| import OAI.NumberTheory.Jacobsthal.Partitions.BinCutGapMovement |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.ActualReferencePrefixes |
| open FinitePathGeometry PrimeHistories ReferenceAdmission StrongSourceFamilies StrongReferenceTransport |
| open PrimeBinMembership LogarithmicBinPartition |
| open ErdosPrimeInputs.PrimePrefixMass |
| attribute [local instance] Classical.propDecidable |
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| 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) |
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| 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 |
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| namespace NumberTheoryLean.SelectionWordSplit |
| open ErdosCofactorChoices ErdosSubsetWord BinCutSelections SourceLabelStrictOrder |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
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| 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 |
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| namespace NumberTheoryLean.BinCutPrefixGeometry |
| open ErdosCofactorChoices ErdosSubsetWord BinCutSelections SelectionWordSplit |
| open PrimeHistories PrimeProductLogCoordinates |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
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| 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 |
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| 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 |
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| namespace NumberTheoryLean.FullBoxPrefixLower |
| open FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport StrongSourceFamilies |
| open ErdosCofactorChoices ErdosSubsetWord SafeSubsetBoxGeometry ActualReferencePrefixes BinCutSelections BinCutPrefixGeometry |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
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| 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 |
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| end Erdos970 |
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| end OAI |
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