File size: 3,124 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
|