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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 | import Mathlib
namespace OAI
section
section
noncomputable section
open scoped BigOperators
namespace CriticalPercolation
lemma collar_error_budget (κ η shells closed : ℝ) (hκ : 0<κ) (hκ1 : κ≤1)
(hη : η ≤ κ*(κ/4)/(4*(Real.log (4/κ)+1)))
(hclosed : 0<closed) (hshells : 4/(κ*closed)<shells) :
1/(shells*closed)+Real.exp (-Real.log (4/κ))+
(Real.log (4/κ)+1)*η/(κ/4)+κ/4 ≤ κ := by
have hratio : 1 ≤ 4/κ := (le_div_iff₀ hκ).mpr (by linarith)
have hK : 0 ≤ Real.log (4/κ) := Real.log_nonneg hratio
have hK1 : 0<Real.log (4/κ)+1 := by linarith
have hlam : 0<κ/4 := by linarith
have hden : 0<κ*closed := mul_pos hκ hclosed
have hs : 0<shells := (div_pos (by norm_num) hden).trans hshells
have hfirst : 1/(shells*closed) ≤ κ/4 := by
apply (div_le_iff₀ (mul_pos hs hclosed)).mpr
have hh := (div_lt_iff₀ hden).mp hshells
nlinarith
have hexp : Real.exp (-Real.log (4/κ))=κ/4 := by
rw [Real.exp_neg,Real.exp_log (div_pos (by norm_num) hκ)]
simp
have hthird : (Real.log (4/κ)+1)*η/(κ/4) ≤ κ/4 := by
apply (div_le_iff₀ hlam).mpr
have hh := (le_div_iff₀ (mul_pos (by norm_num : (0:ℝ)<4) hK1)).mp hη
nlinarith
rw [hexp]
linarith
lemma positive_trial_power (p : ℝ) (hp : 0<p) (m VM : ℕ) :
0<(p/2)^(m+1+VM) := pow_pos (by linarith) _
lemma trial_ceiling_budget (K s : ℝ) (hs : 0<s) :
K ≤ (⌈K/s⌉₊ : ℝ)*s := by
have hh : K/s ≤ (⌈K/s⌉₊ : ℝ) := Nat.le_ceil _
exact (div_le_iff₀ hs).mp hh
end CriticalPercolation
end
end
end
end OAI
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