| import OAI.NumberTheory.Jacobsthal.Paths.ActualThresholdArrival |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace NumberTheoryLean.CanonicalMarkSlack |
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| open _root_.Set _root_.Filter _root_.MeasureTheory ProbabilityTheory |
| open FinitePathMeasures FinitePathGeometry ArrivalKernelGeometry RegeneratingInverseBands |
| open CanonicalCostCoordinates Erdos970Dependency.MarkedVisits |
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| theorem firstArrivalMark_bounds (z : CostState) (hz : firstArrivalMark z = true) : |
| stateSide z.1 = .even ∧ 209/100 ≤ stateRatio z.1 ∧ stateRatio z.1 ≤ 213/100 := by |
| rcases z with ⟨s,T⟩ |
| cases s with |
| | inl s => |
| have h := (sourceMark_true s).mp hz |
| exact ⟨rfl,h.1,h.2⟩ |
| | inr s => exact Bool.noConfusion hz |
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|
| theorem pair_hit_slack {r b₀ b₁ : ℝ} (hr : 0 < r) (hb₀ : 0 < b₀) (hb₁ : 0 < b₁) |
| (a k N : ℕ) (hk : a+2*k+1 ≤ N) (h : RawHistory N) |
| (htime : (h ⟨a+2*k+1,Finset.mem_Iic.mpr hk⟩).2 = |
| (h ⟨a+2*k,Finset.mem_Iic.mpr (by omega)⟩).2 + |
| cost (stateRatio (h ⟨a+2*k+1,Finset.mem_Iic.mpr hk⟩).1)) |
| (hHit : h ∈ pairHitTest a k N hk (Real.log (r/b₁)) (Real.log (b₁/(10*b₀)))) : |
| let z := h ⟨a+2*k+1,Finset.mem_Iic.mpr hk⟩ |
| stateSide z.1 = .even ∧ 209/100 ≤ stateRatio z.1 ∧ stateRatio z.1 ≤ 213/100 ∧ |
| 10*b₀/(313/100) ≤ currentExponent (Real.log r) z ∧ |
| currentExponent (Real.log r) z ≤ b₁/(309/100) ∧ 6*b₀ < gapValue (Real.log r) z := by |
| dsimp only |
| let pre := h ⟨a+2*k,Finset.mem_Iic.mpr (show a+2*k ≤ N by omega)⟩ |
| let z := h ⟨a+2*k+1,Finset.mem_Iic.mpr hk⟩ |
| have hmark := firstArrivalMark_bounds z hHit.2.1 |
| have hwindow := (marked_cost_window hr hb₀ hb₁ pre.2).mpr hHit.2.2 |
| have hslack := marked_draw_slack hb₀ hwindow.1 hwindow.2 hmark.2.1 hmark.2.2 |
| have hz : z = (z.1,pre.2+cost (stateRatio z.1)) := Prod.ext rfl htime |
| have he : currentExponent (Real.log r) z = nextExponent (gapAt r pre.2) (stateRatio z.1) := by |
| calc |
| _ = currentExponent (Real.log r) (z.1,pre.2+cost (stateRatio z.1)) := congrArg _ hz |
| _ = nextExponent (gapValue (Real.log r) pre) (stateRatio z.1) := currentExponent_update _ pre z.1 |
| _ = _ := by rw [gapValue_log_eq_gapAt hr] |
| have hg : gapValue (Real.log r) z = nextGap (gapAt r pre.2) (stateRatio z.1) := by |
| calc |
| _ = gapValue (Real.log r) (z.1,pre.2+cost (stateRatio z.1)) := congrArg _ hz |
| _ = nextGap (gapValue (Real.log r) pre) (stateRatio z.1) := gapValue_update _ pre z.1 |
| _ = _ := by rw [gapValue_log_eq_gapAt hr] |
| exact ⟨hmark.1,hmark.2.1,hmark.2.2,he ▸ hslack.1,he ▸ hslack.2.1,hg ▸ hslack.2.2.2.2.1⟩ |
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| theorem canonical_all_marks_slack {r b₀ b₁ : ℝ} (hr : 0 < r) (hb₀ : 0 < b₀) (hb₁ : 0 < b₁) |
| (s : State) (N : ℕ) : |
| ∀ᵐ h ∂finitePathMeasure s N, ∀ a k : ℕ, ∀ hk : a+2*k+1 ≤ N, |
| h ∈ pairHitTest a k N hk (Real.log (r/b₁)) (Real.log (b₁/(10*b₀))) → |
| let z := h ⟨a+2*k+1,Finset.mem_Iic.mpr hk⟩ |
| stateSide z.1 = .even ∧ 209/100 ≤ stateRatio z.1 ∧ stateRatio z.1 ≤ 213/100 ∧ |
| 10*b₀/(313/100) ≤ currentExponent (Real.log r) z ∧ |
| currentExponent (Real.log r) z ≤ b₁/(309/100) ∧ 6*b₀ < gapValue (Real.log r) z := by |
| have hall : ∀ᵐ h ∂finitePathMeasure s N, ∀ j : Fin N, |
| (h ⟨j.1+1,Finset.mem_Iic.mpr (by have := j.isLt; omega)⟩).2 = |
| (h ⟨j.1,Finset.mem_Iic.mpr j.isLt.le⟩).2 + |
| cost (stateRatio (h ⟨j.1+1,Finset.mem_Iic.mpr (by have := j.isLt; omega)⟩).1) := by |
| apply ae_all_iff.mpr |
| intro j |
| exact raw_cost_coordinate (Nat.zero_le _) j.isLt (fun _ => (s,0)) |
| filter_upwards [hall] with h hh |
| intro a k hk hHit |
| exact pair_hit_slack hr hb₀ hb₁ a k N hk h |
| (hh ⟨a+2*k,by omega⟩) hHit |
|
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| end NumberTheoryLean.CanonicalMarkSlack |
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| end |
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|
| section |
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|
| namespace NumberTheoryLean.CompletedMarkEvents |
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|
| open _root_.Set _root_.MeasureTheory ProbabilityTheory |
| open FinitePathGeometry FinitePathMeasures FiniteHistoryTransport CemeteryKernel |
| open ArrivalKernelGeometry RegeneratingInverseBands ActualCouplingUpdates |
|
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| def strongMark (v b₀ b₁ : ℝ) (z : CostState) : Prop := |
| stateSide z.1 = .even ∧ 209/100 ≤ stateRatio z.1 ∧ stateRatio z.1 ≤ 213/100 ∧ |
| 10*b₀/(313/100) ≤ currentExponent v z ∧ currentExponent v z ≤ b₁/(309/100) |
|
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| theorem strongMark_measurable (v b₀ b₁ : ℝ) : MeasurableSet {z | strongMark v b₀ b₁ z} := |
| (measurable_fst (side_set_measurable .even)).inter |
| ((measurableSet_le measurable_const (stateRatio_measurable.comp measurable_fst)).inter |
| ((measurableSet_le (stateRatio_measurable.comp measurable_fst) measurable_const).inter |
| ((measurableSet_le measurable_const (currentExponent_measurable v)).inter |
| (measurableSet_le (currentExponent_measurable v) measurable_const)))) |
|
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| def markedPoint (v b₀ b₁ : ℝ) : Space CostState → Prop := |
| Sum.elim (strongMark v b₀ b₁) (fun _ => False) |
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| def compactPoint (v R : ℝ) : Space CostState → Prop := |
| Sum.elim (fun z => gapValue v z ≤ R) (fun _ => False) |
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| theorem markedPoint_measurable (v b₀ b₁ : ℝ) : MeasurableSet {z | markedPoint v b₀ b₁ z} := |
| measurableSet_sum_iff.mpr ⟨strongMark_measurable _ _ _,MeasurableSet.empty⟩ |
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| theorem compactPoint_measurable (v R : ℝ) : MeasurableSet {z | compactPoint v R z} := |
| measurableSet_sum_iff.mpr ⟨measurableSet_le (gapValue_measurable v) measurable_const,MeasurableSet.empty⟩ |
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| def noStrongBefore (v b₀ b₁ : ℝ) (N k : ℕ) (h : Hist (Space CostState) N) : Prop := |
| ∀ i : Finset.Iic N,i.1 ≤ k → ¬markedPoint v b₀ b₁ (h i) |
|
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| theorem noStrongBefore_measurable (v b₀ b₁ : ℝ) (N k : ℕ) : |
| MeasurableSet {h | noStrongBefore v b₀ b₁ N k h} := by |
| simp only [noStrongBefore,Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro i |
| apply MeasurableSet.iInter |
| intro _hi |
| exact ((measurable_pi_apply i) (markedPoint_measurable v b₀ b₁)).compl |
|
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| def badCompactPath (v R b₀ b₁ : ℝ) (N : ℕ) : Set (Hist (Space CostState) N) := |
| {h | ∃ j : Finset.Iic N,compactPoint v R (h j) ∧ noStrongBefore v b₀ b₁ N j.1 h} |
|
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| theorem badCompactPath_measurable (v R b₀ b₁ : ℝ) (N : ℕ) : |
| MeasurableSet (badCompactPath v R b₀ b₁ N) := by |
| simp only [badCompactPath,Set.ofPred_exists] |
| apply MeasurableSet.iUnion |
| intro j |
| exact ((measurable_pi_apply j) (compactPoint_measurable v R)).inter |
| (noStrongBefore_measurable v b₀ b₁ N j.1) |
|
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| end NumberTheoryLean.CompletedMarkEvents |
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| end |
|
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| end Erdos970 |
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| end OAI |
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|
