openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /OrbitTransport.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupedRecovery | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.OrbitFibers | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupOrbitData | |
| open AllFieldHistory AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery | |
| open AllFieldGroupOrbitCount JointCoarseHashing PermutationMatching | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| local instance canonicalGroupAction (allocation : Allocation) (m : ℕ) (sigma : Placement) : | |
| MulAction | |
| (HalfClassPermutations (GroupPositions (K := K) (tick := tick) (sigma := sigma) allocation m)) | |
| (JointCanonicalization.CanonicalPairs (groupCounts (K := K) (tick := tick) allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7)) := | |
| completeWordPairMulAction | |
| local instance canonicalGroupFintype (allocation : Allocation) (m : ℕ) (sigma : Placement) : | |
| Fintype | |
| (JointCanonicalization.CanonicalPairs (groupCounts (K := K) (tick := tick) allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7)) := by | |
| letI : ∀ c : GroupClasses K tick sigma, | |
| Fintype (GroupPositions allocation m c) := fun _ => Fin.fintype _ | |
| letI : ∀ c : GroupClasses K tick sigma, | |
| DecidableEq (GroupPositions allocation m c) := fun _ => Classical.decEq _ | |
| letI : ∀ c : GroupClasses K tick sigma, | |
| Fintype (GroupLetters c) := fun _ => Pi.instFintype | |
| exact HistorySymmetry.completeWordPairFintype | |
| def tag (allocation : Allocation) (m : ℕ) (sigma : Placement) (side : Fin 3) : | |
| Raw (K := K) (tick := tick) allocation m sigma ↪ Variable (K := K) (tick := tick) allocation m sigma where | |
| toFun w := (side, w) | |
| inj' := by intro w v h; exact congrArg Prod.snd h | |
| theorem full_eq_tagged_fiber (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) (o : Orbit (K := K) (tick := tick) allocation m sigma) : | |
| full allocation m sigma e o = | |
| (JointOrbitFibers.transportedFiber (groupCoordinates allocation m sigma e) o.2).map | |
| (tag allocation m sigma o.1) := by | |
| ext v | |
| constructor | |
| · intro hv | |
| have hh := (Finset.mem_filter.mp hv).2 | |
| apply Finset.mem_map.mpr | |
| refine ⟨v.2, ?_, ?_⟩ | |
| · exact (JointOrbitFibers.mem_transportedFiber (groupCoordinates allocation m sigma e)).mpr hh.2 | |
| · exact Prod.ext hh.1.symm rfl | |
| · intro hv | |
| obtain ⟨w, hw, rfl⟩ := Finset.mem_map.mp hv | |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, rfl, | |
| (JointOrbitFibers.mem_transportedFiber (groupCoordinates allocation m sigma e)).mp hw⟩ | |
| theorem full_eq_tagged_localOrbit (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) : | |
| full allocation m sigma e (side, orbitOf allocation m sigma e w) = | |
| (localOrbit allocation m sigma e w).map (tag allocation m sigma side) := by | |
| rw [full_eq_tagged_fiber] | |
| dsimp only [orbitOf] | |
| erw [JointOrbitFibers.transportedFiber_mk] | |
| apply congrArg (fun t : Finset (Raw (K := K) (tick := tick) allocation m sigma) => | |
| t.map (tag allocation m sigma side)) | |
| ext v | |
| simp only [localOrbit, GroupOrbitProjection.rawOrbit, GroupOrbitProjection.transportedOrbit, | |
| OrbitCounting.orbitSet, | |
| Finset.mem_image, Finset.mem_univ, true_and] | |
| rfl | |
| @[simp] theorem full_card_localOrbit (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) : | |
| (full allocation m sigma e (side, orbitOf allocation m sigma e w)).card = | |
| (localOrbit allocation m sigma e w).card := by | |
| rw [full_eq_tagged_localOrbit, Finset.card_map] | |
| theorem full_filter_card_localOrbit (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) (bad : Variable (K := K) (tick := tick) allocation m sigma → Prop) : | |
| ((full allocation m sigma e (side, orbitOf allocation m sigma e w)).filter bad).card = | |
| ((localOrbit allocation m sigma e w).filter (fun v => bad (side, v))).card := by | |
| rw [full_eq_tagged_localOrbit, Finset.filter_map] | |
| exact Finset.card_map _ | |
| end MatrixMultiplication.AllFieldGroupOrbitData | |
| end | |
| end OAI | |