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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 34 35 36 37 38 39 40 41 42 43 | import OAI.Analysis.C0Absorption.Aharoni
namespace OAI
open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace
namespace C0Absorption
noncomputable section
theorem BiLip.comp {X Y Z : Type*} [MetricSpace X] [MetricSpace Y] [MetricSpace Z]
{g : Y → Z} {f : X → Y} (hg : BiLip g) (hf : BiLip f) : BiLip (g ∘ f) := by
obtain ⟨cg,Cg,hcg,hcgC,hg⟩ := hg
obtain ⟨cf,Cf,hcf,hcfC,hf⟩ := hf
have hCg : 0≤Cg := hcg.le.trans hcgC
have hCf : 0≤Cf := hcf.le.trans hcfC
refine ⟨cg*cf,Cg*Cf,mul_pos hcg hcf,mul_le_mul hcgC hcfC hcf.le hCg,fun x y => ⟨?_,?_⟩⟩
· calc
cg*cf*dist x y = cg*(cf*dist x y) := mul_assoc _ _ _
_ ≤ cg*dist (f x) (f y) := mul_le_mul_of_nonneg_left (hf x y).1 hcg.le
_ ≤ dist ((g ∘ f) x) ((g ∘ f) y) := (hg _ _).1
· calc
dist ((g ∘ f) x) ((g ∘ f) y) ≤ Cg*dist (f x) (f y) := (hg _ _).2
_ ≤ Cg*(Cf*dist x y) := mul_le_mul_of_nonneg_left (hf x y).2 hCg
_ = Cg*Cf*dist x y := (mul_assoc _ _ _).symm
theorem concrete_metricUniversal : MetricUniversal ConcreteSpace := by
intro M instM instSep
obtain ⟨f,hf⟩ := c0_metricUniversal M
exact ⟨concreteC0Embedding ∘ f,concreteC0Embedding_bilip.comp hf⟩
theorem main_result : MainConclusion := by
refine ⟨ConcreteSpace,inferInstance,inferInstance,inferInstance,inferInstance,
concrete_noLinearC0,⟨concreteAbsorption,concreteAbsorption_surjective,concreteAbsorption_bilip⟩,
⟨concreteC0Embedding,concreteC0Embedding_bilip⟩,concrete_metricUniversal,
concrete_not_linearly_isomorphic⟩
end
end C0Absorption
end OAI
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