openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixFields /Construction /GroupAssignmentRetention.lean
| import Mathlib | |
| import OAI.LinearAlgebra.MatrixFields.Entropy.GroupBranchLaws | |
| import OAI.LinearAlgebra.MatrixFields.Extraction.JointMaskedGoodSelection | |
| import OAI.LinearAlgebra.MatrixFields.Histories.JointScalarOrbitAdmissibility | |
| namespace OAI | |
| namespace MatrixAllFields | |
| open scoped BigOperators Topology Polynomial | |
| section | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupAdmissible | |
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws | |
| open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery AllFieldGroupOrbitData | |
| open JointPopulation JointCanonicalization PermutationMatching | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| abbrev Symmetries (allocation : Allocation) (m : ℕ) (sigma : Placement) := | |
| HalfClassPermutations (ClassPositions (Counts (K := K) (tick := tick) allocation m sigma)) | |
| @[instance_reducible] | |
| def rawAction (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) : | |
| MulAction (Symmetries (K := K) (tick := tick) allocation m sigma) (Raw (K := K) (tick := tick) allocation m sigma) := | |
| EquivOrbitTransport.action (groupCoordinates (K := K) (tick := tick) allocation m sigma e) | |
| theorem admissible_smul (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (side : Fin 3) (e : Targets (K := K) (tick := tick) allocation m sigma) : | |
| letI : MulAction (Symmetries (K := K) (tick := tick) allocation m sigma) (Raw (K := K) (tick := tick) allocation m sigma) := | |
| rawAction (K := K) (tick := tick) allocation m sigma e | |
| ∀ (g : Symmetries (K := K) (tick := tick) allocation m sigma) (w : Raw (K := K) (tick := tick) allocation m sigma), | |
| Admissible (K := K) (tick := tick) allocation m ε sigma side e w → | |
| Admissible (K := K) (tick := tick) allocation m ε sigma side e ((rawAction (K := K) (tick := tick) allocation m sigma e).smul g w) := by | |
| let : MulAction (Symmetries (K := K) (tick := tick) allocation m sigma) (Raw (K := K) (tick := tick) allocation m sigma) := | |
| rawAction (K := K) (tick := tick) allocation m sigma e | |
| intro g w hw | |
| have hweights : ∀ h j, | |
| CWStrands.weight (((rawAction (K := K) (tick := tick) allocation m sigma e).smul g w) h j).1 = (shapeSide (sigma side) ((e h).val j)).val ∧ | |
| CWStrands.weight (((rawAction (K := K) (tick := tick) allocation m sigma e).smul g w) h j).2 = activeParentShape h.val (sigma side) - | |
| (shapeSide (sigma side) ((e h).val j)).val := by | |
| exact JointScalarOrbitAdmissibility.classWeights_smul | |
| (Counts (K := K) (tick := tick) allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun _ => CWStrands.weight) (fun _ => CWStrands.weight) | |
| (fun _ u => (shapeSide (sigma side) u).val) | |
| (fun h u => activeParentShape h.val (sigma side) - (shapeSide (sigma side) u).val) | |
| e g w hw.2 | |
| refine ⟨⟨?_, ?_⟩, hweights⟩ | |
| · intro h j | |
| apply Fin.ext | |
| exact (hweights h j).1 | |
| · exact (JointScalarOrbitAdmissibility.rawWindows_smul | |
| (SL := fun h : ActiveOrder K tick sigma => Statistic h.val) | |
| (SR := fun h : ActiveOrder K tick sigma => Statistic h.val) | |
| (Counts (K := K) (tick := tick) allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun c => statistic c.1.val) (fun c => statistic c.1.val) | |
| (fun c => leftLaw c.1.val c.2 (sigma side)) | |
| (fun c => rightLaw c.1.val c.2 (sigma side)) | |
| (fun c => AllFieldHistoryMasks.childWidth ε c.1.val) | |
| (fun c => AllFieldHistoryMasks.childWidth ε c.1.val) e g w).mpr hw.1.2 | |
| theorem admissible_of_orbitOf_eq (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (side : Fin 3) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (w v : Raw (K := K) (tick := tick) allocation m sigma) (hw : Admissible (K := K) (tick := tick) allocation m ε sigma side e w) | |
| (hv : orbitOf (K := K) (tick := tick) allocation m sigma e v = orbitOf (K := K) (tick := tick) allocation m sigma e w) : | |
| Admissible (K := K) (tick := tick) allocation m ε sigma side e v := by | |
| let : MulAction (Symmetries (K := K) (tick := tick) allocation m sigma) (Raw (K := K) (tick := tick) allocation m sigma) := | |
| rawAction (K := K) (tick := tick) allocation m sigma e | |
| have hrel : MulAction.orbitRel | |
| (Symmetries (K := K) (tick := tick) allocation m sigma) | |
| (CanonicalPairs (Counts (K := K) (tick := tick) allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7)) | |
| (groupCoordinates (K := K) (tick := tick) allocation m sigma e v) | |
| (groupCoordinates (K := K) (tick := tick) allocation m sigma e w) := Quotient.exact hv | |
| obtain ⟨g, hg⟩ := hrel | |
| have hgv : ((rawAction (K := K) (tick := tick) allocation m sigma e).smul g w) = v := by | |
| apply (groupCoordinates (K := K) (tick := tick) allocation m sigma e).injective | |
| exact (EquivOrbitTransport.equiv_smul | |
| (groupCoordinates (K := K) (tick := tick) allocation m sigma e) g w).trans hg | |
| rw [← hgv] | |
| exact admissible_smul allocation m ε sigma side e g w hw | |
| theorem full_admissible (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (ho : o ∈ (data (K := K) (tick := tick) allocation m ε sigma).orbits e) | |
| (v : Variable (K := K) (tick := tick) allocation m sigma) (hv : v ∈ (data (K := K) (tick := tick) allocation m ε sigma).full e o) : | |
| Admissible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 := by | |
| obtain ⟨w, hw, hwo⟩ := (Finset.mem_filter.mp ho).2 | |
| obtain ⟨hside, hvo⟩ := (Finset.mem_filter.mp hv).2 | |
| rw [hside] | |
| exact admissible_of_orbitOf_eq allocation m ε sigma o.1 e w v.2 hw (hvo.trans hwo.symm) | |
| end MatrixMultiplication.AllFieldGroupAdmissible | |
| end | |
| end | |
| end MatrixAllFields | |
| namespace MatrixAllFields | |
| open scoped BigOperators Topology Polynomial | |
| section | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupAdmissible | |
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws | |
| open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery AllFieldGroupOrbitData | |
| open JointPopulation JointCanonicalization PermutationMatching | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| theorem used_project_admissible (F : Type*) [CommRing F] | |
| (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m) | |
| (side : Fin 3) (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) | |
| (hw : AllFieldHistoryRecovery.Used F allocation m ε e (sigma side) w) : | |
| Admissible (K := K) (tick := tick) allocation m ε sigma side | |
| (projectTarget (K := K) (tick := tick) allocation m sigma e) (projectRaw (K := K) (tick := tick) allocation m sigma w) := by | |
| obtain ⟨x, y, z, hq, rfl⟩ := hw | |
| have hi : | |
| AllFieldHistoryRecovery.idealSide allocation m ε 0 e x ∧ | |
| AllFieldHistoryRecovery.idealSide allocation m ε 1 e y ∧ | |
| AllFieldHistoryRecovery.idealSide allocation m ε 2 e z := by | |
| by_contra hn | |
| apply hq | |
| rw [AllFieldHistoryRecovery.ideal_eq_delete_source F allocation m ε e] | |
| exact ite_eq_right hn | |
| have ht := AllFieldHistoryRecovery.ideal_support_weights F allocation m ε e x y z hq | |
| have hs (s : Fin 3) : | |
| AllFieldHistoryRecovery.idealSide allocation m ε s e | |
| (AllFieldHistoryRecovery.sideVariable s x y z) ∧ | |
| AllFieldHistoryRecovery.sideWeights allocation m s e | |
| (AllFieldHistoryRecovery.sideVariable s x y z) := by | |
| fin_cases s | |
| · exact ⟨hi.1, ht.1⟩ | |
| · exact ⟨hi.2.1, ht.2.1⟩ | |
| · exact ⟨hi.2.2, ht.2.2⟩ | |
| refine ⟨?_, ?_⟩ | |
| · exact (idealSide_iff_all_groups allocation m ε | |
| (sigma side) e _).mp (hs (sigma side)).1 sigma | |
| · intro h j | |
| exact (hs (sigma side)).2 h.val j | |
| theorem used_project_covered (F : Type*) [CommRing F] | |
| (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m) | |
| (side : Fin 3) (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) | |
| (hw : AllFieldHistoryRecovery.Used F allocation m ε e (sigma side) w) : | |
| ∃ o ∈ (data (K := K) (tick := tick) allocation m ε sigma).orbits (projectTarget (K := K) (tick := tick) allocation m sigma e), | |
| (side, projectRaw (K := K) (tick := tick) allocation m sigma w) ∈ | |
| (data (K := K) (tick := tick) allocation m ε sigma).full (projectTarget (K := K) (tick := tick) allocation m sigma e) o := | |
| admissible_covered allocation m ε sigma side _ _ | |
| (used_project_admissible F allocation m ε sigma e side w hw) | |
| end MatrixMultiplication.AllFieldGroupAdmissible | |
| end | |
| end | |
| end MatrixAllFields | |
| namespace MatrixAllFields | |
| open scoped BigOperators Topology Polynomial | |
| section | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupAssignmentRetention | |
| open AllFieldHistory AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery | |
| open AllFieldGroupOrbitData JointCoarseHashing | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| def assigned (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) | |
| (v : Variable (K := K) (tick := tick) allocation m sigma) : Option (Triple (Pos (K := K) (tick := tick) allocation m sigma)) := | |
| (data allocation m ε sigma).actualAssignment k U | |
| (actualCompatible allocation m ε sigma) (actualUseful allocation m ε sigma) | |
| (actualKeep allocation m ε sigma) s v | |
| def physicalAssignment (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) : | |
| JointExtraction.Assignment (Raw (K := K) (tick := tick) allocation m sigma) (Raw (K := K) (tick := tick) allocation m sigma) | |
| (Raw (K := K) (tick := tick) allocation m sigma) (Triple (Pos (K := K) (tick := tick) allocation m sigma)) where | |
| x w := assigned allocation m ε sigma k U s (sigma.symm 0, w) | |
| y w := assigned allocation m ε sigma k U s (sigma.symm 1, w) | |
| z w := assigned allocation m ε sigma k U s (sigma.symm 2, w) | |
| theorem assigned_spec (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) | |
| (v : Variable (K := K) (tick := tick) allocation m sigma) (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) | |
| (h : assigned allocation m ε sigma k U s v = some t) : | |
| t ∈ (data allocation m ε sigma).actualEligible k U s ∧ | |
| actualCompatible allocation m ε sigma t v ∧ actualUseful allocation m ε sigma t v ∧ | |
| ∀ f, f ∈ (data allocation m ε sigma).actualEligible k U s → | |
| actualCompatible allocation m ε sigma f v → f = t := by | |
| have hh := JointExtraction.retainAssignment_some _ _ v t h | |
| exact JointCoarseAssignment.assignUseful_spec _ _ _ v t hh | |
| theorem assigned_some_idealSide (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) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) (v : Variable (K := K) (tick := tick) allocation m sigma) | |
| (h : assigned allocation m ε sigma k U s v = some (coarse allocation m sigma e)) : | |
| groupIdealSide allocation m ε (sigma v.1) sigma e v.2 := by | |
| exact (actualUseful_coarse allocation m ε sigma e v).mp | |
| (assigned_spec allocation m ε sigma k U s v _ h).2.2.1 | |
| end MatrixMultiplication.AllFieldGroupAssignmentRetention | |
| end | |
| end | |
| end MatrixAllFields | |
| namespace MatrixAllFields | |
| open scoped BigOperators Topology Polynomial | |
| section | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupCompatibility | |
| open AllFieldHistory AllFieldHistoryChildLaws AllFieldHistoryGroupMasks | |
| open AllFieldHistoryGroupedRecovery AllFieldGroupOrbitData | |
| open JointPopulation JointCanonicalization | |
| attribute [local instance] Classical.propDecidable | |
| variable {K tick : ℕ} | |
| theorem groupIdealSide_ownWord (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (side : Fin 3) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (w : Raw (K := K) (tick := tick) allocation m sigma) | |
| (hw : groupIdealSide allocation m ε (sigma side) sigma e w) : | |
| tripleSide side (AllFieldGroupOrbitData.coarse allocation m sigma e) = | |
| ownWord allocation m sigma w := by | |
| change tripleSide side | |
| (reorder sigma (triple (Counts (K := K) (tick := tick) allocation m sigma) e)) = _ | |
| rw [tripleSide_reorder] | |
| funext p | |
| have hs : tripleSide (sigma side) | |
| (triple (Counts (K := K) (tick := tick) allocation m sigma) e) p = | |
| shapeSide (sigma side) ((e p.1).val p.2) := by | |
| generalize sigma side = s | |
| fin_cases s <;> rfl | |
| rw [hs] | |
| exact (hw.1 p.1 p.2).symm | |
| theorem groupIdealSide_fineCompatible (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (side : Fin 3) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (w : Raw (K := K) (tick := tick) allocation m sigma) | |
| (hw : groupIdealSide allocation m ε (sigma side) sigma e w) : | |
| fineCompatible allocation m ε sigma side e w := by | |
| intro right h u hd | |
| apply JointPopulationCompatibility.shape_count_window_of_typeWindow | |
| (Counts (K := K) (tick := tick) allocation m sigma) e h u | |
| (fun i => statistic h.val (if right then (w h i).2 else (w h i).1)) | |
| (compatibilityLaw right side sigma h u) (AllFieldHistoryMasks.childWidth ε h.val) | |
| intro hu | |
| rw [AllFieldGroupNativeLaws.compatibilityLaw_eq allocation m sigma right side h u hu hd] | |
| cases right | |
| · exact (hw.2 h u hu).1 | |
| · exact (hw.2 h u hu).2 | |
| theorem groupIdealSide_compatible (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (side : Fin 3) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (w : Raw (K := K) (tick := tick) allocation m sigma) | |
| (hw : groupIdealSide allocation m ε (sigma side) sigma e w) : | |
| Compatible allocation m ε sigma side e w := | |
| ⟨groupIdealSide_ownWord allocation m ε sigma side e w hw, | |
| groupIdealSide_fineCompatible allocation m ε sigma side e w hw⟩ | |
| theorem admissible_compatible (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (side : Fin 3) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (w : Raw (K := K) (tick := tick) allocation m sigma) | |
| (hw : Admissible allocation m ε sigma side e w) : | |
| Compatible allocation m ε sigma side e w := | |
| groupIdealSide_compatible allocation m ε sigma side e w hw.1 | |
| end MatrixMultiplication.AllFieldGroupCompatibility | |
| end | |
| end | |
| end MatrixAllFields | |
| namespace MatrixAllFields | |
| open scoped BigOperators Topology Polynomial | |
| section | |
| 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 hf 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 | |
| end MatrixAllFields | |
| namespace MatrixAllFields | |
| open scoped BigOperators Topology Polynomial | |
| section | |
| 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 | |
| end MatrixAllFields | |
| end OAI | |