| import Mathlib.Topology.ContinuousMap.ZeroAtInfty | |
| import Mathlib.Analysis.Normed.Operator.ContinuousLinearMap | |
| namespace OAI | |
| namespace C0Absorption | |
| open scoped ZeroAtInfty | |
| abbrev C0 := C₀(ℕ, ℝ) | |
| def BiLip {X Y : Type*} [MetricSpace X] [MetricSpace Y] (f : X → Y) : Prop := | |
| ∃ c C : ℝ, 0 < c ∧ c ≤ C ∧ | |
| ∀ x y, c * dist x y ≤ dist (f x) (f y) ∧ dist (f x) (f y) ≤ C * dist x y | |
| def NoLinearC0 (Z : Type*) [NormedAddCommGroup Z] [NormedSpace ℝ Z] : Prop := | |
| ∀ T : C0 →L[ℝ] Z, ¬ ∃ c : ℝ, 0 < c ∧ ∀ x, c * ‖x‖ ≤ ‖T x‖ | |
| def MetricUniversal (Z : Type*) [MetricSpace Z] : Prop := | |
| ∀ (M : Type) [MetricSpace M] [TopologicalSpace.SeparableSpace M], | |
| ∃ f : M → Z, BiLip f | |
| def MainConclusion : Prop := | |
| ∃ (Z : Type) (_ : NormedAddCommGroup Z) (_ : NormedSpace ℝ Z), | |
| CompleteSpace Z ∧ TopologicalSpace.SeparableSpace Z ∧ NoLinearC0 Z ∧ | |
| (∃ F : Z × C0 → Z, Function.Surjective F ∧ BiLip F) ∧ | |
| (∃ f : C0 → Z, BiLip f) ∧ MetricUniversal Z ∧ | |
| ¬ Nonempty ((Z × C0) ≃L[ℝ] Z) | |
| end C0Absorption | |
| end OAI | |