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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
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