| import OAI.Geometry.IsometricImmersion.Darboux.QEnergyApplication | |
| import Mathlib.Topology.MetricSpace.Thickening | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic | |
| private theorem real_near_closed_interval {a b e x : ℝ} (hab : a ≤ b) | |
| (he : 0 < e) (hx : x ∈ Ioo (a-e) (b+e)) : | |
| ∃ y ∈ Icc a b, |x-y| < e := by | |
| by_cases hxa : x < a | |
| · refine ⟨a,⟨le_rfl,hab⟩,?_⟩ | |
| rw [abs_of_neg (sub_neg.mpr hxa)] | |
| linarith [hx.1] | |
| by_cases hbx : b < x | |
| · refine ⟨b,⟨hab,le_rfl⟩,?_⟩ | |
| rw [abs_of_pos (sub_pos.mpr hbx)] | |
| linarith [hx.2] | |
| exact ⟨x,⟨le_of_not_gt hxa,le_of_not_gt hbx⟩,by simpa using he⟩ | |
| theorem closedRectangle_compact (l r a b : ℝ) : IsCompact (closedRectangle l r a b) := by | |
| have he : closedRectangle l r a b = Icc (![l,a] : Coord) ![r,b] := by | |
| ext p | |
| constructor | |
| · intro hp | |
| constructor | |
| · intro i | |
| fin_cases i | |
| · exact hp.1.1 | |
| · exact hp.2.1 | |
| · intro i | |
| fin_cases i | |
| · exact hp.1.2 | |
| · exact hp.2.2 | |
| · intro hp | |
| exact ⟨⟨hp.1 0,hp.2 0⟩,⟨hp.1 1,hp.2 1⟩⟩ | |
| rw [he] | |
| exact isCompact_Icc | |
| theorem exists_containing_coordinateRectangle {U : Set Coord} (hU : IsOpen U) | |
| {R a b : ℝ} (hR : 0 < R) (hab : a ≤ b) | |
| (hbox : closedRectangle (-R) R a b ⊆ U) : | |
| ∃ radius lo hi : ℝ, R < radius ∧ lo < a ∧ b < hi ∧ | |
| coordinateRectangle radius lo hi ⊆ U := by | |
| obtain ⟨e,he,hthick⟩ := (closedRectangle_compact (-R) R a b).exists_thickening_subset_open hU hbox | |
| refine ⟨R+e,a-e,b+e,by linarith,by linarith,by linarith,?_⟩ | |
| intro p hp | |
| obtain ⟨x,hx,hxe⟩ := real_near_closed_interval (show -R ≤ R by linarith) he | |
| (show p 0 ∈ Ioo (-R-e) (R+e) by | |
| exact ⟨by linarith [hp.1.1],hp.1.2⟩) | |
| obtain ⟨t,ht,hte⟩ := real_near_closed_interval hab he hp.2 | |
| apply hthick | |
| rw [Metric.mem_thickening_iff] | |
| refine ⟨![x,t],⟨hx,ht⟩,?_⟩ | |
| rw [dist_eq_norm] | |
| apply (pi_norm_lt_iff he).mpr | |
| intro i | |
| fin_cases i | |
| · change ‖p 0-x‖ < e | |
| simpa only [Real.norm_eq_abs] using hxe | |
| · change ‖p 1-t‖ < e | |
| simpa only [Real.norm_eq_abs] using hte | |
| theorem exists_actual_Q_containing_rectangle | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} | |
| (hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U) | |
| {R a b : ℝ} (hR : 0 < R) (hab : a ≤ b) | |
| (hbox : closedRectangle (-R) R a b ⊆ U) | |
| (hxx : ∀ p ∈ closedRectangle (-R) R a b, covHessian g z p 0 0 ≠ 0) : | |
| ∃ radius lo hi : ℝ, R < radius ∧ lo < a ∧ b < hi ∧ | |
| coordinateRectangle radius lo hi ⊆ U ∧ | |
| (∀ p ∈ coordinateRectangle radius lo hi, covHessian g z p 0 0 ≠ 0) := by | |
| let V := U ∩ (fun p => covHessian g z p 0 0) ⁻¹' ({0}ᶜ : Set ℝ) | |
| have hV : IsOpen V := (covHessian_contDiffOn hg hU hz 0 0).continuousOn.isOpen_inter_preimage hU | |
| isClosed_singleton.isOpen_compl | |
| have hsub : closedRectangle (-R) R a b ⊆ V := by | |
| intro p hp | |
| exact ⟨hbox hp,hxx p hp⟩ | |
| obtain ⟨radius,lo,hi,hr,hl,hh,hVU⟩ := exists_containing_coordinateRectangle hV hR hab hsub | |
| exact ⟨radius,lo,hi,hr,hl,hh,fun p hp => (hVU hp).1,fun p hp => (hVU hp).2⟩ | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |