| import OAI.Dynamics.EntropyCounterexample.Embedding | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem340 | |
| def HasLocalC1Extension {X E F : Type*} | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| (e : X → E) (f : X → F) : Prop := | |
| ∀ x, ∃ U : Set E, IsOpen U ∧ e x ∈ U ∧ | |
| ∃ F : E → F, ContDiffOn ℝ 1 F U ∧ ∀ y, e y ∈ U → f y = F (e y) | |
| namespace HasLocalC1Extension | |
| variable {X E F G : Type*} | |
| variable [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| variable [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| variable [NormedAddCommGroup G] [NormedSpace ℝ G] | |
| variable {e : X → E} {f : X → F} {g : X → G} | |
| theorem prodMk (hf : HasLocalC1Extension e f) (hg : HasLocalC1Extension e g) : | |
| HasLocalC1Extension e (fun x => (f x, g x)) := by | |
| intro x | |
| obtain ⟨U, hU, hxU, F, hF, hcommF⟩ := hf x | |
| obtain ⟨V, hV, hxV, G, hG, hcommG⟩ := hg x | |
| refine ⟨U ∩ V, hU.inter hV, ⟨hxU, hxV⟩, fun z => (F z, G z), | |
| (hF.mono Set.inter_subset_left).prodMk (hG.mono Set.inter_subset_right), ?_⟩ | |
| intro y hy | |
| exact Prod.ext (hcommF y hy.1) (hcommG y hy.2) | |
| theorem pi {ι : Type*} [Fintype ι] {F : ι → Type*} | |
| [∀ i, NormedAddCommGroup (F i)] [∀ i, NormedSpace ℝ (F i)] | |
| {f : X → ∀ i, F i} (hf : ∀ i, HasLocalC1Extension e (fun x => f x i)) : | |
| HasLocalC1Extension e f := by | |
| intro x | |
| choose U hU hxU F hF hcomm using fun i => hf i x | |
| refine ⟨⋂ i, U i, isOpen_iInter_of_finite hU, Set.mem_iInter.mpr hxU, | |
| fun z i => F i z, contDiffOn_pi.mpr (fun i => (hF i).mono (Set.iInter_subset U i)), ?_⟩ | |
| intro y hy | |
| funext i | |
| exact hcomm i y (Set.mem_iInter.mp hy i) | |
| theorem of_contDiffOn {U : Set E} (hU : IsOpen U) | |
| (he : ∀ x, e x ∈ U) (F : E → F) (hF : ContDiffOn ℝ 1 F U) | |
| (hcomm : ∀ x, f x = F (e x)) : HasLocalC1Extension e f := | |
| fun x => ⟨U, hU, he x, F, hF, fun y _ => hcomm y⟩ | |
| theorem comp_contDiff (hf : HasLocalC1Extension e f) {g : F → G} | |
| (hg : ContDiff ℝ 1 g) : HasLocalC1Extension e (g ∘ f) := by | |
| intro x | |
| obtain ⟨U, hU, hxU, F, hF, hcomm⟩ := hf x | |
| refine ⟨U, hU, hxU, g ∘ F, hg.comp_contDiffOn hF, ?_⟩ | |
| intro y hy | |
| change g (f y) = g (F (e y)) | |
| rw [hcomm y hy] | |
| end HasLocalC1Extension | |
| theorem isC1SelfMap_iff_hasLocalC1Extension {q : ℕ} | |
| {f : CounterexampleManifold q → CounterexampleManifold q} : | |
| IsC1SelfMap q f ↔ | |
| HasLocalC1Extension (standardEmbedding q) (standardEmbedding q ∘ f) := by | |
| constructor | |
| · exact fun hf => hf.2 | |
| · intro hf | |
| exact ⟨continuous_of_local_contDiff_extension f hf, hf⟩ | |
| theorem isC1SelfMap_of_coordinate_extensions {q : ℕ} | |
| {f : CounterexampleManifold q → CounterexampleManifold q} | |
| (hclock : HasLocalC1Extension (standardEmbedding q) (fun x => (f x).1.1)) | |
| (hsphere : ∀ i, HasLocalC1Extension (standardEmbedding q) (fun x => ((f x).2 i).1)) : | |
| IsC1SelfMap q f := | |
| isC1SelfMap_iff_hasLocalC1Extension.mpr (hclock.prodMk (.pi hsphere)) | |
| theorem isC1SelfMap_of_contDiffOn {q : ℕ} | |
| {f : CounterexampleManifold q → CounterexampleManifold q} | |
| {U : Set (CounterexampleAmbient q)} (hU : IsOpen U) | |
| (he : ∀ x, standardEmbedding q x ∈ U) | |
| (F : CounterexampleAmbient q → CounterexampleAmbient q) | |
| (hF : ContDiffOn ℝ 1 F U) | |
| (hcomm : ∀ y, standardEmbedding q (f y) = F (standardEmbedding q y)) : | |
| IsC1SelfMap q f := | |
| isC1SelfMap_iff_hasLocalC1Extension.mpr | |
| (.of_contDiffOn hU he F hF hcomm) | |
| end Problem340 | |
| end | |
| end OAI | |