| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions | |
| namespace OAI | |
| open Set | |
| noncomputable section | |
| namespace CylinderCovering.HalfheightFinite | |
| lemma profile_nonnegative (r : ℝ) : 0 ≤ profile r := by | |
| unfold profile | |
| positivity | |
| lemma cubicCap_continuous (δ : ℝ) : Continuous (cubicCap δ) := by | |
| unfold cubicCap profile | |
| fun_prop | |
| lemma fractionalCap_continuous (δ : ℝ) : Continuous (fractionalCap δ) := by | |
| unfold fractionalCap profile | |
| fun_prop | |
| lemma cubicCap_positive {δ : ℝ} (hδ : 0 ≤ δ) (r : ℝ) : | |
| 0 < cubicCap δ r := by | |
| unfold cubicCap | |
| have hp := profile_nonnegative r | |
| positivity | |
| lemma fractionalCap_positive {δ : ℝ} (hδ : 0 ≤ δ) (r : ℝ) : | |
| 0 < fractionalCap δ r := by | |
| unfold fractionalCap | |
| have hp := profile_nonnegative r | |
| positivity | |
| end CylinderCovering.HalfheightFinite | |
| end | |
| end OAI | |