openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /InformationLabelConditioning.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.IndependentConditioning | |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.Information | |
| namespace OAI | |
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.DualInformation | |
| open MatrixMultiplication.Foundation CompletionLaws | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {A L X : Type*} [Fintype A] [Fintype L] [Fintype X] | |
| theorem mass_factorization_const_iff (p : FiniteLaw A) | |
| (label : A → L) (x : A → X) : | |
| ConditionalMassFactorization p (fun _ => PUnit.unit) label x ↔ | |
| ∀ l xx, (p.map (fun a => (label a, x a))).mass (l, xx) = | |
| (p.map label).mass l * (p.map x).mass xx := by | |
| classical | |
| constructor | |
| · intro h l xx | |
| have hf := h PUnit.unit l xx | |
| simpa [FiniteLaw.map_mass, Prod.mk.injEq, p.total] using hf | |
| · intro h r l xx | |
| cases r | |
| simpa [FiniteLaw.map_mass, Prod.mk.injEq, p.total] using h l xx | |
| theorem conditionalMassFactorization_condition_label (p : FiniteLaw A) | |
| (label : A → L) (x : A → X) (Q : L → Prop) | |
| (h : 0 < eventMass p (fun a => Q (label a))) | |
| (factor : ConditionalMassFactorization p (fun _ => PUnit.unit) label x) : | |
| ConditionalMassFactorization (condition p (fun a => Q (label a)) h) | |
| (fun _ => PUnit.unit) (fun a => label a.val) (fun a => x a.val) := by | |
| have independent := (mass_factorization_const_iff p label x).mp factor | |
| have swapped : ∀ xx l, | |
| (p.map (fun a => (x a, label a))).mass (xx, l) = | |
| (p.map x).mass xx * (p.map label).mass l := by | |
| intro xx l | |
| have hm : (p.map (fun a => (x a, label a))).mass (xx, l) = | |
| (p.map (fun a => (label a, x a))).mass (l, xx) := by | |
| simp only [FiniteLaw.map_mass, Prod.mk.injEq, and_comm] | |
| rw [hm, independent l xx, mul_comm] | |
| apply (mass_factorization_const_iff _ _ _).mpr | |
| intro l xx | |
| have hj := condition_coordinate_mass p (fun a => (label a, x a)) | |
| (fun v => Q v.1) h (l, xx) | |
| have hl := condition_coordinate_mass p label Q h l | |
| have hx := condition_independent_coordinate_mass p x label Q swapped h xx | |
| rw [hj, hl, hx] | |
| by_cases hQ : Q l | |
| · simp only [hQ, ite_true] | |
| rw [independent l xx] | |
| ring | |
| · simp only [hQ, ite_false, zero_mul] | |
| end MatrixMultiplication.DualInformation | |
| end | |
| end OAI | |