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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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Laws
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionLaws
open MatrixMultiplication.Foundation
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {A X : Type*} [Fintype A] [Fintype X]
theorem sum_subtype_indicator (P : A → Prop) (f : A → ℝ) :
(∑ a : {a // P a}, f a.val) = ∑ a, if P a then f a else 0 := by
rw [← Finset.sum_filter]
exact (Finset.sum_subtype _ (by simp) f).symm
def eventMass (p : FiniteLaw A) (P : A → Prop) : ℝ :=
∑ a : {a // P a}, p.mass a.val
theorem eventMass_nonneg (p : FiniteLaw A) (P : A → Prop) : 0 ≤ eventMass p P :=
Finset.sum_nonneg fun a _ => p.nonneg a.val
def condition (p : FiniteLaw A) (P : A → Prop) (h : 0 < eventMass p P) :
FiniteLaw {a // P a} where
mass a := p.mass a.val / eventMass p P
nonneg a := div_nonneg (p.nonneg a.val) h.le
total := by rw [← Finset.sum_div]; exact div_self h.ne'
theorem condition_mass (p : FiniteLaw A) (P : A → Prop)
(h : 0 < eventMass p P) (a : {a // P a}) :
(condition p P h).mass a = p.mass a.val / eventMass p P := rfl
theorem condition_coordinate_mass (p : FiniteLaw A) (f : A → X) (Q : X → Prop)
(h : 0 < eventMass p (fun a => Q (f a))) (x : X) :
((condition p (fun a => Q (f a)) h).map (fun a => f a.val)).mass x =
if Q x then (p.map f).mass x / eventMass p (fun a => Q (f a)) else 0 := by
rw [FiniteLaw.map_mass]
simp only [condition_mass]
have hsum :
(∑ a : {a // Q (f a)}, if f a.val = x then
p.mass a.val / eventMass p (fun a => Q (f a)) else 0) =
∑ a, if Q (f a) then
(if f a = x then p.mass a / eventMass p (fun a => Q (f a)) else 0) else 0 :=
sum_subtype_indicator (fun a => Q (f a))
(fun a => if f a = x then p.mass a / eventMass p (fun a => Q (f a)) else 0)
rw [hsum]
by_cases hx : Q x
· rw [ite_eq_left hx, FiniteLaw.map_mass, Finset.sum_div]
apply Finset.sum_congr rfl
intro a _
by_cases ha : f a = x
· simp [ha, hx]
· simp [ha]
· rw [ite_eq_right hx]
apply Finset.sum_eq_zero
intro a _
by_cases ha : f a = x
· simp [ha, hx]
· simp [ha]
theorem eventMass_coordinate (p : FiniteLaw A) (f : A → X) (Q : X → Prop) :
eventMass p (fun a => Q (f a)) = eventMass (p.map f) Q := by
rw [eventMass, eventMass, sum_subtype_indicator, sum_subtype_indicator]
simp only [FiniteLaw.map_mass]
calc
(∑ a, if Q (f a) then p.mass a else 0) =
∑ a, ∑ x, if Q x then (if f a = x then p.mass a else 0) else 0 := by
apply Finset.sum_congr rfl
intro a _
rw [Finset.sum_eq_single (f a)]
· simp
· intro x _ hne
simp [Ne.symm hne]
· simp
_ = ∑ x, ∑ a, if Q x then (if f a = x then p.mass a else 0) else 0 :=
Finset.sum_comm
_ = ∑ x, if Q x then ∑ a, if f a = x then p.mass a else 0 else 0 := by
apply Finset.sum_congr rfl
intro x _
split_ifs <;> simp
def transport {B : Type*} [Fintype B] (e : A ≃ B) (p : FiniteLaw A) : FiniteLaw B :=
p.map e
theorem transport_mass {B : Type*} [Fintype B] (e : A ≃ B) (p : FiniteLaw A) (a : A) :
(transport e p).mass (e a) = p.mass a := p.map_mass_apply e e.injective a
theorem transport_entropy {B : Type*} [Fintype B] (e : A ≃ B) (p : FiniteLaw A) :
finiteEntropy (transport e p).mass = finiteEntropy p.mass :=
p.map_entropy_of_injective e e.injective
end MatrixMultiplication.CompletionLaws
end
end OAI
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