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