openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /ContDiffParametricQuadratic.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Contact.CompactMinty | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| lemma contDiffAt_parametric_quadratic_upper {f : E×F → ℝ} {x : E} {a : F} | |
| (hf : ContDiffAt ℝ ∞ f (x,a)) : | |
| ∃ δ>0, ∃ K>0, ∀ z∈Metric.ball x δ, ∀ w∈Metric.ball x δ, | |
| ∀ b∈Metric.ball a δ, | |
| f (w,b)-f (z,b)-fderiv ℝ f (z,b) (w-z,0) ≤ K*‖w-z‖^2 := by | |
| let K := ‖fderiv ℝ (fderiv ℝ f) (x,a)‖+1 | |
| have hK : 0<K := by dsimp [K]; positivity | |
| have hc : ContinuousAt (fun q => ‖fderiv ℝ (fderiv ℝ f) q‖) (x,a) := | |
| ((hf.fderiv_right (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)).continuousAt_fderiv (by norm_num)).norm | |
| have hf2 : ContDiffAt ℝ 2 f (x,a) := hf.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) | |
| have hk : ‖fderiv ℝ (fderiv ℝ f) (x,a)‖<K := by dsimp [K]; linarith | |
| have he : ∀ᶠ q in 𝓝 (x,a), ContDiffAt ℝ 2 f q ∧ | |
| ‖fderiv ℝ (fderiv ℝ f) q‖<K := | |
| (hf2.eventually (by norm_num)).and (hc.preimage_mem_nhds (Iio_mem_nhds hk)) | |
| obtain ⟨δ,hδ,hbound⟩ := Metric.mem_nhds_iff.mp he | |
| refine ⟨δ,hδ,K,hK,?_⟩ | |
| intro z hz w hw b hb | |
| have hzb : (z,b)∈Metric.ball (x,a) δ := by | |
| simpa only [Metric.mem_ball,Prod.dist_eq,max_lt_iff] using And.intro hz hb | |
| have hwb : (w,b)∈Metric.ball (x,a) δ := by | |
| simpa only [Metric.mem_ball,Prod.dist_eq,max_lt_iff] using And.intro hw hb | |
| have H := c2_quadratic_remainder_on_convex hK.le (convex_ball (x,a) δ) | |
| (fun q hq => (hbound hq).1) (fun q hq => (hbound hq).2.le) hzb hwb | |
| have H' := (abs_le.mp (show |f (w,b)-f (z,b)-fderiv ℝ f (z,b) ((w,b)-(z,b))| | |
| ≤ K*‖(w,b)-(z,b)‖^2 from H)).2 | |
| simpa only [Prod.mk_sub_mk,sub_self,Prod.norm_def,norm_zero,max_eq_left (norm_nonneg _)] using H' | |
| end | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace WeakMTWTransport | |
| lemma compact_eventually_uniform_radius_bound {A T : Type*} | |
| [TopologicalSpace A] [TopologicalSpace T] {K : Set A} (hK : IsCompact K) | |
| {t : T} {P : ℝ → ℝ → T → A → Prop} | |
| (hmono : ∀ {r R b B : ℝ}, 0<r → r≤R → b≤B → ∀ s a, P R b s a → P r B s a) | |
| (hlocal : ∀ a∈K, ∃ r>0, ∃ b>0, ∀ᶠ q : T×A in 𝓝 (t,a), P r b q.1 q.2) : | |
| ∃ r>0, ∃ b>0, ∀ᶠ s in 𝓝 t, ∀ a∈K, P r b s a := by | |
| apply hK.induction_on | |
| (p := fun J => ∃ r>0, ∃ b>0, ∀ᶠ s in 𝓝 t, ∀ a∈J, P r b s a) | |
| · exact ⟨1,by norm_num,1,by norm_num,Filter.Eventually.of_forall (by simp)⟩ | |
| · intro J L hJL hL | |
| obtain ⟨r,hr,b,hb,H⟩ := hL | |
| refine ⟨r,hr,b,hb,?_⟩ | |
| filter_upwards [H] with s hs | |
| exact fun a ha => hs a (hJL ha) | |
| · intro J L hJ hL | |
| obtain ⟨r,hr,b,hb,H⟩ := hJ | |
| obtain ⟨R,hR,B,hB,G⟩ := hL | |
| refine ⟨min r R,lt_min hr hR,max b B,lt_max_of_lt_left hb,?_⟩ | |
| filter_upwards [H,G] with s hs gs | |
| intro a ha | |
| rcases ha with ha|ha | |
| · exact hmono (lt_min hr hR) (min_le_left _ _) (le_max_left _ _) s a (hs a ha) | |
| · exact hmono (lt_min hr hR) (min_le_right _ _) (le_max_right _ _) s a (gs a ha) | |
| · intro a ha | |
| obtain ⟨r,hr,b,hb,H⟩ := hlocal a ha | |
| obtain ⟨U,hU,V,hV,hUV⟩ := mem_nhds_prod_iff.mp H | |
| refine ⟨V,mem_nhdsWithin_of_mem_nhds hV,r,hr,b,hb,?_⟩ | |
| filter_upwards [hU] with s hs | |
| intro a ha | |
| exact hUV (show (s,a)∈U×ˢV from ⟨hs,ha⟩) | |
| end WeakMTWTransport | |
| end | |
| end | |
| end | |
| end OAI | |