openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldHistory /GroupedRecovery.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupCore | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Recovery | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.GroupOrbitProjection | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.RecoveryOrbitDecisions | |
| namespace OAI | |
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldHistoryGroupedRecovery | |
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws | |
| open AllFieldHistoryGroupMasks JointPopulation JointCanonicalization JointCanonicalCW | |
| open PermutationMatching | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| theorem coordinates_project (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m) | |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : | |
| GroupOrbitProjection.projectPair | |
| (ClassPositions (activeCounts allocation m)) | |
| AllFieldHistoryMasks.Letters AllFieldHistoryMasks.Letters | |
| (fun h : Active K tick => h.val.physicalOrder = sigma) | |
| (AllFieldHistoryRecovery.coordinates allocation m e w) = | |
| groupCoordinates allocation m sigma (projectTarget allocation m sigma e) | |
| (projectRaw allocation m sigma w) := rfl | |
| theorem idealSide_iff_all_groups (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (side : Fin 3) | |
| (e : AllFieldHistoryRecovery.Targets (K := K) (tick := tick) allocation m) | |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : | |
| AllFieldHistoryRecovery.idealSide allocation m ε side e w ↔ | |
| ∀ sigma, groupIdealSide allocation m ε side sigma | |
| (projectTarget allocation m sigma e) (projectRaw allocation m sigma w) := by | |
| constructor | |
| · intro hw sigma | |
| exact ⟨fun h j => hw.1 h.val j, fun h u hu => hw.2 h.val u hu⟩ | |
| · intro hw | |
| exact ⟨fun h j => (hw h.val.physicalOrder).1 ⟨h, rfl⟩ j, | |
| fun h u hu => (hw h.val.physicalOrder).2 ⟨h, rfl⟩ u hu⟩ | |
| theorem project_joinGroupTargets (allocation : Allocation) (m : ℕ) | |
| (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma) | |
| (sigma : Placement) : | |
| projectTarget allocation m sigma (joinGroupTargets (activeCounts allocation m) e) = | |
| e sigma := by | |
| funext h | |
| rcases h with ⟨h, hh⟩ | |
| cases hh | |
| rfl | |
| def groupedAssignment (allocation : Allocation) (m : ℕ) (E : Placement → Type*) | |
| (a : ∀ sigma, JointExtraction.Assignment | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma)) : | |
| JointExtraction.Assignment (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) | |
| (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) | |
| (∀ sigma, E sigma) where | |
| x w := GroupAssignment.collect (fun sigma => (a sigma).x) | |
| (fun sigma => projectRaw allocation m sigma w) | |
| y w := GroupAssignment.collect (fun sigma => (a sigma).y) | |
| (fun sigma => projectRaw allocation m sigma w) | |
| z w := GroupAssignment.collect (fun sigma => (a sigma).z) | |
| (fun sigma => projectRaw allocation m sigma w) | |
| theorem groupedAssignment_sound (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (E : Placement → Type*) | |
| (a : ∀ sigma, JointExtraction.Assignment | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma)) | |
| (i : ∀ sigma, E sigma) (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma) | |
| (hx : ∀ sigma w, (a sigma).x w = some (i sigma) → | |
| groupIdealSide allocation m ε 0 sigma (e sigma) w) | |
| (hy : ∀ sigma w, (a sigma).y w = some (i sigma) → | |
| groupIdealSide allocation m ε 1 sigma (e sigma) w) | |
| (hz : ∀ sigma w, (a sigma).z w = some (i sigma) → | |
| groupIdealSide allocation m ε 2 sigma (e sigma) w) : | |
| (∀ w, (groupedAssignment allocation m E a).x w = some i → | |
| AllFieldHistoryRecovery.idealSide allocation m ε 0 | |
| (joinGroupTargets (activeCounts allocation m) e) w) ∧ | |
| (∀ w, (groupedAssignment allocation m E a).y w = some i → | |
| AllFieldHistoryRecovery.idealSide allocation m ε 1 | |
| (joinGroupTargets (activeCounts allocation m) e) w) ∧ | |
| (∀ w, (groupedAssignment allocation m E a).z w = some i → | |
| AllFieldHistoryRecovery.idealSide allocation m ε 2 | |
| (joinGroupTargets (activeCounts allocation m) e) w) := by | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro w hw | |
| apply (idealSide_iff_all_groups allocation m ε 0 _ w).2 | |
| intro sigma | |
| rw [project_joinGroupTargets] | |
| exact hx sigma _ ((GroupAssignment.collect_eq_some _ _ _).mp hw sigma) | |
| · intro w hw | |
| apply (idealSide_iff_all_groups allocation m ε 1 _ w).2 | |
| intro sigma | |
| rw [project_joinGroupTargets] | |
| exact hy sigma _ ((GroupAssignment.collect_eq_some _ _ _).mp hw sigma) | |
| · intro w hw | |
| apply (idealSide_iff_all_groups allocation m ε 2 _ w).2 | |
| intro sigma | |
| rw [project_joinGroupTargets] | |
| exact hz sigma _ ((GroupAssignment.collect_eq_some _ _ _).mp hw sigma) | |
| def localOrbit (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : GroupTarget (K := K) (tick := tick) allocation m sigma) | |
| (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Finset (GroupRaw (K := K) (tick := tick) allocation m sigma) := | |
| GroupOrbitProjection.rawOrbit (groupCounts allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) e w | |
| def localAssigned (allocation : Allocation) (m : ℕ) (sigma : Placement) {E : Type*} | |
| (a : JointExtraction.Assignment | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) E) | |
| (i : E) (side : Fin 3) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) : Prop := | |
| AllFieldHistoryRecovery.sideVariable side (a.x w = some i) (a.y w = some i) (a.z w = some i) | |
| theorem grouped_assigned_iff (allocation : Allocation) (m : ℕ) | |
| (E : Placement → Type*) | |
| (a : ∀ sigma, JointExtraction.Assignment | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma)) | |
| (i : ∀ sigma, E sigma) (side : Fin 3) | |
| (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) : | |
| AllFieldHistoryRecovery.sideVariable side | |
| ((groupedAssignment allocation m E a).x w = some i) | |
| ((groupedAssignment allocation m E a).y w = some i) | |
| ((groupedAssignment allocation m E a).z w = some i) ↔ | |
| ∀ sigma, localAssigned allocation m sigma (a sigma) (i sigma) side | |
| (projectRaw allocation m sigma w) := by | |
| fin_cases side <;> exact GroupAssignment.collect_eq_some _ _ _ | |
| theorem filter_eq_rejected {V : Type*} (orbit : Finset V) | |
| (bad : V → Prop) (d : DecidablePred bad) : | |
| @Finset.filter V bad d orbit = rejectedFilter orbit bad := by | |
| ext v | |
| simp only [rejectedFilter, Finset.mem_filter] | |
| theorem grouped_collision_le (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (E : Placement → Type*) | |
| (a : ∀ sigma, JointExtraction.Assignment | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma)) | |
| (i : ∀ sigma, E sigma) (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma) | |
| (side : Fin 3) (w : AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) | |
| (hbound : ∀ sigma, | |
| (((localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).filter | |
| (fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧ | |
| ¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) / | |
| (localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).card ≤ 1 / m) : | |
| letI : MulAction (AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) | |
| (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) := | |
| AllFieldHistoryRecovery.rawAction allocation m (joinGroupTargets (activeCounts allocation m) e) | |
| (((OrbitCounting.orbitSet (G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) w).filter | |
| (fun v => AllFieldHistoryRecovery.completeKeep allocation m ε side v ∧ | |
| ¬AllFieldHistoryRecovery.sideVariable side | |
| ((groupedAssignment allocation m E a).x v = some i) | |
| ((groupedAssignment allocation m E a).y v = some i) | |
| ((groupedAssignment allocation m E a).z v = some i))).card : ℝ) / | |
| (OrbitCounting.orbitSet (G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) w).card ≤ 6 / m := by | |
| let target := joinGroupTargets (activeCounts allocation m) e | |
| let : MulAction (AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) | |
| (AllFieldHistoryRecovery.Raw (K := K) (tick := tick) allocation m) := AllFieldHistoryRecovery.rawAction allocation m target | |
| have hb (sigma : Placement) : | |
| ((rejectedFilter (GroupOrbitProjection.rawOrbit | |
| (GroupOrbitProjection.groupCounts (activeCounts allocation m) | |
| (fun h : Active K tick => h.val.physicalOrder = sigma)) | |
| (fun h => Left activeHalfLength h.val) (fun h => Right activeHalfLength h.val) | |
| (GroupOrbitProjection.projectTarget (activeCounts allocation m) | |
| (fun h => h.val.physicalOrder = sigma) target) | |
| (projectRaw allocation m sigma w)) | |
| (fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧ | |
| ¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) / | |
| (GroupOrbitProjection.rawOrbit | |
| (GroupOrbitProjection.groupCounts (activeCounts allocation m) | |
| (fun h : Active K tick => h.val.physicalOrder = sigma)) | |
| (fun h => Left activeHalfLength h.val) (fun h => Right activeHalfLength h.val) | |
| (GroupOrbitProjection.projectTarget (activeCounts allocation m) | |
| (fun h => h.val.physicalOrder = sigma) target) | |
| (projectRaw allocation m sigma w)).card ≤ 1 / m := by | |
| change ((rejectedFilter | |
| (localOrbit allocation m sigma (projectTarget allocation m sigma target) | |
| (projectRaw allocation m sigma w)) | |
| (fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧ | |
| ¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) / | |
| (localOrbit allocation m sigma (projectTarget allocation m sigma target) | |
| (projectRaw allocation m sigma w)).card ≤ 1 / m | |
| dsimp only [target] | |
| rw [project_joinGroupTargets] | |
| have hb0 := hbound sigma | |
| rw [filter_eq_rejected] at hb0 | |
| exact hb0 | |
| have hh := GroupOrbitProjection.raw_group_passing_rejection_fraction_le | |
| (activeCounts allocation m) (Left activeHalfLength) (Right activeHalfLength) | |
| (fun h : Active K tick => h.val.physicalOrder) target w | |
| (AllFieldHistoryRecovery.completeKeep allocation m ε side) | |
| (fun sigma => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma) | |
| (fun sigma => localAssigned allocation m sigma (a sigma) (i sigma) side) | |
| (fun v hv sigma => completeKeep_project allocation m ε side v hv sigma) | |
| (fun _ => (1 : ℝ) / m) hb | |
| have hcard : Fintype.card Placement = 6 := by | |
| norm_num [Placement, Fintype.card_perm, Nat.factorial] | |
| have horbit : | |
| OrbitCounting.orbitSet | |
| (G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) w = | |
| GroupOrbitProjection.rawOrbit (activeCounts allocation m) | |
| (Left activeHalfLength) (Right activeHalfLength) target w := by | |
| exact RecoveryOrbitDecisions.orbitSet_eq_transportedOrbit | |
| (G := AllFieldHistoryRecovery.Symmetries (K := K) (tick := tick) allocation m) | |
| _ _ _ (AllFieldHistoryRecovery.coordinates allocation m target) w | |
| rw [filter_eq_rejected, horbit] | |
| have hfilter := rejectedFilter_congr | |
| (GroupOrbitProjection.rawOrbit (activeCounts allocation m) | |
| (Left activeHalfLength) (Right activeHalfLength) target w) | |
| (fun v => AllFieldHistoryRecovery.completeKeep allocation m ε side v ∧ | |
| ¬AllFieldHistoryRecovery.sideVariable side | |
| ((groupedAssignment allocation m E a).x v = some i) | |
| ((groupedAssignment allocation m E a).y v = some i) | |
| ((groupedAssignment allocation m E a).z v = some i)) | |
| (fun v => AllFieldHistoryRecovery.completeKeep allocation m ε side v ∧ | |
| ¬∀ sigma, localAssigned allocation m sigma (a sigma) (i sigma) side | |
| (GroupOrbitProjection.projectRaw (activeCounts allocation m) | |
| (Left activeHalfLength) (Right activeHalfLength) | |
| (fun h : Active K tick => h.val.physicalOrder = sigma) v)) | |
| (fun v => and_congr Iff.rfl | |
| (not_congr (grouped_assigned_iff allocation m E a i side v))) | |
| rw [hfilter] | |
| simpa only [Finset.sum_const, Finset.card_univ, hcard, nsmul_eq_mul, Nat.cast_ofNat, mul_one_div] using hh | |
| theorem repair_grouped (F : Type*) [CommRing F] (allocation : Allocation) | |
| {m : ℕ} {ε : ℝ} (hε : 0 < ε) | |
| (hm : AllFieldHistoryRecovery.minimumDilation (K := K) (tick := tick) allocation ε ≤ m) | |
| (E : Placement → Type*) | |
| (a : ∀ sigma, JointExtraction.Assignment | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (GroupRaw (K := K) (tick := tick) allocation m sigma) (GroupRaw (K := K) (tick := tick) allocation m sigma) (E sigma)) | |
| (i : ∀ sigma, E sigma) (e : ∀ sigma, GroupTarget (K := K) (tick := tick) allocation m sigma) | |
| (hx : ∀ sigma w, (a sigma).x w = some (i sigma) → | |
| groupIdealSide allocation m ε 0 sigma (e sigma) w) | |
| (hy : ∀ sigma w, (a sigma).y w = some (i sigma) → | |
| groupIdealSide allocation m ε 1 sigma (e sigma) w) | |
| (hz : ∀ sigma w, (a sigma).z w = some (i sigma) → | |
| groupIdealSide allocation m ε 2 sigma (e sigma) w) | |
| (hbound : ∀ side w, | |
| AllFieldHistoryRecovery.Used F allocation m ε | |
| (joinGroupTargets (activeCounts allocation m) e) side w → ∀ sigma, | |
| (((localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).filter | |
| (fun v => AllFieldHistoryGroupMasks.completeKeep allocation m ε side sigma v ∧ | |
| ¬localAssigned allocation m sigma (a sigma) (i sigma) side v)).card : ℝ) / | |
| (localOrbit allocation m sigma (e sigma) (projectRaw allocation m sigma w)).card ≤ 1 / m) : | |
| InverseLinearRecovery.RecoveredBy | |
| (AllFieldHistoryRecovery.ideal F allocation m ε (joinGroupTargets (activeCounts allocation m) e)) | |
| (JointExtraction.branch | |
| (ExactRecovery.delete (AllFieldHistoryRecovery.rawSource F allocation m) | |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 0) | |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 1) | |
| (AllFieldHistoryRecovery.completeKeep allocation m ε 2)) | |
| (groupedAssignment allocation m E a) i) | |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m) := by | |
| have hs := groupedAssignment_sound allocation m ε E a i e hx hy hz | |
| apply AllFieldHistoryRecovery.repair_selected_branch F allocation hε hm | |
| (groupedAssignment allocation m E a) i (joinGroupTargets (activeCounts allocation m) e) | |
| hs.1 hs.2.1 hs.2.2 | |
| intro side w hw | |
| exact grouped_collision_le allocation m ε E a i e side w (hbound side w hw) | |
| end MatrixMultiplication.AllFieldHistoryGroupedRecovery | |
| end | |
| end OAI | |