File size: 9,760 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Regrouping
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.IncomingMasks
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupMasks
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistorySource
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily
open AllFieldHistoryRegrouping AllFieldHistoryIncomingMasks
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable Classical.decEq
private def deleteWithPred {F : Type*} [Field F] {X Y Z : Type}
[Fintype X] [Fintype Y] [Fintype Z]
(source : Tensor F X Y Z) (px : X → Prop) (py : Y → Prop) (pz : Z → Prop)
[DecidablePred px] [DecidablePred py] [DecidablePred pz] :
LocalMap source (ExactRecovery.delete source px py pz) := by
let d := LocalMap.delete source px py pz
refine ⟨d.x, d.y, d.z, d.coefficient.trans ?_⟩
funext x y z
unfold ExactRecovery.delete
split_ifs <;> rfl
variable {K tick : ℕ} (allocation : Allocation) (m : ℕ)
abbrev Raw := AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m
def toActive (w : Raw (K := K) (tick := tick) allocation m) :
ActiveWords (K := K) (tick := tick) allocation m := fun h => incomingWord allocation m h w
theorem parent_coefficient (F : Type*) [Field F] (h : Active K tick)
(x y z : Raw (K := K) (tick := tick) allocation m) :
Tensor.power
(CWStrands.shapeTensor (Fin (currentLength h.val.1.source))
(currentPhysicalShape (h.val.1.source, h.val.2)))
(population allocation m (h.val.1.source, h.val.2))
(incomingWord allocation m h x) (incomingWord allocation m h y)
(incomingWord allocation m h z) =
∏ i : JointPopulation.Positions (activeCounts allocation m) h,
AllFieldHistoryRecovery.parents F h (x h i) (y h i) (z h i) := by
change (∏ p : Fin (population allocation m (h.val.1.source, h.val.2)),
CWStrands.shapeTensor _ _ (incomingWord allocation m h x p)
(incomingWord allocation m h y p) (incomingWord allocation m h z p)) = _
refine ((sourcePositionsEquiv allocation m h).prod_comp (fun p =>
CWStrands.shapeTensor (Fin (currentLength h.val.1.source))
(currentPhysicalShape (h.val.1.source, h.val.2))
(incomingWord allocation m h x p) (incomingWord allocation m h y p)
(incomingWord allocation m h z p))).symm.trans ?_
apply Finset.prod_congr rfl
intro i _
simp only [incomingWord, Equiv.symm_apply_apply]
erw [shapeTensor_joinHalves]
unfold AllFieldHistoryRecovery.parents JointCanonicalMixed.inputTensor
rw [ite_eq_left rfl]
unfold JointCanonicalCW.parentTensor currentPhysicalShape
rw [source_shape]
rfl
def receivedSource (F : Type*) [Field F] (ε : ℝ) :
Tensor F (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) :=
ExactRecovery.delete (AllFieldHistoryRecovery.rawSource F (K := K) (tick := tick) allocation m)
(received allocation m ε 0) (received allocation m ε 1) (received allocation m ε 2)
theorem received_coefficient (F : Type*) [Field F] (ε : ℝ)
(x y z : Raw (K := K) (tick := tick) allocation m) :
activeTensor (K := K) (tick := tick) allocation m F ε (toActive allocation m x) (toActive allocation m y)
(toActive allocation m z) = receivedSource (K := K) (tick := tick) allocation m F ε x y z := by
change (∏ h : Active K tick, historyTensor F allocation m ε
(h.val.1.source, h.val.2) (incomingWord allocation m h x)
(incomingWord allocation m h y) (incomingWord allocation m h z)) = _
unfold receivedSource ExactRecovery.delete
by_cases hp : received allocation m ε 0 x ∧ received allocation m ε 1 y ∧
received allocation m ε 2 z
· rw [ite_eq_left hp]
change _ = ∏ h : Active K tick, ∏ i : JointPopulation.Positions (activeCounts allocation m) h,
AllFieldHistoryRecovery.parents F h (x h i) (y h i) (z h i)
apply Finset.prod_congr rfl
intro h _
unfold historyTensor ExactRecovery.delete
rw [ite_eq_left ⟨hp.1 h, hp.2.1 h, hp.2.2 h⟩]
exact parent_coefficient allocation m F h x y z
· rw [ite_eq_right hp]
have hh : ∃ h : Active K tick,
¬(residentMask allocation m ε (h.val.1.source, h.val.2) 0 (incomingWord allocation m h x) ∧
residentMask allocation m ε (h.val.1.source, h.val.2) 1 (incomingWord allocation m h y) ∧
residentMask allocation m ε (h.val.1.source, h.val.2) 2 (incomingWord allocation m h z)) := by
by_contra hn
apply hp
push Not at hn
exact ⟨fun h => (hn h).1, fun h => (hn h).2.1, fun h => (hn h).2.2⟩
obtain ⟨h, hh⟩ := hh
apply Finset.prod_eq_zero (Finset.mem_univ h)
exact ite_eq_right hh
def receivedMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (activeTensor (K := K) (tick := tick) allocation m F ε)
(receivedSource (K := K) (tick := tick) allocation m F ε) := by
apply LocalMap.ofExists
refine ⟨_, _, _, (Tensor.pullback_eq_restrict
(toActive (K := K) (tick := tick) allocation m)
(toActive (K := K) (tick := tick) allocation m)
(toActive (K := K) (tick := tick) allocation m) _).symm.trans ?_⟩
funext x y z
exact received_coefficient allocation m F ε x y z
def marginal (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m) : Prop :=
JointCanonicalMixed.sideMask (activeCounts allocation m) activeHalfLength activeHalfLength
AllFieldHistoryRecovery.halfLength_le_eight side w
def preparedSource (F : Type*) [Field F] (ε : ℝ) :
Tensor F (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) (Raw (K := K) (tick := tick) allocation m) :=
ExactRecovery.delete (receivedSource (K := K) (tick := tick) allocation m F ε)
(fun x => marginal (K := K) (tick := tick) allocation m 0 x ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 0 x)
(fun y => marginal (K := K) (tick := tick) allocation m 1 y ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 1 y)
(fun z => marginal (K := K) (tick := tick) allocation m 2 z ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 2 z)
theorem preparedSource_eq_delete (F : Type*) [Field F] (ε : ℝ) :
preparedSource (K := K) (tick := tick) allocation m F ε =
ExactRecovery.delete (AllFieldHistoryRecovery.rawSource F (K := K) (tick := tick) allocation m)
(AllFieldHistoryRecovery.completeKeep allocation m ε 0)
(AllFieldHistoryRecovery.completeKeep allocation m ε 1)
(AllFieldHistoryRecovery.completeKeep allocation m ε 2) := by
funext x y z
unfold preparedSource receivedSource marginal AllFieldHistoryRecovery.completeKeep
ExactRecovery.delete
split_ifs <;> simp_all only [and_true, true_and, not_false_eq_true]
all_goals tauto
def preparedMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (activeTensor (K := K) (tick := tick) allocation m F ε)
(preparedSource (K := K) (tick := tick) allocation m F ε) :=
(receivedMap (K := K) (tick := tick) allocation m F ε).comp (deleteWithPred (receivedSource (K := K) (tick := tick) allocation m F ε)
(fun x => marginal (K := K) (tick := tick) allocation m 0 x ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 0 x)
(fun y => marginal (K := K) (tick := tick) allocation m 1 y ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 1 y)
(fun z => marginal (K := K) (tick := tick) allocation m 2 z ∧ AllFieldHistoryRawMasks.rawPass allocation m ε 2 z))
abbrev GroupWords := ∀ sigma : Placement,
AllFieldHistoryGroupMasks.GroupRaw (K := K) (tick := tick) allocation m sigma
def groupsEquiv : Raw (K := K) (tick := tick) allocation m ≃
GroupWords (K := K) (tick := tick) allocation m :=
activeWordsEquivOrders (fun h : Active K tick =>
JointPopulation.Positions (activeCounts allocation m) h →
((Fin (activeHalfLength h) → Fin 7) × (Fin (activeHalfLength h) → Fin 7)))
@[simp] theorem project_groups_symm
(x : GroupWords (K := K) (tick := tick) allocation m) (sigma : Placement) :
AllFieldHistoryGroupMasks.projectRaw allocation m sigma
((groupsEquiv allocation m).symm x) = x sigma :=
congrFun ((groupsEquiv allocation m).apply_symm_apply x) sigma
theorem grouped_coefficient (F : Type*) [Field F] (ε : ℝ)
(x y z : GroupWords (K := K) (tick := tick) allocation m) :
preparedSource (K := K) (tick := tick) allocation m F ε
((groupsEquiv allocation m).symm x) ((groupsEquiv allocation m).symm y)
((groupsEquiv allocation m).symm z) =
CommonDimensions.familyProduct
(fun sigma => AllFieldHistoryGroupMasks.groupPreparedSource F allocation m ε sigma)
x y z := by
rw [preparedSource_eq_delete, AllFieldHistoryGroupMasks.preparedSource_factor]
simp only [project_groups_symm, CommonDimensions.familyProduct]
def groupsMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (preparedSource (K := K) (tick := tick) allocation m F ε)
(CommonDimensions.familyProduct (fun sigma : Placement =>
AllFieldHistoryGroupMasks.groupPreparedSource F (K := K) (tick := tick)
allocation m ε sigma)) := by
apply LocalMap.ofExists
refine ⟨_, _, _, (Tensor.pullback_eq_restrict
(groupsEquiv (K := K) (tick := tick) allocation m).symm
(groupsEquiv (K := K) (tick := tick) allocation m).symm
(groupsEquiv (K := K) (tick := tick) allocation m).symm _).symm.trans ?_⟩
funext x y z
exact grouped_coefficient allocation m F ε x y z
end MatrixMultiplication.AllFieldHistorySource
end
end OAI
|