openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Probability /FullBoxVarianceGeometry.lean
| import OAI.NumberTheory.Jacobsthal.Partitions.FullBoxTotalLength | |
| namespace OAI | |
| namespace Erdos970 | |
| open scoped _root_.Erdos970 | |
| section | |
| namespace ErdosVarianceWeighted | |
| open NumberTheoryLean FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport SafeSubsetBoxGeometry | |
| ErdosCofactorChoices ErdosSubsetWord SingletonBinSelection LogarithmicBinScale | |
| theorem product_le_Y_of_length (w a : ℝ) (hw : 1 < w) (ha : 0 ≤ a) (Y d : ℕ) | |
| (hY : 0 < Y) (hd : 0 < d) (hlen : a ≤ Real.log ((Y : ℝ)/(d : ℝ))/Real.log w) : d ≤ Y := by | |
| have hdR : (0 : ℝ) < d := by exact_mod_cast hd | |
| have hYR : (0 : ℝ) < Y := by exact_mod_cast hY | |
| have hh : w^a ≤ (Y : ℝ)/(d : ℝ) := (Real.le_logb_iff_rpow_le hw (div_pos hYR hdR)).mp hlen | |
| have hratio : 1 ≤ (Y : ℝ)/(d : ℝ) := (Real.one_le_rpow hw.le ha).trans hh | |
| have hreal := (le_div_iff₀ hdR).mp hratio | |
| have hdle : (d : ℝ) ≤ Y := by simpa only [one_mul] using hreal | |
| exact_mod_cast hdle | |
| theorem full_box_variance_fibres {w top xi B C K a : ℝ} | |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ C) (ha0 : 0 ≤ a) | |
| (hcomp : Real.log B ≤ 2*Real.log w) (Y : ℕ) (hY : 0 < Y) (z : Node) | |
| (hroot : z.gap = Real.log (Y : ℝ)/Real.log w-a+2) | |
| (hs : Valid z.side z.ratio) (hz : Consistent z) (hg : StrongState z) | |
| (hclosed : z.closed = true) (hcap : w^z.cutoff = top) | |
| (m : Fin (binCount w top xi) → ℕ) (hanchor : SafeAnchor hw htop hxi C B K z m) | |
| (i : Fin (binCount w top xi)) (hi : m i = 1) : | |
| ∀ f ∈ selections (globalBins w top xi) (eraseMultiplicity m i),Squarefree (selectionProduct f) ∧ | |
| (∀ t ∈ (selectionProduct f).primeFactors,w < (t : ℝ)) ∧ | |
| (∀ p ∈ globalBins w top xi i,(selectionProduct f).Coprime p ∧ | |
| 3*a/4 ≤ Real.log ((Y : ℝ)/((p : ℝ)*selectionProduct f))/Real.log w) := by | |
| intro f hf | |
| have harith := erased_cofactor_arithmetic hw htop hxi m i f hf | |
| have hlen := erased_cofactor_all_lengths hw htop hxi hC hcomp Y hY z hroot hs hz hg hclosed hcap m hanchor i hi f hf | |
| refine ⟨harith.1,harith.2.1,?_⟩ | |
| intro p hp | |
| exact ⟨harith.2.2 p hp,(by linarith : 3*a/4 ≤ a).trans (hlen p hp)⟩ | |
| end ErdosVarianceWeighted | |
| end | |
| end Erdos970 | |
| end OAI | |