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695bd70 | 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 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Geometry.RuledCovering.HyperbolicLattice
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
lemma HasFiniteCover.of_fintype {ι : Type*} [Fintype ι] {shape : Cylinder → Prop}
{S : Set Space} {A : ℝ} (C : ι → Cylinder) (hshape : ∀ i, shape (C i))
(hcover : S ⊆ ⋃ i, (C i).carrier) (hcost : (∑ i, (C i).area) ≤ A) :
HasFiniteCover shape S A := by
classical
let e := Fintype.equivFin ι
refine ⟨Fintype.card ι, C ∘ e.symm, fun i => hshape _, ?_, ?_⟩
· intro x hx
obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hx)
exact mem_iUnion.mpr ⟨e i, by simpa using hi⟩
· change (∑ i : Fin (Fintype.card ι), (C (e.symm i)).area) ≤ A
rw [e.symm.sum_comp (fun j => (C j).area)]
exact hcost
/-- Convert the finite weighted lattice certificate into actual perpendicular Euclidean cylinders.
The factor is the exact area inflation caused by the derivative remainder. -/
theorem framed_local_cover (M : Coordinates ≃L[ℝ] Plane) {h L R delta : ℝ}
(hh : 0 < h) (hL : 0 ≤ L) (hR : 0 ≤ R) (hd : 0 ≤ delta)
{K U : Set Coordinates} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
(v : Plane → Plane) (A : Coordinates →L[ℝ] Coordinates)
(hmodel : LatticeModel A L R)
(hrem : ∀ p ∈ U, ∀ b ∈ U,
‖M.symm (v (M p)) - M.symm (v (M b)) - A (p - b)‖ ≤ delta * ‖p - b‖)
{F : Coordinates → ℝ} (hF : Continuous F) (hFpos : ∀ p, 0 ≤ F p)
(hFval : ∀ p ∈ U, F p = physicalWeight h (v (M p)))
{eps : ℝ} (heps : 0 < eps) :
HasFiniteCover Cylinder.parallelogram (physicalRuled h (M '' K) v L)
(h * (1 + 2 * L * delta * R) ^ 2 *
(∫ p in M '' K, physicalWeight h (v p)) + eps) := by
classical
let B : ℝ := (1 + 2 * L * delta * R) ^ 2
have hB : 0 < B := by dsimp [B]; positivity
have hscale : 0 < h * frameJacobian M * B := mul_pos (mul_pos hh (frameJacobian_pos M)) hB
obtain ⟨n, hn, S, hS, hSU, hcov, hcost⟩ := local_finite_certificate hK hU hKU
(fun p => M.symm (v (M p))) A hL hd hmodel hrem hF hFpos (div_pos heps hscale)
let : Fintype S := hS.fintype
let a : ℝ := (1 / 2 + L * delta * R) / n
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have ha : 0 < a := by dsimp [a]; positivity
let C : S → Cylinder := fun b => projectedCylinder h (v (M b)) (M '' closedBall (b : Coordinates) a)
have harea (b : S) : (C b).area = h * frameJacobian M * (2 * a) ^ 2 * F b := by
dsimp only [C]
rw [projectedCylinder_area hh, frame_volume_image M isClosed_closedBall.measurableSet,
coordinates_closedBall_volume b ha.le, hFval b (hSU b.property)]
ring
have hBrel : (2 * a) ^ 2 = B / (n : ℝ) ^ 2 := by
dsimp [a, B]
ring
have hintegral : (∫ p in M '' K, physicalWeight h (v p)) =
frameJacobian M * ∫ x in K, F x := by
rw [frame_integral_image M hK.measurableSet]
congr 1
apply setIntegral_congr_fun hK.measurableSet
intro x hx
exact (hFval x (hKU hx)).symm
apply HasFiniteCover.of_fintype C
· intro b
apply projectedCylinder_parallelogram hh
exact (coordinates_closedBall_parallelogram (b : Coordinates) ha).linear_image
M.toLinearMap M.injective
· rintro x ⟨_, ⟨p, hp, rfl⟩, s, hs, rfl⟩
obtain ⟨b, hb, hc⟩ := hcov p hp s hs
apply mem_iUnion.mpr
refine ⟨⟨b, hb⟩, ?_⟩
apply physicalRuled_mem_projectedCylinder
refine ⟨p + s • (M.symm (v (M p)) - M.symm (v (M b))), ?_, ?_⟩
· rw [mem_closedBall, dist_eq_norm]
convert hc using 1
congr 1
module
· simp
· have hsum : (∑ b : S, (C b).area) =
(h * frameJacobian M * B) * ((∑' b : S, F b) / (n : ℝ) ^ 2) := by
simp_rw [harea, hBrel]
rw [tsum_fintype, ← Finset.mul_sum]
ring
rw [hsum, hintegral]
have hc := (mul_lt_mul_of_pos_left hcost hscale).le
calc
_ ≤ (h * frameJacobian M * B) *
((∫ x in K, F x) + eps / (h * frameJacobian M * B)) := hc
_ = _ := by
rw [mul_add, mul_div_cancel₀ _ hscale.ne']
dsimp [B]
ring
end RuledApproximation
end
end OAI
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