| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Coordinates.CompactConfigurations | |
| import OAI.Analysis.BiholderTransport.Regularity.FiniteSimplexRepresentation | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace WeakMTWTransport | |
| lemma fixed_simplex_representation {E : Type*} [AddCommGroup E] [Module ℝ E] | |
| [FiniteDimensional ℝ E] {S : Set E} {x : E} (hx : x∈convexHull ℝ S) : | |
| ∃ z : Fin (Module.finrank ℝ E+1) → E, ∃ w : Fin (Module.finrank ℝ E+1) → ℝ, | |
| (∀ i,z i∈S) ∧ (∀ i,0≤w i) ∧ ∑ i,w i=1 ∧ ∑ i,w i • z i=x := by | |
| classical | |
| obtain ⟨m,hm,hN,z,w,hz,hw,hs,hb⟩ := finite_simplex_representation hx | |
| obtain ⟨r,he⟩ := Nat.exists_eq_add_of_le hN | |
| rw [he] | |
| refine ⟨Fin.addCases z (fun _ : Fin r => z ⟨0,hm⟩), | |
| Fin.addCases w (fun _ : Fin r => 0),?_,?_,?_,?_⟩ | |
| · intro i | |
| refine Fin.addCases (fun j=>?_) (fun j=>?_) i | |
| · simpa only [Fin.addCases_left] using hz j | |
| · simpa only [Fin.addCases_right] using hz ⟨0,hm⟩ | |
| · intro i | |
| refine Fin.addCases (fun j=>?_) (fun j=>?_) i | |
| · simpa only [Fin.addCases_left] using hw j | |
| · simp only [Fin.addCases_right,le_refl] | |
| · simpa only [Fin.sum_univ_add,Fin.addCases_left,Fin.addCases_right,Finset.sum_const_zero, | |
| add_zero] using hs | |
| · simpa only [Fin.sum_univ_add,Fin.addCases_left,Fin.addCases_right,zero_smul, | |
| Finset.sum_const_zero,add_zero] using hb | |
| lemma compact_barycentric_limit {E : Type*} [NormedAddCommGroup E] | |
| [NormedSpace ℝ E] [FiniteDimensional ℝ E] {C : Set (E×E)} (hC : IsCompact C) | |
| {a v : ℕ → E} {a₀ v₀ : E} (ha : Tendsto a atTop (𝓝 a₀)) | |
| (hv : Tendsto v atTop (𝓝 v₀)) | |
| (hrep : ∀ k,v k∈convexHull ℝ {p | (a k,p)∈C}) : | |
| ∃ z : ℕ → Fin (Module.finrank ℝ E+1) → E, | |
| ∃ w : ℕ → Fin (Module.finrank ℝ E+1) → ℝ, | |
| ∃ z₀ : Fin (Module.finrank ℝ E+1) → E, | |
| ∃ w₀ : Fin (Module.finrank ℝ E+1) → ℝ, | |
| ∃ σ : ℕ → ℕ, StrictMono σ ∧ | |
| (∀ k,(∀ i,(a k,z k i)∈C) ∧ (∀ i,0≤w k i) ∧ | |
| ∑ i,w k i=1 ∧ ∑ i,w k i • z k i=v k) ∧ | |
| Tendsto (z ∘ σ) atTop (𝓝 z₀) ∧ Tendsto (w ∘ σ) atTop (𝓝 w₀) ∧ | |
| (∀ i,(a₀,z₀ i)∈C) ∧ (∀ i,0≤w₀ i) ∧ ∑ i,w₀ i=1 ∧ ∑ i,w₀ i • z₀ i=v₀ := by | |
| classical | |
| choose z w hz hw hs hb using (fun k=>fixed_simplex_representation (hrep k)) | |
| let d : ℕ → EnvelopeData E (Fin (Module.finrank ℝ E+1)) := | |
| fun k=>((a k,z k),(w k,0)) | |
| have hcompact := compact_common_base_configurations | |
| (ι := Fin (Module.finrank ℝ E+1)) hC 0 | |
| have hd : ∀ k,d k∈{q : EnvelopeData E (Fin (Module.finrank ℝ E+1)) | | |
| (∀ i,(q.1.1,q.1.2 i)∈C) ∧ (∀ i,0≤q.2.1 i) ∧ ∑ i,q.2.1 i=1 ∧ q.2.2∈Icc 0 0} := | |
| fun k=>⟨hz k,hw k,hs k,le_refl _,le_refl _⟩ | |
| obtain ⟨q,hq,σ,hσ,hlim⟩ := hcompact.tendsto_subseq hd | |
| have ha' : Tendsto (a ∘ σ) atTop (𝓝 q.1.1) := hlim.fst_nhds.fst_nhds | |
| have hae : q.1.1=a₀ := tendsto_nhds_unique ha' (ha.comp hσ.tendsto_atTop) | |
| refine ⟨z,w,q.1.2,q.2.1,σ,hσ,(fun k=>⟨hz k,hw k,hs k,hb k⟩), | |
| hlim.fst_nhds.snd_nhds,hlim.snd_nhds.fst_nhds,?_,hq.2.1,hq.2.2.1,?_⟩ | |
| · simpa only [hae] using hq.1 | |
| · have hsum : Tendsto (fun k=>∑ i,w (σ k) i • z (σ k) i) atTop | |
| (𝓝 (∑ i,q.2.1 i • q.1.2 i)) := by | |
| apply tendsto_finsetSum | |
| intro i _ | |
| exact ((tendsto_pi_nhds.mp hlim.snd_nhds.fst_nhds) i).smul | |
| ((tendsto_pi_nhds.mp hlim.fst_nhds.snd_nhds) i) | |
| simp only [hb] at hsum | |
| exact tendsto_nhds_unique hsum (hv.comp hσ.tendsto_atTop) | |
| lemma positive_barycentric_restriction {E : Type*} [AddCommGroup E] [Module ℝ E] | |
| {ι : Type*} [Fintype ι] (p : ι → E) (w : ι → ℝ) | |
| (hw : ∀ i,0≤w i) (hs : ∑ i,w i=1) : | |
| let κ := {i : ι // 0<w i} | |
| Nonempty κ ∧ (∑ i:κ,w i=1) ∧ | |
| (∑ i:κ,w i • p i=∑ i,w i • p i) ∧ | |
| ∀ f : ι → ℝ,∑ i:κ,w i*f i=∑ i,w i*f i := by | |
| classical | |
| dsimp only | |
| have he (f : ι → E) : (∑ i:{i : ι // 0<w i},w i • f i)=∑ i,w i • f i := by | |
| rw [←Finset.sum_subtype (p := fun i=>0<w i) (Finset.univ.filter (fun i=>0<w i)) (by simp) (fun i=>w i • f i)] | |
| exact Finset.sum_subset (Finset.filter_subset _ _) (by | |
| intro i hi hn | |
| have hi0 : w i=0 := le_antisymm (not_lt.mp (by simpa using hn)) (hw i) | |
| simp only [hi0,zero_smul]) | |
| have hr (f : ι → ℝ) : (∑ i:{i : ι // 0<w i},w i*f i)=∑ i,w i*f i := by | |
| rw [←Finset.sum_subtype (p := fun i=>0<w i) (Finset.univ.filter (fun i=>0<w i)) (by simp) (fun i=>w i*f i)] | |
| exact Finset.sum_subset (Finset.filter_subset _ _) (by | |
| intro i hi hn | |
| have hi0 : w i=0 := le_antisymm (not_lt.mp (by simpa using hn)) (hw i) | |
| simp only [hi0,zero_mul]) | |
| have hsum : (∑ i:{i : ι // 0<w i},w i)=1 := by | |
| simpa only [mul_one] using (hr (fun _=>1)).trans (by simpa only [mul_one] using hs) | |
| refine ⟨?_,hsum,he p,hr⟩ | |
| by_contra hn | |
| have : IsEmpty {i : ι // 0<w i} := not_nonempty_iff.mp hn | |
| simp at hsum | |
| end WeakMTWTransport | |
| end | |
| end OAI | |