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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.BranchWeights
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryMaskLaws
open AllFieldHistory AllFieldHistoryChildLaws AllFieldHistorySupport
open AllFieldHistoryMasks AllFieldHistoryBranchWeights AllFieldParameters
open PermutationMatching
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
def parentCenter (allocation : Allocation) (side : Fin 3) (h : Active K tick)
(a b : Statistic h) : ℝ :=
∑ u : JointPopulation.Shape,
(activeCounts allocation 1 h u : ℝ) /
population allocation 1 (h.val.1.source, h.val.2) *
(leftLaw h u side a * rightLaw h u side b)
theorem center_eq_parentCenter (allocation : Allocation) (side : Fin 3)
(i : MaskIndex K tick) :
center allocation side i = parentCenter allocation side i.1 (i.2.1 i.1) (i.2.2 i.1) := by
have hw (c : testedClasses allocation i) :
classWeight (fun c : testedClasses allocation i => Positions allocation 1 c.val) c =
(activeCounts allocation 1 c.val.1 c.val.2 : ℝ) /
population allocation 1 (i.1.val.1.source, i.1.val.2) := by
rw [selected_classWeight_formula allocation (by decide : 0 < (1 : ℕ))]
simp only [selectedBasePopulation, testedClasses, positiveClasses_baseCount]
unfold center classProductMixture
simp_rw [hw]
let f : Classes K tick → ℝ := fun c =>
(activeCounts allocation 1 c.1 c.2 : ℝ) /
population allocation 1 (i.1.val.1.source, i.1.val.2) *
(leftProbability side i c * rightProbability side i c)
change (∑ c : testedClasses allocation i, f c.val) = _
rw [Finset.sum_coe_sort]
have hzero (c : Classes K tick) :
(if 0 < activeCounts allocation 1 c.1 c.2 then f c else 0) = f c := by
split_ifs with hc
· rfl
· have hz : activeCounts allocation 1 c.1 c.2 = 0 := Nat.eq_zero_of_not_pos hc
simp [f, hz]
simp only [testedClasses, positiveClasses, Finset.sum_filter, ite_and, hzero]
rw [Fintype.sum_prod_type, Fintype.sum_eq_single i.1]
· simp [f, parentCenter, leftProbability, rightProbability, leftSymbol, rightSymbol]
· intro h hh
apply Finset.sum_eq_zero
intro u _
exact ite_eq_right hh
def stageBActive (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick) :
Active K tick := ⟨(.stageB h, phi), ht⟩
theorem stageB_leftLaw_branch (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick)
(k : BSplit h) (side : Fin 3) (a : Fin 6) :
leftLaw (stageBActive h phi ht) (branchShape (.stageB h, phi) k) side a =
(littleLaw (aShape h.val) (bSplit ⟨h, k, false⟩) (phi.symm side) a : ℝ) := by
change (placedChildLawRat (.stageB h, phi) (branchShape (.stageB h, phi) k)
side false a : ℝ) = _
rw [placedChildLawRat_branchShape (.stageB h, phi) k side false]
simp [Work.childLaw, Work.splitShape, halfShape]
theorem stageB_rightLaw_branch (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick)
(k : BSplit h) (side : Fin 3) (b : Fin 6) :
rightLaw (stageBActive h phi ht) (branchShape (.stageB h, phi) k) side b =
(littleLaw (aShape h.val) (complement (aShape h.val) (bSplit ⟨h, k, false⟩))
(phi.symm side) b : ℝ) := by
change (placedChildLawRat (.stageB h, phi) (branchShape (.stageB h, phi) k)
side true b : ℝ) = _
rw [placedChildLawRat_branchShape (.stageB h, phi) k side true]
simp [Work.childLaw, Work.parentShape, Work.splitShape, halfShape]
theorem stageB_sum_laws (h : APositive K) (side : Fin 3) (a b : Fin 6) :
(∑ k : BSplit h,
stageBLaw (aShape h.val) (bSplit ⟨h, k, false⟩) *
littleLaw (aShape h.val) (bSplit ⟨h, k, false⟩) side a *
littleLaw (aShape h.val) (complement (aShape h.val) (bSplit ⟨h, k, false⟩)) side b) =
halfLaw (aShape h.val) side (a, b) := by
rw [halfLaw_positive_parent _ h.property]
unfold positiveHalfLaw
change (∑ k : Fin (below (aShape h.val)).length,
stageBLaw (aShape h.val) (below (aShape h.val))[k.val] *
littleLaw (aShape h.val) (below (aShape h.val))[k.val] side a *
littleLaw (aShape h.val) (complement (aShape h.val) (below (aShape h.val))[k.val]) side b) = _
exact Fin.sum_univ_fun_getElem (below (aShape h.val))
(fun u => stageBLaw (aShape h.val) u * littleLaw (aShape h.val) u side a *
littleLaw (aShape h.val) (complement (aShape h.val) u) side b)
theorem parentCenter_stageB (allocation : Allocation) (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick)
(side : Fin 3) (a b : Fin 6) :
parentCenter allocation side (stageBActive h phi ht) a b =
(halfLaw (aShape h.val) (phi.symm side) (a, b) : ℝ) := by
let w : PlacedWork K := (.stageB h, phi)
let f : JointPopulation.Shape → ℝ := fun u =>
leftLaw (stageBActive h phi ht) u side a * rightLaw (stageBActive h phi ht) u side b
calc
parentCenter allocation side (stageBActive h phi ht) a b =
(∑ u, (jointCounts allocation 1 w u : ℝ) * f u) /
population allocation 1 (.afterA h.val, phi) := by
unfold parentCenter
rw [Finset.sum_div]
apply Finset.sum_congr rfl
intro u _
dsimp only [activeCounts, stageBActive, w, f, Work.source]
ring
_ = (∑ k : BSplit h, (branchPopulation allocation 1 w k : ℝ) * f (branchShape w k)) /
population allocation 1 (.afterA h.val, phi) := by
rw [jointCounts, shapeCounts_weighted_sum]
rfl
_ = ∑ k : BSplit h,
(stageBLaw (aShape h.val) (bSplit ⟨h, k, false⟩) : ℝ) *
(littleLaw (aShape h.val) (bSplit ⟨h, k, false⟩) (phi.symm side) a : ℝ) *
(littleLaw (aShape h.val) (complement (aShape h.val) (bSplit ⟨h, k, false⟩))
(phi.symm side) b : ℝ) := by
rw [Finset.sum_div]
apply Finset.sum_congr rfl
intro k _
dsimp only [f, w]
rw [stageB_leftLaw_branch, stageB_rightLaw_branch]
rw [mul_div_right_comm,
stageB_branchWeight_real allocation (by decide : 0 < (1 : ℕ)) h k phi]
ring
_ = (halfLaw (aShape h.val) (phi.symm side) (a, b) : ℝ) := by
exact_mod_cast stageB_sum_laws h (phi.symm side) a b
theorem center_stageB (allocation : Allocation) (h : APositive K) (phi : Placement)
(ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick)
(side : Fin 3) (a b : SymbolChoice K tick) :
center allocation side (stageBActive h phi ht, a, b) =
(halfLaw (aShape h.val) (phi.symm side)
(a (stageBActive h phi ht), b (stageBActive h phi ht)) : ℝ) := by
rw [center_eq_parentCenter]
exact parentCenter_stageB allocation h phi ht side _ _
end MatrixMultiplication.AllFieldHistoryMaskLaws
end
end OAI
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