| import OAI.Geometry.CylinderCovering.Caps.Symmetry |
| import OAI.Geometry.CylinderCovering.Footprints.Footprints |
| import OAI.Geometry.CylinderCovering.Radial.SetProjection |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace CylinderCovering.HalfheightFinite |
|
|
| lemma halfheightFootprint_compact (ρ e M m : ℝ) : |
| IsCompact (RadialFootprints.halfheightFootprint ρ e M m) := by |
| have heq : RadialFootprints.halfheightFootprint ρ e M m = |
| {p : ℝ × ℝ | p.1 ∈ Icc (-e) (M+e) ∧ |
| p.2 ∈ Icc (m*p.1-ρ*(p.1+2*e)) (m*p.1+ρ*(p.1+2*e))} := by |
| ext p |
| simp only [RadialFootprints.halfheightFootprint, mem_ofPred_eq, mem_Icc, abs_le] |
| constructor |
| · rintro ⟨⟨h1,h2⟩,h3,h4⟩ |
| exact ⟨⟨h1,h2⟩, by constructor <;> linarith⟩ |
| · rintro ⟨⟨h1,h2⟩,h3,h4⟩ |
| exact ⟨⟨h1,h2⟩, by constructor <;> linarith⟩ |
| rw [heq] |
| exact Footprints.isCompact_closedBand |
| (f := fun u => m*u-ρ*(u+2*e)) (g := fun u => m*u+ρ*(u+2*e)) |
| (by fun_prop) (by fun_prop) _ _ |
|
|
| lemma halfheightFootprint_convex (ρ e M m : ℝ) : |
| Convex ℝ (RadialFootprints.halfheightFootprint ρ e M m) := by |
| intro x hx y hy a b ha hb hab |
| simp only [RadialFootprints.halfheightFootprint, mem_ofPred_eq, mem_Icc, abs_le, |
| Prod.fst_add, Prod.snd_add, Prod.smul_fst, Prod.smul_snd, smul_eq_mul] at * |
| have he : a*e+b*e=e := by rw [← add_mul, hab, one_mul] |
| have hM : a*(M+e)+b*(M+e)=M+e := by rw [← add_mul, hab, one_mul] |
| have hρe : a*(2*ρ*e)+b*(2*ρ*e)=2*ρ*e := by rw [← add_mul, hab, one_mul] |
| refine ⟨⟨?_,?_⟩,?_,?_⟩ |
| · nlinarith only [mul_le_mul_of_nonneg_left hx.1.1 ha, |
| mul_le_mul_of_nonneg_left hy.1.1 hb, he] |
| · linarith only [mul_le_mul_of_nonneg_left hx.1.2 ha, |
| mul_le_mul_of_nonneg_left hy.1.2 hb, hM] |
| · nlinarith only [mul_le_mul_of_nonneg_left hx.2.1 ha, |
| mul_le_mul_of_nonneg_left hy.2.1 hb, hρe] |
| · nlinarith only [mul_le_mul_of_nonneg_left hx.2.2 ha, |
| mul_le_mul_of_nonneg_left hy.2.2 hb, hρe] |
|
|
| lemma firstAxis_norm (δ r : ℝ) : ‖firstAxis δ r‖ = 1 := by |
| apply TriangularCovering.unit_norm |
| intro hz |
| have := congrArg (fun z : Space => z 0) hz |
| simp [firstRaw] at this |
|
|
| lemma firstFoot_coordinateAffine : |
| firstFoot = RadialProjection.coordinateAffine (0 : Space) |
| (!₂[0,0,height]) (!₂[0,1/2,0]) := by |
| funext p |
| ext i |
| fin_cases i <;> simp [firstFoot, RadialProjection.coordinateAffine] <;> ring |
|
|
| theorem firstBase_properties (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : |
| BaseProperties (firstAxis δ (midpoint N i)) (firstBase δ R N i) := by |
| have hfoot : IsCompact (foot δ R N i) := |
| halfheightFootprint_compact _ _ _ _ |
| have hvfoot : Convex ℝ (foot δ R N i) := |
| halfheightFootprint_convex _ _ _ _ |
| have hc : IsCompact (firstBase δ R N i) := by |
| unfold firstBase |
| rw [firstFoot_coordinateAffine] |
| exact RadialProjection.affine_projection_isCompact _ _ _ _ hfoot |
| refine ⟨firstAxis_norm _ _, hc, ?_, hc.measurableSet, ?_, ?_⟩ |
| · unfold firstBase |
| rw [firstFoot_coordinateAffine] |
| exact (hvfoot.affine_image _).linear_image (TriangularCovering.projLinear _) |
| · unfold firstBase |
| rw [firstFoot_coordinateAffine] |
| exact RadialProjection.affine_projection_finite _ _ _ _ (firstAxis_norm _ _) hfoot |
| · rintro z ⟨p,hp,rfl⟩ |
| exact TriangularCovering.projection_perp (firstAxis_norm _ _) p |
|
|
| theorem secondBase_properties (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : |
| BaseProperties (secondAxis δ (midpoint N i)) (secondBase δ R N i) := |
| secondBase_properties_of_first (firstBase_properties δ R N i) |
|
|
| theorem base_properties (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : |
| BaseProperties (firstAxis δ (midpoint N i)) (firstBase δ R N i) ∧ |
| BaseProperties (secondAxis δ (midpoint N i)) (secondBase δ R N i) := |
| ⟨firstBase_properties δ R N i, secondBase_properties δ R N i⟩ |
|
|
| end CylinderCovering.HalfheightFinite |
|
|
| end |
|
|
| end OAI |
|
|
