openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /InformationConditioning.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Conditioning | |
| 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 R L X : Type*} [Fintype A] [Fintype R] [Fintype L] [Fintype X] | |
| theorem conditionalMassFactorization_condition (p : FiniteLaw A) | |
| (prior : A → R) (label : A → L) (x : A → X) (Q : R → Prop) | |
| (h : 0 < eventMass p (fun a => Q (prior a))) | |
| (factor : ConditionalMassFactorization p prior label x) : | |
| ConditionalMassFactorization (condition p (fun a => Q (prior a)) h) | |
| (fun a => prior a.val) (fun a => label a.val) (fun a => x a.val) := by | |
| intro r l xx | |
| have hj := condition_coordinate_mass p (fun a => (prior a, (label a, x a))) | |
| (fun v => Q v.1) h (r, (l, xx)) | |
| have hr := condition_coordinate_mass p prior Q h r | |
| have hl := condition_coordinate_mass p (fun a => (prior a, label a)) | |
| (fun v => Q v.1) h (r, l) | |
| have hx := condition_coordinate_mass p (fun a => (prior a, x a)) | |
| (fun v => Q v.1) h (r, xx) | |
| change | |
| ((condition p (fun a => Q (prior a)) h).map | |
| (fun a => (prior a.val, (label a.val, x a.val)))).mass (r, (l, xx)) * | |
| ((condition p (fun a => Q (prior a)) h).map (fun a => prior a.val)).mass r = | |
| ((condition p (fun a => Q (prior a)) h).map | |
| (fun a => (prior a.val, label a.val))).mass (r, l) * | |
| ((condition p (fun a => Q (prior a)) h).map | |
| (fun a => (prior a.val, x a.val))).mass (r, xx) | |
| rw [hj, hr, hl, hx] | |
| by_cases hQ : Q r | |
| · simp only [hQ, ite_true] | |
| rw [div_mul_div_comm, div_mul_div_comm, factor r l xx] | |
| · simp only [hQ, ite_false, zero_mul] | |
| theorem conditionalIndependent_condition_of_mass_factorization (p : FiniteLaw A) | |
| (prior : A → R) (label : A → L) (x : A → X) (Q : R → Prop) | |
| (h : 0 < eventMass p (fun a => Q (prior a))) | |
| (factor : ConditionalMassFactorization p prior label x) : | |
| ConditionalIndependent (condition p (fun a => Q (prior a)) h) | |
| (fun a => prior a.val) (fun a => label a.val) (fun a => x a.val) := | |
| conditionalIndependent_of_mass_factorization _ _ _ _ | |
| (conditionalMassFactorization_condition p prior label x Q h factor) | |
| theorem conditionalIndependent_condition (p : FiniteLaw A) | |
| (prior : A → R) (label : A → L) (x : A → X) (Q : R → Prop) | |
| (h : 0 < eventMass p (fun a => Q (prior a))) | |
| (independent : ConditionalIndependent p prior label x) : | |
| ConditionalIndependent (condition p (fun a => Q (prior a)) h) | |
| (fun a => prior a.val) (fun a => label a.val) (fun a => x a.val) := | |
| conditionalIndependent_condition_of_mass_factorization p prior label x Q h | |
| (mass_factorization_of_conditionalIndependent p prior label x independent) | |
| end MatrixMultiplication.DualInformation | |
| end | |
| end OAI | |