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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitTransport
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Admissible
import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AssignmentLoss
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_local_failure_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 allocation m ε sigma).Good k U N e s)
(side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma)
(hw : Admissible allocation m ε sigma side e w) :
(((localOrbit allocation m sigma e w).filter (fun v =>
completeKeep allocation m ε (sigma side) sigma v ∧
assigned allocation m ε sigma k U s (side, v) ≠ some (coarse allocation m sigma e))).card : ℝ) /
((localOrbit allocation m sigma e w).card : ℝ) ≤ 1 / N := by
have ho : (side, orbitOf allocation m sigma e w) ∈ (data allocation m ε sigma).orbits e :=
Finset.mem_filter.mpr ⟨Finset.mem_univ _, w, hw, rfl⟩
have hh := good_failure_fraction_le allocation m ε sigma k U hAP N s e hg _ ho
change (((passing allocation m sigma ε e (side, orbitOf allocation m sigma e w)).filter
(fun v => assigned allocation m ε sigma k U s v ≠ some (coarse allocation m sigma e))).card : ℝ) /
((full allocation m sigma e (side, orbitOf allocation m sigma e w)).card : ℝ) ≤ 1 / N at hh
rw [passing, Finset.filter_filter] at hh
simp only [full_eq_tagged_localOrbit, Finset.filter_map, Finset.card_map] at hh
rw [Finset.filter_congr_decidable] at hh
exact hh
theorem localAssigned_iff (allocation : Allocation) (m : ℕ) (ε : ℝ)
(sigma : Placement) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(s : Sample (Pos (K := K) (tick := tick) allocation m sigma) k)
(t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) :
localAssigned allocation m sigma (physicalAssignment allocation m ε sigma k U s) t side w ↔
assigned allocation m ε sigma k U s (sigma.symm side, w) = some t := by
fin_cases side <;> rfl
theorem good_physical_local_failure_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 allocation m ε sigma).Good k U N e s)
(side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma)
(hw : Admissible allocation m ε sigma (sigma.symm side) e w) :
(((localOrbit allocation m sigma e w).filter (fun v =>
completeKeep allocation m ε side sigma v ∧
¬localAssigned allocation m sigma (physicalAssignment allocation m ε sigma k U s)
(coarse allocation m sigma e) side v)).card : ℝ) /
((localOrbit allocation m sigma e w).card : ℝ) ≤ 1 / N := by
simpa only [sigma.apply_symm_apply, localAssigned_iff] using
good_local_failure_le allocation m ε sigma k U hAP N s e hg (sigma.symm side) w hw
theorem good_projected_failure_le (F : Type*) [CommRing F]
(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 : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m)
(hg : (data allocation m ε sigma).Good k U N (projectTarget allocation m sigma e) s)
(side : Fin 3) (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m)
(hw : AllFieldHistoryRecovery.Used F allocation m ε e side w) :
(((localOrbit allocation m sigma (projectTarget allocation m sigma e)
(projectRaw allocation m sigma w)).filter (fun v =>
completeKeep allocation m ε side sigma v ∧
¬localAssigned allocation m sigma (physicalAssignment allocation m ε sigma k U s)
(coarse allocation m sigma (projectTarget allocation m sigma e)) side v)).card : ℝ) /
((localOrbit allocation m sigma (projectTarget allocation m sigma e)
(projectRaw allocation m sigma w)).card : ℝ) ≤ 1 / N := by
apply good_physical_local_failure_le allocation m ε sigma k U hAP N s _ hg
exact AllFieldGroupAdmissible.used_project_admissible F allocation m ε sigma e
(sigma.symm side) w (by simpa only [sigma.apply_symm_apply] using hw)
end MatrixMultiplication.AllFieldGroupAssignmentRetention
end
end OAI
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