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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | 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
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