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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 99 100 101 102 103 104 105 106 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorEntropy
import OAI.LinearAlgebra.MatrixMultiplication.Completion.Entropy
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.IndependentConditioning
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionTerminationLaw
open MatrixMultiplication.Foundation CompletionLabels CompletionColorLaws CompletionLaws
open scoped BigOperators
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
universe u
variable {X Y Z : Type u} [Fintype X] [Fintype Y] [Fintype Z]
theorem retained_mass_one (S : FlaggedTensor X Y Z) (first second : Color)
(pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second))
(a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) :
eventMass (colorMix S first second pf ps a ha ha')
(fun w => leafColor S w = first ∨ leafColor S w = second) = 1 := by
rw [eventMass, sum_subtype_indicator]
calc
_ = ∑ w, (colorMix S first second pf ps a ha ha').mass w := by
apply Finset.sum_congr rfl
intro w _
split_ifs with h
· rfl
· exact (colorMix_mass_zero S first second pf ps a ha ha' w
(fun h' => h (Or.inl h')) (fun h' => h (Or.inr h'))).symm
_ = _ := (colorMix S first second pf ps a ha ha').total
def retainedEquiv (S : FlaggedTensor X Y Z) (first second : Color) :
{w : Leaf S // leafColor S w = first ∨ leafColor S w = second} ≃
RetainedLeaf S first second := Equiv.refl _
theorem retained_sum (S : FlaggedTensor X Y Z) (first second : Color) (f : Leaf S → ℝ) :
(∑ w : RetainedLeaf S first second, f w.val) =
∑ w : {w : Leaf S // leafColor S w = first ∨ leafColor S w = second}, f w.val := by
exact ((retainedEquiv S first second).sum_comp
(fun w : RetainedLeaf S first second => f w.val)).symm
def law (S : FlaggedTensor X Y Z) (first second : Color)
(pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second))
(a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) :
FiniteLaw (RetainedLeaf S first second) where
mass w := (colorMix S first second pf ps a ha ha').mass w.val
nonneg w := (colorMix S first second pf ps a ha ha').nonneg w.val
total := by
rw [retained_sum]
exact retained_mass_one S first second pf ps a ha ha'
theorem law_mass (S : FlaggedTensor X Y Z) (first second : Color)
(pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second))
(a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) (w : RetainedLeaf S first second) :
(law S first second pf ps a ha ha').mass w =
(colorMix S first second pf ps a ha ha').mass w.val := rfl
theorem law_map_mass {U : Type*} [Fintype U]
(S : FlaggedTensor X Y Z) (first second : Color)
(pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second))
(a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) (f : Leaf S → U) (u : U) :
((law S first second pf ps a ha ha').map (fun w => f w.val)).mass u =
((colorMix S first second pf ps a ha ha').map f).mass u := by
rw [FiniteLaw.map_mass]
simp only [law_mass]
rw [retained_sum S first second
(fun w => if f w = u then (colorMix S first second pf ps a ha ha').mass w else 0)]
rw [sum_subtype_indicator (fun w : Leaf S =>
leafColor S w = first ∨ leafColor S w = second)
(fun w => if f w = u then (colorMix S first second pf ps a ha ha').mass w else 0)]
rw [FiniteLaw.map_mass]
apply Finset.sum_congr rfl
intro w _
by_cases h : leafColor S w = first ∨ leafColor S w = second
· simp [h]
· have hz := colorMix_mass_zero S first second pf ps a ha ha' w
(fun h' => h (Or.inl h')) (fun h' => h (Or.inr h'))
simp [h, hz]
theorem law_entropy (S : FlaggedTensor X Y Z) (first second : Color)
(hne : first ≠ second)
(pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second))
(a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) :
finiteEntropy (law S first second pf ps a ha ha').mass =
entropyTerm a + entropyTerm (1 - a) +
a * finiteEntropy pf.mass + (1 - a) * finiteEntropy ps.mass := by
have hmap : ((law S first second pf ps a ha ha').map Subtype.val).mass =
(colorMix S first second pf ps a ha ha').mass := by
funext w
exact (law_map_mass S first second pf ps a ha ha' id w).trans
((colorMix S first second pf ps a ha ha').map_mass_apply id Function.injective_id w)
rw [← (law S first second pf ps a ha ha').map_entropy_of_injective
Subtype.val Subtype.val_injective, hmap]
exact colorMix_entropy S first second hne pf ps a ha ha'
end MatrixMultiplication.CompletionTerminationLaw
end
end OAI
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