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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.ChildLaws
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Support
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.HistorySymmetry
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskAction
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskOrbitFilter
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryMasks
open AllFieldHistory AllFieldHistoryChildLaws AllFieldHistorySupport
open JointCanonicalization InheritedMasks PermutationMatching HistorySymmetry
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
theorem testedClasses_parent (allocation : Allocation) (i : MaskIndex K tick)
(c : Classes K tick) (hc : c ∈ testedClasses allocation i) : c.1 = i.1 :=
(Finset.mem_filter.mp hc).2.1
theorem rightProbability_le_one (allocation : Allocation) (side : Fin 3)
(i : MaskIndex K tick) (c : Classes K tick) (hc : c ∈ testedClasses allocation i) :
rightProbability side i c ≤ 1 := by
have hsum := rightLaw_normalized allocation 1 c.1 c.2
(positiveClasses_positive allocation i.1 c hc) side
rw [← hsum]
exact Finset.single_le_sum (fun a _ => rightLaw_nonnegative c.1 c.2 side a)
(Finset.mem_univ _)
theorem left_density_le (allocation : Allocation) {m : ℕ} (hm : 0 < m)
(ε : ℝ) (side : Fin 3) (w : Words (K := K) (tick := tick) allocation m)
(hw : childWindows allocation m ε side w) (i : MaskIndex K tick)
(c : Classes K tick) (hc : c ∈ testedClasses allocation i) :
|classDensity (wordPairLeftPositions w classStatistic (leftSymbol i)) c -
leftProbability side i c| ≤ childWidth ε i.1 := by
have hp : 0 < activeCounts allocation m c.1 c.2 :=
(activeCounts_pos_iff allocation hm c.1 c.2).2
(positiveClasses_positive allocation i.1 c hc)
have ht := (hw c).1 (leftSymbol i c)
have hh := testedClasses_parent allocation i c hc
have ht' : |empiricalLaw (classStatistic c ∘ w.left c) (leftSymbol i c) -
leftProbability side i c| ≤ childWidth ε c.1 := by
simpa only [activeLaw, activeWidth, ite_eq_left hp, leftProbability] using ht
have hd := classDensity_statisticPositions w.left classStatistic (leftSymbol i) c
change classDensity (wordPairLeftPositions w classStatistic (leftSymbol i)) c = _ at hd
rw [hd]
exact ht'.trans_eq (congrArg (childWidth ε) hh)
theorem right_density_le (allocation : Allocation) {m : ℕ} (hm : 0 < m)
(ε : ℝ) (side : Fin 3) (w : Words (K := K) (tick := tick) allocation m)
(hw : childWindows allocation m ε side w) (i : MaskIndex K tick)
(c : Classes K tick) (hc : c ∈ testedClasses allocation i) :
|classDensity (wordPairRightPositions w classStatistic (rightSymbol i)) c -
rightProbability side i c| ≤ childWidth ε i.1 := by
have hp : 0 < activeCounts allocation m c.1 c.2 :=
(activeCounts_pos_iff allocation hm c.1 c.2).2
(positiveClasses_positive allocation i.1 c hc)
have ht := (hw c).2 (rightSymbol i c)
have hh := testedClasses_parent allocation i c hc
have ht' : |empiricalLaw (classStatistic c ∘ w.right c) (rightSymbol i c) -
rightProbability side i c| ≤ childWidth ε c.1 := by
simpa only [activeLaw, activeWidth, ite_eq_left hp, rightProbability] using ht
have hd := classDensity_statisticPositions w.right classStatistic (rightSymbol i) c
change classDensity (wordPairRightPositions w classStatistic (rightSymbol i)) c = _ at hd
rw [hd]
exact ht'.trans_eq (congrArg (childWidth ε) hh)
theorem mixture_error_zero (allocation : Allocation) {m : ℕ} (hm : 0 < m)
(side : Fin 3) (i : MaskIndex K tick) :
|classProductMixture (fun c : testedClasses allocation i => Positions allocation m c.val)
(fun c => leftProbability side i c.val) (fun c => rightProbability side i c.val) -
center allocation side i| ≤ (0 : ℝ) := by
rw [selected_classProductMixture_eq_one allocation hm]
simp only [center, sub_self, abs_zero, le_refl]
theorem orbitSet_as_filter {G V : Type*} [Group G] [Fintype G] [MulAction G V]
[Fintype V] [DecidableEq V] (v : V) :
OrbitCounting.orbitSet (G := G) v =
Finset.univ.filter (fun w => ∃ g : G, g • v = w) := by
ext w
simp only [OrbitCounting.orbitSet, Finset.mem_image, Finset.mem_univ,
true_and, Finset.mem_filter]
attribute [local irreducible] activeCounts
theorem full_filter_loss_le (allocation : Allocation) {m : ℕ} (hm : 2 ≤ m)
{ε : ℝ} (hε : 0 < ε) (side : Fin 3)
(w : Words (K := K) (tick := tick) allocation m)
(hw : childWindows allocation m ε side w) :
((rejectedFilter
(fullOrbitFilter (HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w)
(fun v => ∃ i : MaskIndex K tick,
selectedWordPairMaskBad (testedClasses allocation i) classStatistic classStatistic
(leftSymbol i) (rightSymbol i) (pairWidth ε i.1) (center allocation side i) v)).card : ℝ) /
(fullOrbitFilter
(HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w).card ≤
lossConstant (K := K) (tick := tick) allocation ε / m := by
have hmpos : 0 < m := lt_of_lt_of_le (by decide : 0 < 2) hm
have hsize (i : MaskIndex K tick) :
populationSize (fun c : testedClasses allocation i => Positions allocation m c.val) =
selectedBasePopulation allocation (testedClasses allocation i) * m :=
selected_populationSize allocation m (testedClasses allocation i)
have hn (i : MaskIndex K tick) (c : Classes K tick)
(hc : c ∈ testedClasses allocation i) : 2 ≤ Fintype.card (Positions allocation m c) := by
exact selected_two_le allocation hm (testedClasses allocation i)
(positiveClasses_positive allocation i.1) ⟨c, hc⟩
have hmass (i : MaskIndex K tick) :
0 < selectedBasePopulation allocation (testedClasses allocation i) :=
positiveClasses_basePopulation_pos allocation i.1
have hright0 (i : MaskIndex K tick) (c : Classes K tick)
(_hc : c ∈ testedClasses allocation i) : 0 ≤ rightProbability side i c :=
rightLaw_nonnegative c.1 c.2 side _
have h := finite_selectedWordPair_full_filter_inverse_linear_of_windows
(C := Classes K tick) (P := Positions (K := K) (tick := tick) allocation m)
(X := Letters) (Y := Letters) (I := MaskIndex K tick)
(SL := fun _ c => Statistic c.1) (SR := fun _ c => Statistic c.1)
(testedClasses allocation) w (fun _ => classStatistic) (fun _ => classStatistic)
leftSymbol rightSymbol hn
(fun i => selectedBasePopulation allocation (testedClasses allocation i)) (m : ℝ)
(by exact_mod_cast hmpos) hmass hsize
(leftProbability side) (rightProbability side)
(fun i => pairWidth ε i.1) (fun i => childWidth ε i.1) (fun _ => 0)
(center allocation side) (fun i => pairWidth_pos hε i.1)
hright0 (rightProbability_le_one allocation side)
(left_density_le allocation hmpos ε side w hw)
(right_density_le allocation hmpos ε side w hw)
(mixture_error_zero allocation hmpos side) (fun _ => le_rfl)
(fun i => (div_pos (pairWidth_pos hε i.1) (by norm_num : (0 : ℝ) < 4)).le)
simpa only [lossConstant] using h
theorem orbit_loss_le (allocation : Allocation) {m : ℕ} (hm : 2 ≤ m)
{ε : ℝ} (hε : 0 < ε) (side : Fin 3)
(w : Words (K := K) (tick := tick) allocation m)
(hw : childWindows allocation m ε side w) :
(((OrbitCounting.orbitSet
(G := HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w).filter
(fun v => ¬passes allocation m ε side v)).card : ℝ) /
(OrbitCounting.orbitSet
(G := HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w).card ≤
lossConstant (K := K) (tick := tick) allocation ε / m := by
have heq := orbit_bad_fraction_eq_fullOrbitFilter_of_iff
(G := HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w
(fun v => ¬passes allocation m ε side v)
(fun v => ∃ i : MaskIndex K tick,
selectedWordPairMaskBad (testedClasses allocation i) classStatistic classStatistic
(leftSymbol i) (rightSymbol i) (pairWidth ε i.1) (center allocation side i) v)
(fun v => by simp only [passes, not_forall, not_not])
exact heq.trans_le (full_filter_loss_le allocation hm hε side w hw)
end MatrixMultiplication.AllFieldHistoryMasks
end
end OAI
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