File size: 3,032 Bytes
d4317d7 | 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Tactic
namespace OAI
/-! # Scalar dyadic selection for comparable positive displacements
The arithmetic is independent of dimension, angular energy and distance-set
estimates.
-/
namespace PlanarFalconer
open Set Metric
theorem exists_displacement_depth {d : ℝ} (hd : 0 < d) :
∃ n : ℤ, (2:ℝ)^(-(n:ℝ)-1) < d ∧ d ≤ (2:ℝ)^(-(n:ℝ)) := by
obtain ⟨m,hm⟩ := exists_mem_Ioc_zpow hd (by norm_num : (1:ℝ) < 2)
refine ⟨-(m+1),?_,?_⟩
· have he : -((-(m+1):ℤ):ℝ)-1 = (m:ℝ) := by push_cast; ring
rw [he,Real.rpow_intCast]
exact hm.1
· have he : -((-(m+1):ℤ):ℝ) = ((m+1:ℤ):ℝ) := by push_cast; ring
rw [he,Real.rpow_intCast]
exact hm.2
theorem comparable_displacement_depth {d₁ d₂ : ℝ} {a h : ℕ}
(hupper : max d₁ d₂ ≤ 2*(2:ℝ)^(-(a:ℝ)))
(hfar : 16*(2:ℝ)^(-(h:ℝ)) < max d₁ d₂)
(hc₁ : max d₁ d₂ ≤ 2*d₁) (hc₂ : max d₁ d₂ ≤ 2*d₂) :
∃ b : ℕ, a+9 ≤ b ∧ b ≤ h+5 ∧
(256*(2:ℝ)^(-(b:ℝ)) < d₁ ∧ d₁ ≤ 1024*(2:ℝ)^(-(b:ℝ))) ∧
(256*(2:ℝ)^(-(b:ℝ)) < d₂ ∧ d₂ ≤ 1024*(2:ℝ)^(-(b:ℝ))) := by
have hd : 0 < max d₁ d₂ := (by positivity : 0 < 16*(2:ℝ)^(-(h:ℝ))).trans hfar
obtain ⟨n,hnlo,hnhi⟩ := exists_displacement_depth hd
have haexp : (2:ℝ)^(-(n:ℝ)-1) < (2:ℝ)^(1-(a:ℝ)) := by
have he : (2:ℝ)^(1-(a:ℝ)) = 2*(2:ℝ)^(-(a:ℝ)) := by
rw [sub_eq_add_neg,Real.rpow_add (by norm_num : (0:ℝ) < 2),Real.rpow_one]
rw [he]
exact hnlo.trans_le hupper
have haR := (Real.rpow_lt_rpow_left_iff (by norm_num : (1:ℝ) < 2)).mp haexp
have hhExp : (2:ℝ)^(4-(h:ℝ)) < (2:ℝ)^(-(n:ℝ)) := by
have he : (2:ℝ)^(4-(h:ℝ)) = 16*(2:ℝ)^(-(h:ℝ)) := by
rw [sub_eq_add_neg,Real.rpow_add (by norm_num : (0:ℝ) < 2)]
norm_num
rw [he]
exact hfar.trans_le hnhi
have hhR := (Real.rpow_lt_rpow_left_iff (by norm_num : (1:ℝ) < 2)).mp hhExp
have haZ : (a:ℤ)-2 < n := by exact_mod_cast (show (a:ℝ)-2 < (n:ℝ) by linarith)
have hhZ : n < (h:ℤ)-4 := by exact_mod_cast (show (n:ℝ) < (h:ℝ)-4 by linarith)
have hb0 : 0 ≤ n+10 := by omega
let b := (n+10).toNat
have hbZ : (b:ℤ) = n+10 := Int.toNat_of_nonneg hb0
have hbR : (b:ℝ) = (n:ℝ)+10 := by exact_mod_cast hbZ
have hpow : (2:ℝ)^(-(n:ℝ)) = 1024*(2:ℝ)^(-(b:ℝ)) := by
rw [hbR,show -(n:ℝ) = 10+(-((n:ℝ)+10)) by ring,
Real.rpow_add (by norm_num : (0:ℝ) < 2)]
norm_num
have hhalf : (2:ℝ)^(-(n:ℝ)-1) = 512*(2:ℝ)^(-(b:ℝ)) := by
rw [hbR,show -(n:ℝ)-1 = 9+(-((n:ℝ)+10)) by ring,
Real.rpow_add (by norm_num : (0:ℝ) < 2)]
norm_num
rw [hpow] at hnhi
rw [hhalf] at hnlo
refine ⟨b,?_,?_,?_,?_⟩
· omega
· omega
· exact ⟨by linarith, (le_max_left _ _).trans hnhi⟩
· exact ⟨by linarith, (le_max_right _ _).trans hnhi⟩
end PlanarFalconer
end OAI
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