| import OAI.Probability.RadialTransition.BoundedCore |
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| namespace OAI |
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| universe uE uX uY |
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| noncomputable section |
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| open scoped BigOperators NNReal ContDiff ENNReal |
| open Filter MeasureTheory Set |
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| namespace RadialTransition |
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| section CompactEnvelope |
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| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
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| theorem support_norm_bound {F : E → ℝ} {D : Set E} {x v : E} {r M : ℝ} |
| (hr : 0 < r) (hM : 0 ≤ M) (hball : Metric.closedBall x r ⊆ D) |
| (hbound : ∀ y ∈ Metric.closedBall x r, |F y| ≤ M) |
| (hv : SupportsOn F D x v) : ‖v‖ ≤ 2 * M / r := by |
| apply norm_le_of_unit_inner_le _ (by positivity) |
| intro z hz |
| have hy : x + r • z ∈ Metric.closedBall x r := by |
| simp [Metric.mem_closedBall, dist_eq_norm, norm_smul, abs_of_pos hr, hz] |
| have hx : x ∈ Metric.closedBall x r := Metric.mem_closedBall_self hr.le |
| have hh := hv (x + r • z) (hball hy) |
| simp only [add_sub_cancel_left, real_inner_smul_right] at hh |
| have hlo := (abs_le.mp (hbound x hx)).1 |
| have hhi := (abs_le.mp (hbound _ hy)).2 |
| apply (le_div_iff₀ hr).mpr |
| nlinarith |
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| theorem supportsOn_of_tendsto {F : ℕ → E → ℝ} {F₀ : E → ℝ} {D : Set E} |
| {x v : ℕ → E} {x₀ v₀ : E} |
| (hx : Tendsto x atTop (nhds x₀)) (hv : Tendsto v atTop (nhds v₀)) |
| (hbase : Tendsto (fun n => F n (x n)) atTop (nhds (F₀ x₀))) |
| (hpoint : ∀ y ∈ D, Tendsto (fun n => F n y) atTop (nhds (F₀ y))) |
| (hs : ∀ n, SupportsOn (F n) D (x n) (v n)) : |
| SupportsOn F₀ D x₀ v₀ := by |
| intro y hy |
| exact le_of_tendsto_of_tendsto |
| (hbase.add (hv.inner (tendsto_const_nhds.sub hx))) (hpoint y hy) |
| (Eventually.of_forall fun n => hs n y hy) |
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| theorem continuousOn_tendsto_sequence {X : Type uX} {Y : Type uY} [TopologicalSpace X] |
| [TopologicalSpace Y] {f : X → Y} {S : Set X} (hf : ContinuousOn f S) |
| {a : X} (ha : a ∈ S) {s : ℕ → X} (hs : ∀ n, s n ∈ S) |
| (hsa : Tendsto s atTop (nhds a)) : Tendsto (fun n => f (s n)) atTop (nhds (f a)) := by |
| apply (hf.continuousWithinAt ha).tendsto.comp |
| exact tendsto_nhdsWithin_iff.mpr ⟨hsa, Eventually.of_forall hs⟩ |
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| theorem supportSet_envelope_closedHull [FiniteDimensional ℝ E] |
| {F : ℝ → E → ℝ} {P : E → ℝ} {D : Set E} {K : Set ℝ} {x : E} {r : ℝ} |
| (hD : IsOpen D) (hK : IsCompact K) (hr : 0 < r) |
| (hball : Metric.closedBall x (2 * r) ⊆ D) |
| (hF : ∀ i ∈ K, ConvexOn ℝ D (F i)) |
| (hFc : ContinuousOn (fun z : ℝ × E => F z.1 z.2) (K ×ˢ D)) |
| (hPc : ContinuousOn P D) |
| (hle : ∀ i ∈ K, ∀ y ∈ D, F i y ≤ P y) |
| (hactive : ∀ y ∈ Metric.closedBall x r, ∃ i ∈ K, F i y = P y) : |
| supportSet P D x = closure (convexHull ℝ |
| {v | ∃ i ∈ K, F i x = P x ∧ SupportsOn (F i) D x v}) := by |
| classical |
| let G : Set E := {v | ∃ i ∈ K, F i x = P x ∧ SupportsOn (F i) D x v} |
| have hx : x ∈ D := hball (Metric.mem_closedBall_self (by positivity)) |
| have hG : G ⊆ supportSet P D x := by |
| rintro v ⟨i, hi, heq, hv⟩ y hy |
| rw [← heq] |
| exact (hv y hy).trans (hle i hi y hy) |
| have hback : closure (convexHull ℝ G) ⊆ supportSet P D x := |
| closure_minimal (convexHull_min hG (convex_supportSet P D x)) |
| (isClosed_supportSet P D x) |
| apply Set.Subset.antisymm ?_ hback |
| intro v hv |
| by_contra hvout |
| obtain ⟨ell, c, hsep, hvc⟩ := geometric_hahn_banach_closed_point |
| (convex_convexHull ℝ G).closure isClosed_closure hvout |
| let z : E := (InnerProductSpace.toDual ℝ E).symm ell |
| have hell (w : E) : inner ℝ w z = ell w := by |
| rw [real_inner_comm] |
| exact InnerProductSpace.toDual_symm_apply |
| obtain ⟨M₀, hM₀⟩ := (hK.prod (isCompact_closedBall x (2 * r))).exists_bound_of_continuousOn |
| (hFc.mono (Set.prod_mono Subset.rfl hball)) |
| let M := max 0 M₀ |
| have hM : 0 ≤ M := le_max_left _ _ |
| have hbound (i : ℝ) (hi : i ∈ K) (y : E) (hy : y ∈ Metric.closedBall x (2 * r)) : |
| |F i y| ≤ M := (hM₀ (i, y) ⟨hi, hy⟩).trans (le_max_right _ _) |
| let s : ℕ → ℝ := fun n => (r / (‖z‖ + 1)) * (1 / ((n : ℝ) + 1)) |
| have hspos (n : ℕ) : 0 < s n := by dsimp [s]; positivity |
| have hsbound (n : ℕ) : s n * ‖z‖ ≤ r := by |
| have hn : 1 ≤ (n : ℝ) + 1 := by have := Nat.cast_nonneg (α := ℝ) n; linarith |
| have hden : 0 < ‖z‖ + 1 := by positivity |
| have hdiv : 1 / ((n : ℝ) + 1) ≤ 1 := (div_le_one (by positivity)).mpr hn |
| have hq : r / (‖z‖ + 1) * ‖z‖ ≤ r := by |
| rw [div_mul_eq_mul_div] |
| exact (div_le_iff₀ hden).mpr (show r * ‖z‖ ≤ r * (‖z‖ + 1) by nlinarith) |
| calc |
| s n * ‖z‖ = (r / (‖z‖ + 1) * ‖z‖) * (1 / ((n : ℝ) + 1)) := by dsimp [s]; ring |
| _ ≤ (r / (‖z‖ + 1) * ‖z‖) * 1 := |
| mul_le_mul_of_nonneg_left hdiv (by positivity) |
| _ ≤ r := by simpa using hq |
| have hs0 : Tendsto s atTop (nhds 0) := by |
| simpa [s] using tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (r / (‖z‖ + 1)) |
| let y : ℕ → E := fun n => x + s n • z |
| have hymem (n : ℕ) : y n ∈ Metric.closedBall x r := by |
| simpa [y, Metric.mem_closedBall, dist_eq_norm, norm_smul, abs_of_pos (hspos n)] using hsbound n |
| have hyD (n : ℕ) : y n ∈ D := |
| hball (Metric.closedBall_subset_closedBall (by linarith) (hymem n)) |
| have hy0 : Tendsto y atTop (nhds x) := by |
| simpa [y] using tendsto_const_nhds.add (hs0.smul (tendsto_const_nhds (x := z))) |
| choose i hi hia using fun n => hactive (y n) (hymem n) |
| choose w hw using fun n => supportSet_nonempty hD (hF (i n) (hi n)) (hyD n) |
| have hwbound (n : ℕ) : ‖w n‖ ≤ 2 * M / r := by |
| have hb : Metric.closedBall (y n) r ⊆ Metric.closedBall x (2 * r) := by |
| intro q hq |
| apply (dist_triangle q (y n) x).trans |
| have hqn := Metric.mem_closedBall.mp hq |
| have hyn := Metric.mem_closedBall.mp (hymem n) |
| linarith |
| exact support_norm_bound hr hM (hb.trans hball) |
| (fun q hq => hbound (i n) (hi n) q (hb hq)) (hw n) |
| obtain ⟨iw, hiw, σ, hσ, hlim⟩ := |
| (hK.prod (isCompact_closedBall (0 : E) (2 * M / r))).tendsto_subseq |
| (x := fun n => (i n, w n)) (fun n => ⟨hi n, by simpa using hwbound n⟩) |
| have hiwK : iw.1 ∈ K := hiw.1 |
| have hilim : Tendsto (fun n => i (σ n)) atTop (nhds iw.1) := |
| continuous_fst.tendsto iw |>.comp hlim |
| have hwlim : Tendsto (fun n => w (σ n)) atTop (nhds iw.2) := |
| continuous_snd.tendsto iw |>.comp hlim |
| have hylim : Tendsto (fun n => y (σ n)) atTop (nhds x) := hy0.comp hσ.tendsto_atTop |
| have hbase : Tendsto (fun n => F (i (σ n)) (y (σ n))) atTop (nhds (F iw.1 x)) := |
| continuousOn_tendsto_sequence hFc (a := (iw.1, x)) ⟨hiwK, hx⟩ |
| (fun n => ⟨hi (σ n), hyD (σ n)⟩) (hilim.prodMk_nhds hylim) |
| have hPbase : Tendsto (fun n => P (y (σ n))) atTop (nhds (P x)) := |
| continuousOn_tendsto_sequence hPc hx (fun n => hyD (σ n)) hylim |
| have hia0 : F iw.1 x = P x := by |
| apply tendsto_nhds_unique hbase |
| simpa only [hia] using hPbase |
| have hiws : SupportsOn (F iw.1) D x iw.2 := by |
| apply supportsOn_of_tendsto hylim hwlim hbase ?_ (fun n => hw (σ n)) |
| intro q hq |
| exact continuousOn_tendsto_sequence hFc (a := (iw.1, q)) ⟨hiwK, hq⟩ |
| (fun n => ⟨hi (σ n), hq⟩) (hilim.prodMk_nhds tendsto_const_nhds) |
| have hiwG : iw.2 ∈ G := ⟨iw.1, hiwK, hia0, hiws⟩ |
| have hendpoint (n : ℕ) : inner ℝ v z ≤ inner ℝ (w n) z := by |
| have hleft := hv (y n) (hyD n) |
| have hright := hw n x hx |
| have hdom := hle (i n) (hi n) x hx |
| have hyx : y n - x = s n • z := by simp [y] |
| have hxy : x - y n = -(s n • z) := by simp [y] |
| rw [hyx, real_inner_smul_right] at hleft |
| rw [hxy, inner_neg_right, real_inner_smul_right, hia n] at hright |
| exact (mul_le_mul_iff_right₀ (hspos n)).mp (by nlinarith [hleft, hright, hdom]) |
| have hfinal : inner ℝ v z ≤ inner ℝ iw.2 z := |
| le_of_tendsto_of_tendsto tendsto_const_nhds (hwlim.inner tendsto_const_nhds) |
| (Eventually.of_forall fun n => hendpoint (σ n)) |
| rw [hell, hell] at hfinal |
| have hh := hsep iw.2 (subset_closure (subset_convexHull ℝ G hiwG)) |
| linarith |
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| end CompactEnvelope |
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| end RadialTransition |
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| end |
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| end OAI |
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