| import Mathlib |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| open scoped BigOperators |
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| namespace CriticalPercolation |
| lemma sum_two_pow (N : ℕ) : ∑ k ∈ Finset.range N, (2:ℝ)^k = 2^N-1 := by |
| induction N with |
| | zero => simp |
| | succ N ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring |
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| lemma sum_two_inv_pow (N : ℕ) : ∑ k ∈ Finset.range N, 1/(2:ℝ)^k = 2-2/2^N := by |
| induction N with |
| | zero => norm_num |
| | succ N ih => |
| rw [Finset.sum_range_succ,ih,pow_succ] |
| field_simp |
| ring |
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| lemma dyadic_sum_bound (N K : ℕ) : |
| (∑ k ∈ Finset.range N, |
| min ((2:ℝ)^(k+1)/4^K) ((2:ℝ)^(k+1)/4^k)) ≤ 6/2^K := by |
| have hfour (k : ℕ) : (4:ℝ)^k = (2^k)^2 := by |
| rw [← pow_mul,mul_comm k 2,pow_mul] |
| norm_num |
| have hnonneg (k : ℕ) : 0 ≤ min ((2:ℝ)^(k+1)/4^K) ((2:ℝ)^(k+1)/4^k) := by positivity |
| calc |
| _ ≤ ∑ k ∈ Finset.range (K+N), |
| min ((2:ℝ)^(k+1)/4^K) ((2:ℝ)^(k+1)/4^k) := by |
| apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono (by omega)) |
| intro k _ _ |
| exact hnonneg k |
| _ = (∑ k ∈ Finset.range K, |
| min ((2:ℝ)^(k+1)/4^K) ((2:ℝ)^(k+1)/4^k)) + |
| ∑ k ∈ Finset.range N, |
| min ((2:ℝ)^(K+k+1)/4^K) ((2:ℝ)^(K+k+1)/4^(K+k)) := |
| Finset.sum_range_add _ _ _ |
| _ ≤ (∑ k ∈ Finset.range K, (2:ℝ)^(k+1)/4^K) + |
| ∑ k ∈ Finset.range N, (2:ℝ)^(K+k+1)/4^(K+k) := by |
| apply add_le_add |
| · exact Finset.sum_le_sum (fun k _ => min_le_left _ _) |
| · exact Finset.sum_le_sum (fun k _ => min_le_right _ _) |
| _ ≤ 6/2^K := by |
| have h1 : (∑ k ∈ Finset.range K, (2:ℝ)^(k+1)/4^K) = |
| 2*(2^K-1)/4^K := by |
| simp only [pow_succ,mul_div_assoc] |
| rw [← Finset.sum_mul,sum_two_pow] |
| ring |
| have h2 : (∑ k ∈ Finset.range N, (2:ℝ)^(K+k+1)/4^(K+k)) = |
| 2/2^K * (2-2/2^N) := by |
| have he (k : ℕ) : (2:ℝ)^(K+k+1)/4^(K+k) = (2/2^K)*(1/2^k) := by |
| rw [hfour,pow_succ,pow_add] |
| field_simp |
| simp_rw [he] |
| rw [← Finset.mul_sum,sum_two_inv_pow] |
| rw [h1,h2,hfour] |
| have hpowK : (0:ℝ)<2^K := by positivity |
| have hpowN : (0:ℝ)<2^N := by positivity |
| have h1' : 2*((2:ℝ)^K-1)/((2:ℝ)^K)^2 ≤ 2/2^K := by |
| apply (div_le_iff₀ (sq_pos_of_pos hpowK)).mpr |
| field_simp |
| nlinarith |
| have h2' : (2/(2:ℝ)^K) * (2-2/2^N) ≤ 4/2^K := by |
| calc _ ≤ (2/(2:ℝ)^K)*2 := mul_le_mul_of_nonneg_left (by have := (div_nonneg (by norm_num : (0:ℝ)≤2) hpowN.le); linarith) (by positivity) |
| _ = _ := by ring |
| calc _ ≤ 2/(2:ℝ)^K+4/2^K := add_le_add h1' h2' |
| _ = _ := by ring |
|
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| end CriticalPercolation |
| end |
| end |
| end |
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| end OAI |
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