File size: 3,408 Bytes
135052a | 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 75 76 77 78 79 80 81 82 83 84 85 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.ThermodynamicLimit
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
def finiteRateConstant (β : ℝ) : ℝ :=
(β^2/4)*(60+6*(3/β^2)+2*(3/β^2)^2)
theorem finiteRateConstant_pos {β : ℝ} (hβ : 0 < β) : 0 < finiteRateConstant β := by
unfold finiteRateConstant
positivity
theorem finiteComparisonError_rate {β r : ℝ} (hβ : 0 < β) (hr : 2 ≤ r)
{N k : ℕ} (hN : (N:ℝ) = r^12) (hk0 : r ≤ (k:ℝ)) (hk1 : (k:ℝ) ≤ 2*r) :
(k:ℝ)/Real.sqrt ((N:ℝ)*(β^2*(1/(β^2*r^4)))) ≤ 1 ∧
finiteComparisonError β k (1/(β^2*r^4)) N ≤
finiteRateConstant β*(1/Real.sqrt r) := by
have hr0 : 0 < r := by linarith
have hr1 : 1 ≤ r := by linarith
have hb : β ≠ 0 := hβ.ne'
have hrn : r ≠ 0 := hr0.ne'
have hs : Real.sqrt ((N:ℝ)*(β^2*(1/(β^2*r^4)))) = r^4 := by
rw [hN]
have he : r^12*(β^2*(1/(β^2*r^4))) = (r^4)^2 := by field_simp
rw [he,Real.sqrt_sq (by positivity)]
have hr3 : 1 ≤ r^3 := one_le_pow₀ hr1
have hr38 : (8:ℝ) ≤ r^3 := by nlinarith [pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 2) hr 3]
have hrs : 1 ≤ Real.sqrt r := by simpa using Real.sqrt_le_sqrt hr1
have hspos : 0 < Real.sqrt r := Real.sqrt_pos.mpr hr0
have hsr : Real.sqrt r ≤ r := by
apply (Real.sqrt_le_iff).mpr
exact ⟨hr0.le,by nlinarith⟩
have hrr3 : r ≤ r^3 := le_self_pow₀ hr1 (by decide)
have hq : 1/r^3 ≤ 1/Real.sqrt r := one_div_le_one_div_of_le hspos (hsr.trans hrr3)
have hrho : 1/Real.sqrt r ≤ 1 := (div_le_one hspos).mpr hrs
have HR : (k:ℝ)/r^4 ≤ 2/r^3 := by
calc
_ ≤ (2*r)/r^4 := div_le_div_of_nonneg_right hk1 (by positivity)
_ = _ := by field_simp
have HRsmall : (k:ℝ)/r^4 ≤ 1 := HR.trans ((div_le_one (by positivity)).mpr (by linarith))
have HRmesh : (k:ℝ)/r^4 ≤ 2*(1/Real.sqrt r) := by
apply HR.trans
simpa only [mul_one_div] using mul_le_mul_of_nonneg_left hq (by norm_num : (0:ℝ) ≤ 2)
have HM : 1/Real.sqrt (k:ℝ) ≤ 1/Real.sqrt r :=
one_div_le_one_div_of_le hspos (Real.sqrt_le_sqrt hk0)
let d : ℝ := 3/β^2
let δ : ℝ := ((k+1:ℕ):ℝ)*(1/(β^2*r^4))
have hd : 0 < d := by dsimp [d]; positivity
have hδ0 : 0 ≤ δ := by dsimp [δ]; positivity
have hδ : δ ≤ d*(1/Real.sqrt r) := by
have HK : ((k+1:ℕ):ℝ) ≤ 3*r := by push_cast; linarith
calc
_ ≤ (3*r)*(1/(β^2*r^4)) := mul_le_mul_of_nonneg_right HK (by positivity)
_ = d*(1/r^3) := by dsimp [d]; field_simp
_ ≤ _ := mul_le_mul_of_nonneg_left hq hd.le
have hδd : δ ≤ d := hδ.trans (mul_le_of_le_one_right hd.le hrho)
have HP : δ*(1+δ) ≤ (d*(1/Real.sqrt r))*(1+d) :=
mul_le_mul hδ (by linarith) (by positivity) (by positivity)
refine ⟨by simpa only [hs] using HRsmall,?_⟩
rw [finiteComparisonError,hs]
change (β^2/4)*(20*((k:ℝ)/r^4+1/Real.sqrt (k:ℝ))+2*δ*(1+δ)+4*δ) ≤ _
calc
_ ≤ (β^2/4)*(20*(2*(1/Real.sqrt r)+1/Real.sqrt r)+
2*((d*(1/Real.sqrt r))*(1+d))+4*(d*(1/Real.sqrt r))) := by
apply mul_le_mul_of_nonneg_left _ (by positivity)
nlinarith [HRmesh,HM,hδ,HP]
_ = finiteRateConstant β*(1/Real.sqrt r) := by dsimp [finiteRateConstant,d]; ring
end SK.Analytic
end
end
end
end OAI
|