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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 | import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformConditioning
import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformProducts
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
universe uCoord
namespace MatrixMultiplication.DualUniformUpdates
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws
open CompletionLaws CompletionProductLaws DualUniformConditioning DualUniformProducts
open scoped BigOperators
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
variable {X Y Z : Type uCoord} [Fintype X] [Fintype Y] [Fintype Z]
theorem conditionalCompletionLaw_uniform_independent
(S : FlaggedTensor X Y Z) (center output : Color)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(m : ℕ) (hm : 0 < m) (α : ℝ) (hα : 0 ≤ α) (hα' : α < 1)
(P : X → Prop)
(hu : UniformCoordinate (colorMix S center output pc po α hα hα'.le)
(fun a => a.val.1) P)
(hi : ∀ x c,
((colorMix S center output pc po α hα hα'.le).map
(fun a => (a.val.1, leafColor S a))).mass (x,c) =
((colorMix S center output pc po α hα hα'.le).map (fun a => a.val.1)).mass x *
((colorMix S center output pc po α hα hα'.le).map (leafColor S)).mass c) :
UniformCoordinate
(conditionalCompletionLaw S center output pc po m hm α hα hα')
(fun a => a.val.val.1) (fun x => ∀ i, P (x i)) := by
let p := iidColorLaw S center output pc po m α hα hα'.le
let f : (Fin m → Leaf S) → (Fin m → X) := fun w i => (w i).val.1
let g : (Fin m → Leaf S) → (Fin m → Color) := fun w i => leafColor S (w i)
let Q : (Fin m → Color) → Prop := fun w =>
raisedFlag center output (fun i => w i = output) ∧
∀ i, w i = center ∨ w i = output
have hprod : UniformCoordinate p f (fun x => ∀ i, P (x i)) :=
independentProduct_uniformCoordinate
(fun _ : Fin m => colorMix S center output pc po α hα hα'.le)
(fun _ a => a.val.1) (fun _ => P) (fun _ => hu)
have hind : ∀ x c, (p.map (fun a => (f a, g a))).mass (x,c) =
(p.map f).mass x * (p.map g).mass c :=
independentProduct_independent
(fun _ : Fin m => colorMix S center output pc po α hα hα'.le)
(fun _ a => a.val.1) (fun _ => leafColor S) (fun _ => hi)
have hcond := uniform_coordinate_independent_condition p f g
(fun x => ∀ i, P (x i)) Q hprod hind
(completionEvent_mass_pos S center output pc po m hm α hα hα')
intro x
rw [conditionalCompletionLaw_X_mass]
exact hcond x
theorem conditionalCompletionLaw_uniform_incident
(S : FlaggedTensor X Y Z) (center output : Color)
(hc : center = .B ∨ output = .B)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(m : ℕ) (hm : 0 < m) (α : ℝ) (hα : 0 ≤ α) (hα' : α < 1)
(hu : UniformCoordinate (colorMix S center output pc po α hα hα'.le)
(fun a => a.val.1) (fun _ => True)) :
UniformCoordinate
(conditionalCompletionLaw S center output pc po m hm α hα hα')
(fun a => a.val.val.1)
(fun x => if output = .B then (complete S center m).flagB x
else ¬ (complete S center m).flagB x) := by
let p := iidColorLaw S center output pc po m α hα hα'.le
let f : (Fin m → Leaf S) → (Fin m → X) := fun w i => (w i).val.1
let Q : (Fin m → X) → Prop := fun x => if output = .B then
(complete S center m).flagB x else ¬ (complete S center m).flagB x
have hprod : UniformCoordinate p f (fun _ => True) := by
apply uniform_coordinate_pred_congr p f (fun x => ∀ _i : Fin m, True)
(fun _ => True) (fun _ => by simp)
exact independentProduct_uniformCoordinate
(fun _ : Fin m => colorMix S center output pc po α hα hα'.le)
(fun _ a => a.val.1) (fun _ _ => True) (fun _ => hu)
have heq : ∀ w, p.mass w ≠ 0 →
(completionEvent S center output m w ↔ Q (f w)) := by
intro w hw
have ha := iidColorLaw_allowed S center output pc po m α hα hα'.le w hw
by_cases ho : output = .B
· subst output
simpa only [Q, f, ite_eq_left rfl, ite_true] using completionEvent_outputB S center m w ha
· have hcenter : center = .B := hc.resolve_right ho
subst center
simpa only [Q, f, ite_eq_right ho] using completionEvent_centerB S output ho m w ha
have hcond := uniform_coordinate_condition_supported p f (fun _ => True) Q
(completionEvent S center output m) hprod heq
(completionEvent_mass_pos S center output pc po m hm α hα hα')
have hcond' := uniform_coordinate_pred_congr _ _ (fun x => True ∧ Q x) Q
(fun _ => ⟨And.right, fun h => ⟨True.intro, h⟩⟩) hcond
intro x
rw [conditionalCompletionLaw_X_mass]
exact hcond' x
end MatrixMultiplication.DualUniformUpdates
end
end OAI
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