openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /IdealBranches.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CoarseAssignment | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.ExactRecovery | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointIdealBranches | |
| open MatrixMultiplication.Foundation JointExtraction JointCoarseAssignment | |
| attribute [local instance] Classical.propDecidable | |
| variable {F X Y Z A B C : Type*} [CommSemiring F] | |
| def ideal (T : Tensor F X Y Z) | |
| (cx : X → A) (cy : Y → B) (cz : Z → C) | |
| (usefulX : Triple A B C → X → Prop) | |
| (usefulY : Triple A B C → Y → Prop) | |
| (usefulZ : Triple A B C → Z → Prop) (e : Triple A B C) : Tensor F X Y Z := | |
| ExactRecovery.delete T (fun x => cx x = e.1 ∧ usefulX e x) | |
| (fun y => cy y = e.2.1 ∧ usefulY e y) | |
| (fun z => cz z = e.2.2 ∧ usefulZ e z) | |
| theorem branch_eq_delete_ideal | |
| (T : Tensor F X Y Z) (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (cx : X → A) (cy : Y → B) (cz : Z → C) | |
| (compatibleY : Triple A B C → Y → Prop) | |
| (compatibleZ : Triple A B C → Z → Prop) | |
| (usefulX : Triple A B C → X → Prop) | |
| (usefulY : Triple A B C → Y → Prop) | |
| (usefulZ : Triple A B C → Z → Prop) (e : Triple A B C) : | |
| let a := assignments ambient targets hx hy hz cx cy cz | |
| compatibleY compatibleZ usefulX usefulY usefulZ | |
| branch T a e = ExactRecovery.delete (ideal T cx cy cz usefulX usefulY usefulZ e) | |
| (fun x => a.x x = some e) (fun y => a.y y = some e) (fun z => a.z z = some e) := by | |
| classical | |
| dsimp only | |
| funext x y z | |
| by_cases h : | |
| (assignments ambient targets hx hy hz cx cy cz compatibleY compatibleZ | |
| usefulX usefulY usefulZ).x x = some e ∧ | |
| (assignments ambient targets hx hy hz cx cy cz compatibleY compatibleZ | |
| usefulX usefulY usefulZ).y y = some e ∧ | |
| (assignments ambient targets hx hy hz cx cy cz compatibleY compatibleZ | |
| usefulX usefulY usefulZ).z z = some e | |
| · have hX := assignUseful_spec (eligible ambient targets hx hy hz) | |
| (fun e x => cx x = e.1) usefulX x e h.1 | |
| have hY := assignUseful_spec (eligible ambient targets hx hy hz) | |
| (fun e y => cy y = e.2.1 ∧ compatibleY e y) usefulY y e h.2.1 | |
| have hZ := assignUseful_spec (eligible ambient targets hx hy hz) | |
| (fun e z => cz z = e.2.2 ∧ compatibleZ e z) usefulZ z e h.2.2 | |
| simp [branch, ExactRecovery.delete, ideal, h, hX.2.1, hX.2.2.1, | |
| hY.2.1.1, hY.2.2.1, hZ.2.1.1, hZ.2.2.1] | |
| · simp [branch, ExactRecovery.delete, h] | |
| theorem ideal_delete (T : Tensor F X Y Z) | |
| (px : X → Prop) (py : Y → Prop) (pz : Z → Prop) | |
| (cx : X → A) (cy : Y → B) (cz : Z → C) | |
| (usefulX : Triple A B C → X → Prop) | |
| (usefulY : Triple A B C → Y → Prop) | |
| (usefulZ : Triple A B C → Z → Prop) (e : Triple A B C) : | |
| ideal (ExactRecovery.delete T px py pz) cx cy cz usefulX usefulY usefulZ e = | |
| ExactRecovery.delete (ideal T cx cy cz usefulX usefulY usefulZ e) px py pz := by | |
| funext x y z | |
| simp only [ideal, ExactRecovery.delete] | |
| split_ifs <;> simp_all | |
| theorem branch_masked_eq_delete_ideal | |
| (T : Tensor F X Y Z) (px : X → Prop) (py : Y → Prop) (pz : Z → Prop) | |
| (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (cx : X → A) (cy : Y → B) (cz : Z → C) | |
| (compatibleY : Triple A B C → Y → Prop) | |
| (compatibleZ : Triple A B C → Z → Prop) | |
| (usefulX : Triple A B C → X → Prop) | |
| (usefulY : Triple A B C → Y → Prop) | |
| (usefulZ : Triple A B C → Z → Prop) (e : Triple A B C) : | |
| let a := assignments ambient targets hx hy hz cx cy cz | |
| compatibleY compatibleZ usefulX usefulY usefulZ | |
| branch (ExactRecovery.delete T px py pz) a e = | |
| ExactRecovery.delete (ideal T cx cy cz usefulX usefulY usefulZ e) | |
| (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) := by | |
| dsimp only | |
| rw [branch_eq_delete_ideal, ideal_delete] | |
| funext x y z | |
| simp only [ExactRecovery.delete] | |
| split_ifs <;> simp_all | |
| theorem masked_source_restriction [Fintype X] [Fintype Y] [Fintype Z] | |
| [DecidableEq A] [DecidableEq B] [DecidableEq C] | |
| (T : Tensor F X Y Z) (a : Assignment X Y Z (Triple A B C)) | |
| (hcoherent : Coherent T a) (px : X → Prop) (py : Y → Prop) (pz : Z → Prop) : | |
| Tensor.restrict (assignmentMatrix a.x) (assignmentMatrix a.y) (assignmentMatrix a.z) | |
| (ExactRecovery.delete T px py pz) = | |
| Tensor.directSum (branch (ExactRecovery.delete T px py pz) a) := by | |
| apply restrict_assignments_eq_directSum | |
| apply coherent_of_support_subset T _ a hcoherent | |
| intro x y z h ht | |
| exact h (by simp [ExactRecovery.delete, ht]) | |
| end MatrixMultiplication.JointIdealBranches | |
| end | |
| end OAI | |