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