openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Analysis /SourceBadCompactOccurrence.lean
| import OAI.NumberTheory.Jacobsthal.Paths.NearPrefixOccurrence | |
| import OAI.NumberTheory.Jacobsthal.Probability.CompletedMarkEvents | |
| namespace OAI | |
| namespace Erdos970 | |
| open scoped _root_.Erdos970 | |
| section | |
| namespace NumberTheoryLean.SourceBadCompactOccurrence | |
| open _root_.Set _root_.Filter _root_.MeasureTheory ProbabilityTheory | |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain PrimeBinMembership | |
| open ActualCoupledHistories ActualSuccessfulHistories ActualFlagInvariant ActualProcessCoupling | |
| open FiniteHistoryTransport PriorPrimeWindow CompactPrefixOccurrence CompletedMarkEvents | |
| open RegeneratingInverseBands NearbyParentGeometry | |
| variable {w ell S R b₀ b₁ : ℝ} {start : Node} | |
| variable (hw : normalizationThreshold ≤ w) (hell : 1 ≤ ell) (hS0 : 0 ≤ S) | |
| variable (hS : S ≤ (Real.log w)^3) (hr : 0 < start.gap) | |
| variable (hs : Valid start.side start.ratio) (hsS : start.ratio ≤ S) | |
| theorem missing_occurrence_inclusion (hc : Consistent start) (hb₀ : 0 < b₀) | |
| {mesh : ℝ} (hm : 0 < mesh) (hmesh : mesh ≤ 1/100) (N : ℕ) | |
| (herror : 4*(N:ℝ)*mesh ≤ Real.log (11/10)) (E : Set (List ℕ)) | |
| (hE : ∀ p : History w ell S start,p.primes ∈ E → | |
| p.node.gap ≤ R ∧ ¬hasPrimeWindow w b₀ b₁ start p.primes) : | |
| ∀ᵐ h ∂sourceHistoryLaw hw hell hS0 hS hr hs hsS mesh N, | |
| occurs E N h → (last N h).2 = true ∨ | |
| mapHist (fun q => q.1.2) N h ∈ badCompactPath (Real.log start.gap) ((11/10)*R) b₀ b₁ N := by | |
| filter_upwards [sourceHistory_success_good hw hell hS0 hS hr hs hsS hm N, | |
| sourceHistory_prior_mark hw hell hS0 hS hr hs hsS hc hm hmesh N herror] with h hg hprior | |
| intro hocc | |
| by_cases hf : (last N h).2 = true | |
| · exact Or.inl hf | |
| have hfalse : (last N h).2 = false := by cases hval : (last N h).2 <;> simp_all | |
| obtain ⟨j,hjE⟩ := hocc | |
| let j' : Finset.Iic N := ⟨j,Finset.mem_Iic.mpr j.isLt.le⟩ | |
| cases hp : (h j').1.1 with | |
| | none => | |
| change selectedNode E (h j').1.1 at hjE | |
| rw [hp] at hjE | |
| exact False.elim hjE | |
| | some p => | |
| change selectedNode E (h j').1.1 at hjE | |
| rw [hp] at hjE | |
| have hpE := hE p hjE | |
| have hgj : GoodAt (Real.log start.gap) mesh j'.1 (some p,(h j').1.2) := by | |
| simpa only [← hp] using hg hfalse j' | |
| cases hz : (h j').1.2 with | |
| | inr u => rw [hz] at hgj; exact False.elim hgj | |
| | inl z => | |
| rw [hz] at hgj | |
| have hn : (j'.1:ℝ) ≤ N := by exact_mod_cast Finset.mem_Iic.mp j'.2 | |
| have herr : 4*(j'.1:ℝ)*mesh ≤ Real.log (11/10) := | |
| (by nlinarith : 4*(j'.1:ℝ)*mesh ≤ 4*(N:ℝ)*mesh).trans herror | |
| have he : Real.exp (4*(j'.1:ℝ)*mesh) ≤ (11/10:ℝ) := by | |
| rw [← Real.exp_log (by norm_num : 0 < (11/10:ℝ))] | |
| exact Real.exp_le_exp.mpr herr | |
| have hpPos := terminal_gap_positive (by linarith : 0 ≤ ell) hr hs p.admissible | |
| have hrel := log_gap_relative hpPos (Real.exp_pos _) hgj.2.2.2 | |
| have hgap : gapValue (Real.log start.gap) z ≤ (11/10)*R := | |
| hrel.2.trans ((mul_le_mul_of_nonneg_right he hpPos.le).trans | |
| (mul_le_mul_of_nonneg_left hpE.1 (by norm_num))) | |
| apply Or.inr | |
| refine ⟨j',?_,?_⟩ | |
| · change compactPoint (Real.log start.gap) ((11/10)*R) (h j').1.2 | |
| rw [hz] | |
| exact hgap | |
| · intro i hij hMark | |
| change markedPoint (Real.log start.gap) b₀ b₁ (h i).1.2 at hMark | |
| cases hi : (h i).1.2 with | |
| | inr u => rw [hi] at hMark; exact False.elim hMark | |
| | inl y => | |
| rw [hi] at hMark | |
| exact hpE.2 (hprior hfalse i j' hij p hp y hi b₀ b₁ hb₀ | |
| hMark.1 hMark.2.1 hMark.2.2.1 hMark.2.2.2.1 hMark.2.2.2.2) | |
| end NumberTheoryLean.SourceBadCompactOccurrence | |
| end | |
| end Erdos970 | |
| end OAI | |