File size: 3,909 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Conditioning
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionLaws
open MatrixMultiplication.Foundation
open scoped BigOperators
attribute [local instance 10000] Classical.propDecidable Classical.decEq
variable {A : Type*} [Fintype A]
theorem entropy_scaled_law (p : FiniteLaw A) (s : ℝ) :
finiteEntropy (fun a => s * p.mass a) = entropyTerm s + s * finiteEntropy p.mass := by
simp only [finiteEntropy, entropyTerm_mul, Finset.sum_add_distrib,
← Finset.sum_mul, ← Finset.mul_sum, p.total, one_mul]
theorem mass_eq_event_mul_condition (p : FiniteLaw A) (P : A → Prop)
(h : 0 < eventMass p P) (a : {a // P a}) :
p.mass a.val = eventMass p P * (condition p P h).mass a := by
rw [condition_mass, mul_div_cancel₀ _ h.ne']
theorem event_entropy (p : FiniteLaw A) (P : A → Prop)
(h : 0 < eventMass p P) :
(∑ a : {a // P a}, entropyTerm (p.mass a.val)) =
entropyTerm (eventMass p P) + eventMass p P * finiteEntropy (condition p P h).mass := by
conv_lhs => enter [2, a]; rw [mass_eq_event_mul_condition p P h a]
exact entropy_scaled_law (condition p P h) (eventMass p P)
theorem condition_entropy_mul (p : FiniteLaw A) (P : A → Prop)
(h : 0 < eventMass p P) :
eventMass p P * finiteEntropy (condition p P h).mass =
(∑ a : {a // P a}, entropyTerm (p.mass a.val)) - entropyTerm (eventMass p P) := by
linarith [event_entropy p P h]
theorem eventMass_add_complement (p : FiniteLaw A) (P : A → Prop) :
eventMass p P + eventMass p (fun a => ¬ P a) = 1 := by
rw [eventMass, eventMass, sum_subtype_indicator, sum_subtype_indicator,
← Finset.sum_add_distrib]
calc
(∑ a, ((if P a then p.mass a else 0) + (if ¬ P a then p.mass a else 0))) =
∑ a, p.mass a := by
apply Finset.sum_congr rfl
intro a _
by_cases h : P a <;> simp [h]
_ = _ := p.total
theorem entropy_sum_event_complement (p : FiniteLaw A) (P : A → Prop) :
(∑ a : {a // P a}, entropyTerm (p.mass a.val)) +
(∑ a : {a // ¬ P a}, entropyTerm (p.mass a.val)) = finiteEntropy p.mass := by
rw [sum_subtype_indicator P (fun a => entropyTerm (p.mass a)),
sum_subtype_indicator (fun a => ¬ P a) (fun a => entropyTerm (p.mass a)),
← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro a _
by_cases h : P a <;> simp [h]
theorem entropy_event_split (p : FiniteLaw A) (P : A → Prop)
(hP : 0 < eventMass p P) (hQ : 0 < eventMass p (fun a => ¬ P a)) :
finiteEntropy p.mass =
entropyTerm (eventMass p P) + entropyTerm (eventMass p (fun a => ¬ P a)) +
eventMass p P * finiteEntropy (condition p P hP).mass +
eventMass p (fun a => ¬ P a) * finiteEntropy (condition p (fun a => ¬ P a) hQ).mass := by
rw [← entropy_sum_event_complement p P, event_entropy p P hP,
event_entropy p (fun a => ¬ P a) hQ]
ring
theorem condition_entropy_of_removed_event (p : FiniteLaw A) (P : A → Prop)
(hP : 0 < eventMass p P) (removedMass removedEntropy : ℝ)
(hmass : eventMass p (fun a => ¬ P a) = removedMass)
(hremoved : (∑ a : {a // ¬ P a}, entropyTerm (p.mass a.val)) =
entropyTerm removedMass + removedMass * removedEntropy) :
finiteEntropy (condition p P hP).mass =
(finiteEntropy p.mass - entropyTerm removedMass -
entropyTerm (1 - removedMass) - removedMass * removedEntropy) /
(1 - removedMass) := by
have hevent : eventMass p P = 1 - removedMass := by
linarith [eventMass_add_complement p P]
have hsum := entropy_sum_event_complement p P
have hc := condition_entropy_mul p P hP
rw [hremoved] at hsum
rw [hevent] at hc
apply (eq_div_iff (by linarith : 1 - removedMass ≠ 0)).mpr
nlinarith
end MatrixMultiplication.CompletionLaws
end
end OAI
|