| import Mathlib |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
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| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
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| namespace LogConcaveSampling |
| open Set Filter Metric Function |
| open scoped Topology |
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| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| lemma hasFDerivAt_of_time_partial {f g : ℝ → E → F} {s : Set (ℝ × E)} |
| (hs : IsOpen s) {t : ℝ} {x : E} (hp : (t,x)∈s) |
| (hg : ContinuousAt (uncurry g) (t,x)) |
| (ht : ∀ p∈s,HasDerivAt (fun u => f u p.2) (g p.1 p.2) p.1) |
| {D : E →L[ℝ] F} (hx : HasFDerivAt (f t) D x) : |
| HasFDerivAt (uncurry f) |
| (((ContinuousLinearMap.fst ℝ ℝ E).smulRight (g t x))+ |
| D.comp (ContinuousLinearMap.snd ℝ ℝ E)) (t,x) := by |
| apply HasFDerivAt.of_isLittleO |
| rw [Asymptotics.isLittleO_iff] |
| intro ε hε |
| have hgε : ∀ᶠ q in 𝓝 (t,x),q∈s ∧ ‖g q.1 q.2-g t x‖<ε/2 := by |
| filter_upwards [hs.mem_nhds hp,(Metric.tendsto_nhds.mp hg) (ε/2) (by positivity)] with q hq hq' |
| exact ⟨hq,by simpa [dist_eq_norm,Function.uncurry] using hq'⟩ |
| obtain ⟨δ,hδ,hball⟩ := Metric.mem_nhds_iff.mp hgε |
| have hxε := (Asymptotics.isLittleO_iff.mp hx.isLittleO) (by positivity : 0<ε/2) |
| filter_upwards [Metric.ball_mem_nhds (t,x) hδ, |
| (continuous_snd.tendsto (t,x)) hxε] with q hq hqx |
| have hqt : |q.1-t|<δ := lt_of_le_of_lt (le_max_left _ _) (by |
| simpa [mem_ball,dist_eq_norm,Prod.norm_def,Real.norm_eq_abs] using hq) |
| have hqxδ : ‖q.2-x‖<δ := lt_of_le_of_lt (le_max_right _ _) (by |
| simpa [mem_ball,dist_eq_norm,Prod.norm_def,Real.norm_eq_abs] using hq) |
| have hc (u : ℝ) (hu : u∈closedBall t |q.1-t|) : |
| (u,q.2)∈s ∧ ‖g u q.2-g t x‖<ε/2 := by |
| apply hball |
| have hu' : |u-t|≤|q.1-t| := by simpa [mem_closedBall,dist_eq_norm,Real.norm_eq_abs] using hu |
| simpa [mem_ball,dist_eq_norm,Prod.norm_def,Real.norm_eq_abs] using |
| (max_lt (hu'.trans_lt hqt) hqxδ) |
| have hd (u : ℝ) (hu : u∈closedBall t |q.1-t|) : |
| HasFDerivWithinAt (fun v => f v q.2-v •g t x) |
| ((1 : ℝ →L[ℝ] ℝ).smulRight (g u q.2-g t x)) |
| (closedBall t |q.1-t|) u := by |
| convert! ((ht (u,q.2) (hc u hu).1).sub |
| ((hasDerivAt_id u).smul_const (g t x))).hasFDerivAt.hasFDerivWithinAt using 1 |
| ext |
| simp |
| have htime := (convex_closedBall t |q.1-t|).norm_image_sub_le_of_norm_hasFDerivWithin_le |
| hd (fun u hu => by simpa using (hc u hu).2.le) |
| (mem_closedBall_self (abs_nonneg _)) |
| (show q.1∈closedBall t |q.1-t| by simp [mem_closedBall,dist_eq_norm,Real.norm_eq_abs]) |
| have htime' : ‖f q.1 q.2-f t q.2-(q.1-t) •g t x‖≤(ε/2)*|q.1-t| := by |
| have he : f q.1 q.2-f t q.2-(q.1-t) •g t x= |
| f q.1 q.2-q.1 •g t x-(f t q.2-t •g t x) := by rw [sub_smul]; abel |
| simpa only [he,Real.norm_eq_abs] using htime |
| have he : f q.1 q.2-f t x-((q.1-t) •g t x+D (q.2-x))= |
| (f q.1 q.2-f t q.2-(q.1-t) •g t x)+(f t q.2-f t x-D (q.2-x)) := by abel |
| change ‖f q.1 q.2-f t x-((q.1-t) •g t x+D (q.2-x))‖≤ε*‖q-(t,x)‖ |
| rw [he] |
| apply (norm_add_le _ _).trans |
| apply (add_le_add htime' hqx).trans |
| change ε/2*|q.1-t|+ε/2*‖q.2-x‖≤ε*max |q.1-t| ‖q.2-x‖ |
| have he0 : 0≤ε/2 := by positivity |
| nlinarith [mul_le_mul_of_nonneg_left (le_max_left |q.1-t| ‖q.2-x‖) he0, |
| mul_le_mul_of_nonneg_left (le_max_right |q.1-t| ‖q.2-x‖) he0] |
| end LogConcaveSampling |
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| end UpperProof |
| end |
| end |
| end |
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| end OAI |
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