openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Analysis /NormalizedCollisionSmallness.lean
| import Mathlib | |
| import OAI.NumberTheory.Jacobsthal.Estimates.UniformCollisionError | |
| namespace OAI | |
| namespace Erdos970 | |
| section | |
| namespace ErdosLineCollision | |
| theorem normalized_collision_bound (I K C n R N L : ℝ) | |
| (hK0 : 0 ≤ K) (hC : 0 ≤ C) (hn : 0 < n) (hN : 0 < N) (hL : 0 < L) | |
| (hcollision : I^2 ≤ K*(C*N^2+I)) (hthin : K ≤ n*R/L^6) (hI : I ≤ n*N) : | |
| (I/(n*N))^2 ≤ C*R/(n*L^6)+R/(N*L^6) := by | |
| have hup : I^2 ≤ (n*R/L^6)*(C*N^2+n*N) := by | |
| calc | |
| _ ≤ K*(C*N^2+I) := hcollision | |
| _ ≤ K*(C*N^2+n*N) := mul_le_mul_of_nonneg_left (add_le_add le_rfl hI) hK0 | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hthin (by positivity) | |
| rw [div_pow] | |
| calc | |
| _ ≤ ((n*R/L^6)*(C*N^2+n*N))/(n*N)^2 := | |
| div_le_div_of_nonneg_right hup (sq_nonneg _) | |
| _ = _ := by field_simp | |
| end ErdosLineCollision | |
| end | |
| section | |
| open Filter | |
| open scoped Topology | |
| namespace ErdosSourceCollision | |
| open ErdosLineCollision | |
| theorem uniform_normalized_collision (C eta eps : ℝ) (hC : 0 ≤ C) (heta : 0 < eta) (heps : 0 < eps) : | |
| ∀ᶠ z : ℝ in atTop, 1 < z ∧ ∀ inc K R N n : ℝ, | |
| 1 < R → R ≤ z^((1 : ℝ)/100) → z^((93 : ℝ)/100) ≤ N → | |
| eta*R/(4*Real.log R) ≤ n → 0 ≤ K → | |
| inc^2 ≤ K*(C*N^2+inc) → K ≤ n*R/(Real.log z)^6 → inc ≤ n*N → | |
| inc ≤ eps*n*N := by | |
| filter_upwards [uniform_collision_error C eta eps hC heta heps] with z hz | |
| refine ⟨hz.1,?_⟩ | |
| intro inc K R N n hR hRz hN hcount hK hcoll hthin hI | |
| have hz0 : 0 < z := by linarith [hz.1] | |
| have hR0 : 0 < R := by linarith | |
| have hlogR : 0 < Real.log R := Real.log_pos hR | |
| have hn : 0 < n := (div_pos (mul_pos heta hR0) (mul_pos (by norm_num) hlogR)).trans_le hcount | |
| have hNp : 0 < N := (Real.rpow_pos_of_pos hz0 _).trans_le hN | |
| have hL : 0 < Real.log z := by linarith [hz.2.1] | |
| have hsq := (normalized_collision_bound inc K C n R N (Real.log z) hK hC hn hNp hL hcoll hthin hI).trans | |
| (hz.2.2 R N n hR hRz hN hcount) | |
| have hratio : inc/(n*N) ≤ eps := by nlinarith [sq_nonneg (inc/(n*N)-eps)] | |
| have hh := (div_le_iff₀ (mul_pos hn hNp)).mp hratio | |
| nlinarith | |
| end ErdosSourceCollision | |
| end | |
| end Erdos970 | |
| end OAI | |