| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Conditioning |
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| namespace OAI |
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| /-! Readable tensor completion and its finite arithmetic realization. -/ |
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| noncomputable section |
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| namespace MatrixMultiplication.CompletionLaws |
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| open MatrixMultiplication.Foundation |
| open scoped BigOperators |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq |
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| variable {A : Type*} [Fintype A] |
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| 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] |
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| 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'] |
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|
| 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] |
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|
| 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 |
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| 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 |
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|
| 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 |
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| end MatrixMultiplication.CompletionLaws |
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| end |
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| end OAI |
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