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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryRawMasks
open AllFieldHistory AllFieldHistoryChildLaws AllFieldHistorySupport AllFieldHistoryMasks
open JointCanonicalization PermutationMatching
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
def rawPairCount (allocation : Allocation) (m : ℕ) (i : MaskIndex K tick)
(w : ActiveRawPairs (K := K) (tick := tick) allocation m) : ℝ :=
∑ j : JointPopulation.Positions (activeCounts allocation m) i.1,
if statistic i.1 (w i.1 j).1 = i.2.1 i.1 ∧
statistic i.1 (w i.1 j).2 = i.2.2 i.1 then 1 else 0
def rawPass (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3)
(w : ActiveRawPairs (K := K) (tick := tick) allocation m) : Prop :=
∀ i : MaskIndex K tick,
|rawPairCount allocation m i w /
(population allocation m (i.1.val.1.source, i.1.val.2) : ℝ) -
center allocation side i| ≤ pairWidth ε i.1
theorem sum_positiveClasses (allocation : Allocation) (m : ℕ) (h : Active K tick)
(f : ∀ c : Classes K tick, Positions allocation m c → ℝ) :
(∑ c : positiveClasses allocation h, ∑ j, f c.val j) =
∑ u : JointPopulation.Shape, ∑ j, f (h, u) j := by
have hzero (c : Classes K tick) (hc : ¬0 < activeCounts allocation 1 c.1 c.2) :
(∑ j, f c j) = 0 := by
have hcount : activeCounts allocation m c.1 c.2 = 0 := by
rw [activeCounts_dilation, Nat.eq_zero_of_not_pos hc, mul_zero]
apply Finset.sum_eq_zero
intro j _
have hj : j.val < 0 := by simpa only [hcount] using j.isLt
exact (Nat.not_lt_zero _ hj).elim
rw [Finset.sum_coe_sort (positiveClasses allocation h)
(fun c : Classes K tick => ∑ j, f c j)]
unfold positiveClasses
rw [Finset.sum_filter]
calc
(∑ c : Classes K tick,
if c.1 = h ∧ 0 < activeCounts allocation 1 c.1 c.2 then ∑ j, f c j else 0) =
∑ c : Classes K tick, if c.1 = h then ∑ j, f c j else 0 := by
apply Finset.sum_congr rfl
intro c _
by_cases hc : 0 < activeCounts allocation 1 c.1 c.2
· simp only [hc, and_true]
· rw [hzero c hc]
simp
_ = ∑ u : JointPopulation.Shape, ∑ j, f (h, u) j := by
rw [Fintype.sum_prod_type, Fintype.sum_eq_single h]
· simp only [ite_true]
· intro h' hh'
apply Finset.sum_eq_zero
intro u _
exact ite_eq_right hh'
theorem selectedWordPairCount_pairEquiv (allocation : Allocation) (m : ℕ)
(e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m))
(i : MaskIndex K tick) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) :
selectedWordPairCount (testedClasses allocation i)
(pairEquiv (activeCounts allocation m)
(JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w)
classStatistic classStatistic (leftSymbol i) (rightSymbol i) =
rawPairCount allocation m i w := by
let v : Words (K := K) (tick := tick) allocation m :=
pairEquiv (activeCounts allocation m)
(JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w
have hsum := sum_positiveClasses allocation m i.1 (fun c j =>
if classStatistic c (v.left c j) = leftSymbol i c ∧
classStatistic c (v.right c j) = rightSymbol i c then (1 : ℝ) else 0)
unfold selectedWordPairCount
rw [testedClasses, hsum]
unfold rawPairCount
rw [← Fintype.sum_sigma']
apply Fintype.sum_equiv (positionEquiv (activeCounts allocation m) e i.1).symm
rintro ⟨u, j⟩
rfl
theorem testedClasses_populationSize (allocation : Allocation) (m : ℕ)
(i : MaskIndex K tick) :
populationSize (fun c : testedClasses allocation i => Positions allocation m c.val) =
(population allocation m (i.1.val.1.source, i.1.val.2) : ℝ) := by
change populationSize (selectedPositions allocation m (positiveClasses allocation i.1)) = _
rw [positiveClasses_populationSize, population_dilation allocation m, Nat.cast_mul]
exact mul_comm _ _
theorem raw_population_pos (allocation : Allocation) {m : ℕ} (hm : 0 < m)
(h : Active K tick) :
0 < (population allocation m (h.val.1.source, h.val.2) : ℝ) := by
exact_mod_cast work_source_population_pos allocation hm h.val
theorem passes_pairEquiv_iff_rawPass (allocation : Allocation) (m : ℕ) (ε : ℝ)
(side : Fin 3)
(e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m))
(w : ActiveRawPairs (K := K) (tick := tick) allocation m) :
passes allocation m ε side
(pairEquiv (activeCounts allocation m)
(JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w) ↔
rawPass allocation m ε side w := by
simp only [passes, selectedWordPairMaskBad, selectedWordPairCount_pairEquiv,
testedClasses_populationSize, rawPass, not_lt]
theorem passes_pairEquiv_target_independent (allocation : Allocation) (m : ℕ) (ε : ℝ)
(side : Fin 3)
(e f : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m))
(w : ActiveRawPairs (K := K) (tick := tick) allocation m) :
passes allocation m ε side
(pairEquiv (activeCounts allocation m)
(JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w) ↔
passes allocation m ε side
(pairEquiv (activeCounts allocation m)
(JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) f w) :=
(passes_pairEquiv_iff_rawPass allocation m ε side e w).trans
(passes_pairEquiv_iff_rawPass allocation m ε side f w).symm
end MatrixMultiplication.AllFieldHistoryRawMasks
end
end OAI
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