| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Caps.Caps |
| import OAI.Geometry.CylinderCovering.Radial.Riemann |
| import OAI.Geometry.CylinderCovering.Radial.PartitionCover |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology InnerProductSpace |
| namespace CylinderCovering.HalfheightFinite |
|
|
| lemma halfwidth_nonneg (N : ℕ) : 0 ≤ halfwidth N := by |
| unfold halfwidth |
| positivity |
|
|
| lemma midpoint_distance {N : ℕ} (i : Fin N) {r : ℝ} |
| (hr : r ∈ RadialRiemann.cell (-1) 1 N i) : |
| |r - midpoint N i| ≤ halfwidth N := by |
| have hs := RadialRiemann.point_succ (-1) 1 N (i : ℕ) |
| change RadialRiemann.point (-1) 1 N i ≤ r ∧ |
| r ≤ RadialRiemann.point (-1) 1 N (i+1) at hr |
| have hd : RadialRiemann.step (-1) 1 N = 2 * halfwidth N := by |
| unfold RadialRiemann.step halfwidth |
| ring |
| rw [hd] at hs |
| unfold midpoint |
| apply abs_le.mpr |
| constructor <;> linarith [hr.1, hr.2] |
|
|
| lemma intercept_mem {ρ e M m r u σ : ℝ} |
| (hρ : 0 ≤ ρ) (he : 0 ≤ e) (hu : 0 ≤ u) (huM : u ≤ M) |
| (hr : |r-m| ≤ ρ) (hσ : |σ| ≤ e) : |
| (u+σ, u*r+σ*(r+m)/2) ∈ RadialFootprints.halfheightFootprint ρ e M m := by |
| have hs := abs_le.mp hσ |
| refine ⟨⟨by linarith, by linarith⟩, ?_⟩ |
| have hid : u*r+σ*(r+m)/2-m*(u+σ) = (r-m)*(u+σ/2) := by ring |
| change |u*r+σ*(r+m)/2-m*(u+σ)| ≤ ρ*(u+σ+2*e) |
| rw [hid, abs_mul] |
| have hb : |u+σ/2| ≤ u+e/2 := by |
| calc |
| |u+σ/2| ≤ |u|+|σ/2| := abs_add_le _ _ |
| _ = u+|σ|/2 := by rw [abs_of_nonneg hu, abs_div, abs_of_pos (by norm_num : (0:ℝ)<2)] |
| _ ≤ u+e/2 := by linarith |
| calc |
| |r-m| * |u+σ/2| ≤ ρ*(u+e/2) := |
| mul_le_mul hr hb (abs_nonneg _) hρ |
| _ ≤ ρ*(u+σ+2*e) := mul_le_mul_of_nonneg_left (by linarith) hρ |
|
|
| lemma firstRaw_ne (δ r : ℝ) : firstRaw δ r ≠ 0 := by |
| intro h |
| have := congrArg (fun z : Space => z 0) h |
| norm_num [firstRaw] at this |
|
|
| lemma secondRaw_ne (δ r : ℝ) : secondRaw δ r ≠ 0 := by |
| intro h |
| have := congrArg (fun z : Space => z 1) h |
| norm_num [secondRaw] at this |
|
|
| lemma normalized_cylinder_mem {v z q : Space} {F : Set Space} |
| (hv : v ≠ 0) (hq : q ∈ F) (t : ℝ) (hz : z = q+t • v) : |
| z ∈ RadialSweep.cylinder (‖v‖⁻¹ • v) |
| (TriangularCovering.projection (‖v‖⁻¹ • v) '' F) := by |
| let a : Space := ‖v‖⁻¹ • v |
| have hnorm : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv |
| have hva : ‖v‖ • a = v := by |
| dsimp [a] |
| rw [smul_smul, mul_inv_cancel₀ hnorm, one_smul] |
| refine ⟨TriangularCovering.projection a q, ⟨q,hq,rfl⟩, |
| inner ℝ a q+t*‖v‖, ?_⟩ |
| change z = (q-(inner ℝ a q) • a)+(inner ℝ a q+t*‖v‖) • a |
| rw [add_smul, mul_smul, hva] |
| rw [hz] |
| abel |
|
|
| lemma first_intercept (δ r m u X : ℝ) : |
| point X (u*r+δ*X*alpha r) (u+δ*X*beta r) = |
| firstFoot (u+δ*X*(r-m)/2, |
| u*r+(δ*X*(r-m)/2)*(r+m)/2) + (X/2) • firstRaw δ m := by |
| ext i |
| fin_cases i <;> simp [point, firstFoot, firstRaw, alpha, beta] <;> ring |
|
|
| lemma second_intercept (δ s m v Y : ℝ) : |
| point (v*s-δ*Y*alpha s) Y (1-v+δ*Y*beta s) = |
| secondFoot (v+(-δ*Y*(s-m)/2), |
| v*s+(-δ*Y*(s-m)/2)*(s+m)/2) + (Y/2) • secondRaw δ m := by |
| ext i |
| fin_cases i <;> simp [point, secondFoot, secondRaw, alpha, beta] <;> ring |
|
|
| lemma first_member {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 : 0 ≤ u) (huR : u ≤ 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 |
| have hdist := midpoint_distance i hr |
| have hs : |δ*X*(r-midpoint N i)/2| ≤ collar δ N := by |
| rw [abs_div, abs_mul, abs_mul, abs_of_nonneg hδ] |
| norm_num |
| have hm := mul_le_mul hX hdist (abs_nonneg _) (by norm_num : (0:ℝ)≤1) |
| have hm' := mul_le_mul_of_nonneg_left hm hδ |
| unfold collar |
| nlinarith |
| have huM : u ≤ maxCap R N i := huR.trans |
| (RadialRiemann.le_capMax (by norm_num) hR hr) |
| have hm := intercept_mem (halfwidth_nonneg N) |
| (show 0 ≤ collar δ N by unfold collar halfwidth; positivity) hu huM hdist hs |
| apply normalized_cylinder_mem (firstRaw_ne δ (midpoint N i)) |
| (show firstFoot (u+δ*X*(r-midpoint N i)/2, |
| u*r+(δ*X*(r-midpoint N i)/2)*(r+midpoint N i)/2) ∈ firstFoot '' foot δ R N i from |
| ⟨_,hm,rfl⟩) (X/2) |
| exact first_intercept δ r (midpoint N i) u X |
|
|
| lemma second_member {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N) |
| {s v Y : ℝ} (hs : s ∈ RadialRiemann.cell (-1) 1 N i) |
| (hv : 0 ≤ v) (hvR : v ≤ R s) (hY : |Y| ≤ 1) : |
| point (v*s-δ*Y*alpha s) Y (1-v+δ*Y*beta s) ∈ |
| RadialSweep.cylinder (secondAxis δ (midpoint N i)) (secondBase δ R N i) := by |
| have hdist := midpoint_distance i hs |
| have hσ : |-δ*Y*(s-midpoint N i)/2| ≤ collar δ N := by |
| rw [abs_div, abs_mul, abs_mul, abs_neg, abs_of_nonneg hδ] |
| norm_num |
| have hm := mul_le_mul hY hdist (abs_nonneg _) (by norm_num : (0:ℝ)≤1) |
| have hm' := mul_le_mul_of_nonneg_left hm hδ |
| unfold collar |
| nlinarith |
| have hvM : v ≤ maxCap R N i := hvR.trans |
| (RadialRiemann.le_capMax (by norm_num) hR hs) |
| have hm := intercept_mem (halfwidth_nonneg N) |
| (show 0 ≤ collar δ N by unfold collar halfwidth; positivity) hv hvM hdist hσ |
| apply normalized_cylinder_mem (secondRaw_ne δ (midpoint N i)) |
| (show secondFoot (v+(-δ*Y*(s-midpoint N i)/2), |
| v*s+(-δ*Y*(s-midpoint N i)/2)*(s+midpoint N i)/2) ∈ secondFoot '' foot δ R N i from |
| ⟨_,hm,rfl⟩) (Y/2) |
| exact second_intercept δ s (midpoint N i) v Y |
|
|
| theorem finite_coverage {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,huR,hYr,hZr⟩ | |
| ⟨s,v,hs,hv,hvR,hXs,hZs⟩ |
| · obtain ⟨i,hi⟩ := RadialRiemann.exists_cell hN (abs_le.mp hr) |
| left |
| refine ⟨i,?_⟩ |
| change Y = u*r+δ*X*alpha r at hYr |
| change Z = u+δ*X*beta r at hZr |
| rw [hYr,hZr] |
| exact first_member hδ hR i hi hu huR hX1 |
| · obtain ⟨i,hi⟩ := RadialRiemann.exists_cell hN (abs_le.mp hs) |
| right |
| refine ⟨i,?_⟩ |
| change X = v*s-δ*Y*alpha s at hXs |
| change 1-Z = v-δ*Y*beta s at hZs |
| have hZ : Z = 1-v+δ*Y*beta s := by linarith |
| rw [hXs,hZ] |
| exact second_member hδ hR i hi hv hvR hY1 |
|
|
| lemma firstAxis_unit (δ r : ℝ) : ‖firstAxis δ r‖ = 1 := by |
| unfold firstAxis |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (inv_nonneg.mpr (norm_nonneg _)), |
| inv_mul_cancel₀ (norm_ne_zero_iff.mpr (firstRaw_ne δ r))] |
|
|
| lemma secondAxis_unit (δ r : ℝ) : ‖secondAxis δ r‖ = 1 := by |
| unfold secondAxis |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (inv_nonneg.mpr (norm_nonneg _)), |
| inv_mul_cancel₀ (norm_ne_zero_iff.mpr (secondRaw_ne δ r))] |
|
|
| theorem cubic_finite_coverage {δ : ℝ} (hδ : 0 < δ) {N : ℕ} (hN : 0<N) : |
| ∀ z ∈ tetrahedron, |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder (firstAxis δ (midpoint N i)) (firstBase δ (cubicCap δ) N i)) ∨ |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder (secondAxis δ (midpoint N i)) (secondBase δ (cubicCap δ) N i)) := by |
| apply finite_coverage hδ.le (by unfold cubicCap profile; fun_prop) _ hN |
| exact HalfHeightCaps.cubic_covers hδ |
|
|
| theorem fractional_finite_coverage {δ : ℝ} (hδ : 0 < δ) (hsmall : δ ≤ 1) |
| {N : ℕ} (hN : 0<N) : |
| ∀ z ∈ tetrahedron, |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder (firstAxis δ (midpoint N i)) (firstBase δ (fractionalCap δ) N i)) ∨ |
| (∃ i : Fin N, z ∈ RadialSweep.cylinder (secondAxis δ (midpoint N i)) (secondBase δ (fractionalCap δ) N i)) := by |
| apply finite_coverage hδ.le (by unfold fractionalCap profile; fun_prop) _ hN |
| exact HalfHeightCaps.fractional_covers hδ hsmall |
|
|
| end CylinderCovering.HalfheightFinite |
|
|
| end |
|
|
| end OAI |
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