| import OAI.NumberTheory.Jacobsthal.Renewal.CompletedCycleFirstArrival |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
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|
| section |
|
|
| namespace Erdos970Dependency.MarkedVisits |
| open Filter Set MeasureTheory ProbabilityTheory |
| open scoped ProbabilityTheory ENNReal Classical |
| open NumberTheoryLean.FinitePathMeasures |
|
|
| def appendRawState (a : ℕ) (past : RawHistory a) (s : CostState) : RawHistory (a+1) := |
| fun i => if hi : i.1 ≤ a then past ⟨i.1,Finset.mem_Iic.mpr hi⟩ else s |
|
|
| lemma appendRawState_measurable (a : ℕ) (past : RawHistory a) : Measurable (appendRawState a past) := by |
| apply Measurable.of_eval |
| intro i |
| by_cases hi : i.1 ≤ a |
| · simp only [appendRawState,dite_eq_left hi] |
| exact measurable_const |
| · simp only [appendRawState,dite_eq_right hi] |
| exact measurable_id |
|
|
| lemma appendRawState_reconstruct (a : ℕ) (past : RawHistory a) (h : RawHistory (a+1)) |
| (hp : rawPrefix (show a ≤ a+1 by omega) h=past) : appendRawState a past (rawLast (a+1) h)=h := by |
| funext i |
| by_cases hi : i.1 ≤ a |
| · rw [appendRawState,dite_eq_left hi] |
| exact (congrFun hp (⟨i.1,Finset.mem_Iic.mpr hi⟩ : Finset.Iic a)).symm |
| · rw [appendRawState,dite_eq_right hi] |
| have hin : i.1=a+1 := by have := Finset.mem_Iic.mp i.2; omega |
| have he : i=⟨a+1,by simp⟩ := Subtype.ext hin |
| rw [he] |
| rfl |
|
|
| theorem oneStep_measure_reconstruct (a : ℕ) (past : RawHistory a) (μ : Measure (RawHistory (a+1))) |
| (hp : ∀ᵐ h ∂μ, rawPrefix (show a ≤ a+1 by omega) h=past) : |
| μ=(μ.map (rawLast (a+1))).map (appendRawState a past) := by |
| rw [Measure.map_map (appendRawState_measurable a past) (rawLast_measurable (a+1))] |
| have he : appendRawState a past ∘ rawLast (a+1) =ᵐ[μ] id := by |
| filter_upwards [hp] with h hh |
| exact appendRawState_reconstruct a past h hh |
| rw [Measure.map_congr he,Measure.map_id] |
|
|
| lemma oneStep_measure_eq_of_last (a : ℕ) (past : RawHistory a) |
| (μ ν : Measure (RawHistory (a+1))) |
| (hμ : ∀ᵐ h ∂μ, rawPrefix (show a ≤ a+1 by omega) h=past) |
| (hν : ∀ᵐ h ∂ν, rawPrefix (show a ≤ a+1 by omega) h=past) |
| (he : μ.map (rawLast (a+1))=ν.map (rawLast (a+1))) : μ=ν := by |
| rw [oneStep_measure_reconstruct a past μ hμ,oneStep_measure_reconstruct a past ν hν,he] |
|
|
| end Erdos970Dependency.MarkedVisits |
|
|
| end |
|
|
| end Erdos970 |
|
|
| end OAI |
|
|
