openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /GeometricFourth.lean
| 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 | |