| import Mathlib |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
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| section |
| section |
| open Filter Topology |
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| namespace CannonNode |
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| def radius (ell : ℝ) (k : ℕ) : ℝ := Real.exp (-ell*k) |
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| lemma radius_pos (ell : ℝ) (k : ℕ) : 0 < radius ell k := Real.exp_pos _ |
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| lemma radius_add (ell : ℝ) (k d : ℕ) : radius ell (k+d) = radius ell d*radius ell k := by |
| simp only [radius,Nat.cast_add, mul_add,Real.exp_add] |
| ring |
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| lemma radius_ratio (ell : ℝ) (k d : ℕ) : radius ell k/radius ell (k+d) = Real.exp (ell*d) := by |
| simp only [radius, ← Real.exp_sub, Nat.cast_add] |
| congr 1 |
| ring |
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| lemma radius_sub_ratio (ell : ℝ) (k d e : ℕ) (hed : e ≤ d) : |
| radius ell (k+d)/radius ell (k+e) = radius ell (d-e) := by |
| simp only [radius,← Real.exp_sub,Nat.cast_add,Nat.cast_sub hed] |
| congr 1 |
| ring |
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| lemma radius_antitone {ell : ℝ} (hell : 0 ≤ ell) : Antitone (radius ell) := by |
| intro i j hij |
| apply Real.exp_le_exp.mpr |
| have hh : (i:ℝ) ≤ j := by exact_mod_cast hij |
| exact mul_le_mul_of_nonpos_left hh (neg_nonpos.mpr hell) |
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| lemma radius_eq_power (ell : ℝ) (k : ℕ) : radius ell k = (Real.exp ell^k)⁻¹ := by |
| rw [← Real.exp_nat_mul,← Real.exp_neg] |
| unfold radius |
| congr 1 |
| ring |
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| lemma radius_tendsto {ell : ℝ} (hell : 0 < ell) : Tendsto (radius ell) atTop (𝓝 0) := by |
| have he : Real.exp (-ell) < 1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hell) |
| have ht := tendsto_pow_atTop_nhds_zero_of_lt_one (Real.exp_pos (-ell)).le he |
| change Tendsto (fun n : ℕ => Real.exp (-ell*n)) atTop (𝓝 0) |
| simpa only [radius,← Real.exp_nat_mul, mul_comm] using ht |
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| lemma nat_mul_tendsto {d : ℕ} (hd : 0 < d) : |
| Tendsto (fun n : ℕ => d*n) atTop atTop := by |
| apply tendsto_atTop_mono (f := fun n : ℕ => n) |
| · intro n |
| exact Nat.le_mul_of_pos_left n hd |
| · exact tendsto_id |
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| theorem eventually_flow_scales (qdim : ℕ) (ell c₀ H C : ℝ) (hell : 0 < ell) : |
| ∀ᶠ n : ℕ in atTop, |
| 0 < n ∧ radius ell (160*(qdim+1)*n) < 1/4 ∧ |
| c₀*radius ell (160*(qdim+1)*n) ≤ 1 ∧ |
| 2*(H+3)*radius ell (160*(qdim+1)*n) ≤ 1 ∧ |
| radius ell (2*n) ≤ 1/4 ∧ |
| C*radius ell ((160*(qdim+1)-2)*n) ≤ 1 := by |
| have hd : 0 < 160*(qdim+1) := by omega |
| have he : 0 < 160*(qdim+1)-2 := by omega |
| have hdl : Tendsto (fun n : ℕ => radius ell (160*(qdim+1)*n)) atTop (𝓝 0) := |
| (radius_tendsto hell).comp (nat_mul_tendsto hd) |
| have hel : Tendsto (fun n : ℕ => radius ell ((160*(qdim+1)-2)*n)) atTop (𝓝 0) := |
| (radius_tendsto hell).comp (nat_mul_tendsto he) |
| have h₂l : Tendsto (fun n : ℕ => radius ell (2*n)) atTop (𝓝 0) := |
| (radius_tendsto hell).comp (nat_mul_tendsto (by norm_num : 0 < (2:ℕ))) |
| have hcl : Tendsto (fun n : ℕ => c₀*radius ell (160*(qdim+1)*n)) atTop (𝓝 0) := by |
| simpa only [mul_zero] using hdl.const_mul c₀ |
| have hHl : Tendsto (fun n : ℕ => 2*(H+3)*radius ell (160*(qdim+1)*n)) atTop (𝓝 0) := by |
| simpa only [mul_zero] using hdl.const_mul (2*(H+3)) |
| have hCl : Tendsto (fun n : ℕ => C*radius ell ((160*(qdim+1)-2)*n)) atTop (𝓝 0) := by |
| simpa only [mul_zero] using hel.const_mul C |
| filter_upwards [eventually_gt_atTop (0:ℕ), |
| hdl.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1/4)), |
| hcl.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1)), |
| hHl.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1)), |
| h₂l.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1/4)), |
| hCl.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1))] with n hn hdn hcn hHn h₂n hCn |
| exact ⟨hn,hdn,hcn.le,hHn.le,h₂n.le,hCn.le⟩ |
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| end CannonNode |
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| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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