| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AdmissibilityCore | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupedRecovery | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| 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 OAI | |