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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.NumberTheory.Jacobsthal.Partitions.BinCutGapMovement
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace NumberTheoryLean.ActualReferencePrefixes
open FinitePathGeometry PrimeHistories ReferenceAdmission StrongSourceFamilies StrongReferenceTransport
open PrimeBinMembership LogarithmicBinPartition
open ErdosPrimeInputs.PrimePrefixMass
attribute [local instance] Classical.propDecidable
theorem reference_prefix_mem (w : ℝ) (P : Finset ℕ) (i : Side) (r : ℝ) (ps pre : List ℕ)
(hp : ps ∈ referencePrefixes w P i r) (hpre : pre ∈ ps.inits) : pre ∈ referencePrefixes w P i r := by
obtain ⟨tail,he⟩ := (List.mem_inits pre ps).mp hpre
obtain ⟨hd,ha⟩ := Finset.mem_filter.mp hp
obtain ⟨horder,hmem⟩ := mem_decreasingPrefixes.mp hd
rw [← he] at horder ha
apply Finset.mem_filter.mpr
refine ⟨mem_decreasingPrefixes.mpr ⟨(List.pairwise_append.mp horder).1,?_⟩,
((admitted_append w r i pre tail).mp ha).1⟩
intro p hp
exact hmem p (by rw [← he]; exact List.mem_append_left _ hp)
theorem reference_prefix_gap_floor {w top : ℝ} (hw : 1 < w) (htop : w < top) (z : Node)
(hs : Valid z.side z.ratio) (hz : Consistent z) (hg : StrongState z)
(hclosed : z.closed=true) (hcap : w^z.cutoff=top) (ps pre : List ℕ)
(hp : ps ∈ referencePrefixes w (sourcePrimeSet w top) z.side z.gap) (hpre : pre ∈ ps.inits) :
2 ≤ (terminal w z pre).gap :=
(source_reference_transport hw htop z hs hz hg hclosed hcap pre (reference_prefix_mem w _ _ _ ps pre hp hpre)).2
end NumberTheoryLean.ActualReferencePrefixes
namespace NumberTheoryLean.SelectionWordSplit
open ErdosCofactorChoices ErdosSubsetWord BinCutSelections SourceLabelStrictOrder
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition
theorem selected_word_split {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi)
(m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ)
(hf : f ∈ selections (globalBins w top xi) m) (j : Fin (binCount w top xi))
(u : ℕ) (hfu : f j={u}) :
descendingWord f=descendingWord (aboveSelection f j)++u::descendingWord (belowSelection f j) := by
have hu : u ∈ f j := by rw [hfu]; simp
have hul := selected_prime_label hw htop hxi m f hf j hu
have humem : u ∈ descendingWord f := (mem_descendingWord f u).mpr ⟨j,hu⟩
have hsource := word_source_membership hw htop hxi m f hf
have hsmaller : ∀ p ∈ descendingWord (belowSelection f j),p<u := by
intro p hp
have hh := (below_word_membership hw htop hxi m f hf j p).mp hp
apply prime_lt_of_label_lt hw htop hxi (hsource p hh.1) (hsource u humem)
rw [hul]
exact hh.2
have hgreater : ∀ p ∈ descendingWord (aboveSelection f j),u<p := by
intro p hp
have hh := (above_word_membership hw htop hxi m f hf j p).mp hp
apply prime_lt_of_label_lt hw htop hxi (hsource u humem) (hsource p hh.1)
rw [hul]
exact hh.2
have hright : (descendingWord (aboveSelection f j)++u::descendingWord (belowSelection f j)).Pairwise (· > ·) := by
rw [List.pairwise_append,List.pairwise_cons]
refine ⟨descendingWord_strict _,⟨hsmaller,descendingWord_strict _⟩,?_⟩
intro a ha b hb
rcases List.mem_cons.mp hb with rfl | hb
· exact hgreater a ha
· exact (hsmaller b hb).trans (hgreater a ha)
apply (descendingWord_strict f).eq_of_mem_iff hright
intro p
constructor
· intro hp
obtain ⟨k,hpk⟩ := (mem_descendingWord f p).mp hp
have hpl := selected_prime_label hw htop hxi m f hf k hpk
rcases lt_trichotomy k j with hlt | rfl | hgt
· exact List.mem_append_right _ (List.mem_cons_of_mem _ ((below_word_membership hw htop hxi m f hf j p).mpr ⟨hp,by rwa [hpl]⟩))
· have he : p=u := by simpa only [hfu,Finset.mem_singleton] using hpk
subst p
simp
· exact List.mem_append_left _ ((above_word_membership hw htop hxi m f hf j p).mpr ⟨hp,by rwa [hpl]⟩)
· intro hp
rcases List.mem_append.mp hp with hp | hp
· exact ((above_word_membership hw htop hxi m f hf j p).mp hp).1
· rcases List.mem_cons.mp hp with rfl | hp
· exact humem
· exact ((below_word_membership hw htop hxi m f hf j p).mp hp).1
end NumberTheoryLean.SelectionWordSplit
namespace NumberTheoryLean.BinCutPrefixGeometry
open ErdosCofactorChoices ErdosSubsetWord BinCutSelections SelectionWordSplit
open PrimeHistories PrimeProductLogCoordinates
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition
theorem above_word_is_prefix {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi)
(m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ)
(hf : f ∈ selections (globalBins w top xi) m) (j : Fin (binCount w top xi))
(u : ℕ) (hfu : f j={u}) :
descendingWord (aboveSelection f j) ∈ (descendingWord f).inits ∧
descendingWord (aboveSelection f j)++[u] ∈ (descendingWord f).inits := by
have he := selected_word_split hw htop hxi m f hf j u hfu
constructor
· exact (List.mem_inits _ _).mpr ⟨u::descendingWord (belowSelection f j),he.symm⟩
· apply (List.mem_inits _ _).mpr
refine ⟨descendingWord (belowSelection f j),?_⟩
simpa only [List.append_assoc,List.singleton_append] using he.symm
theorem above_word_product {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi)
(m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ)
(hf : f ∈ selections (globalBins w top xi) m) (j : Fin (binCount w top xi)) :
(descendingWord (aboveSelection f j)).prod=selectionProduct (aboveSelection f j) := by
apply descendingWord_product (globalBins w top xi)
(fun i k hik => bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi i k hik)
intro k
exact ((mem_selections _ _ _).mp (above_selection_mem _ m f hf j) k).1
end NumberTheoryLean.BinCutPrefixGeometry
namespace NumberTheoryLean.FullBoxPrefixLower
open FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport StrongSourceFamilies
open ErdosCofactorChoices ErdosSubsetWord SafeSubsetBoxGeometry ActualReferencePrefixes BinCutSelections BinCutPrefixGeometry
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition
theorem full_box_child_gap_floor {w top xi B C K : ℝ}
(hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ C)
(hcomp : Real.log B ≤ 2*Real.log w) (z : Node)
(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) (ha : SafeAnchor hw htop hxi C B K z m)
(f : Fin (binCount w top xi) → Finset ℕ) (hf : f ∈ selections (globalBins w top xi) m)
(j : Fin (binCount w top xi)) (u : ℕ) (hu : f j={u}) :
2 ≤ (terminal w z (descendingWord (aboveSelection f j)++[u])).gap := by
have hfull := (anchored_choice_geometry hw htop hxi hC hcomp z m ha f hf).1
have hprefix := (above_word_is_prefix hw htop hxi m f hf j u hu).2
exact reference_prefix_gap_floor hw htop z hs hz hg hclosed hcap _ _ hfull hprefix
end NumberTheoryLean.FullBoxPrefixLower
end Erdos970
end OAI
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