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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 | import Mathlib
namespace OAI
section
noncomputable section
open scoped ENNReal
namespace CriticalPercolation
lemma geometric_fourth_bound (I : ℝ≥0∞) (A : ℝ) (hA : 0≤A) (n : ℕ)
(h : I^4≤ENNReal.ofReal (A/(16:ℝ)^n)) :
I≤ENNReal.ofReal ((A+1)*(1/2:ℝ)^n) := by
apply (ENNReal.pow_le_pow_left_iff (by decide : 4≠0)).mp
rw [←ENNReal.ofReal_pow (by positivity)]
apply h.trans (ENNReal.ofReal_le_ofReal _)
have hpow : ((1/2:ℝ)^n)^4 = (1/16:ℝ)^n := by
rw [←pow_mul,Nat.mul_comm n 4,pow_mul]
norm_num
rw [mul_pow,hpow,one_div_pow,mul_one_div]
apply div_le_div_of_nonneg_right _ (by positivity)
have hsq : A≤(A+1)^2 := by nlinarith
have ht : 1≤(A+1)^2 := by nlinarith
nlinarith [sq_nonneg ((A+1)^2-1)]
end CriticalPercolation
end
end
end OAI
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