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720fa25 | 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 | import Mathlib
import OAI.Geometry.BallPacking.Continuation.ProperPaths
namespace OAI
noncomputable section
open scoped ContDiff Topology
open Set Function Filter
namespace HigherDimensionalBallPacking
open Rigidity Rigidity.HolderCompletion
theorem adapted_affine_line {n : ℕ} {J : Phase n → End n}
(hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x))
(hJc : HasCompactSupport (fun x => J x-standardJ n))
(hinside : tsupport (fun x => J x-standardJ n)⊆openBall n 1)
(p q : Phase n) (hpq : p≠q) : ∃ u,AffineLineCurve J p q u := by
let : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance
let : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance
let : NormedAddCommGroup (HMap ℂ (Phase n)) := inferInstance
let : NormedSpace ℝ (HMap ℂ (Phase n)) := inferInstance
obtain ⟨β,β₁,M,δ,hpath,hend⟩ := adapted_line_continuation_data hJs hJ hJc hinside p q hpq
let : NormedAddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance
let : NormedSpace ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance
let : NormedAddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance
let : NormedSpace ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance
obtain ⟨w,hw,hz⟩ := hpath.endpoint_exists
exact ⟨_,hend w hw hz⟩
theorem normalized_two_ball_bound (n : ℕ) (_ : 3≤n)
(a b : ℝ) (ha : 0<a) (hb : 0<b) (hp : HasPacking n 2 1 ![a,b]) : a+b≤1 := by
obtain ⟨J,p,q,hJs,hJ,hJc,hinside,hpq,harea⟩ := packing_two_line_reduction ha hb hp
obtain ⟨u,hu⟩ := adapted_affine_line hJs hJ hJc hinside p q hpq
exact harea u hu
end HigherDimensionalBallPacking
end
end OAI
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