openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /Compatibility.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.NativeLaws | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| 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 OAI | |