| import OAI.NumberTheory.Jacobsthal.Paths.RawFullHitIncreasing | |
| namespace OAI | |
| namespace Erdos970 | |
| section | |
| namespace Erdos970Dependency.MarkedVisits | |
| open Filter Set MeasureTheory ProbabilityTheory | |
| open scoped ProbabilityTheory ENNReal | |
| open NumberTheoryLean.FinitePathMeasures NumberTheoryLean.FinitePathGeometry | |
| open NumberTheoryLean.FirstHitKernels | |
| lemma gapAt_antitone_cost {r T U : ℝ} (hr : 0 ≤ r) (hTU : T ≤ U) : gapAt r U ≤ gapAt r T := | |
| mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (neg_le_neg hTU)) hr | |
| lemma increasingCosts_before_smallGap {a b : ℕ} {h : RawHistory b} | |
| (hh : h ∈ increasingCosts a b) {r K : ℝ} (hr : 0 ≤ r) | |
| (i j : Finset.Iic b) (haj : a ≤ j.1) | |
| (hi : K < gapAt r (h i).2) (hj : gapAt r (h j).2 ≤ K) : i.1 < j.1 := by | |
| by_contra hn | |
| have hji : j.1 ≤ i.1 := by omega | |
| have hc := hh j i haj hji | |
| have hg := gapAt_antitone_cost hr hc | |
| exact (not_lt_of_ge (hg.trans hj)) hi | |
| lemma rawEvenHit_actual_anchor (a : ℕ) (v H : ℝ) (h0 : RawHistory a) : | |
| ∀ᵐ z ∂rawEvenMarkedHitKernel a v H h0, | |
| let q := rawEvenFinalCycle a z | |
| q.2.2 ⟨q.1,Finset.mem_Iic.mpr (by omega)⟩ ∈ regenerationSet := | |
| rawEvenHit_anchor_regenerates a v H h0 | |
| theorem rawEvenHit_before_compact {r b0 b1 K : ℝ} (hr : 0 < r) (hb0 : 0 < b0) | |
| (hb1 : 0 < b1) (hK : K ≤ 6*b0) (a : ℕ) (h0 : RawHistory a) : | |
| ∀ᵐ z ∂rawEvenMarkedHitKernel a (Real.log (r/b1)) (Real.log (b1/(10*b0))) h0, | |
| let q := rawEvenFinalCycle a z | |
| ∀ j : Finset.Iic (q.1+2*(q.2.1+1)), a ≤ j.1 → | |
| gapAt r (q.2.2 j).2 ≤ K → q.1+1 < j.1 := by | |
| filter_upwards [rawEvenHit_ae_geometric hr hb0 hb1 a h0, | |
| rawEvenHit_increasing a (Real.log (r/b1)) (Real.log (b1/(10*b0))) h0] with z hG hI | |
| obtain ⟨tau,_ht0,_ht1,_hF,_hg0,_hg1,_hx0,_hx1,_hx2,_hx3,hgap,_hratio⟩ := hG | |
| have hR := rawEvenFinalCycle_record a z | |
| rw [hR] at hgap | |
| rcases he : rawEvenFinalCycle a z with ⟨t,n,h⟩ | |
| change rawEvenFinalCycle a z ∈ selectedIncreasing a at hI | |
| rw [he] at hI hgap | |
| change h ∈ increasingCosts a (t+2*(n+1)) at hI | |
| change 6*b0 < gapAt r (h ⟨t+1,Finset.mem_Iic.mpr (by omega)⟩).2 at hgap | |
| change ∀ j : Finset.Iic (t+2*(n+1)), a ≤ j.1 → gapAt r (h j).2 ≤ K → t+1 < j.1 | |
| intro j haj hj | |
| exact increasingCosts_before_smallGap hI hr.le ⟨t+1,Finset.mem_Iic.mpr (by omega)⟩ j haj | |
| (hK.trans_lt hgap) hj | |
| theorem rawEvenHit_before_compact_extension {r b0 b1 K : ℝ} (hr : 0 < r) (hb0 : 0 < b0) | |
| (hb1 : 0 < b1) (hK : K ≤ 6*b0) (a : ℕ) (h0 : RawHistory a) : | |
| ∀ᵐ z ∂rawEvenMarkedHitKernel a (Real.log (r/b1)) (Real.log (b1/(10*b0))) h0, | |
| let q := rawEvenFinalCycle a z | |
| ∀ (c : ℕ) (_hc : q.1+2*(q.2.1+1) ≤ c), | |
| ∀ᵐ y ∂rawExtension (q.1+2*(q.2.1+1)) c q.2.2, | |
| ∀ j : Finset.Iic c, a ≤ j.1 → gapAt r (y j).2 ≤ K → q.1+1 < j.1 := by | |
| filter_upwards [rawEvenHit_ae_geometric hr hb0 hb1 a h0, | |
| rawEvenHit_increasing a (Real.log (r/b1)) (Real.log (b1/(10*b0))) h0] with z hG hI | |
| obtain ⟨tau,_ht0,_ht1,_hF,_hg0,_hg1,_hx0,_hx1,_hx2,_hx3,hgap,_hratio⟩ := hG | |
| have hR := rawEvenFinalCycle_record a z | |
| rw [hR] at hgap | |
| rcases he : rawEvenFinalCycle a z with ⟨t,n,h⟩ | |
| change rawEvenFinalCycle a z ∈ selectedIncreasing a at hI | |
| rw [he] at hI hgap | |
| change h ∈ increasingCosts a (t+2*(n+1)) at hI | |
| change 6*b0 < gapAt r (h ⟨t+1,Finset.mem_Iic.mpr (by omega)⟩).2 at hgap | |
| change ∀ (c : ℕ) (hc : t+2*(n+1) ≤ c), ∀ᵐ y ∂rawExtension (t+2*(n+1)) c h, | |
| ∀ j : Finset.Iic c, a ≤ j.1 → gapAt r (y j).2 ≤ K → t+1 < j.1 | |
| intro c hc | |
| filter_upwards [rawExtension_preserves_increasing hc h hI,rawExtension_retains_prefix hc h] with y hy hp | |
| have heval := congrFun hp (⟨t+1,Finset.mem_Iic.mpr (by omega)⟩ : Finset.Iic (t+2*(n+1))) | |
| change y ⟨t+1,Finset.mem_Iic.mpr (by omega)⟩=h ⟨t+1,Finset.mem_Iic.mpr (by omega)⟩ at heval | |
| intro j haj hj | |
| apply increasingCosts_before_smallGap hy hr.le ⟨t+1,Finset.mem_Iic.mpr (by omega)⟩ j haj _ hj | |
| rw [heval] | |
| exact hK.trans_lt hgap | |
| end Erdos970Dependency.MarkedVisits | |
| end | |
| end Erdos970 | |
| end OAI | |