| import OAI.Analysis.ForestSpace.Exhaustions |
|
|
| namespace OAI |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| namespace ForestSpace |
|
|
| noncomputable section |
| attribute [local instance] Classical.propDecidable |
|
|
| / |
| is the full continuous dual of the specified segment-norm completion, not the |
| coordinate completion in place of that dual. -/ |
| theorem main_theorem : |
| (¬FiniteDimensional ℝ (FullDual Vertex)) ∧ |
| TopologicalSpace.SeparableSpace (FullDual Vertex) ∧ |
| Reflexive (FullDual Vertex) ∧ |
| (∀ (α β : ℝ) (R : EquivalentNorm (FullDual Vertex) α β), ¬IsAUC R.N) ∧ |
| (∀ t : ℝ, 0 < t → |
| Real.sqrt (1+t^2/12)-1 ≤ averageModulus (FullDual Vertex) norm t) ∧ |
| (∀ (k : ℕ) (f : (Diamond.diamond k).V → FullDual Vertex) (C : ℝ), |
| Diamond.EmbeddingBound k f C → Real.sqrt (1+(k:ℝ)/12) ≤ C) := by |
| exact ⟨infiniteDimensional_main_fullDual, separable_main_fullDual, |
| reflexive_main_fullDual, fun _ _ R => main_no_equivalent_AUC R, |
| main_average_modulus, main_diamond_distortion⟩ |
|
|
| / |
| ancestor set is a chain; finite ancestor sets provide the rooted-forest |
| structure. No reflexivity or full-dual identification is assumed here. -/ |
| theorem general_forest_main (V : Type) [PartialOrder V] [Countable V] [Infinite V] |
| (hF : ForestOrder V) (hAnc : ∀ v : V, (Set.Iic v).Finite) : |
| (∀ (H : Finset V) (hH : IsLowerSet (H : Set V)) (x y : X V), |
| x = PX H hH x → y = QX H hH y → |
| Real.sqrt (‖x‖^2+‖y‖^2/12) ≤ (‖x+y‖+‖x-y‖)/2) ∧ |
| (∀ (H : Finset V) (hH : IsLowerSet (H : Set V)) (x y : X V), |
| x = PX H hH x → y = QX H hH y → ∀ (s t : ℝ), 0 < s → 0 < t → |
| (‖x‖+s*t*‖y‖)/Real.sqrt (1+12*s^2) ≤ (‖x+t•y‖+‖x-t•y‖)/2) ∧ |
| (∀ t : ℝ, 0 < t → Real.sqrt (1+t^2/12)-1 ≤ averageModulus (X V) norm t) := by |
| exact ⟨paired_X hF hAnc, paired_X_two_parameters hF hAnc, forest_average_modulus hF hAnc⟩ |
|
|
| / |
| scale `alpha/(2 beta)` and every equivalent norm. -/ |
| theorem word_forest_main (α β : ℝ) (R : EquivalentNorm (X Word) α β) : |
| aucModulus (X Word) R.N (α/(2*β)) = 0 := |
| word_auc_modulus_zero R |
|
|
| end |
|
|
| end ForestSpace |
|
|
| end OAI |
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|
