| 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 |
|
|
