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f89a930 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | 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
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