| import OAI.Analysis.IrrationalRotation.Model.CircleDigitCoupling |
| import OAI.Analysis.IrrationalRotation.Analysis.UniformQuantization |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| namespace Invariant233.CircleDigitCoupling |
|
|
| lemma bin_subset_unit {r : ℕ} [NeZero r] (d : Fin r) : |
| bin r d ⊆ Ico (0 : ℝ) 1 := by |
| have hr : (0 : ℝ) < r := by exact_mod_cast Nat.pos_of_ne_zero (NeZero.ne r) |
| intro x hx |
| have hd : (d : ℝ) + 1 ≤ r := by exact_mod_cast Nat.succ_le_of_lt d.isLt |
| exact ⟨(div_nonneg (by positivity) hr.le).trans hx.1, |
| hx.2.trans_le ((div_le_iff₀ hr).mpr (by simpa using hd))⟩ |
|
|
| lemma bin_length {r : ℕ} (d : Fin r) : |
| ((d : ℝ) + 1) / r - (d : ℝ) / r = 1 / r := by ring |
|
|
| lemma bin_subset_top_left {r : ℕ} [NeZero r] (d : Fin r) |
| {z u w : ℝ} (hz : z ∈ bin r d) (hzu : z < u) (hu : u ≤ 1 - w) |
| (hsmall : 1 / (r : ℝ) < w / 2) : |
| bin r d ⊆ Ico (0 : ℝ) (1 - w / 2) := by |
| intro x hx |
| have hx0 := (bin_subset_unit d hx).1 |
| have hlen := bin_length d |
| exact ⟨hx0, by dsimp [bin] at hz hx; linarith [hz.1, hx.2]⟩ |
|
|
| lemma bin_subset_bottom_right {r : ℕ} [NeZero r] (d : Fin r) |
| {z u w : ℝ} (hz : z ∈ bin r d) (hzu : u ≤ z) (hu : w ≤ u) |
| (hsmall : 1 / (r : ℝ) < w / 2) : |
| bin r d ⊆ Ico (w / 2) (1 : ℝ) := by |
| intro x hx |
| have hx1 := (bin_subset_unit d hx).2 |
| have hlen := bin_length d |
| exact ⟨by dsimp [bin] at hz hx; linarith [hz.2, hx.1], hx1⟩ |
|
|
| lemma abs_sub_lt_bin_width {r : ℕ} (d : Fin r) {x y : ℝ} |
| (hx : x ∈ bin r d) (hy : y ∈ bin r d) : |
| |x - y| < 1 / (r : ℝ) := by |
| rw [abs_lt] |
| have hlen := bin_length d |
| constructor <;> dsimp [bin] at hx hy <;> linarith [hx.1, hx.2, hy.1, hy.2] |
|
|
| lemma top_forces_crossing {r : ℕ} {Y y u w : ℝ} |
| (hY : 1 - w / 2 ≤ Y) (hround : |y - Y| ≤ 1 / (r : ℝ)) |
| (hsmall : 1 / (r : ℝ) < w / 2) (hu : u ≤ 1 - w) : u ≤ y := by |
| have h := (abs_le.mp hround).1 |
| linarith |
|
|
| lemma bottom_forces_crossing {r : ℕ} {X x u w : ℝ} |
| (hX : X < w / 2) (hround : |x - X| ≤ 1 / (r : ℝ)) |
| (hsmall : 1 / (r : ℝ) < w / 2) (hu : w ≤ u) : x < u := by |
| have h := (abs_le.mp hround).2 |
| linarith |
|
|
| end Invariant233.CircleDigitCoupling |
|
|
| end |
|
|
| end OAI |
|
|
