File size: 3,009 Bytes
f205175 | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace CriticalPercolation
def interpolationConstant (c ε : ℝ) : ℝ :=
(4*(16/(2*ε*ε))+6)*(1+Real.log 8+max 0 (Real.log (1/c))) +
(2*(16/(2*ε*ε))+5)*Real.log 4
lemma interpolationConstant_pos (c : ℝ) {ε : ℝ} (hε : 0<ε) :
0 < interpolationConstant c ε := by
have h8 : 0 < Real.log 8 := Real.log_pos (by norm_num)
have h4 : 0 < Real.log 4 := Real.log_pos (by norm_num)
have hm : 0 ≤ max 0 (Real.log (1/c)) := le_max_left _ _
unfold interpolationConstant
positivity
lemma interpolation_uniform_algebra {p c ε δ : ℝ} (hp : p ∈ Set.Ioo 0 1)
(hc : 0<c) (hc1 : c≤1) (hε : 0<ε) (hεp : ε≤p) (hεq : ε≤1-p)
(hδ : 0<δ) (hδ1 : δ≤1) (n : ℕ) :
(4*(16/(2*p*(1-p)))+6)*(Real.log (1/δ)+n*(Real.log 8+Real.log (1/c))) +
(2*(16/(2*p*(1-p)))+5)*n*Real.log 4 ≤
interpolationConstant c ε * (Real.log (1/δ)+n+1) := by
have h4 : 0 ≤ Real.log 4 := (Real.log_pos (by norm_num : (1:ℝ)<4)).le
have h8 : 0 ≤ Real.log 8 := (Real.log_pos (by norm_num : (1:ℝ)<8)).le
have hr : 0 ≤ Real.log (1/δ) := by
rw [one_div,Real.log_inv]
exact neg_nonneg.mpr (Real.log_nonpos hδ.le hδ1)
have hc' : 0 ≤ Real.log (1/c) := by
rw [one_div,Real.log_inv]
exact neg_nonneg.mpr (Real.log_nonpos hc.le hc1)
have hK : 16/(2*p*(1-p)) ≤ 16/(2*ε*ε) := by
apply div_le_div_of_nonneg_left (by norm_num) (by positivity)
have hh := mul_le_mul hεp hεq hε.le hp.1.le
nlinarith
let A : ℝ := 4*(16/(2*ε*ε))+6
let B : ℝ := 2*(16/(2*ε*ε))+5
let l : ℝ := Real.log 8+max 0 (Real.log (1/c))
let r : ℝ := Real.log (1/δ)
let T : ℝ := r+n+1
have hA : 0 ≤ A := by dsimp [A]; positivity
have hB : 0 ≤ B := by dsimp [B]; positivity
have hl : 0 ≤ l := add_nonneg h8 (le_max_left _ _)
have hcoef : 0 ≤ A*l+B*Real.log 4 := add_nonneg (mul_nonneg hA hl) (mul_nonneg hB h4)
have hn : (0:ℝ) ≤ n := Nat.cast_nonneg _
have hrT : r≤T := by dsimp [T]; linarith
have hnT : (n:ℝ)≤T := by dsimp [T,r]; linarith
calc
_ ≤ A*(r+n*(Real.log 8+Real.log (1/c)))+B*(n*Real.log 4) := by
rw [mul_assoc _ (n:ℝ) (Real.log 4)]
apply add_le_add
· apply mul_le_mul_of_nonneg_right
· dsimp [A]; linarith
· exact add_nonneg hr (mul_nonneg hn (add_nonneg h8 hc'))
· apply mul_le_mul_of_nonneg_right
· dsimp [B]; linarith
· exact mul_nonneg hn h4
_ ≤ A*(r+n*l)+B*(n*Real.log 4) := by
apply add_le_add ?_ le_rfl
apply mul_le_mul_of_nonneg_left _ hA
apply add_le_add le_rfl
exact mul_le_mul_of_nonneg_left (add_le_add le_rfl (le_max_right _ _)) hn
_ = A*r+(A*l+B*Real.log 4)*n := by ring
_ ≤ A*T+(A*l+B*Real.log 4)*T :=
add_le_add (mul_le_mul_of_nonneg_left hrT hA) (mul_le_mul_of_nonneg_left hnT hcoef)
_ = _ := by dsimp [A,B,l,r,T,interpolationConstant]; ring
end CriticalPercolation
end
end
end
end OAI
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