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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
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
open Set Filter Metric Function
open scoped Topology
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
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
end UpperProof
end
end
end
end OAI
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