| import OAI.Probability.ContinuumTransition.ConstructionBounds |
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| namespace OAI |
|
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| section |
| open scoped Topology |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
|
|
| lemma uniform_square_error_tendsto {F : Parameter → ℝ} {Q : ℕ → Parameter → ℝ} |
| {a : ℕ → ℝ} (ha : Tendsto a atTop (𝓝 0)) {x : Parameter} {r : ℝ} |
| (happ : ∀ n, ∀ y ∈ ball x r, |Q n y-F y| ≤ (a n)^2) |
| {Y : ℕ → Parameter} {y : Parameter} (hy : y ∈ ball x r) |
| (hY : Tendsto Y atTop (𝓝 y)) (hF : ContinuousAt F y) : |
| Tendsto (fun n => Q n (Y n)) atTop (𝓝 (F y)) := by |
| have he : Tendsto (fun n => Q n (Y n)-F (Y n)) atTop (𝓝 0) := by |
| apply (tendsto_zero_iff_abs_tendsto_zero _).mpr |
| exact squeeze_zero' (Eventually.of_forall fun n => abs_nonneg (Q n (Y n)-F (Y n))) |
| ((hY.eventually (isOpen_ball.mem_nhds hy)).mono fun n hn => happ n _ hn) (by simpa using ha.pow 2) |
| have hh := he.add (hF.tendsto.comp hY) |
| simpa only [Function.comp_def,sub_add_cancel,zero_add] using hh |
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| lemma approximate_support_limit {F : Parameter → ℝ} (hF : ConvexOn ℝ pressureDomain F) |
| {Q : ℕ → Parameter → ℝ} {a : ℕ → ℝ} (ha : Tendsto a atTop (𝓝 0)) |
| {x : Parameter} (hx : x ∈ pressureDomain) {r : ℝ} (hr : 0 < r) |
| (hball : ball x r ⊆ pressureDomain) |
| (happ : ∀ n, ∀ y ∈ ball x r, |Q n y-F y| ≤ (a n)^2) |
| {Y V : ℕ → Parameter} {v : Parameter} |
| (hY : Tendsto Y atTop (𝓝 x)) (hV : Tendsto V atTop (𝓝 v)) |
| (hs : ∀ n, SupportOn pressureDomain (Q n) (Y n) (V n)) : |
| SupportOn pressureDomain F x v := by |
| apply supportOn_of_ball hF hx hr |
| intro y hy |
| have hcont := hF.continuousOn isOpen_pressureDomain |
| have hqx := uniform_square_error_tendsto ha happ (mem_ball_self hr) hY |
| (hcont.continuousAt (isOpen_pressureDomain.mem_nhds hx)) |
| have hqy := uniform_square_error_tendsto ha happ hy (tendsto_const_nhds (x := y)) |
| (hcont.continuousAt (isOpen_pressureDomain.mem_nhds (hball hy))) |
| have hd : Tendsto (fun n => pDot (V n) (y-Y n)) atTop (𝓝 (pDot v (y-x))) := by |
| exact (show Continuous (fun p : Parameter × Parameter => pDot p.1 (y-p.2)) by unfold pDot; fun_prop).tendsto |
| (v,x) |>.comp (hV.prodMk_nhds hY) |
| exact le_of_tendsto_of_tendsto (hqx.add hd) hqy (Eventually.of_forall fun n => hs n y (hball hy)) |
|
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| end ContinuumTemperature |
| end |
| end |
|
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| end OAI |
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