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9655094 | 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 | 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
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