openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /AssignmentRetention.lean
| 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 | |