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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 | import OAI.LinearAlgebra.MatrixMultiplication.Duality.Information
import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorLaws
import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformProducts
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualProgramIndependence
universe u v w z
open MatrixMultiplication.Foundation DualInformation CompletionProductLaws
open CompletionColorLaws CompletionLabels RecursiveCompletion DualEntropyHelpers
open DualUniformProducts
open scoped BigOperators
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
variable {I : Type*} [Fintype I] [DecidableEq I]
{A : I → Type u} {R : I → Type v} {L : I → Type w} {X : I → Type z}
[∀ i, Fintype (A i)] [∀ i, Fintype (R i)]
[∀ i, Fintype (L i)] [∀ i, Fintype (X i)]
theorem independentProduct_triple_mass
(p : ∀ i, FiniteLaw (A i)) (prior : ∀ i, A i → R i)
(label : ∀ i, A i → L i) (x : ∀ i, A i → X i)
(r : ∀ i, R i) (l : ∀ i, L i) (xx : ∀ i, X i) :
((independentProduct p).map (fun a => ((fun i => prior i (a i)),
((fun i => label i (a i)), (fun i => x i (a i)))))).mass (r, (l, xx)) =
∏ i, ((p i).map (fun a => (prior i a, (label i a, x i a)))).mass
(r i, (l i, xx i)) := by
rw [← independentProduct_map_mass p
(fun i a => (prior i a, (label i a, x i a)))
(fun i => (r i, (l i, xx i)))]
simp only [FiniteLaw.map_mass]
apply Finset.sum_congr rfl
intro a _
simp only [Prod.mk.injEq, funext_iff, forall_and]
theorem independentProduct_factorization
(p : ∀ i, FiniteLaw (A i)) (prior : ∀ i, A i → R i)
(label : ∀ i, A i → L i) (x : ∀ i, A i → X i)
(h : ∀ i, ConditionalMassFactorization (p i) (prior i) (label i) (x i)) :
ConditionalMassFactorization (independentProduct p)
(fun a i => prior i (a i)) (fun a i => label i (a i))
(fun a i => x i (a i)) := by
intro r l xx
rw [independentProduct_triple_mass p prior label x r l xx,
independentProduct_map_mass p prior r,
independentProduct_joint_mass p prior label r l,
independentProduct_joint_mass p prior x r xx,
← Finset.prod_mul_distrib, ← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro i _
exact h i (r i) (l i) (xx i)
section Constant
variable {B : Type u} {P : Type v} {C : Type w} {Y : Type z}
[Fintype B] [Fintype P] [Fintype C] [Fintype Y]
theorem factorization_constant_label (p : FiniteLaw B) (prior : B → P)
(c : C) (x : B → Y) :
ConditionalMassFactorization p prior (fun _ => c) x := by
intro r l xx
by_cases h : c = l
· subst l
simp only [FiniteLaw.map_mass, Prod.mk.injEq, true_and, and_true]
ring
· simp only [FiniteLaw.map_mass, Prod.mk.injEq, h, false_and, and_false,
ite_false, Finset.sum_const_zero, zero_mul]
end Constant
section ColorMixture
variable {SX SY SZ : Type u} {P : Type v} {C : Type w} {V : Type z}
[Fintype SX] [Fintype SY] [Fintype SZ]
[Fintype P] [Fintype C] [Fintype V]
theorem colorSlice_context_mass (S : FlaggedTensor SX SY SZ) (color c : Color)
(p : FiniteLaw (ColorSlice S color)) (f : Leaf S → V) (v : V) :
((p.map Subtype.val).map (fun a => (leafColor S a, f a))).mass (c, v) =
if color = c then (p.map (fun a => f a.val)).mass v else 0 := by
rw [map_comp_mass]
simp only [FiniteLaw.map_mass, Prod.mk.injEq]
by_cases h : color = c
· simp only [show ∀ a : ColorSlice S color, leafColor S a.val = color from
fun a => a.property, h, true_and, ite_true]
· simp only [show ∀ a : ColorSlice S color, leafColor S a.val = color from
fun a => a.property, h, false_and, ite_false, Finset.sum_const_zero]
theorem colorMix_context_mass (S : FlaggedTensor SX SY SZ) (center output c : Color)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(α : ℝ) (hα : 0 ≤ α) (hα' : α ≤ 1) (f : Leaf S → V) (v : V) :
((colorMix S center output pc po α hα hα').map
(fun a => (leafColor S a, f a))).mass (c, v) =
α * (if center = c then (pc.map (fun a => f a.val)).mass v else 0) +
(1 - α) * (if output = c then (po.map (fun a => f a.val)).mass v else 0) := by
rw [colorMix_map_mass, colorSlice_context_mass, colorSlice_context_mass]
theorem colorMix_refined_context_mass (S : FlaggedTensor SX SY SZ)
(center output c : Color)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(α : ℝ) (hα : 0 ≤ α) (hα' : α ≤ 1)
(prior : Leaf S → P) (f : Leaf S → V) (r : P) (v : V) :
((colorMix S center output pc po α hα hα').map
(fun a => ((leafColor S a, prior a), f a))).mass ((c, r), v) =
α * (if center = c then
(pc.map (fun a => (prior a.val, f a.val))).mass (r, v) else 0) +
(1 - α) * (if output = c then
(po.map (fun a => (prior a.val, f a.val))).mass (r, v) else 0) := by
rw [← colorMix_context_mass S center output c pc po α hα hα'
(fun a => (prior a, f a)) (r, v)]
simp only [FiniteLaw.map_mass, Prod.mk.injEq, and_assoc]
theorem colorMix_factorization (S : FlaggedTensor SX SY SZ) (center output : Color)
(hne : center ≠ output)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(α : ℝ) (hα : 0 ≤ α) (hα' : α ≤ 1)
(prior : Leaf S → P) (label : Leaf S → C) (x : Leaf S → V)
(hc : ConditionalMassFactorization pc (fun a => prior a.val)
(fun a => label a.val) (fun a => x a.val))
(ho : ConditionalMassFactorization po (fun a => prior a.val)
(fun a => label a.val) (fun a => x a.val)) :
ConditionalMassFactorization (colorMix S center output pc po α hα hα')
(fun a => (leafColor S a, prior a)) label x := by
rintro ⟨c, r⟩ l xx
rw [colorMix_refined_context_mass, colorMix_context_mass,
colorMix_refined_context_mass, colorMix_refined_context_mass]
by_cases hcc : center = c
· subst c
have hoc : output ≠ center := Ne.symm hne
simp only [ite_true, ite_eq_right hoc, mul_zero, add_zero]
calc
_ = α ^ 2 *
((pc.map (fun a => (prior a.val, (label a.val, x a.val)))).mass (r, (l, xx)) *
(pc.map (fun a => prior a.val)).mass r) := by ring
_ = _ := by rw [hc r l xx]; ring
· by_cases hoc : output = c
· subst c
simp only [ite_eq_right hne, ite_true, mul_zero, zero_add]
calc
_ = (1 - α) ^ 2 *
((po.map (fun a => (prior a.val, (label a.val, x a.val)))).mass (r, (l, xx)) *
(po.map (fun a => prior a.val)).mass r) := by ring
_ = _ := by rw [ho r l xx]; ring
· simp only [ite_eq_right hcc, ite_eq_right hoc, mul_zero, zero_add]
theorem colorMix_recorded_factorization (S : FlaggedTensor SX SY SZ)
(center output : Color)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(α : ℝ) (hα : 0 ≤ α) (hα' : α ≤ 1) (hbranch : center ≠ output ∨ α = 0)
(prior : Leaf S → P) (label : Leaf S → C) (x : Leaf S → V)
(hc : ConditionalMassFactorization pc (fun a => prior a.val)
(fun a => label a.val) (fun a => x a.val))
(ho : ConditionalMassFactorization po (fun a => prior a.val)
(fun a => label a.val) (fun a => x a.val)) :
ConditionalMassFactorization (colorMix S center output pc po α hα hα')
(fun a => (leafColor S a, prior a)) label x := by
rcases hbranch with hne | hz
· exact colorMix_factorization S center output hne pc po α hα hα' prior label x hc ho
· subst α
rintro ⟨c, r⟩ l xx
rw [colorMix_refined_context_mass, colorMix_context_mass,
colorMix_refined_context_mass, colorMix_refined_context_mass]
by_cases hoc : output = c
· simp only [zero_mul, zero_add, sub_zero, one_mul, ite_eq_left hoc]
exact ho r l xx
· simp only [zero_mul, zero_add, sub_zero, one_mul, ite_eq_right hoc]
end ColorMixture
section ProgramInvariant
variable {SX SY SZ : Type u} [Fintype SX] [Fintype SY] [Fintype SZ]
def ProgramLawsIndependent (S : FlaggedTensor SX SY SZ) (r : ReadableTensor S)
(p : ∀ c, FiniteLaw (ColorSlice S c)) : Prop :=
∀ c, ∀ n : Fin r.depth, r.program.pair n.val = .yz →
ConditionalMassFactorization (p c)
(fun a => labelRecordOf r.program.labels n.val a.val)
(fun a => r.program.labels n.val a.val)
(fun a => a.val.val.1)
end ProgramInvariant
end MatrixMultiplication.DualProgramIndependence
end
end OAI
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