File size: 4,126 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupSourceSupport
open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistorySupport
open AllFieldHistoryChildLaws AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery
open AllFieldGroupOrbitData JointPopulation JointCanonicalization JointCanonicalCW
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {K tick : ℕ}
theorem masks_of_prepared (F : Type*) [CommRing F] (allocation : Allocation)
(m : ℕ) (ε : ℝ) (sigma : Placement)
(x y z : GroupRaw (K := K) (tick := tick) allocation m sigma)
(hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) :
completeKeep allocation m ε 0 sigma x ∧
completeKeep allocation m ε 1 sigma y ∧
completeKeep allocation m ε 2 sigma z := by
by_contra hn
apply hT
exact ite_eq_right hn
theorem raw_nonzero (F : Type*) [CommRing F] (allocation : Allocation)
(m : ℕ) (ε : ℝ) (sigma : Placement)
(x y z : GroupRaw (K := K) (tick := tick) allocation m sigma)
(hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) :
groupRawSource F allocation m sigma x y z ≠ 0 := by
have hm := masks_of_prepared F allocation m ε sigma x y z hT
simpa only [groupPreparedSource, ExactRecovery.delete, ite_eq_left hm] using hT
theorem marginal_sideMask (allocation : Allocation) (m : ℕ) (side : Fin 3)
(sigma : Placement) (w : GroupRaw (K := K) (tick := tick) allocation m sigma)
(hm : marginal allocation m side sigma w) :
JointCanonicalMixed.sideMask (groupCounts allocation m sigma)
(fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val)
(fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) side w := by
intro h a
exact (hm h a).trans
(marginal_population_eq (groupCounts allocation m sigma)
(projectTarget allocation m sigma (canonicalTarget (activeCounts allocation m)))
(canonicalTarget (groupCounts allocation m sigma)) h side a)
theorem marginal_source_nonzero (F : Type*) [CommRing F] (allocation : Allocation)
(m : ℕ) (ε : ℝ) (sigma : Placement)
(x y z : GroupRaw (K := K) (tick := tick) allocation m sigma)
(hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) :
JointCanonicalMixed.maskedSource (F := F) (groupCounts allocation m sigma)
(fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val)
(fun h => activeParentShape h.val) (fun _ => true)
(fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) x y z ≠ 0 := by
have hm := masks_of_prepared F allocation m ε sigma x y z hT
have hx := marginal_sideMask allocation m 0 sigma x hm.1.1
have hy := marginal_sideMask allocation m 1 sigma y hm.2.1.1
have hz := marginal_sideMask allocation m 2 sigma z hm.2.2.1
rw [JointCanonicalMixed.maskedSource, ExactRecovery.delete,
ite_eq_left (And.intro hx (And.intro hy hz))]
exact raw_nonzero F allocation m ε sigma x y z hT
theorem prepared_source_support (F : Type*) [CommRing F] (allocation : Allocation)
(m : ℕ) (ε : ℝ) (sigma : Placement)
(x y z : GroupRaw (K := K) (tick := tick) allocation m sigma)
(hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) :
reorder sigma
(ownWord allocation m sigma x, ownWord allocation m sigma y,
ownWord allocation m sigma z) ∈ ambient allocation m sigma := by
apply Finset.mem_image.mpr
refine ⟨(ownWord allocation m sigma x, ownWord allocation m sigma y,
ownWord allocation m sigma z), ?_, rfl⟩
exact JointCanonicalMixed.maskedSource_support (groupCounts allocation m sigma)
(fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val)
(fun h => activeParentShape h.val) (fun _ => true)
(fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) x y z
(marginal_source_nonzero F allocation m ε sigma x y z hT)
end MatrixMultiplication.AllFieldGroupSourceSupport
end
end OAI
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