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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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
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