openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Graphs /DenseGraphComparableParameters.lean
| import OAI.MeasureTheory.Falconer.Graphs.DenseGraphParameters | |
| namespace OAI | |
| /-! Additional polynomial room for comparable coordinate counts. -/ | |
| namespace Falconer | |
| theorem dense_graph_comparable_parameters {θ : ℝ} (hθ : 0 < θ) : | |
| ∃ γ : ℝ, ∃ N0 : ℕ, 0 < γ ∧ 0 < N0 ∧ ∀ N : ℕ, N0 ≤ N → | |
| let S := Real.exp (γ*(N:ℝ)*Real.log 2) | |
| let R := Real.exp (θ*(N:ℝ)*Real.log 2) | |
| 1 ≤ S ∧ S ≤ R ∧ 67108864*S^15 ≤ R ∧ R⁻¹ ≤ (S⁻¹)^2/64 := by | |
| obtain ⟨N0,hN0,hbounds⟩ := scalar_fixed_losses hθ | |
| (by norm_num : (0:ℝ) < 67108864) (by norm_num : (0:ℝ) < 64) | |
| let γ := θ/100 | |
| have hγ : 0 < γ := by dsimp [γ]; positivity | |
| refine ⟨γ,N0,hγ,hN0,?_⟩ | |
| intro N hN | |
| dsimp only | |
| let S := Real.exp (γ*(N:ℝ)*Real.log 2) | |
| let R := Real.exp (θ*(N:ℝ)*Real.log 2) | |
| have hSpos : 0 < S := Real.exp_pos _ | |
| have hRpos : 0 < R := Real.exp_pos _ | |
| have hlog : 0 ≤ (N:ℝ)*Real.log 2 := by positivity | |
| have hS : 1 ≤ S := Real.one_le_exp_iff.mpr (by positivity) | |
| have hSR : S ≤ R := by | |
| apply Real.exp_le_exp.mpr | |
| dsimp [γ] | |
| nlinarith | |
| have hp (n : ℕ) (hn : (n:ℝ) ≤ 25) : S^n ≤ Real.exp ((θ/4)*(N:ℝ)*Real.log 2) := by | |
| rw [show S = Real.exp (γ*(N:ℝ)*Real.log 2) from rfl,← Real.exp_nat_mul] | |
| apply Real.exp_le_exp.mpr | |
| dsimp [γ] | |
| have hh := mul_le_mul_of_nonneg_right hn (mul_nonneg hθ.le hlog) | |
| nlinarith | |
| have h15 : 67108864*S^15 ≤ R := | |
| (mul_le_mul_of_nonneg_left (hp 15 (by norm_num)) (by norm_num)).trans (hbounds N hN).1 | |
| have h2 : 64*S^2 ≤ R := | |
| (mul_le_mul_of_nonneg_left (hp 2 (by norm_num)) (by norm_num)).trans (hbounds N hN).2 | |
| refine ⟨hS,hSR,h15,?_⟩ | |
| have hi := one_div_le_one_div_of_le (by positivity : 0 < 64*S^2) h2 | |
| have he : (S⁻¹)^2/64 = 1/(64*S^2) := by field_simp | |
| change R⁻¹ ≤ (S⁻¹)^2/64 | |
| rw [he] | |
| simpa only [one_div] using hi | |
| end Falconer | |
| end OAI | |