openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialExact.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialStep | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AssignmentLoss | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldInitialExact | |
| open MatrixMultiplication.Foundation AllFieldFiniteFamily AllFieldInitialStep | |
| open JointPopulation JointCoarseAssignment JointExtraction JointAssignmentLoss | |
| open scoped BigOperators Classical | |
| attribute [local instance] Classical.propDecidable | |
| private def assignedDeletion {F X Y Z A B C : Type*} [CommSemiring F] | |
| (T : Tensor F X Y Z) (a : Assignment X Y Z (Triple A B C)) (e : Triple A B C) | |
| (px : X → Prop) (py : Y → Prop) (pz : Z → Prop) : Tensor F X Y Z := | |
| ExactRecovery.delete T (fun x => a.x x = some e ∧ px x) | |
| (fun y => a.y y = some e ∧ py y) (fun z => a.z z = some e ∧ pz z) | |
| private theorem assignedDeletion_eq_self_of_support {F X Y Z A B C : Type*} | |
| [CommSemiring F] (T : Tensor F X Y Z) | |
| (a : Assignment X Y Z (Triple A B C)) (e : Triple A B C) | |
| (px : X → Prop) (py : Y → Prop) (pz : Z → Prop) | |
| (support : ∀ x y z, T x y z ≠ 0 → | |
| (a.x x = some e ∧ px x) ∧ (a.y y = some e ∧ py y) ∧ (a.z z = some e ∧ pz z)) : | |
| assignedDeletion T a e px py pz = T := by | |
| funext x y z | |
| unfold assignedDeletion ExactRecovery.delete | |
| by_cases ht : T x y z = 0 | |
| · simp only [ht, ite_self] | |
| · exact ite_eq_left (support x y z ht) | |
| variable {K N : ℕ} | |
| (counts : Fin K → Shape → ℕ) | |
| (hx hy hz : CoarseWord counts → Prop) | |
| def Isolated (e : Target counts) : Prop := | |
| survives hx hy hz (triple counts e) ∧ | |
| ∀ f ∈ ambientSet counts (fun _ => 16), survives hx hy hz f → | |
| JointCoarseHashing.SharesSide (triple counts e) f → f = triple counts e | |
| section Coefficients | |
| variable {F : Type*} [CommRing F] | |
| theorem assignments_on_ideal (e : Target counts) | |
| (ha : triple counts e ∈ ambientSet counts (fun _ => 16)) | |
| (isolated : Isolated counts hx hy hz e) | |
| (x y z : JointCanonicalInitial.RawWords counts) | |
| (nonzero : JointCanonicalInitial.ideal (F := F) counts e x y z ≠ 0) : | |
| (ordinaryAssignments counts hx hy hz).x x = some (triple counts e) ∧ | |
| (ordinaryAssignments counts hx hy hz).y y = some (triple counts e) ∧ | |
| (ordinaryAssignments counts hx hy hz).z z = some (triple counts e) := by | |
| have hc : JointCanonicalInitial.coarseWord counts x = (triple counts e).1 ∧ | |
| JointCanonicalInitial.coarseWord counts y = (triple counts e).2.1 ∧ | |
| JointCanonicalInitial.coarseWord counts z = (triple counts e).2.2 := by | |
| by_contra hn | |
| simp [JointCanonicalInitial.ideal, JointIdealBranches.ideal, ExactRecovery.delete, hn] | |
| at nonzero | |
| have ht : triple counts e ∈ targetSet counts := | |
| Finset.mem_image.mpr ⟨e, Finset.mem_univ _, rfl⟩ | |
| have he : triple counts e ∈ eligible (ambientSet counts (fun _ => 16)) | |
| (targetSet counts) hx hy hz := by | |
| apply (mem_eligible_iff _ _ _ _ _ _).2 | |
| exact ⟨ha, ht, isolated.1, fun f hf hs hfx => isolated.2 f hf hs (Or.inl hfx)⟩ | |
| constructor | |
| · apply assignUseful_of_unique _ _ _ _ _ he hc.1 trivial | |
| intro f hf hfx | |
| have hm := (mem_eligible_iff _ _ _ _ _ _).1 hf | |
| exact isolated.2 f hm.1 hm.2.2.1 (Or.inl (hfx.symm.trans hc.1)) | |
| constructor | |
| · apply assignUseful_of_unique _ _ _ _ _ he ⟨hc.2.1, trivial⟩ trivial | |
| intro f hf hfy | |
| have hm := (mem_eligible_iff _ _ _ _ _ _).1 hf | |
| exact isolated.2 f hm.1 hm.2.2.1 (Or.inr (Or.inl (hfy.1.symm.trans hc.2.1))) | |
| · apply assignUseful_of_unique _ _ _ _ _ he ⟨hc.2.2, trivial⟩ trivial | |
| intro f hf hfz | |
| have hm := (mem_eligible_iff _ _ _ _ _ _).1 hf | |
| exact isolated.2 f hm.1 hm.2.2.1 (Or.inr (Or.inr (hfz.1.symm.trans hc.2.2))) | |
| private theorem branch_eq_masked_ideal (e : Target counts) : | |
| branch (JointCanonicalInitial.maskedSource (F := F) counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts e) = | |
| assignedDeletion (JointCanonicalInitial.ideal counts e) | |
| (ordinaryAssignments counts hx hy hz) (triple counts e) | |
| (JointCanonicalInitial.sideMask counts 0) (JointCanonicalInitial.sideMask counts 1) | |
| (JointCanonicalInitial.sideMask counts 2) := | |
| JointIdealBranches.branch_masked_eq_delete_ideal | |
| (JointCanonicalInitial.rawSource (F := F) counts) | |
| (JointCanonicalInitial.sideMask counts 0) (JointCanonicalInitial.sideMask counts 1) | |
| (JointCanonicalInitial.sideMask counts 2) (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) (triple counts e) | |
| theorem branch_eq_ideal (e : Target counts) | |
| (ha : triple counts e ∈ ambientSet counts (fun _ => 16)) | |
| (isolated : Isolated counts hx hy hz e) : | |
| branch (JointCanonicalInitial.maskedSource (F := F) counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts e) = | |
| JointCanonicalInitial.ideal counts e := by | |
| have hdelete : assignedDeletion (JointCanonicalInitial.ideal (F := F) counts e) | |
| (ordinaryAssignments counts hx hy hz) (triple counts e) | |
| (JointCanonicalInitial.sideMask counts 0) (JointCanonicalInitial.sideMask counts 1) | |
| (JointCanonicalInitial.sideMask counts 2) = | |
| JointCanonicalInitial.ideal counts e := by | |
| apply assignedDeletion_eq_self_of_support | |
| intro x y z hh | |
| have hm := JointCanonicalInitial.ideal_support_masks counts e x y z hh | |
| have ha := assignments_on_ideal counts hx hy hz e ha isolated x y z hh | |
| exact ⟨⟨ha.1, hm.1⟩, ⟨ha.2.1, hm.2.1⟩, ⟨ha.2.2, hm.2.2⟩⟩ | |
| exact (branch_eq_masked_ideal counts hx hy hz e).trans hdelete | |
| end Coefficients | |
| section Execution | |
| variable {F : Type*} [Field F] | |
| def selectedExecution (total : ∀ h, ∑ u, counts h u = N) | |
| {I : Type} [Fintype I] (target : I → Target counts) | |
| (injective : Function.Injective target) | |
| (ambient : ∀ i, triple counts (target i) ∈ ambientSet counts (fun _ => 16)) | |
| (isolated : ∀ i, Isolated counts hx hy hz (target i)) : | |
| Execution (cwSource F K N) | |
| (Tensor.directSum (fun _ : I => JointCanonicalInitial.canonical counts)) := by | |
| let input := maskedInitialExecution (F := F) counts total | |
| let extracted := input.restrict _ | |
| (LocalMap.extract (JointCanonicalInitial.maskedSource counts) | |
| (ordinaryAssignments counts hx hy hz) (ordinaryAssignments_coherent counts hx hy hz)) | |
| let selected := extracted.restrict _ | |
| (LocalMap.selectBranches | |
| (branch (JointCanonicalInitial.maskedSource counts) (ordinaryAssignments counts hx hy hz)) | |
| (fun i => triple counts (target i)) ((triple_injective counts).comp injective)) | |
| have heq : Tensor.directSum (fun i => branch (JointCanonicalInitial.maskedSource (F := F) counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts (target i))) = | |
| Tensor.directSum (fun i => JointCanonicalInitial.ideal counts (target i)) := by | |
| apply congrArg Tensor.directSum | |
| funext i | |
| exact branch_eq_ideal counts hx hy hz (target i) (ambient i) (isolated i) | |
| let idMap := LocalMap.identity (Tensor.directSum (fun i => | |
| branch (JointCanonicalInitial.maskedSource (F := F) counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts (target i)))) | |
| let sameCoordinates : LocalMap | |
| (Tensor.directSum (fun i => branch (JointCanonicalInitial.maskedSource (F := F) counts) | |
| (ordinaryAssignments counts hx hy hz) (triple counts (target i)))) | |
| (Tensor.directSum (fun i => JointCanonicalInitial.ideal counts (target i))) := | |
| { x := idMap.x | |
| y := idMap.y | |
| z := idMap.z | |
| coefficient := idMap.coefficient.trans heq } | |
| exact AllFieldInitialStep.canonicalize counts target (selected.restrict _ sameCoordinates) | |
| @[simp] theorem selectedExecution_copies (total : ∀ h, ∑ u, counts h u = N) | |
| {I : Type} [Fintype I] (target : I → Target counts) | |
| (injective : Function.Injective target) | |
| (ambient : ∀ i, triple counts (target i) ∈ ambientSet counts (fun _ => 16)) | |
| (isolated : ∀ i, Isolated counts hx hy hz (target i)) : | |
| Fintype.card (selectedExecution (F := F) counts hx hy hz total target | |
| injective ambient isolated).Copies = 1 := rfl | |
| end Execution | |
| end MatrixMultiplication.AllFieldInitialExact | |
| end | |
| end OAI | |