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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 | import OAI.NumberTheory.Jacobsthal.Paths.GeometricWords
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace NumberTheoryLean.GeometricIsolatedBin
open PrimeHistories GeometricRegularWords GeometricBoxImages GeometricWordsAudit ActualWordSelection ActualRegularBoxes
open ErdosSubsetWord SafeSubsetBoxGeometry FirstIsolatedPosition IsolatedBinGeometry IsolatedRealPrimeRange
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinEndpoints LogarithmicBinPartition
open ErdosPrimeInputs.PrimePrefixMass
attribute [local instance] Classical.propDecidable
theorem geometric_box_isolated_singleton {w top xi C B R L alpha beta : ℝ}
(hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hpower : w^B=top)
(hsearch : w^((1/4:ℝ)) ≤ alpha*B) (z : Node)
(m : Fin (binCount w top xi) → ℕ) (hm : m ∈ geometricBoxes hw htop hxi C B R L alpha beta z) :
∃ i : Fin (binCount w top xi),m i=1 ∧ isolatedBin w top xi B alpha beta i ∧ top^alpha ≤ lower w top xi i ∧
upper w top xi i ≤ top^beta ∧ ActualSourceTags.searchBin w (lower w top xi i) := by
obtain ⟨ps,hps,rfl⟩ := Finset.mem_image.mp hm
obtain ⟨pre,p,tail,he,hisol,_hfirst⟩ := geometric_first_isolated hw htop hxi z ps hps
have hrange := isolated_real_range hw htop hxi hpower hisol
have hlo := hrange.1
have hhi := hrange.2
have hs := isolated_bin_search hisol hsearch
let i := label (zero_lt_one.trans hw) htop hxi p
have hp : p ∈ ps := by rw [he]; simp
have hreg := (Finset.mem_filter.mp hps).1
obtain ⟨hd,_hl,_hc,_hg,hsingle⟩ := Finset.mem_filter.mp hreg
have hn : ps.Nodup := (mem_decreasingPrefixes.mp hd).1.imp (fun h => ne_of_gt h)
have hcount : 0 < (ps.map (label (zero_lt_one.trans hw) htop hxi)).count i :=
List.count_pos_iff.mpr (List.mem_map.mpr ⟨p,hp,rfl⟩)
have hle := hsingle i hs
refine ⟨i,?_,hisol,hlo,hhi,hs⟩
rw [wordMultiplicity_count _ ps hn]
omega
end NumberTheoryLean.GeometricIsolatedBin
namespace NumberTheoryLean.GeometricBinOrdering
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinEndpoints PrimeBinRepresentatives
open IsolatedBinGeometry BoundedEdgeBins BoundedEdgePrimeRange
theorem left_exponent_mono {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi)
(i j : Fin (binCount w top xi)) (hij : i ≤ j) :
leftExponent w (lower w top xi i) ≤ leftExponent w (lower w top xi j) := by
have hlow := endpoint_mono (zero_lt_one.trans hw) htop hxi (show i.1 ≤ j.1 from hij)
exact div_le_div_of_nonneg_right (Real.log_le_log (endpoint_pos (zero_lt_one.trans hw) i.1) hlow) (Real.log_pos hw).le
theorem isolated_before_bounded_edge {w top xi B alpha beta M X : ℝ} {Y : ℕ}
(hw : 1 < w) (htop : w < top) (hxi : 0 < xi)
(m : Fin (binCount w top xi) → ℕ) (i j : Fin (binCount w top xi))
(hi : isolatedBin w top xi B alpha beta i) (hj : boundedEdgeBin w top xi Y m M X j)
(hlarge : X < alpha*B) : j < i := by
by_contra! hle
have hh := left_exponent_mono hw htop hxi i j hle
have hb := bounded_edge_lower_exponent hw htop hxi m j hj
linarith [hi.1]
end NumberTheoryLean.GeometricBinOrdering
namespace NumberTheoryLean.GeometricInverseGeometry
open FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport
open ErdosCofactorChoices ErdosSubsetWord GeometricBoxImages GeometricIsolatedBin
open GeometricBinOrdering BoundedEdgeBins BoundedEdgePrimeRange ParentCofactorChoices CofactorAllChoiceBounds
open SafeSubsetBoxGeometry BinCutSelections
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition LogarithmicBinEndpoints
theorem geometric_inverse_geometry {w top xi B C K L alpha a M X : ℝ}
(hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 1 ≤ C) (hlog : 1 ≤ Real.log w)
(hcomp : Real.log B ≤ 2*Real.log w) (hDelta : 2*C*xi ≤ 1) (hB : 3 ≤ B)
(hpower : w^B=top) (hsearch : w^((1/4:ℝ)) ≤ alpha*B) (hlarge : X < alpha*B)
(Y : ℕ) (hY : 0 < Y) (z : Node) (hroot : z.gap=Real.log (Y:ℝ)/Real.log w-a+2)
(hi : z.side=.even) (h199 : 199/100 ≤ z.ratio) (hz : Consistent z) (hcut : z.cutoff=B) (hclosed : z.closed=true)
(m : Fin (binCount w top xi) → ℕ) (hm : m ∈ geometricBoxes hw htop hxi C B K L alpha (1/100) z) :
∃ i : Fin (binCount w top xi),m i=1 ∧ top^alpha ≤ lower w top xi i ∧ lower w top xi i ≤ top^((1:ℝ)/100) ∧
∀ j : Fin (binCount w top xi),boundedEdgeBin w top xi Y m M X j →
i ≠ j ∧ m j=1 ∧
parentCofactorMultiplicity m i j i=0 ∧ parentCofactorMultiplicity m i j j=0 ∧
((∑ k,parentCofactorMultiplicity m i j k : ℕ):ℝ) ≤ C*Real.log B ∧
w ≤ lower w top xi j ∧ lower w top xi j ≤ w^X ∧
(∀ q ∈ cofactorChoices (globalBins w top xi) (parentCofactorMultiplicity m i j),
∀ p ∈ globalBins w top xi i,Real.log ((Y:ℝ)/((p:ℝ)*(q:ℝ)))/Real.log w ≤ M+1) ∧
(∀ q ∈ cofactorChoices (globalBins w top xi) (parentCofactorMultiplicity m i j),
∀ p ∈ globalBins w top xi i,∀ u ∈ globalBins w top xi j,
a ≤ Real.log ((Y:ℝ)/((p:ℝ)*(q:ℝ)*(u:ℝ)))/Real.log w) := by
obtain ⟨i,hmi,hisol,hilo,hihi,_hsearch⟩ := geometric_box_isolated_singleton hw htop hxi hpower hsearch z m hm
have hsmall : 2*(xi/Real.log w) ≤ 6*(2*C*xi) := by
have hh := div_le_self hxi.le hlog
nlinarith
have ha := geometric_box_anchor hw htop hxi (by linarith : 0 ≤ C) hcomp hsmall z m hm
have hlen := anchor_total_length hw htop hxi z m ha
have hs : Valid z.side z.ratio := by rw [hi]; change 198/100 ≤ z.ratio; linarith
have hg := source_strong_state hB z hi h199 hz hcut
have hcap : w^z.cutoff=top := by rwa [hcut]
refine ⟨i,hmi,hilo,(bin_source_bounds (zero_lt_one.trans hw) htop hxi i).2.1.le.trans hihi,?_⟩
intro j hj
have hji := isolated_before_bounded_edge hw htop hxi m i j hisol hj hlarge
have hzero := cofactor_slots_zero m i j
have htotal : ((∑ k,parentCofactorMultiplicity m i j k : ℕ):ℝ) ≤ C*Real.log B :=
(Nat.cast_le.mpr (cofactor_total_le m i j)).trans hlen
have hrange := bounded_edge_prime_range hw htop hxi m j hj
have hall := cofactor_all_choice_lengths hw htop hxi (by linarith : 0 ≤ C) hcomp Y hY z hroot
hs hz hg hclosed hcap m ha i j hji hmi hj
refine ⟨ne_of_gt hji,bounded_edge_singleton m j hj,hzero.1,hzero.2,htotal,hrange.1,hrange.2,?_,hall.2⟩
intro q hq p hp
exact (hall.1 q hq p hp).trans (by linarith)
end NumberTheoryLean.GeometricInverseGeometry
end Erdos970
end OAI
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