openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /ConstructionData.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialChoice | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Iteration | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FinishedWitness | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionRates | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldConstructionData | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily | |
| open scoped BigOperators Classical | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K : ℕ} | |
| structure Data (allocation : Allocation) (dilation : ℕ) (width slack : ℝ) where | |
| initial : AllFieldInitialChoice.Choice (K := K) allocation dilation slack | |
| groups : AllFieldHistoryIteration.ScheduleChoices | |
| (K := K) allocation dilation width slack | |
| threshold : ∀ t : Fin (K + 2), | |
| AllFieldHistoryRecovery.minimumDilation (K := K) (tick := t.val) allocation width ≤ dilation | |
| variable (allocation : Allocation) (dilation : ℕ) (width slack : ℝ) | |
| abbrev Tags (data : Data (K := K) allocation dilation width slack) := | |
| AllFieldHistoryIteration.Tags allocation dilation width slack | |
| (AllFieldInitialChoice.Tags allocation dilation slack data.initial) data.groups | |
| instance tagsFintype (data : Data (K := K) allocation dilation width slack) : | |
| Fintype (Tags allocation dilation width slack data) := inferInstance | |
| def execution (F : Type*) [Field F] (hwidth : 0 < width) | |
| (data : Data (K := K) allocation dilation width slack) : | |
| Execution (cwSource F K (populationLength (K := K) allocation dilation)) | |
| (Tensor.directSum (fun _ : Tags allocation dilation width slack data => | |
| stateTensor F (K := K) allocation dilation width (K + 2))) := | |
| AllFieldHistoryIteration.execution allocation dilation width slack F hwidth | |
| data.threshold data.groups | |
| (AllFieldInitialChoice.execution allocation dilation slack data.initial F width) | |
| theorem card_tags (data : Data (K := K) allocation dilation width slack) : | |
| Fintype.card (Tags allocation dilation width slack data) = | |
| Fintype.card (AllFieldConstructionRates.SelectedLabels | |
| (K := K) allocation slack dilation) := by | |
| rw [AllFieldHistoryIteration.card_tags, AllFieldInitialChoice.card_tags, | |
| AllFieldConstructionRates.card_selected] | |
| congr 1 | |
| apply Finset.prod_congr rfl | |
| intro t _ | |
| apply Finset.prod_congr rfl | |
| intro sigma _ | |
| exact (AllFieldConstructionRates.groupCount_eq_native allocation slack dilation t sigma).symm | |
| theorem execution_copies (F : Type*) [Field F] (hwidth : 0 < width) | |
| (data : Data (K := K) allocation dilation width slack) : | |
| Fintype.card (execution allocation dilation width slack F hwidth data).Copies = | |
| Fintype.card (AllFieldConstructionRates.Copies | |
| (K := K) allocation width dilation) := by | |
| rw [execution, AllFieldHistoryIteration.execution_copies, | |
| AllFieldInitialChoice.execution_copies, one_mul] | |
| simp only [AllFieldConstructionRates.Copies, Fintype.card_pi, | |
| InverseLinearRecovery.MaskRectangles, ExactRecovery.card_mask_rectangles, Fintype.card_fin] | |
| theorem tags_pos | |
| (hslack : ∀ slot : AllFieldConstructionRates.Slot K, | |
| slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) * | |
| AllFieldConstructionRates.capacity allocation slot) | |
| (data : Data (K := K) allocation dilation width slack) : | |
| 0 < Fintype.card (Tags allocation dilation width slack data) := by | |
| rw [card_tags] | |
| exact AllFieldConstructionRates.selected_pos allocation slack hslack dilation | |
| variable {F : Type*} [Field F] (m : ℕ) | |
| def witness (hm : 0 < m) (hwidth : 0 < width) | |
| (hslack : ∀ slot : AllFieldConstructionRates.Slot K, | |
| slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) * | |
| AllFieldConstructionRates.capacity allocation slot) | |
| (data : Data (K := K) allocation | |
| (AllFieldTerminalRates.terminalDilation K m) width slack) : AllFieldWitness F := | |
| AllFieldFinishedWitness.witness allocation m hm hwidth.le | |
| (execution allocation (AllFieldTerminalRates.terminalDilation K m) width slack F hwidth data) | |
| (tags_pos allocation (AllFieldTerminalRates.terminalDilation K m) width slack hslack data) | |
| @[simp] theorem witness_volume (hm : 0 < m) (hwidth : 0 < width) | |
| (hslack : ∀ slot : AllFieldConstructionRates.Slot K, | |
| slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) * | |
| AllFieldConstructionRates.capacity allocation slot) | |
| (data : Data (K := K) allocation | |
| (AllFieldTerminalRates.terminalDilation K m) width slack) : | |
| (witness (F := F) allocation width slack m hm hwidth hslack data).volume = | |
| AllFieldTerminalRates.terminalVolume (K := K) allocation m := | |
| AllFieldFinishedWitness.witness_volume _ _ _ _ _ _ | |
| @[simp] theorem witness_multiplicity (hm : 0 < m) (hwidth : 0 < width) | |
| (hslack : ∀ slot : AllFieldConstructionRates.Slot K, | |
| slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) * | |
| AllFieldConstructionRates.capacity allocation slot) | |
| (data : Data (K := K) allocation | |
| (AllFieldTerminalRates.terminalDilation K m) width slack) : | |
| (witness (F := F) allocation width slack m hm hwidth hslack data).multiplicity = | |
| Fintype.card (AllFieldConstructionRates.SelectedLabels | |
| (K := K) allocation slack (AllFieldTerminalRates.terminalDilation K m)) := by | |
| change Fintype.card (Tags _ _ _ _ data) = _ | |
| exact card_tags _ _ _ _ data | |
| @[simp] theorem witness_rankBound (hm : 0 < m) (hwidth : 0 < width) | |
| (hslack : ∀ slot : AllFieldConstructionRates.Slot K, | |
| slack ≤ (AllFieldConstructionRates.denominator (K := K) allocation : ℝ) * | |
| AllFieldConstructionRates.capacity allocation slot) | |
| (data : Data (K := K) allocation | |
| (AllFieldTerminalRates.terminalDilation K m) width slack) : | |
| (witness (F := F) allocation width slack m hm hwidth hslack data).rankBound = | |
| AllFieldSource.rankBudget | |
| (AllFieldConstructionRates.Copies (K := K) allocation width | |
| (AllFieldTerminalRates.terminalDilation K m)) K | |
| (AllFieldTerminalRates.terminalLength allocation K m) := by | |
| rw [witness] | |
| erw [AllFieldFinishedWitness.witness_rankBound] | |
| unfold AllFieldSource.rankBudget | |
| rw [execution_copies] | |
| end MatrixMultiplication.AllFieldConstructionData | |
| end | |
| end OAI | |