openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /CostGeometry /ProjectionContinuation.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.CostGeometry.UniformLocalGap | |
| import OAI.Analysis.BiholderTransport.CostGeometry.LimitIsolation | |
| import OAI.Analysis.BiholderTransport.CostGeometry.PrecutExtension | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| lemma WeakMTW.extend_injective_graphProjection | |
| (hmtw : WeakMTW (n := n) (M := M)) {v : M → ℝ} (hv : Continuous v) | |
| {T : ℝ} (hT : 0<T) (hT1 : T<1) | |
| (hinj : ∀ t ∈ Ioo (0:ℝ) T, Function.Injective (graphProjection (n := n) (cTransform v) t)) : | |
| ∃ R, T<R ∧ R<1 ∧ ∀ t ∈ Ioc (0:ℝ) R, | |
| Function.Injective (graphProjection (n := n) (cTransform v) t) := by | |
| let : CompactSpace (subgradientGraph (n := n) (cTransform v)) := | |
| isCompact_iff_compactSpace.mp (isCompact_subgradientGraph hv) | |
| have hTI := hmtw.injective_graphProjection_at_left_limit hv hT hT1 hinj | |
| have hID := hmtw.active_hull_precut_at_left_limit hv hT hT1 hinj | |
| obtain ⟨S,hTS,hS1,hSID⟩ := exists_extended_uniform_precut hv hT hT1 hID | |
| obtain ⟨δ,hδ,hsep⟩ := hmtw.uniform_graph_fiber_separation hv hS1 hSID | |
| have hnear : ∀ᶠ t in 𝓝 T, Function.Injective (graphProjection (n := n) (cTransform v) t) := by | |
| apply eventually_injective_of_uniform_local_injective | |
| (graphProjection (n := n) (cTransform v)) (continuous_graphProjection_family _) | |
| (fun z => z.1.1) ((FiberBundle.continuous_proj _ _).comp continuous_subtype_val) hδ hTI | |
| filter_upwards [eventually_gt_nhds hT,eventually_lt_nhds hTS] with t ht htS | |
| exact hsep t ⟨ht,htS.le⟩ | |
| obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp hnear | |
| let R := T+min (ε/2) ((1-T)/2) | |
| have hTR : T<R := by dsimp [R]; have : 0 < min (ε/2) ((1-T)/2) := lt_min (by linarith) (by linarith); linarith | |
| have hR1 : R<1 := by have := min_le_right (ε/2) ((1-T)/2); dsimp [R]; linarith | |
| have hRε : R<T+ε := by have := min_le_left (ε/2) ((1-T)/2); dsimp [R]; linarith | |
| refine ⟨R,hTR,hR1,?_⟩ | |
| intro t ht | |
| by_cases htT : t<T | |
| · exact hinj t ⟨ht.1,htT⟩ | |
| · apply hball | |
| rw [Metric.mem_ball,Real.dist_eq,abs_of_nonneg (sub_nonneg.mpr (le_of_not_gt htT))] | |
| linarith [ht.2] | |
| lemma WeakMTW.injective_graphProjection | |
| (hmtw : WeakMTW (n := n) (M := M)) {v : M → ℝ} (hv : Continuous v) | |
| {t : ℝ} (ht : 0<t) (ht1 : t<1) : | |
| Function.Injective (graphProjection (n := n) (cTransform v) t) := by | |
| let A : Set ℝ := {T | 0<T ∧ T<1 ∧ ∀ s ∈ Ioc (0:ℝ) T, | |
| Function.Injective (graphProjection (n := n) (cTransform v) s)} | |
| obtain ⟨τ,hτ,hτ1,hτI⟩ := hmtw.exists_short_injective_graphProjection hv | |
| have hτA : τ∈A := ⟨hτ,hτ1,hτI⟩ | |
| have hA : A.Nonempty := ⟨τ,hτA⟩ | |
| have hAb : BddAbove A := ⟨1,fun _ h => h.2.1.le⟩ | |
| have hB : 0<sSup A := hτ.trans_le (le_csSup hAb hτA) | |
| have hB1 : sSup A≤1 := csSup_le hA (fun _ h => h.2.1.le) | |
| have hBI : ∀ s ∈ Ioo (0:ℝ) (sSup A), | |
| Function.Injective (graphProjection (n := n) (cTransform v) s) := by | |
| intro s hs | |
| obtain ⟨R,hRA,hsR⟩ := exists_lt_of_lt_csSup hA hs.2 | |
| exact hRA.2.2 s ⟨hs.1,hsR.le⟩ | |
| have heq : sSup A=1 := by | |
| apply le_antisymm hB1 | |
| by_contra hn | |
| obtain ⟨R,hBR,hR1,hRI⟩ := hmtw.extend_injective_graphProjection hv hB (lt_of_not_ge hn) hBI | |
| have hRA : R∈A := ⟨hB.trans hBR,hR1,hRI⟩ | |
| exact (not_lt_of_ge (le_csSup hAb hRA)) hBR | |
| apply hBI t | |
| exact ⟨ht,by rwa [heq]⟩ | |
| lemma WeakMTW.active_hull_precut | |
| (hmtw : WeakMTW (n := n) (M := M)) {v : M → ℝ} (hv : Continuous v) | |
| {t : ℝ} (ht : 0<t) (ht1 : t<1) : | |
| ∀ x : M, ∀ p ∈ convexHull ℝ (activeLogs (n := n) v x), t • p ∈ injectivityDomain x := | |
| hmtw.active_hull_precut_at_left_limit hv ht ht1 | |
| (fun _ hs => hmtw.injective_graphProjection hv hs.1 (hs.2.trans ht1)) | |
| end WeakMTWTransport | |
| end | |
| end OAI | |