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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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.UniformCoordinate
namespace OAI
/-! Finite entropy, rate estimates and ordered asymptotic limits. -/
noncomputable section
namespace MatrixMultiplication.CompletionLaws
open MatrixMultiplication.Foundation
open scoped BigOperators
attribute [local instance 10000] Classical.propDecidable Classical.decEq
variable {A B X : Type*} [Fintype A] [Fintype B] [Fintype X]
theorem condition_map_mass (p : FiniteLaw A) (P : A → Prop)
(h : 0 < eventMass p P) (f : A → X) (x : X) :
((condition p P h).map (fun a => f a.val)).mass x =
(∑ a : {a // P a}, if f a.val = x then p.mass a.val else 0) / eventMass p P := by
rw [FiniteLaw.map_mass, Finset.sum_div]
apply Finset.sum_congr rfl
intro a _
rw [condition_mass]
split_ifs <;> simp
theorem joint_event_sum (p : FiniteLaw A) (f : A → X) (g : A → B)
(Q : B → Prop) (x : X) :
(∑ a : {a // Q (g a)}, if f a.val = x then p.mass a.val else 0) =
∑ b : {b // Q b}, (p.map (fun a => (f a, g a))).mass (x, b.val) := by
rw [sum_subtype_indicator (fun a => Q (g a))
(fun a => if f a = x then p.mass a else 0),
sum_subtype_indicator Q
(fun b => (p.map (fun a => (f a, g a))).mass (x, b))]
simp only [FiniteLaw.map_mass, Prod.mk.injEq]
calc
(∑ a, if Q (g a) then (if f a = x then p.mass a else 0) else 0) =
∑ a, ∑ b, if Q b then (if f a = x ∧ g a = b then p.mass a else 0) else 0 := by
apply Finset.sum_congr rfl
intro a _
rw [Finset.sum_eq_single (g a)]
· simp
· intro b _ hb
simp [Ne.symm hb]
· simp
_ = ∑ b, ∑ a, if Q b then (if f a = x ∧ g a = b then p.mass a else 0) else 0 :=
Finset.sum_comm
_ = ∑ b, if Q b then ∑ a, if f a = x ∧ g a = b then p.mass a else 0 else 0 := by
apply Finset.sum_congr rfl
intro b _
split_ifs <;> simp
theorem condition_independent_coordinate_mass (p : FiniteLaw A)
(f : A → X) (g : A → B) (Q : B → Prop)
(independent : ∀ x b, (p.map (fun a => (f a, g a))).mass (x, b) =
(p.map f).mass x * (p.map g).mass b)
(h : 0 < eventMass p (fun a => Q (g a))) (x : X) :
((condition p (fun a => Q (g a)) h).map (fun a => f a.val)).mass x =
(p.map f).mass x := by
rw [condition_map_mass, joint_event_sum]
simp_rw [independent]
rw [← Finset.mul_sum]
change ((p.map f).mass x * eventMass (p.map g) Q) /
eventMass p (fun a => Q (g a)) = _
rw [← eventMass_coordinate p g Q, mul_div_cancel_right₀ _ h.ne']
theorem uniform_coordinate_independent_condition (p : FiniteLaw A)
(f : A → X) (g : A → B) (P : X → Prop) (Q : B → Prop)
(uniform : UniformCoordinate p f P)
(independent : ∀ x b, (p.map (fun a => (f a, g a))).mass (x, b) =
(p.map f).mass x * (p.map g).mass b)
(h : 0 < eventMass p (fun a => Q (g a))) :
UniformCoordinate (condition p (fun a => Q (g a)) h)
(fun a => f a.val) P := by
intro x
rw [condition_independent_coordinate_mass p f g Q independent h]
exact uniform x
end MatrixMultiplication.CompletionLaws
end
end OAI
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