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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 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.NumberTheory.Jacobsthal.Estimates.UncappedHighRemoval
import OAI.NumberTheory.Jacobsthal.Probability.RetainedCompactEvent
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
namespace NumberTheoryLean.PrimeFamilyOccupation
attribute [local instance] Classical.propDecidable
open FinitePathGeometry PrimeHistories PrimeTiltGeometry
open ActualPrimeHigh
theorem retained_no_high {w ell S : ℝ} {start : Node} {ps : List ℕ}
(hsS : start.ratio ≤ S) (hp : allowed w ell S start ps) :
¬actualHasHigh w S start ps := by
rintro ⟨pre,suf,he,hh⟩
have hpre := (allowed_append w ell S start pre suf).mp (he ▸ hp) |>.1
exact (not_lt_of_ge (terminal_ratio_le hsS hpre)) hh
theorem allowed_of_uncapped_prefix_ceiling {w ell S : ℝ} {start : Node} {ps : List ℕ}
(hp : uncappedAllowed w ell start ps)
(hbound : ∀ pre suf : List ℕ, ps = pre++suf → (terminal w start pre).ratio ≤ S) :
allowed w ell S start ps := by
classical
induction ps generalizing start with
| nil => exact allowed_nil w ell S start
| cons p ps ih =>
obtain ⟨hc,ht⟩ := hp
have hpS : (step w start p).ratio ≤ S := by
simpa only [terminal_cons,terminal_nil] using hbound [p] ps rfl
have hb : ∀ pre suf : List ℕ, ps = pre++suf →
(terminal w (step w start p) pre).ratio ≤ S := by
intro pre suf he
exact hbound (p::pre) suf (by rw [List.cons_append,he])
obtain ⟨hprime,hlo,hcap,hmin⟩ := Finset.mem_filter.mp hc
apply (allowed_cons w ell S start p ps).mpr
exact ⟨Finset.mem_filter.mpr ⟨hprime,hlo,hcap,hmin,hpS⟩,ih ht hb⟩
theorem allowed_iff_uncapped_no_high {w ell S : ℝ} {start : Node} (hsS : start.ratio ≤ S)
(ps : List ℕ) : allowed w ell S start ps ↔
uncappedAllowed w ell start ps ∧ ¬actualHasHigh w S start ps := by
constructor
· intro hp
exact ⟨(uncapped_iff_exists_ceiling w ell start ps).mpr ⟨S,hsS,hp⟩,retained_no_high hsS hp⟩
· rintro ⟨hp,hh⟩
apply allowed_of_uncapped_prefix_ceiling hp
intro pre suf he
exact le_of_not_gt (fun h => hh ⟨pre,suf,he,h⟩)
theorem retained_eq_nonhigh {w ell S : ℝ} (hw : 1 < w) {start : Node} (hsS : start.ratio ≤ S) :
retainedPrefixes w ell S start =
(uncappedPrefixes w ell start).filter (fun ps => ¬actualHasHigh w S start ps) := by
classical
ext ps
rw [mem_retainedPrefixes hw,Finset.mem_filter,mem_uncappedPrefixes hw]
exact allowed_iff_uncapped_no_high hsS ps
theorem uncapped_sum_partition {w ell S : ℝ} (hw : 1 < w) {start : Node}
(hsS : start.ratio ≤ S) (F : List ℕ → ℝ) :
(∑ ps ∈ uncappedPrefixes w ell start,F ps) =
(∑ ps ∈ retainedPrefixes w ell S start,F ps) +
∑ ps ∈ (uncappedPrefixes w ell start).filter (actualHasHigh w S start),F ps := by
classical
rw [retained_eq_nonhigh hw hsS]
exact (Finset.sum_filter_not_add_sum_filter _ _ _).symm
end NumberTheoryLean.PrimeFamilyOccupation
end
section
namespace NumberTheoryLean.CompactSelectedPartition
open _root_.Set _root_.Finset
open FinitePathGeometry PrimeHistories PrimeFamilyOccupation ActualPrimeHigh
open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail
attribute [local instance] Classical.propDecidable
theorem selected_uncapped_partition {w ell S : ℝ} (hw : 1 < w) {start : Node}
(hsS : start.ratio ≤ S) (E : Set (List ℕ)) :
(∑ ps ∈ (uncappedPrefixes w ell start).filter (· ∈ E),prefixWeight ps) =
(∑ ps ∈ (retainedPrefixes w ell S start).filter (· ∈ E),prefixWeight ps) +
∑ ps ∈ ((uncappedPrefixes w ell start).filter (actualHasHigh w S start)).filter (· ∈ E),
prefixWeight ps := by
have h := uncapped_sum_partition (ell:=ell) hw hsS (fun ps => if ps ∈ E then prefixWeight ps else 0)
simpa only [Finset.sum_filter] using h
theorem selected_uncapped_le {w ell S R : ℝ} (hw : 1 < w) {start : Node}
(hsS : start.ratio ≤ S) (E : Set (List ℕ))
(hcompact : ∀ ps ∈ uncappedPrefixes w ell start,ps ∈ E → (terminal w start ps).gap ≤ R) :
(∑ ps ∈ (uncappedPrefixes w ell start).filter (· ∈ E),prefixWeight ps) ≤
(∑ ps ∈ (retainedPrefixes w ell S start).filter (· ∈ E),prefixWeight ps) +
uncappedLowGapHighMass w ell start S R := by
rw [selected_uncapped_partition hw hsS E]
apply add_le_add le_rfl
unfold uncappedLowGapHighMass
apply Finset.sum_le_sum_of_subset_of_nonneg
· intro ps hp
obtain ⟨hhigh,hE⟩ := Finset.mem_filter.mp hp
obtain ⟨hps,hh⟩ := Finset.mem_filter.mp hhigh
exact Finset.mem_filter.mpr ⟨hps,hcompact ps hps hE,hh⟩
· intro ps _ _
exact prefixWeight_nonneg ps
end NumberTheoryLean.CompactSelectedPartition
end
end Erdos970
end OAI
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