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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
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