openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialChoice.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialFamily | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionLabels | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialChoice | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily Filter | |
| open scoped BigOperators Topology Classical | |
| variable {K : ℕ} | |
| structure Choice (allocation : Allocation) (m : ℕ) (δ : ℝ) where | |
| sample : AllFieldInitialFamily.Sample (K := K) allocation | |
| (δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) m | |
| selected : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation m)) | |
| card_selected : selected.card = AllFieldConstructionRates.initialCount (K := K) allocation δ m | |
| good : ∀ e ∈ selected, AllFieldInitialFamily.Good allocation | |
| (δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) m sample e | |
| theorem initial_count_eq (allocation : Allocation) (m : ℕ) (δ : ℝ) : | |
| ⌊Real.exp ((populationLength (K := K) allocation m : ℝ) * | |
| ((K : ℝ) * AllFieldInitialEntropy.nativeH0 - | |
| δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)))⌋₊ = | |
| AllFieldConstructionRates.initialCount (K := K) allocation δ m := by | |
| rw [AllFieldConstructionRates.initialCount_eq, | |
| AllFieldInitialFamily.populationLength_eq_mul_grid, Nat.cast_mul] | |
| congr 2 | |
| change ((m : ℝ) * (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) * | |
| ((K : ℝ) * AllFieldInitialEntropy.nativeH0 - | |
| δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) = | |
| (m : ℝ) * ((AllFieldInitialFamily.grid (K := K) allocation : ℝ) * | |
| ((K : ℝ) * AllFieldInitialEntropy.nativeH0) - δ) | |
| have hD : (AllFieldInitialFamily.grid (K := K) allocation : ℝ) ≠ 0 := | |
| Nat.cast_ne_zero.mpr (AllFieldInitialFamily.grid_pos allocation).ne' | |
| field_simp [hD] | |
| theorem eventually_nonempty (allocation : Allocation) {δ : ℝ} (hδ : 0 < δ) : | |
| ∀ᶠ m : ℕ in atTop, Nonempty (Choice (K := K) allocation m δ) := by | |
| have hs : 0 < δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ) := | |
| div_pos hδ (Nat.cast_pos.mpr (AllFieldInitialFamily.grid_pos allocation)) | |
| have h := AllFieldInitialFamily.eventually_exists_execution_with_width | |
| (K := K) allocation ℚ 0 hs | |
| filter_upwards [h] with m hm | |
| obtain ⟨sample, selected, hcard, hgood, _execution⟩ := hm | |
| exact ⟨{ | |
| sample := sample | |
| selected := selected | |
| card_selected := hcard.trans (initial_count_eq allocation m δ) | |
| good := hgood }⟩ | |
| abbrev Tags (allocation : Allocation) (m : ℕ) (δ : ℝ) | |
| (choice : Choice (K := K) allocation m δ) := | |
| AllFieldInitialHash.SelectedTargets (initialJointCounts (K := K) allocation m) | |
| choice.selected | |
| @[simp] theorem card_tags (allocation : Allocation) (m : ℕ) (δ : ℝ) | |
| (choice : Choice (K := K) allocation m δ) : | |
| Fintype.card (Tags allocation m δ choice) = | |
| AllFieldConstructionRates.initialCount (K := K) allocation δ m := by | |
| rw [AllFieldInitialHash.selectedTargets_card] | |
| exact choice.card_selected | |
| def execution (allocation : Allocation) (m : ℕ) (δ : ℝ) | |
| (choice : Choice (K := K) allocation m δ) (F : Type*) [Field F] (ε : ℝ) : | |
| Execution (cwSource F K (populationLength (K := K) allocation m)) | |
| (Tensor.directSum (fun _ : Tags allocation m δ choice => | |
| stateTensor F (K := K) allocation m ε 0)) := | |
| AllFieldInitialFamily.executionWithWidth allocation F | |
| (δ / (AllFieldInitialFamily.grid (K := K) allocation : ℝ)) ε m | |
| choice.sample choice.selected choice.good | |
| @[simp] theorem execution_copies (allocation : Allocation) (m : ℕ) (δ : ℝ) | |
| (choice : Choice (K := K) allocation m δ) (F : Type*) [Field F] (ε : ℝ) : | |
| Fintype.card (execution allocation m δ choice F ε).Copies = 1 := rfl | |
| end MatrixMultiplication.AllFieldInitialChoice | |
| end | |
| end OAI | |