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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AssignmentCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.SourceSupport
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.CompatibilityTransfer
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupAssignments
open AllFieldHistory AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery
open AllFieldGroupOrbitData AllFieldGroupAssignmentRetention JointCoarseHashing
attribute [local instance] Classical.propDecidable
variable {K tick : ℕ}
theorem actualCompatible_own (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) (t : Triple (Pos (K := K) (tick := tick) allocation m sigma))
(v : Variable (K := K) (tick := tick) allocation m sigma)
(hc : actualCompatible allocation m ε sigma t v) :
tripleSide v.1 t = ownWord allocation m sigma v.2 := by
obtain ⟨_, rfl, hcompatibility⟩ := hc
exact hcompatibility.1
theorem physicalAssignment_coherent (F : Type*) [Field F]
(allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement)
(k : ℕ) (U : Finset (ZMod (37 ^ k)))
(sample : Sample (Pos (K := K) (tick := tick) allocation m sigma) k) :
JointExtraction.Coherent (groupPreparedSource F allocation m ε sigma)
(physicalAssignment allocation m ε sigma k U sample) := by
intro x y z tx ty tz hT htx hty htz
let a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma := ![x, y, z]
let physicalTag : Fin 3 → Triple (Pos (K := K) (tick := tick) allocation m sigma) :=
![tx, ty, tz]
let b : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma := fun r => a (sigma r)
let tag : Fin 3 → Triple (Pos (K := K) (tick := tick) allocation m sigma) :=
fun r => physicalTag (sigma r)
have hphysical (r : Fin 3) :
assigned allocation m ε sigma k U sample (sigma.symm r, a r) =
some (physicalTag r) := by
fin_cases r
· exact htx
· exact hty
· exact htz
have hassign (r : Fin 3) :
assigned allocation m ε sigma k U sample (r, b r) = some (tag r) := by
simpa only [b, tag, sigma.symm_apply_apply] using hphysical (sigma r)
have hspec (r : Fin 3) :=
assigned_spec allocation m ε sigma k U sample (r, b r) (tag r) (hassign r)
let d := data (K := K) (tick := tick) allocation m ε sigma
let hx := JointMaskedSelection.OrbitData.xFlag k U sample
let hy := JointMaskedSelection.OrbitData.yFlag k U sample
let hz := d.zFlag k U sample
have heligible (r : Fin 3) :=
(JointCoarseAssignment.mem_eligible_iff d.ambient d.actualTargets hx hy hz (tag r)).mp
(hspec r).1
let observed : Triple (Pos (K := K) (tick := tick) allocation m sigma) :=
reorder sigma (ownWord allocation m sigma x,
ownWord allocation m sigma y, ownWord allocation m sigma z)
have hword (r : Fin 3) :
ownWord allocation m sigma (a r) =
tripleSide r (ownWord allocation m sigma x,
ownWord allocation m sigma y, ownWord allocation m sigma z) := by
fin_cases r <;> rfl
have hobserved (r : Fin 3) : tripleSide r observed = tripleSide r (tag r) := by
calc
tripleSide r observed = ownWord allocation m sigma (b r) := by
rw [show observed = reorder sigma (ownWord allocation m sigma x,
ownWord allocation m sigma y, ownWord allocation m sigma z) from rfl,
tripleSide_reorder]
exact (hword (sigma r)).symm
_ = tripleSide r (tag r) :=
(actualCompatible_own allocation m ε sigma (tag r) (r, b r) (hspec r).2.1).symm
have hobsX : observed.1 = (tag 0).1 := hobserved 0
have hobsY : observed.2.1 = (tag 1).2.1 := hobserved 1
have hobsZ : observed.2.2 = (tag 2).2.2 := hobserved 2
have hsurvives : JointCoarseAssignment.survives hx hy hz observed := by
refine ⟨?_, ?_, ?_⟩
· rw [hobsX]
exact (heligible 0).2.2.1.1
· rw [hobsY]
exact (heligible 1).2.2.1.2.1
· rw [hobsZ]
exact (heligible 2).2.2.1.2.2
have hsupport : observed ∈ d.ambient :=
AllFieldGroupSourceSupport.prepared_source_support F allocation m ε sigma x y z hT
have hobs : observed = tag 0 :=
(heligible 0).2.2.2 observed hsupport hsurvives hobsX
obtain ⟨e, he, huseX⟩ := (hspec 0).2.2.1
have hcoarse : reorder sigma (ownWord allocation m sigma (a 0),
ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) =
coarse allocation m sigma e := hobs.trans he.symm
have hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0 :=
AllFieldGroupSourceSupport.raw_nonzero F allocation m ε sigma x y z hT
have hcompY : actualCompatible allocation m ε sigma (tag 0) (1, b 1) :=
⟨e, he, AllFieldGroupCompatibilityTransfer.transferY F allocation m ε sigma e a
hraw hcoarse huseX⟩
have h01 : tag 0 = tag 1 := (hspec 1).2.2.2 (tag 0) (hspec 0).1 hcompY
have huseY : groupIdealSide allocation m ε (sigma 1) sigma e (b 1) := by
apply (actualUseful_coarse allocation m ε sigma e (1, b 1)).mp
rw [he, h01]
exact (hspec 1).2.2.1
have hcompZ : actualCompatible allocation m ε sigma (tag 0) (2, b 2) :=
⟨e, he, AllFieldGroupCompatibilityTransfer.transferZ F allocation m ε sigma e a
hraw hcoarse huseX huseY⟩
have h02 : tag 0 = tag 2 := (hspec 2).2.2.2 (tag 0) (hspec 0).1 hcompZ
have htag (r : Fin 3) : tag r = tag 0 := by
fin_cases r
· rfl
· exact h01.symm
· exact h02.symm
have hphysicalTag (r : Fin 3) : physicalTag r = tag 0 := by
simpa only [tag, sigma.apply_symm_apply] using htag (sigma.symm r)
exact ⟨(hphysicalTag 0).trans (hphysicalTag 1).symm,
(hphysicalTag 0).trans (hphysicalTag 2).symm⟩
end MatrixMultiplication.AllFieldGroupAssignments
end
end OAI
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