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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AssignmentCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AdmissibilityCore
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Compatibility
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.MaskedGoodSelection
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupAssignmentRetention
open AllFieldHistory AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery
open AllFieldGroupOrbitData JointCoarseHashing
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
theorem good_failure_fraction_le (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(hAP : ∀ x ∈ U, ∀ y ∈ U, ∀ z ∈ U, x + y = 2 * z → x = z ∧ y = z)
(N : ℝ) (s : Sample (Pos (K := K) (tick := tick) allocation m sigma) k)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(hg : (data (K := K) (tick := tick) allocation m ε sigma).Good k U N e s)
(o : Orbit (K := K) (tick := tick) allocation m sigma) (ho : o ∈ (data (K := K) (tick := tick) allocation m ε sigma).orbits e) :
((((data (K := K) (tick := tick) allocation m ε sigma).passing e o).filter
(fun v => assigned (K := K) (tick := tick) allocation m ε sigma k U s v ≠ some (coarse (K := K) (tick := tick) allocation m sigma e))).card : ℝ) /
(((data (K := K) (tick := tick) allocation m ε sigma).full e o).card : ℝ) ≤ 1 / N := by
have ha (v : Variable (K := K) (tick := tick) allocation m sigma)
(hv : v ∈ (data (K := K) (tick := tick) allocation m ε sigma).passing e o) :
Admissible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 :=
AllFieldGroupAdmissible.full_admissible (K := K) (tick := tick) allocation m ε sigma e o ho v
((mem_passing (K := K) (tick := tick) allocation m ε sigma e o v).mp hv).1
have hc (v : Variable (K := K) (tick := tick) allocation m sigma)
(hv : v ∈ (data (K := K) (tick := tick) allocation m ε sigma).passing e o) :
Compatible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 :=
AllFieldGroupCompatibility.admissible_compatible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 (ha v hv)
have hh := (data (K := K) (tick := tick) allocation m ε sigma).good_passing_actual_failure_fraction_le k U hAP N e o ho s hg
(actualCompatible (K := K) (tick := tick) allocation m ε sigma) (actualUseful (K := K) (tick := tick) allocation m ε sigma)
(actualKeep (K := K) (tick := tick) allocation m ε sigma)
(fun v hv => (actualCompatible_coarse (K := K) (tick := tick) allocation m ε sigma e v).mpr (hc v hv))
(fun v hv => (actualUseful_coarse (K := K) (tick := tick) allocation m ε sigma e v).mpr (ha v hv).1)
(fun v hv => ((mem_passing (K := K) (tick := tick) allocation m ε sigma e o v).mp hv).2)
(fun v hv f _ hn hcomp => competitor_coverage (K := K) (tick := tick) allocation m ε sigma e o v (hc v hv) f hn hcomp)
rw [Finset.filter_congr_decidable] at hh
exact hh
end MatrixMultiplication.AllFieldGroupAssignmentRetention
end
end OAI
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