| import OAI.Probability.InvariantIsing.Magnetic.MagneticGridProjection | |
| namespace OAI | |
| /-! The normalized radius of the magnetic grid projection converges to its mesh. -/ | |
| noncomputable section | |
| open Filter | |
| open scoped Topology | |
| namespace InvariantIsing | |
| lemma magnetic_cover_distance_tendsto (N : ℕ → ℕ) (hN : ∀ k, 0<N k) | |
| (hNlim : Tendsto N atTop atTop) (A : ℕ) {δ : ℝ} (hδ : 0<δ) : | |
| Tendsto (fun k => (⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)) atTop (𝓝 δ) := by | |
| have he k : 0≤(⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)-δ ∧ | |
| (⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)-δ≤((A:ℝ)+1)/(N k:ℝ) := by | |
| have hp : (0:ℝ)<N k := Nat.cast_pos.mpr (hN k) | |
| have hx : 0≤δ*N k+A := by positivity | |
| have hl := Nat.le_ceil (δ*N k+A) | |
| have hu := Nat.ceil_lt_add_one hx | |
| constructor | |
| · apply sub_nonneg.mpr | |
| apply (le_div_iff₀ hp).mpr | |
| linarith [(Nat.cast_nonneg A : (0:ℝ)≤A)] | |
| · have hdiff : (⌈δ*N k+A⌉₊:ℝ)/(N k:ℝ)-δ= | |
| ((⌈δ*N k+A⌉₊:ℝ)-δ*N k)/(N k:ℝ) := by field_simp | |
| rw [hdiff] | |
| exact div_le_div_of_nonneg_right (by linarith) hp.le | |
| have hz := squeeze_zero (fun k => (he k).1) (fun k => (he k).2) | |
| (tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_atTop.comp hNlim)) | |
| simpa only [sub_add_cancel,zero_add] using hz.add_const δ | |
| end InvariantIsing | |
| end | |
| end OAI | |