openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Estimates /AssemblySmallParameters.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SuffixDescendants | |
| namespace OAI | |
| /-! Quantitative estimates for the distance-set construction. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND | |
| open Set MeasureTheory | |
| open scoped Classical | |
| lemma assembly_small_parameters (n : ℕ) {chi Cd : ℝ} (hchi : 0<chi) (hCd : 0≤Cd) : | |
| ∃ beta : ℝ, 0<beta ∧ beta< min (chi/10) (1/10) ∧ | |
| ∃ kap : ℝ, 0<kap ∧ kap≤1/1000 ∧ | |
| ∃ sig : ℝ, 0<sig ∧ sig<kap/1000 ∧ beta+(n+4:ℝ)*sig<chi/4 ∧ | |
| ∃ omega : ℝ, 0<omega ∧ 4*(n+1:ℝ)*omega<beta/8 ∧ | |
| -beta+Cd*(kap+sig/kap)+beta/8+4*(n+1:ℝ)*omega ≤ -beta/2 := by | |
| obtain ⟨beta,hbeta,hbetasm⟩ := exists_between | |
| (show (0:ℝ)< min (chi/10) (1/10) from lt_min (by positivity) (by norm_num)) | |
| have hCb : 0<Cd+1 := by linarith | |
| obtain ⟨kap,hkap,hkapub⟩ := exists_between | |
| (show (0:ℝ)< min (1/1000) (beta/(32*(Cd+1))) from lt_min (by norm_num) (by positivity)) | |
| have hkap1 : kap≤1/1000 := (hkapub.trans_le (min_le_left _ _)).le | |
| have hkapCd : Cd*kap<beta/32 := by | |
| have hh := (lt_div_iff₀ (by positivity : 0<32*(Cd+1))).mp | |
| (hkapub.trans_le (min_le_right _ _)) | |
| nlinarith | |
| have hdiff : 0<chi/4-beta := by | |
| have := hbetasm.trans_le (min_le_left _ _) | |
| linarith | |
| have hn4 : 0<(n+4:ℝ) := by positivity | |
| obtain ⟨sig,hsig,hsigub⟩ := exists_between | |
| (show (0:ℝ)< min (kap/1000) (min (kap*beta/(32*(Cd+1))) ((chi/4-beta)/(n+4:ℝ))) from | |
| lt_min (by positivity) (lt_min (by positivity) (by positivity))) | |
| have hsigkap : sig<kap/1000 := hsigub.trans_le (min_le_left _ _) | |
| have hsig2 := hsigub.trans_le (min_le_right _ _) | |
| have hsigCd : Cd*(sig/kap)<beta/32 := by | |
| have hh := (lt_div_iff₀ (by positivity : 0<32*(Cd+1))).mp | |
| (hsig2.trans_le (min_le_left _ _)) | |
| rw [← mul_div_assoc] | |
| apply (div_lt_iff₀ hkap).mpr | |
| nlinarith | |
| have hsmall : beta+(n+4:ℝ)*sig<chi/4 := by | |
| have hh := (lt_div_iff₀ hn4).mp (hsig2.trans_le (min_le_right _ _)) | |
| linarith | |
| obtain ⟨omega,homega,homegaub⟩ := exists_between | |
| (show (0:ℝ)<beta/(32*(n+1:ℝ)) from by positivity) | |
| have homegasm : 4*(n+1:ℝ)*omega<beta/8 := by | |
| have hh := (lt_div_iff₀ (by positivity : 0<32*(n+1:ℝ))).mp homegaub | |
| linarith | |
| refine ⟨beta,hbeta,hbetasm,kap,hkap,hkap1,sig,hsig,hsigkap,hsmall,omega,homega,homegasm,?_⟩ | |
| nlinarith | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |