| import Mathlib | |
| namespace OAI | |
| namespace CylinderCovering.AngularEqualPartition | |
| lemma left_endpoint (m : ℕ) : (-1 : ℝ) + 2 * (0 : ℕ) / m = -1 := by | |
| simp | |
| lemma right_endpoint {m : ℕ} (hm : 0 < m) : | |
| (-1 : ℝ) + 2 * (m : ℝ) / m = 1 := by | |
| have hm0 : (m : ℝ) ≠ 0 := by positivity | |
| field_simp | |
| ring | |
| lemma width (m j : ℕ) : | |
| ((-1 : ℝ) + 2 * ((j + 1 : ℕ) : ℝ) / m) - | |
| ((-1 : ℝ) + 2 * (j : ℝ) / m) = 2 / m := by | |
| push_cast | |
| ring | |
| lemma strictly_increasing {m : ℕ} (hm : 0 < m) (j : ℕ) : | |
| ((-1 : ℝ) + 2 * (j : ℝ) / m) < | |
| ((-1 : ℝ) + 2 * ((j + 1 : ℕ) : ℝ) / m) := by | |
| have hw := width m j | |
| have hp : (0 : ℝ) < 2 / m := by positivity | |
| linarith only [hw, hp] | |
| lemma absolute_bound {m j : ℕ} (hm : 0 < m) (hj : j ≤ m) : | |
| |(-1 : ℝ) + 2 * (j : ℝ) / m| ≤ 1 := by | |
| have hmR : (0 : ℝ) < m := by exact_mod_cast hm | |
| have hjR : (j : ℝ) ≤ m := by exact_mod_cast hj | |
| have hl : (0 : ℝ) ≤ 2 * (j : ℝ) / m := by positivity | |
| have hu : 2 * (j : ℝ) / m ≤ 2 := by | |
| apply (div_le_iff₀ hmR).2 | |
| linarith only [hjR] | |
| exact abs_le.mpr ⟨by linarith only [hl], by linarith only [hu]⟩ | |
| theorem partition_laws {m : ℕ} {M τ : ℝ} (hm : 0 < m) | |
| (hmesh : (2 : ℝ) / m ≤ M * τ ^ 2) : | |
| (fun j : ℕ => (-1 : ℝ) + 2 * (j : ℝ) / m) 0 = -1 ∧ | |
| (fun j : ℕ => (-1 : ℝ) + 2 * (j : ℝ) / m) m = 1 ∧ | |
| (∀ j < m, (-1 : ℝ) + 2 * (j : ℝ) / m < | |
| (-1 : ℝ) + 2 * ((j+1 : ℕ) : ℝ) / m) ∧ | |
| (∀ j ≤ m, |(-1 : ℝ) + 2 * (j : ℝ) / m| ≤ 1) ∧ | |
| ∀ j < m, ((-1 : ℝ) + 2 * ((j+1 : ℕ) : ℝ) / m) - | |
| ((-1 : ℝ) + 2 * (j : ℝ) / m) ≤ M * τ ^ 2 := by | |
| refine ⟨left_endpoint m, right_endpoint hm, | |
| fun j _ => strictly_increasing hm j, fun j hj => absolute_bound hm hj, ?_⟩ | |
| intro j _ | |
| rw [width] | |
| exact hmesh | |
| end CylinderCovering.AngularEqualPartition | |
| end OAI | |