| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Caps.Caps |
| import OAI.Geometry.CylinderCovering.Footprints.Containment |
| import OAI.Geometry.CylinderCovering.Radial.Riemann |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory Filter |
| open scoped Topology InnerProductSpace |
| noncomputable section |
| namespace CylinderCovering.HalfheightFinite.CoverageLemmas |
|
|
| lemma firstRaw_ne_zero (δ r : ℝ) : firstRaw δ r ≠ 0 := by |
| intro h |
| have h0 := congrArg (fun z : Space => z 0) h |
| norm_num [firstRaw] at h0 |
|
|
| lemma secondRaw_ne_zero (δ r : ℝ) : secondRaw δ r ≠ 0 := by |
| intro h |
| have h1 := congrArg (fun z : Space => z 1) h |
| norm_num [secondRaw] at h1 |
|
|
| lemma mem_projected_cylinder_of_intercept {v p z : Space} (hv : v ≠ 0) |
| {S : Set Space} (hp : p ∈ S) (s : ℝ) (hz : z = p + s • v) : |
| z ∈ RadialSweep.cylinder (‖v‖⁻¹ • v) |
| (TriangularCovering.projection (‖v‖⁻¹ • v) '' S) := by |
| let u := ‖v‖⁻¹ • v |
| have hn : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv |
| have hscale : (s * ‖v‖) • u = s • v := by |
| dsimp [u] |
| rw [smul_smul, mul_assoc, mul_inv_cancel₀ hn, mul_one] |
| refine ⟨TriangularCovering.projection u p, ⟨p, hp, rfl⟩, |
| inner ℝ u p + s * ‖v‖, ?_⟩ |
| rw [hz, TriangularCovering.projection, add_smul, hscale] |
| module |
|
|
| lemma halfwidth_nonneg (N : ℕ) : 0 ≤ halfwidth N := by |
| unfold halfwidth |
| positivity |
|
|
| lemma halfwidth_eq_step (N : ℕ) : |
| halfwidth N = RadialRiemann.step (-1) 1 N / 2 := by |
| unfold halfwidth RadialRiemann.step |
| ring |
|
|
| lemma cell_midpoint_dist {N : ℕ} {i : Fin N} {r : ℝ} |
| (hr : r ∈ RadialRiemann.cell (-1) 1 N i) : |
| |r - midpoint N i| ≤ halfwidth N := by |
| rw [halfwidth_eq_step, abs_le] |
| have hstep := RadialRiemann.point_succ (-1) 1 N i |
| unfold midpoint |
| constructor <;> linarith [hr.1, hr.2] |
|
|
| lemma midpoint_tagged (N : ℕ) : RadialRiemann.IsTagged (-1) 1 N (midpoint N) := by |
| intro i |
| have hstep := RadialRiemann.point_succ (-1) 1 N i |
| have hpos := RadialRiemann.step_pos (by norm_num : (-1:ℝ) < 1) |
| (Nat.zero_lt_of_lt i.isLt) |
| change _ ≤ midpoint N i ∧ midpoint N i ≤ _ |
| unfold midpoint |
| constructor <;> linarith |
|
|
| lemma first_intercept_identity (δ r u X p : ℝ) : |
| point X (u*r+δ*X*alpha r) (u+δ*X*beta r) = |
| firstFoot (u+δ*X*(r-p)/2, u*r+δ*X*(r-p)*(r+p)/4) + |
| (X/2) • firstRaw δ p := by |
| ext j |
| fin_cases j <;> simp [point, firstFoot, firstRaw, alpha, beta] <;> ring |
|
|
| lemma second_intercept_identity (δ r u Y p : ℝ) : |
| point (u*r-δ*Y*alpha r) Y (1-u+δ*Y*beta r) = |
| secondFoot (u-δ*Y*(r-p)/2, u*r-δ*Y*(r-p)*(r+p)/4) + |
| (Y/2) • secondRaw δ p := by |
| ext j |
| fin_cases j <;> simp [point, secondFoot, secondRaw, alpha, beta] <;> ring |
|
|
| lemma first_intercept_mem {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N) |
| {r u X : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i) |
| (hu : u ∈ Icc (0:ℝ) (R r)) (hX : |X| ≤ 1) : |
| (u+δ*X*(r-midpoint N i)/2, |
| u*r+δ*X*(r-midpoint N i)*(r+midpoint N i)/4) ∈ foot δ R N i := by |
| have hd := cell_midpoint_dist hr |
| have he : 0 ≤ collar δ N := |
| div_nonneg (mul_nonneg hδ (halfwidth_nonneg N)) (by norm_num) |
| have hσ := FootprintContainment.halfheight_displacement hδ hX hd |
| have hM : u ≤ maxCap R N i := |
| hu.2.trans (RadialRiemann.le_capMax (by norm_num) hR hr) |
| have h := FootprintContainment.halfheight_trapezoid |
| (halfwidth_nonneg N) he hd hu.1 hM hσ |
| (v := u*r+δ*X*(r-midpoint N i)*(r+midpoint N i)/4) |
| (by ring) rfl |
| exact ⟨⟨h.1, h.2.1⟩, h.2.2⟩ |
|
|
| lemma second_intercept_mem {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N) |
| {r u Y : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i) |
| (hu : u ∈ Icc (0:ℝ) (R r)) (hY : |Y| ≤ 1) : |
| (u-δ*Y*(r-midpoint N i)/2, |
| u*r-δ*Y*(r-midpoint N i)*(r+midpoint N i)/4) ∈ foot δ R N i := by |
| have hd := cell_midpoint_dist hr |
| have he : 0 ≤ collar δ N := |
| div_nonneg (mul_nonneg hδ (halfwidth_nonneg N)) (by norm_num) |
| have hσ : |-(δ*Y*(r-midpoint N i)/2)| ≤ collar δ N := by |
| rw [abs_neg] |
| exact FootprintContainment.halfheight_displacement hδ hY hd |
| have hM : u ≤ maxCap R N i := |
| hu.2.trans (RadialRiemann.le_capMax (by norm_num) hR hr) |
| have h := FootprintContainment.halfheight_trapezoid |
| (halfwidth_nonneg N) he hd hu.1 hM hσ |
| (w := u-δ*Y*(r-midpoint N i)/2) |
| (v := u*r-δ*Y*(r-midpoint N i)*(r+midpoint N i)/4) |
| (by ring) (by ring) |
| exact ⟨⟨h.1, h.2.1⟩, h.2.2⟩ |
|
|
| lemma first_point_mem_cylinder {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N) |
| {r u X : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i) |
| (hu : u ∈ Icc (0:ℝ) (R r)) (hX : |X| ≤ 1) : |
| point X (u*r+δ*X*alpha r) (u+δ*X*beta r) ∈ |
| RadialSweep.cylinder (firstAxis δ (midpoint N i)) (firstBase δ R N i) := by |
| apply mem_projected_cylinder_of_intercept (firstRaw_ne_zero δ (midpoint N i)) |
| (s := X/2) |
| · exact ⟨_, first_intercept_mem hδ hR i hr hu hX, rfl⟩ |
| · exact first_intercept_identity δ r u X (midpoint N i) |
|
|
| lemma second_point_mem_cylinder {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N) |
| {r u Y : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i) |
| (hu : u ∈ Icc (0:ℝ) (R r)) (hY : |Y| ≤ 1) : |
| point (u*r-δ*Y*alpha r) Y (1-u+δ*Y*beta r) ∈ |
| RadialSweep.cylinder (secondAxis δ (midpoint N i)) (secondBase δ R N i) := by |
| apply mem_projected_cylinder_of_intercept (secondRaw_ne_zero δ (midpoint N i)) |
| (s := Y/2) |
| · exact ⟨_, second_intercept_mem hδ hR i hr hu hY, rfl⟩ |
| · exact second_intercept_identity δ r u Y (midpoint N i) |
|
|
| theorem family_covers_of_covered {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) (hcover : HalfHeightCaps.Covered δ R) |
| {N : ℕ} (hN : 0 < N) : |
| ∀ z ∈ tetrahedron, |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder |
| (firstAxis δ (midpoint N i)) (firstBase δ R N i)) ∨ |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder |
| (secondAxis δ (midpoint N i)) (secondBase δ R N i)) := by |
| rintro z ⟨X,Y,Z,hX,hY,rfl⟩ |
| have hX1 : |X| ≤ 1 := by linarith [abs_nonneg Y] |
| have hY1 : |Y| ≤ 1 := by linarith [abs_nonneg X] |
| rcases hcover X Y Z hX hY with ⟨r,u,hr,hu,hRu,hYr,hZr⟩ | |
| ⟨r,u,hr,hu,hRu,hXr,hZr⟩ |
| · obtain ⟨i,hi⟩ := RadialRiemann.exists_cell hN (abs_le.mp hr) |
| left |
| refine ⟨i, ?_⟩ |
| rw [hYr,hZr] |
| exact first_point_mem_cylinder hδ hR i hi ⟨hu,hRu⟩ hX1 |
| · obtain ⟨i,hi⟩ := RadialRiemann.exists_cell hN (abs_le.mp hr) |
| right |
| refine ⟨i, ?_⟩ |
| have hZ : Z = 1-u+δ*Y*HalfHeightCaps.beta r := by linarith only [hZr] |
| rw [hXr,hZ] |
| exact second_point_mem_cylinder hδ hR i hi ⟨hu,hRu⟩ hY1 |
|
|
| end CylinderCovering.HalfheightFinite.CoverageLemmas |
|
|
| end |
|
|
| end OAI |
|
|
