openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialFamily.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialHash | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialState | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialGibbs | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialFamily | |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory AllFieldFiniteFamily | |
| open AllFieldInitialEntropy JointOrdinaryPopulationSelection Filter | |
| open scoped BigOperators Topology | |
| attribute [local instance] Classical.propDecidable | |
| variable {K : ℕ} (allocation : Allocation) | |
| def grid : ℕ := populationLength (K := K) allocation 1 | |
| theorem grid_pos : 0 < grid (K := K) allocation := | |
| AllFieldPopulationCounts.blockLength_pos (amount (K := K) allocation) (by decide) | |
| theorem populationLength_eq_mul_grid (dilation : ℕ) : | |
| populationLength (K := K) allocation dilation = dilation * grid (K := K) allocation := by | |
| simp only [grid, populationLength, AllFieldPopulationCounts.blockLength, one_mul] | |
| theorem initialJointCounts_dilation_ratio {dilation : ℕ} (hd : 0 < dilation) | |
| (j : Fin K) (u : JointPopulation.Shape) : | |
| (initialJointCounts allocation dilation j u : ℝ) / (dilation : ℝ) = | |
| (grid (K := K) allocation : ℝ) * orderedInitialLaw.mass u := by | |
| rw [initialJointCounts_cast, populationLength_eq_mul_grid, Nat.cast_mul, mul_assoc] | |
| have hd' : (dilation : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hd) | |
| exact mul_div_cancel_left₀ _ hd' | |
| theorem initialJointCounts_dilation_rate (j : Fin K) (u : JointPopulation.Shape) : | |
| Tendsto (fun dilation : ℕ => | |
| (initialJointCounts allocation dilation j u : ℝ) / (dilation : ℝ)) atTop | |
| (𝓝 ((grid (K := K) allocation : ℝ) * orderedInitialLaw.mass u)) := by | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with dilation hd | |
| exact (initialJointCounts_dilation_ratio allocation hd j u).symm | |
| theorem native_dilation_capacity : | |
| entropyRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) | |
| (fun _ => orderedInitialLaw) - | |
| degreeRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) | |
| (fun _ => orderedInitialLaw) = | |
| (grid (K := K) allocation : ℝ) * ((K : ℝ) * nativeH0) := by | |
| simp only [entropyRate, degreeRate, sideDegreeRate, orderedInitialLaw_side_entropy, | |
| Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, max_self] | |
| ring | |
| def hashRate (ε : ℝ) : ℝ := | |
| selectionHashRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) | |
| (fun _ => orderedInitialLaw) ((grid (K := K) allocation : ℝ) * ε) | |
| theorem selection_exponent (ε : ℝ) (dilation : ℕ) : | |
| (dilation : ℝ) * | |
| (entropyRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) | |
| (fun _ => orderedInitialLaw) - | |
| degreeRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) | |
| (fun _ => orderedInitialLaw) - (grid (K := K) allocation : ℝ) * ε) = | |
| (populationLength (K := K) allocation dilation : ℝ) * ((K : ℝ) * nativeH0 - ε) := by | |
| rw [native_dilation_capacity, populationLength_eq_mul_grid, Nat.cast_mul] | |
| ring | |
| abbrev Sample (ε : ℝ) (dilation : ℕ) := | |
| JointCoarseHashing.Sample (JointPopulation.Position | |
| (initialJointCounts (K := K) allocation dilation)) | |
| (JointOrdinarySelection.hashLevel (hashRate (K := K) allocation ε) dilation) | |
| def Good (ε : ℝ) (dilation : ℕ) | |
| (s : Sample (K := K) allocation ε dilation) | |
| (e : JointPopulation.Target (initialJointCounts (K := K) allocation dilation)) : Prop := | |
| JointOrdinarySelection.Good | |
| (JointOrdinarySelection.hashLevel (hashRate (K := K) allocation ε) dilation) | |
| (JointOrdinarySelection.hashSet (hashRate (K := K) allocation ε) dilation) | |
| (JointPopulation.ambientSet (initialJointCounts allocation dilation) (fun _ => 16)) | |
| (JointPopulation.triple (initialJointCounts allocation dilation) e) s | |
| def executionWithWidth (F : Type*) [Field F] (ε residentWidth : ℝ) (dilation : ℕ) | |
| (s : Sample (K := K) allocation ε dilation) | |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) | |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : | |
| Execution (cwSource F K (populationLength (K := K) allocation dilation)) | |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets | |
| (initialJointCounts (K := K) allocation dilation) G => | |
| stateTensor F (K := K) allocation dilation residentWidth 0)) := | |
| AllFieldInitialState.execution allocation dilation F residentWidth | |
| (AllFieldInitialHash.selectedExecution (F := F) | |
| (initialJointCounts allocation dilation) (initialJointCounts_sum allocation dilation) | |
| (initialJointCounts_support allocation dilation) | |
| (JointOrdinarySelection.hashLevel (hashRate (K := K) allocation ε) dilation) | |
| (JointOrdinarySelection.hashSet (hashRate (K := K) allocation ε) dilation) | |
| (JointOrdinarySelection.hashSet_AP (hashRate (K := K) allocation ε) dilation) | |
| s G hgood) | |
| @[simp] theorem executionWithWidth_copies (F : Type*) [Field F] | |
| (ε residentWidth : ℝ) (dilation : ℕ) | |
| (s : Sample (K := K) allocation ε dilation) | |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) | |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : | |
| Fintype.card | |
| (executionWithWidth allocation F ε residentWidth dilation s G hgood).Copies = 1 := rfl | |
| def execution (F : Type*) [Field F] (ε : ℝ) (dilation : ℕ) | |
| (s : Sample (K := K) allocation ε dilation) | |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) | |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : | |
| Execution (cwSource F K (populationLength (K := K) allocation dilation)) | |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets | |
| (initialJointCounts (K := K) allocation dilation) G => | |
| stateTensor F (K := K) allocation dilation ε 0)) := | |
| executionWithWidth allocation F ε ε dilation s G hgood | |
| @[simp] theorem execution_copies (F : Type*) [Field F] (ε : ℝ) (dilation : ℕ) | |
| (s : Sample (K := K) allocation ε dilation) | |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) | |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : | |
| Fintype.card (execution allocation F ε dilation s G hgood).Copies = 1 := rfl | |
| theorem eventually_exists_execution_with_width (F : Type*) [Field F] | |
| (residentWidth : ℝ) {ε : ℝ} (hε : 0 < ε) : | |
| ∀ᶠ dilation in atTop, | |
| ∃ (s : Sample (K := K) allocation ε dilation) | |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))), | |
| G.card = ⌊Real.exp ((populationLength (K := K) allocation dilation : ℝ) * | |
| ((K : ℝ) * nativeH0 - ε))⌋₊ ∧ | |
| (∀ e ∈ G, Good allocation ε dilation s e) ∧ | |
| ∃ E : Execution (cwSource F K (populationLength (K := K) allocation dilation)) | |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets | |
| (initialJointCounts (K := K) allocation dilation) G => | |
| stateTensor F (K := K) allocation dilation residentWidth 0)), | |
| Fintype.card E.Copies = 1 := by | |
| have hslack : 0 < (grid (K := K) allocation : ℝ) * ε := | |
| mul_pos (by exact_mod_cast grid_pos (K := K) allocation) hε | |
| have hselection := JointOrdinaryPopulationSelection.eventually_exists_selection | |
| (fun dilation => initialJointCounts (K := K) allocation dilation) (fun _ => 16) | |
| (fun _ : Fin K => (grid (K := K) allocation : ℝ)) (fun _ => orderedInitialLaw) | |
| (fun _ => AllFieldInitialGibbs.initialPotential) | |
| (fun _ => AllFieldInitialGibbs.initialPotential) | |
| (fun _ => AllFieldInitialGibbs.initialPotential) | |
| (fun _ => AllFieldInitialGibbs.initialLogNormalizer) | |
| (fun _ => Nat.cast_nonneg _) | |
| (fun dilation => initialJointCounts_support allocation dilation) | |
| (fun _ => AllFieldInitialGibbs.orderedInitialLaw_positive) | |
| (fun _ => AllFieldInitialGibbs.orderedInitialLaw_log) | |
| (initialJointCounts_dilation_rate allocation) hslack | |
| filter_upwards [hselection] with dilation hd | |
| obtain ⟨s, G, hcard, hgood, _hpairwise⟩ := hd | |
| rw [selection_exponent allocation ε dilation] at hcard | |
| exact ⟨s, G, hcard, hgood, | |
| executionWithWidth allocation F ε residentWidth dilation s G hgood, | |
| executionWithWidth_copies allocation F ε residentWidth dilation s G hgood⟩ | |
| theorem eventually_exists_execution (F : Type*) [Field F] {ε : ℝ} (hε : 0 < ε) : | |
| ∀ᶠ dilation in atTop, | |
| ∃ (s : Sample (K := K) allocation ε dilation) | |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))), | |
| G.card = ⌊Real.exp ((populationLength (K := K) allocation dilation : ℝ) * | |
| ((K : ℝ) * nativeH0 - ε))⌋₊ ∧ | |
| (∀ e ∈ G, Good allocation ε dilation s e) ∧ | |
| ∃ E : Execution (cwSource F K (populationLength (K := K) allocation dilation)) | |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets | |
| (initialJointCounts (K := K) allocation dilation) G => | |
| stateTensor F (K := K) allocation dilation ε 0)), | |
| Fintype.card E.Copies = 1 := | |
| eventually_exists_execution_with_width allocation F ε hε | |
| end MatrixMultiplication.AllFieldInitialFamily | |
| end | |
| end OAI | |