| import Mathlib | |
| import OAI.Probability.LogConcave.Dynamics.Curve | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal NNReal Topology | |
| section UpperProof | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal RealInnerProductSpace Topology | |
| open Function MeasureTheory Set Filter | |
| open scoped Topology NNReal | |
| namespace LogConcaveSampling.GlobalODE | |
| open Set Function Filter Metric | |
| open scoped Topology | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] | |
| {a b : ℝ} (hab : a≤b) {f : ℝ → E → E} (hc : Continuous (uncurry f)) | |
| {K : ℝ≥0} (hL : ∀ t∈Icc a b,LipschitzWith K (f t)) | |
| def spaceTimeForward (p : ℝ × E) : ℝ × E := | |
| (p.1,flow hL hc ⟨a,le_rfl,hab⟩ p.2 (projIcc a b hab p.1)) | |
| def spaceTimeBackward (p : ℝ × E) : ℝ × E := | |
| (p.1,flow hL hc (projIcc a b hab p.1) p.2 a) | |
| lemma spaceTimeForward_continuous : Continuous (spaceTimeForward hab hc hL) := | |
| continuous_fst.prodMk ((flow_joint_continuous hL hc ⟨a,le_rfl,hab⟩).comp | |
| (continuous_snd.prodMk ((continuous_projIcc (h:=hab)).comp continuous_fst))) | |
| lemma spaceTime_left_inverse : LeftInverse (spaceTimeBackward hab hc hL) (spaceTimeForward hab hc hL) := by | |
| intro p | |
| apply Prod.ext | |
| · rfl | |
| change flow hL hc (projIcc a b hab p.1) | |
| (flow hL hc ⟨a,le_rfl,hab⟩ p.2 (projIcc a b hab p.1)) a=p.2 | |
| rw [flow_cocycle hL hc ⟨a,le_rfl,hab⟩ (projIcc a b hab p.1) ⟨a,le_rfl,hab⟩,flow_initial] | |
| lemma spaceTime_right_inverse : RightInverse (spaceTimeBackward hab hc hL) (spaceTimeForward hab hc hL) := by | |
| intro p | |
| apply Prod.ext | |
| · rfl | |
| change flow hL hc ⟨a,le_rfl,hab⟩ (flow hL hc (projIcc a b hab p.1) p.2 a) | |
| (projIcc a b hab p.1)=p.2 | |
| rw [flow_cocycle hL hc (projIcc a b hab p.1) ⟨a,le_rfl,hab⟩ (projIcc a b hab p.1),flow_initial] | |
| lemma spaceTimeBackward_continuous : Continuous (spaceTimeBackward hab hc hL) := by | |
| apply continuous_fst.prodMk | |
| rw [continuous_iff_continuousAt] | |
| intro p | |
| let z := (spaceTimeBackward hab hc hL p).2 | |
| let B := Real.exp ((K:ℝ)*(b-a)) | |
| have hB : 0<B := Real.exp_pos _ | |
| have he : flow hL hc ⟨a,le_rfl,hab⟩ z (projIcc a b hab p.1)=p.2 := | |
| congrArg Prod.snd (spaceTime_right_inverse hab hc hL p) | |
| have hh : Continuous (fun q : ℝ × E => | |
| q.2-flow hL hc ⟨a,le_rfl,hab⟩ z (projIcc a b hab q.1)) := | |
| continuous_snd.sub ((flow_continuous hL hc ⟨a,le_rfl,hab⟩ z).comp | |
| (continuous_subtype_val.comp ((continuous_projIcc (h:=hab)).comp continuous_fst))) | |
| apply Metric.tendsto_nhds.mpr | |
| intro ε hε | |
| have hn : ∀ᶠ q in 𝓝 p,‖q.2-flow hL hc ⟨a,le_rfl,hab⟩ z (projIcc a b hab q.1)‖<ε/B := by | |
| have he0 : p.2-flow hL hc ⟨a,le_rfl,hab⟩ z (projIcc a b hab p.1)=0 := by rw [he,sub_self] | |
| simpa only [he0,dist_zero_right] using (Metric.tendsto_nhds.mp (hh.tendsto p)) (ε/B) (div_pos hε hB) | |
| filter_upwards [hn] with q hq | |
| have hdist := (flow_lipschitz hL hc (projIcc a b hab q.1) ⟨a,le_rfl,hab⟩).dist_le_mul q.2 | |
| (flow hL hc ⟨a,le_rfl,hab⟩ z (projIcc a b hab q.1)) | |
| rw [flow_cocycle hL hc ⟨a,le_rfl,hab⟩ (projIcc a b hab q.1) ⟨a,le_rfl,hab⟩,flow_initial] at hdist | |
| change dist (flow hL hc (projIcc a b hab q.1) q.2 a) z<ε | |
| apply hdist.trans_lt | |
| have hmax : Real.exp ((K:ℝ)*|a-(projIcc a b hab q.1:ℝ)|)≤B := by | |
| apply Real.exp_le_exp.mpr | |
| apply mul_le_mul_of_nonneg_left _ K.coe_nonneg | |
| rw [abs_of_nonpos (sub_nonpos.mpr (projIcc a b hab q.1).2.1)] | |
| linarith [(projIcc a b hab q.1).2.2] | |
| apply (mul_le_mul_of_nonneg_right hmax dist_nonneg).trans_lt | |
| rw [dist_eq_norm,mul_comm] | |
| exact (lt_div_iff₀ hB).mp hq | |
| def spaceTimeHomeomorph : (ℝ × E) ≃ₜ (ℝ × E) where | |
| toFun := spaceTimeForward hab hc hL | |
| invFun := spaceTimeBackward hab hc hL | |
| left_inv := spaceTime_left_inverse hab hc hL | |
| right_inv := spaceTime_right_inverse hab hc hL | |
| continuous_toFun := spaceTimeForward_continuous hab hc hL | |
| continuous_invFun := spaceTimeBackward_continuous hab hc hL | |
| end LogConcaveSampling.GlobalODE | |
| end UpperProof | |
| end | |
| end | |
| end | |
| end OAI | |