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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
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