openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Partitions /GeometricInverseGeometry.lean
| 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 | |