openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialLeafRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.TerminalRateFormulas | |
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWWordDimensionParameters | |
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWWindowedLeaves | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ShapeEncoding | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialLeafRates | |
| open AllFieldHistory AllFieldParameters AllFieldTerminalRates | |
| open CWWindowedLeaves Filter | |
| open scoped BigOperators Topology | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| abbrev InitialZero (K : ℕ) := | |
| {h : Initial K // initialShape h ∉ positiveInitial} | |
| abbrev PopulationHistory (K : ℕ) := InitialZero K × Placement | |
| theorem shapeMax_lt {K : ℕ} (h : InitialZero K) : | |
| shapeMax (initialShape h.val) < 17 := by | |
| have hb := (root_shape_spec _ (initialShape_mem h.val)).1 | |
| exact Nat.lt_succ_of_le (max_le (hb 0) (max_le (hb 1) (hb 2))) | |
| def shapeIndex {K : ℕ} (h : InitialZero K) : Fin 17 := | |
| ⟨shapeMax (initialShape h.val), shapeMax_lt h⟩ | |
| theorem card_alphabet {K : ℕ} (h : InitialZero K) : | |
| Fintype.card (Alphabet 8 (shapeMax (initialShape h.val))) = | |
| Contractions.coefficients[shapeMax (initialShape h.val)]?.getD 0 := by | |
| have hc := CWWordDimensionParameters.table_eq_card (shapeIndex h) | |
| rw [Fintype.card_eq_nat_card] at hc ⊢ | |
| exact hc.symm | |
| theorem card_alphabet_pos {K : ℕ} (h : InitialZero K) : | |
| 0 < Fintype.card (Alphabet 8 (shapeMax (initialShape h.val))) := by | |
| have hc := CWWordDimensionParameters.word_card_positive (shapeIndex h) | |
| rw [Fintype.card_eq_nat_card] at hc ⊢ | |
| exact hc | |
| abbrev Words {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| (h : PopulationHistory K) := | |
| Fin (population allocation dilation (.initial h.1.val, h.2)) → | |
| Alphabet 8 (shapeMax (initialShape h.1.val)) | |
| theorem card_words {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| (h : PopulationHistory K) : | |
| Fintype.card (Words allocation dilation h) = | |
| (Contractions.coefficients[shapeMax (initialShape h.1.val)]?.getD 0) ^ | |
| population allocation dilation (.initial h.1.val, h.2) := by | |
| simp only [Words, Fintype.card_fun, Fintype.card_fin, card_alphabet] | |
| theorem card_words_pos {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| (h : PopulationHistory K) : | |
| 0 < Fintype.card (Words allocation dilation h) := by | |
| simp only [Words, Fintype.card_fun, Fintype.card_fin] | |
| exact pow_pos (card_alphabet_pos h.1) _ | |
| theorem log_card_words {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| (h : PopulationHistory K) : | |
| Real.log (Fintype.card (Words allocation dilation h) : ℝ) = | |
| (population allocation dilation (.initial h.1.val, h.2) : ℝ) * | |
| initialRate (initialShape h.1.val) := by | |
| simp only [card_words, Nat.cast_pow, Real.log_pow, initialRate] | |
| abbrev ProductWords {K : ℕ} (allocation : Allocation) (dilation : ℕ) := | |
| ∀ h : PopulationHistory K, Words allocation dilation h | |
| def volume {K : ℕ} (allocation : Allocation) (dilation : ℕ) : ℕ := | |
| Fintype.card (ProductWords (K := K) allocation dilation) | |
| theorem volume_eq_prod {K : ℕ} (allocation : Allocation) (dilation : ℕ) : | |
| volume (K := K) allocation dilation = | |
| ∏ h : PopulationHistory K, Fintype.card (Words allocation dilation h) := by | |
| rw [volume, Fintype.card_pi] | |
| theorem volume_pos {K : ℕ} (allocation : Allocation) (dilation : ℕ) : | |
| 0 < volume (K := K) allocation dilation := by | |
| rw [volume_eq_prod] | |
| exact Finset.prod_pos (fun h _ => card_words_pos allocation dilation h) | |
| theorem log_volume {K : ℕ} (allocation : Allocation) (dilation : ℕ) : | |
| Real.log (volume (K := K) allocation dilation : ℝ) = | |
| (populationLength (K := K) allocation dilation : ℝ) * | |
| ∑ h : PopulationHistory K, | |
| (amount allocation (.initial h.1.val, h.2) : ℝ) * | |
| initialRate (initialShape h.1.val) := by | |
| rw [volume_eq_prod, Nat.cast_prod, Real.log_prod (s := Finset.univ) | |
| (f := fun h : PopulationHistory K => (Fintype.card (Words allocation dilation h) : ℝ)) | |
| (fun h _ => Nat.cast_ne_zero.mpr | |
| (Nat.ne_of_gt (card_words_pos allocation dilation h)))] | |
| simp only [log_card_words] | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro h _ | |
| have hc : (population allocation dilation (.initial h.1.val, h.2) : ℝ) = | |
| (populationLength (K := K) allocation dilation : ℝ) * | |
| (amount allocation (.initial h.1.val, h.2) : ℝ) := by | |
| exact_mod_cast population_cast allocation dilation (.initial h.1.val, h.2) | |
| rw [hc, mul_assoc] | |
| theorem sum_populationHistory_rate {K : ℕ} (allocation : Allocation) : | |
| (∑ h : PopulationHistory K, | |
| (amount allocation (.initial h.1.val, h.2) : ℝ) * | |
| initialRate (initialShape h.1.val)) = | |
| ∑ h : InitialZero K, | |
| (initialAmount h.val : ℝ) * initialRate (initialShape h.val) := by | |
| rw [Fintype.sum_prod_type] | |
| apply Finset.sum_congr rfl | |
| intro h _ | |
| dsimp only | |
| rw [← Finset.sum_mul] | |
| have hm : (∑ phi : Placement, (amount allocation (.initial h.val, phi) : ℝ)) = | |
| (initialAmount h.val : ℝ) := by | |
| exact_mod_cast sum_placement_amount allocation (.initial h.val) | |
| rw [hm] | |
| theorem log_volume_eq_S0 {K : ℕ} (allocation : Allocation) (dilation : ℕ) : | |
| Real.log (volume (K := K) allocation dilation : ℝ) = | |
| (populationLength (K := K) allocation dilation : ℝ) * ((K : ℝ) * S0) := by | |
| rw [log_volume, sum_populationHistory_rate, initial_zero_history_rate] | |
| theorem normalized_log_volume_eq_S0 {K : ℕ} (allocation : Allocation) | |
| (dilation : ℕ) (hd : 0 < dilation) : | |
| Real.log (volume (K := K) allocation dilation : ℝ) / | |
| populationLength (K := K) allocation dilation = (K : ℝ) * S0 := by | |
| rw [log_volume_eq_S0] | |
| have hn : (populationLength (K := K) allocation dilation : ℝ) ≠ 0 := | |
| Nat.cast_ne_zero.mpr | |
| (Nat.ne_of_gt (AllFieldPopulationCounts.blockLength_pos _ hd)) | |
| exact mul_div_cancel_left₀ _ hn | |
| theorem tendsto_normalized_log_volume {K : ℕ} (allocation : Allocation) : | |
| Tendsto (fun dilation : ℕ => | |
| Real.log (volume (K := K) allocation dilation : ℝ) / | |
| populationLength (K := K) allocation dilation) atTop (𝓝 ((K : ℝ) * S0)) := by | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with dilation hd | |
| exact (normalized_log_volume_eq_S0 allocation dilation hd).symm | |
| end MatrixMultiplication.AllFieldInitialLeafRates | |
| end | |
| end OAI | |