z : ι → ℝ) | |
| (j : GridBoxIndex lo hi η) (main E : ι → ℝ) | |
| (F : (ι → (ZMod M)ˣ) → ℂ) {ε B : ℝ} (hε : 0 ≤ ε) (hB : 1 ≤ B) | |
| (hE : ∀ i, E i ≤ ε) (hbound : ∀ i, |main i| + E i ≤ B) | |
| (hmass : ∀ i (u : (ZMod M)ˣ), | |
| |assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) - main i| ≤ E i) : | |
| ‖assignedJointComplexTestSum N M lo hi η Z j F - | |
| (∏ i, (main i : ℂ)) * ∑ u : ι → (ZMod M)ˣ, F u‖ ≤ | |
| Fintype.card ι * ε * B ^ Fintype.card ι * ∑ u : ι → (ZMod M)ˣ, ‖F u‖ := by | |
| classical | |
| have hμ : ∀ i (u : (ZMod M)ˣ), | |
| ‖(assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) : ℂ)‖ ≤ B := by | |
| intro i u | |
| have ht := abs_add_le | |
| (assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) - main i) (main i) | |
| rw [sub_add_cancel] at ht | |
| simp only [Complex.norm_real, Real.norm_eq_abs] | |
| linarith [hmass i u, hbound i] | |
| have hν : ∀ i (u : (ZMod M)ˣ), ‖(main i : ℂ)‖ ≤ B := by | |
| intro i u | |
| have he0 : 0 ≤ E i := (abs_nonneg _).trans (hmass i 1) | |
| simp only [Complex.norm_real, Real.norm_eq_abs] | |
| linarith [hbound i] | |
| have he : ∀ i (u : (ZMod M)ˣ), | |
| ‖(assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) : ℂ) - (main i : ℂ)‖ ≤ ε := by | |
| intro i u | |
| simpa only [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs] using | |
| (hmass i u).trans (hE i) | |
| have h := joint_product_test_error | |
| (fun i (u : (ZMod M)ˣ) => | |
| (assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) : ℂ)) | |
| (fun i (_ : (ZMod M)ˣ) => (main i : ℂ)) F hε hB hμ hν he | |
| rw [assignedJointComplexTestSum_eq] | |
| simpa only [Finset.mul_sum] using h | |
| theorem exists_assigned_joint_primeCell_replacement_constants : | |
| ∃ d K L₀ : ℝ, 0 < d ∧ 0 < K ∧ 1 ≤ L₀ ∧ | |
| ∀ (ι : Type*) [Fintype ι] [DecidableEq ι] (N : ι → ℕ) (M : ℕ) [NeZero M] | |
| (lo hi η Z : ι → ℝ), | |
| (∀ i, L₀ ≤ lo i ∧ lo i ≤ hi i ∧ 0 < η i ∧ η i ≤ 1 ∧ | |
| ⌊Real.exp (hi i)⌋₊ ≤ N i ∧ 0 < Z i ∧ | |
| (M : ℝ) ≤ Real.exp (d * (lo i) ^ (1 / 3 : ℝ))) → | |
| ∀ (j : GridBoxIndex lo hi η) (ε B : ℝ), 0 ≤ ε → 1 ≤ B → | |
| (∀ i, (K / Z i) * Real.exp (-d * (gridPoint (lo i) (hi i) (η i) (j i).val) ^ (1 / 3 : ℝ)) + | |
| (Z i*Real.exp (gridPoint (lo i) (hi i) (η i) (j i).val))⁻¹ ≤ ε) → | |
| (∀ i, |harmonicIntegral M (gridPoint (lo i) (hi i) (η i) (j i).val) | |
| (gridPoint (lo i) (hi i) (η i) ((j i).val+1)) / Z i| + | |
| ((K / Z i) * Real.exp (-d * (gridPoint (lo i) (hi i) (η i) (j i).val) ^ (1 / 3 : ℝ)) + | |
| (Z i*Real.exp (gridPoint (lo i) (hi i) (η i) (j i).val))⁻¹) ≤ B) → | |
| ∀ F : (ι → (ZMod M)ˣ) → ℂ, | |
| ‖assignedJointComplexTestSum N M lo hi η Z j F - | |
| (∏ i, ((harmonicIntegral M (gridPoint (lo i) (hi i) (η i) (j i).val) | |
| (gridPoint (lo i) (hi i) (η i) ((j i).val+1)) / Z i : ℝ) : ℂ)) * | |
| ∑ u : ι → (ZMod M)ˣ, F u‖ ≤ | |
| Fintype.card ι * ε * B ^ Fintype.card ι * | |
| ∑ u : ι → (ZMod M)ˣ, ‖F u‖ := by | |
| obtain ⟨d,K,L₀,hd,hK,hL₀,h⟩ := exists_assigned_residueMass_error_constants | |
| refine ⟨d,K,L₀,hd,hK,hL₀,?_⟩ | |
| intro ι _ _ N M _ lo hi η Z hcell j ε B hε hB hE hbound F | |
| apply assignedJointComplexTestSum_error N M lo hi η Z j | |
| (fun i => harmonicIntegral M (gridPoint (lo i) (hi i) (η i) (j i).val) | |
| (gridPoint (lo i) (hi i) (η i) ((j i).val+1)) / Z i) | |
| (fun i => (K / Z i) * Real.exp (-d * (gridPoint (lo i) (hi i) (η i) (j i).val) ^ (1 / 3 : ℝ)) + | |
| (Z i*Real.exp (gridPoint (lo i) (hi i) (η i) (j i).val))⁻¹) F hε hB hE hbound | |
| intro i u | |
| obtain ⟨hlo,hlohi,hη,hη1,hN,hZ,hmod⟩ := hcell i | |
| exact h (lo i) (hi i) (η i) hlo hlohi hη hη1 (N i) M (Z i) hN hZ hmod (j i) u u.isUnit | |
| end Ostmann.Arithmetic.LogCellPartition | |
| end | |
| end OAI | |