| import OAI.Probability.ContinuumTransition.ConvexSupport |
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| namespace OAI |
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| universe u_1 u_2 |
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| section |
| open scoped Topology NNReal |
| open Filter Set Metric |
| noncomputable section |
| namespace ContinuumTemperature |
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| variable {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {l : Filter ι} [l.NeBot] {S : Set E} {F : ι → E → ℝ} {f : E → ℝ} |
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| omit [FiniteDimensional ℝ E] in |
| lemma convexOn_of_pointwise_tendsto (hconv : ∀ᶠ i in l, ConvexOn ℝ S (F i)) |
| (hlim : ∀ x ∈ S, Tendsto (fun i => F i x) l (𝓝 (f x))) : ConvexOn ℝ S f := by |
| obtain ⟨i,hi⟩ := hconv.exists |
| refine ⟨hi.1,?_⟩ |
| intro x hx y hy a b ha hb hab |
| have hz := hi.1 hx hy ha hb hab |
| exact le_of_tendsto_of_tendsto (hlim _ hz) |
| (((hlim x hx).const_mul a).add ((hlim y hy).const_mul b)) |
| (hconv.mono fun i hi => hi.2 hx hy ha hb hab) |
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| omit [l.NeBot] in |
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| lemma eventually_convex_lipschitz (hS : IsOpen S) |
| (hconv : ∀ᶠ i in l, ConvexOn ℝ S (F i)) |
| (hlim : ∀ x ∈ S, Tendsto (fun i => F i x) l (𝓝 (f x))) {x : E} (hx : x ∈ S) : |
| ∃ r > 0, ball x r ⊆ S ∧ ∃ C : ℝ≥0, ∀ᶠ i in l, LipschitzOnWith C (F i) (ball x r) := by |
| obtain ⟨b,hxb,hbS⟩ := exists_mem_interior_convexHull_affineBasis (hS.mem_nhds hx) |
| obtain ⟨M,hM⟩ := (finite_range (fun j => f (b j))).bddAbove |
| let C : ℝ := |M|+|f x|+2 |
| have hC : 0 < C := by dsimp [C] |
| have hv : ∀ᶠ i in l, ∀ j, F i (b j) ≤ C := by |
| apply eventually_all.mpr |
| intro j |
| have hb : b j ∈ S := hbS (subset_convexHull ℝ _ (mem_range_self j)) |
| filter_upwards [(hlim _ hb).eventually_lt_const (show f (b j) < f (b j)+1 by linarith)] with i hi |
| have hm := hM (mem_range_self j) |
| have hMle := le_abs_self M |
| have hxf := abs_nonneg (f x) |
| dsimp [C] |
| linarith |
| have hxlower : ∀ᶠ i in l, -C ≤ F i x := by |
| filter_upwards [(hlim _ hx).eventually_const_lt (show f x-1 < f x by linarith)] with i hi |
| have hxabs := neg_abs_le (f x) |
| have hm := abs_nonneg M |
| dsimp [C] |
| linarith |
| obtain ⟨r,hr,hrb⟩ := Metric.mem_nhds_iff.mp (isOpen_interior.mem_nhds hxb) |
| have hrh : ball x r ⊆ convexHull ℝ (range b) := hrb.trans interior_subset |
| refine ⟨r/2,by positivity,(ball_subset_ball (by linarith : r/2 ≤ r)).trans (hrh.trans hbS), |
| (2*(3*C)/(r/2)).toNNReal,?_⟩ |
| filter_upwards [hconv,hv,hxlower] with i hi hiv hix |
| have hub : ∀ y ∈ convexHull ℝ (range b), F i y ≤ C := by |
| have hh : convexHull ℝ (range b) ⊆ {y ∈ S | F i y ≤ C} := |
| convexHull_min (by rintro y ⟨j,rfl⟩; exact ⟨hbS (subset_convexHull ℝ _ (mem_range_self j)),hiv j⟩) |
| (hi.convex_le C) |
| exact fun y hy => (hh hy).2 |
| have habs : ∀ y, dist y x < r → |F i y| ≤ 3*C := by |
| intro y hy |
| have hyb := hrh hy |
| have hreflect : 2 • x-y ∈ ball x r := by |
| rw [mem_ball,dist_eq_norm_sub,show 2 • x-y-x = x-y by module,norm_sub_rev] |
| simpa only [dist_eq_norm_sub] using hy |
| have huby := hub y hyb |
| have hubr := hub _ (hrh hreflect) |
| have hmid := hi.2 (hbS hyb) (hbS (hrh hreflect)) |
| (show (0:ℝ) ≤ 1/2 by norm_num) (show (0:ℝ) ≤ 1/2 by norm_num) |
| (show (1/2:ℝ)+1/2=1 by norm_num) |
| have he : (1/2:ℝ) • y + (1/2:ℝ) • (2 • x-y) = x := by module |
| rw [he] at hmid |
| simp only [smul_eq_mul] at hmid |
| exact abs_le.mpr ⟨by linarith,by linarith⟩ |
| have hL := (hi.subset (hrh.trans hbS) (convex_ball x r)).lipschitzOnWith_of_abs_le |
| (show 0 < r/2 by positivity) habs |
| simpa only [show r-r/2=r/2 by ring] using hL |
| |
| theorem convex_pointwise_locally_uniform (hS : IsOpen S) |
| (hconv : ∀ᶠ i in l, ConvexOn ℝ S (F i)) |
| (hlim : ∀ x ∈ S, Tendsto (fun i => F i x) l (𝓝 (f x))) : |
| TendstoLocallyUniformlyOn F f l S := by |
| have hf := (convexOn_of_pointwise_tendsto hconv hlim).continuousOn hS |
| apply Metric.tendstoLocallyUniformlyOn_iff.mpr |
| intro ε hε x hx |
| obtain ⟨r,hr,_,C,hC⟩ := eventually_convex_lipschitz hS hconv hlim hx |
| have hfnear : ∀ᶠ y in 𝓝 x, dist (f y) (f x) < ε/3 := |
| (hf.continuousAt (hS.mem_nhds hx)).eventually (ball_mem_nhds _ (by positivity)) |
| let T := ball x r ∩ ball x (ε/(3*((C:ℝ)+1))) ∩ {y | dist (f y) (f x) < ε/3} |
| have hT : T ∈ 𝓝 x := inter_mem (inter_mem (ball_mem_nhds _ hr) |
| (ball_mem_nhds _ (by positivity))) hfnear |
| refine ⟨T,nhdsWithin_le_nhds hT,?_⟩ |
| filter_upwards [hC,(hlim x hx).eventually (ball_mem_nhds _ (show 0 < ε/3 by positivity))] |
| with i hi hxlim |
| intro y hy |
| have hd := lipschitzOnWith_iff_dist_le_mul.mp hi x (mem_ball_self hr) y hy.1.1 |
| have hεC : 0 < 3*((C:ℝ)+1) := by positivity |
| have hyd : dist x y < ε/(3*((C:ℝ)+1)) := by simpa only [mem_ball,dist_comm] using hy.1.2 |
| have hdd : (C:ℝ)*dist x y < ε/3 := by |
| have he := (lt_div_iff₀ hεC).mp hyd |
| have hnn := dist_nonneg (x := x) (y := y) |
| nlinarith |
| have hb : dist (f x) (F i x) < ε/3 := by simpa only [mem_ball,dist_comm] using hxlim |
| have hyf : dist (f y) (f x) < ε/3 := hy.2 |
| calc |
| dist (f y) (F i y) ≤ dist (f y) (f x)+dist (f x) (F i x)+dist (F i x) (F i y) := dist_triangle4 _ _ _ _ |
| _ < ε := by linarith only [hyf,hb,hd,hdd] |
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| end ContinuumTemperature |
| end |
| end |
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| end OAI |
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