| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Counts |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.HistorySupport |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
| namespace MatrixMultiplication.AllFieldHistory |
|
|
| open AllFieldParameters |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| theorem work_source_amount_pos {K : ℕ} (allocation : Allocation) (w : PlacedWork K) : |
| 0 < amount allocation (w.1.source, w.2) := by |
| apply div_pos _ (by norm_num) |
| cases w.1 with |
| | stageA h => exact initialAmount_pos h.val |
| | stageB h => exact aAmount_pos h.val |
| | stageC h => exact partAmount_pos allocation h |
|
|
| theorem work_source_population_pos {K : ℕ} (allocation : Allocation) {dilation : ℕ} |
| (hd : 0 < dilation) (w : PlacedWork K) : |
| 0 < population allocation dilation (w.1.source, w.2) := |
| (AllFieldPopulationCounts.count_pos_iff _ (amount_nonneg allocation) hd _).mpr |
| (work_source_amount_pos allocation w) |
|
|
| abbrev PositiveBranch {K : ℕ} (allocation : Allocation) (w : PlacedWork K) := |
| {b : w.1.Branch // 0 < amount allocation (w.1.child b false, w.2)} |
|
|
| theorem branchPopulation_pos_iff {K : ℕ} (allocation : Allocation) {dilation : ℕ} |
| (hd : 0 < dilation) (w : PlacedWork K) (b : w.1.Branch) : |
| 0 < branchPopulation allocation dilation w b ↔ |
| 0 < amount allocation (w.1.child b false, w.2) := |
| AllFieldPopulationCounts.count_pos_iff _ (amount_nonneg allocation) hd _ |
|
|
| theorem positive_branch_minimum {K : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (w : PlacedWork K) (b : PositiveBranch allocation w) : |
| dilation ≤ branchPopulation allocation dilation w b.val := |
| AllFieldPopulationCounts.dilation_le_count _ (amount_nonneg allocation) dilation _ b.property |
|
|
| theorem positive_joint_class_minimum {K : ℕ} (allocation : Allocation) {dilation : ℕ} |
| (hd : 0 < dilation) (w : PlacedWork K) (u : JointPopulation.Shape) |
| (hu : 0 < jointCounts allocation dilation w u) : dilation ≤ jointCounts allocation dilation w u := by |
| obtain ⟨b, rfl, hb⟩ := shapeCounts_positive _ _ u hu |
| rw [jointCounts_at] |
| exact positive_branch_minimum allocation dilation w |
| ⟨b, (branchPopulation_pos_iff allocation hd w b).mp hb⟩ |
|
|
| def activeLengthRate {K tick : ℕ} (allocation : Allocation) : ℚ := |
| ∑ h : Active K tick, amount allocation (h.val.1.source, h.val.2) * |
| (2 * h.val.1.halfLength : ℕ) |
|
|
| def activePositions {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (h : Active K tick) : Type := |
| Fin (population allocation dilation (h.val.1.source, h.val.2)) |
|
|
| instance {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) (h : Active K tick) : |
| Fintype (activePositions allocation dilation h) := inferInstanceAs (Fintype (Fin _)) |
|
|
| theorem active_totalLength_cast {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) : |
| (HistorySupport.totalLength (activePositions (K := K) (tick := tick) allocation dilation) |
| activeHalfLength activeHalfLength : ℚ) = |
| (populationLength (K := K) allocation dilation : ℚ) * |
| activeLengthRate (K := K) (tick := tick) allocation := by |
| simp only [HistorySupport.totalLength, Nat.cast_sum, Nat.cast_mul, |
| activeHalfLength, activeLengthRate, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro h hh |
| have hcard : Fintype.card (activePositions allocation dilation h) = |
| population allocation dilation (h.val.1.source, h.val.2) := Fintype.card_fin _ |
| rw [hcard, population_cast] |
| push_cast |
| ring |
|
|
| def activeLengthBudget {K tick : ℕ} (allocation : Allocation) : ℕ := |
| Nat.ceil (activeLengthRate (K := K) (tick := tick) allocation) |
|
|
| theorem active_totalLength_le {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) : |
| HistorySupport.totalLength (activePositions (K := K) (tick := tick) allocation dilation) |
| activeHalfLength activeHalfLength ≤ |
| activeLengthBudget (K := K) (tick := tick) allocation * |
| populationLength (K := K) allocation dilation := by |
| have h := active_totalLength_cast (K := K) (tick := tick) allocation dilation |
| have hc : activeLengthRate (K := K) (tick := tick) allocation ≤ |
| (activeLengthBudget (K := K) (tick := tick) allocation : ℚ) := Nat.le_ceil _ |
| have hb := mul_le_mul_of_nonneg_left hc |
| (Nat.cast_nonneg (populationLength (K := K) allocation dilation) : |
| (0 : ℚ) ≤ populationLength (K := K) allocation dilation) |
| rw [← h] at hb |
| exact_mod_cast hb.trans_eq (mul_comm _ _) |
|
|
| theorem active_cw5_support_bound {F : Type*} [Zero F] {K tick : ℕ} |
| (allocation : Allocation) (dilation : ℕ) |
| (Q : HistorySupport.HistoryWords |
| (activePositions (K := K) (tick := tick) allocation dilation) |
| activeHalfLength activeHalfLength 7 → |
| HistorySupport.HistoryWords |
| (activePositions (K := K) (tick := tick) allocation dilation) |
| activeHalfLength activeHalfLength 7 → |
| HistorySupport.HistoryWords |
| (activePositions (K := K) (tick := tick) allocation dilation) |
| activeHalfLength activeHalfLength 7 → F) : |
| ((RecoverySupport.support Q).card : ℝ) ≤ |
| Real.exp (RecoverySupport.cw5SupportRate |
| (activeLengthBudget (K := K) (tick := tick) allocation) * |
| (populationLength (K := K) allocation dilation : ℝ)) := |
| HistorySupport.cw5_support_card_le_exp _ _ _ Q _ _ |
| (active_totalLength_le allocation dilation) |
|
|
| end MatrixMultiplication.AllFieldHistory |
|
|
| end |
|
|
| end OAI |
|
|
