openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialStep.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialSource | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalInitial | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialStep | |
| open MatrixMultiplication.Foundation AllFieldFiniteFamily AllFieldInitialSource | |
| open JointPopulation | |
| open scoped BigOperators Classical | |
| variable {F : Type*} [Field F] {K N : ℕ} | |
| (counts : Fin K → Shape → ℕ) (total : ∀ h, ∑ u, counts h u = N) | |
| def toLots (w : JointCanonicalInitial.RawWords counts) : Words K N := | |
| fun h i => w h ((finCongr (total h)).symm i) | |
| def positionMap : | |
| LocalMap (groupedSource F K N) (JointCanonicalInitial.rawSource counts) where | |
| x := fun x s => if s = toLots counts total x then 1 else 0 | |
| y := fun y s => if s = toLots counts total y then 1 else 0 | |
| z := fun z s => if s = toLots counts total z then 1 else 0 | |
| coefficient := by | |
| rw [← Tensor.pullback_eq_restrict, groupedSource_eq_lots] | |
| funext x y z | |
| change (∏ h : Fin K, ∏ i : Fin N, CWStrands.strand (F := F) (Fin 8) | |
| (x h ((finCongr (total h)).symm i)) | |
| (y h ((finCongr (total h)).symm i)) | |
| (z h ((finCongr (total h)).symm i))) = | |
| ∏ h : Fin K, ∏ i : Positions counts h, CWStrands.strand (F := F) (Fin 8) | |
| (x h i) (y h i) (z h i) | |
| apply Finset.prod_congr rfl | |
| intro h _ | |
| apply Fintype.prod_equiv (finCongr (total h)).symm | |
| intro i | |
| rfl | |
| def initialExecution : | |
| Execution (cwSource F K N) (JointCanonicalInitial.rawSource counts) := | |
| (AllFieldInitialSource.initialExecution F K N).restrict _ (positionMap counts total) | |
| @[simp] theorem initialExecution_copies : | |
| Fintype.card (initialExecution (F := F) counts total).Copies = 1 := rfl | |
| def maskedInitialExecution : | |
| Execution (cwSource F K N) (JointCanonicalInitial.maskedSource counts) := | |
| (initialExecution counts total).restrict _ | |
| (LocalMap.delete (JointCanonicalInitial.rawSource counts) | |
| (JointCanonicalInitial.sideMask counts 0) (JointCanonicalInitial.sideMask counts 1) | |
| (JointCanonicalInitial.sideMask counts 2)) | |
| abbrev CoarseWord := Position counts → Fin 17 | |
| def ordinaryAssignments (hx hy hz : CoarseWord counts → Prop) : | |
| JointExtraction.Assignment (JointCanonicalInitial.RawWords counts) | |
| (JointCanonicalInitial.RawWords counts) (JointCanonicalInitial.RawWords counts) | |
| (JointCoarseHashing.Triple (Position counts)) := | |
| JointCoarseAssignment.assignments (ambientSet counts (fun _ => 16)) (targetSet counts) | |
| hx hy hz (JointCanonicalInitial.coarseWord counts) (JointCanonicalInitial.coarseWord counts) | |
| (JointCanonicalInitial.coarseWord counts) (fun _ _ => True) (fun _ _ => True) | |
| (fun _ _ => True) (fun _ _ => True) (fun _ _ => True) | |
| theorem ordinaryAssignments_coherent (hx hy hz : CoarseWord counts → Prop) : | |
| JointExtraction.Coherent (JointCanonicalInitial.maskedSource (F := F) counts) | |
| (ordinaryAssignments counts hx hy hz) := by | |
| apply JointCoarseAssignment.assignments_coherent | |
| · exact JointCanonicalInitial.maskedSource_support counts | |
| · intro _ _ _ _ _ _ _ | |
| trivial | |
| · intro _ _ _ _ _ _ _ _ | |
| trivial | |
| def initialCanonicalMap (e : Target counts) : | |
| LocalMap (JointCanonicalInitial.ideal (F := F) counts e) | |
| (JointCanonicalInitial.canonical counts) where | |
| x := JointCanonicalInitial.coordinateMatrix counts e | |
| y := JointCanonicalInitial.coordinateMatrix counts e | |
| z := JointCanonicalInitial.coordinateMatrix counts e | |
| coefficient := JointCanonicalInitial.ideal_restrict counts e | |
| def canonicalize {I : Type} [Fintype I] (target : I → Target counts) | |
| (E : Execution (cwSource F K N) | |
| (Tensor.directSum (fun i => JointCanonicalInitial.ideal counts (target i)))) : | |
| Execution (cwSource F K N) | |
| (Tensor.directSum (fun _ : I => JointCanonicalInitial.canonical counts)) := | |
| E.restrict _ (LocalMap.directSum _ _ (fun i => initialCanonicalMap counts (target i))) | |
| @[simp] theorem canonicalize_copies {I : Type} [Fintype I] (target : I → Target counts) | |
| (E : Execution (cwSource F K N) | |
| (Tensor.directSum (fun i => JointCanonicalInitial.ideal counts (target i)))) : | |
| Fintype.card (canonicalize counts target E).Copies = Fintype.card E.Copies := rfl | |
| def repairedInitialExecution {I : Type} [Fintype I] | |
| (hx hy hz : CoarseWord counts → Prop) (target : I → Target counts) | |
| (injective : Function.Injective target) (shifts : ℕ) | |
| (repair : ∀ i, InverseLinearRecovery.RecoveredBy | |
| (JointCanonicalInitial.ideal (F := F) counts (target i)) | |
| (JointExtraction.branch (JointCanonicalInitial.maskedSource counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts (target i))) shifts) : | |
| Execution (cwSource F K N) | |
| (Tensor.directSum (fun _ : I => JointCanonicalInitial.canonical counts)) := | |
| canonicalize counts target | |
| ((maskedInitialExecution counts total).extractSelectedAndRepair | |
| (ordinaryAssignments counts hx hy hz) (ordinaryAssignments_coherent counts hx hy hz) | |
| (fun i => triple counts (target i)) ((triple_injective counts).comp injective) | |
| (fun i => JointCanonicalInitial.ideal counts (target i)) shifts repair) | |
| @[simp] theorem repairedInitialExecution_copies {I : Type} [Fintype I] | |
| (hx hy hz : CoarseWord counts → Prop) (target : I → Target counts) | |
| (injective : Function.Injective target) (shifts : ℕ) | |
| (repair : ∀ i, InverseLinearRecovery.RecoveredBy | |
| (JointCanonicalInitial.ideal (F := F) counts (target i)) | |
| (JointExtraction.branch (JointCanonicalInitial.maskedSource counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts (target i))) shifts) : | |
| Fintype.card | |
| (repairedInitialExecution counts total hx hy hz target injective shifts repair).Copies = | |
| 2 ^ (3 * shifts) := by | |
| change Fintype.card (InverseLinearRecovery.MaskRectangles shifts × PUnit) = _ | |
| rw [Fintype.card_prod, ExactRecovery.card_mask_rectangles, Fintype.card_fin, | |
| Fintype.card_punit] | |
| exact Nat.mul_one _ | |
| end MatrixMultiplication.AllFieldInitialStep | |
| end | |
| end OAI | |