| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFSuffixSemigroup |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def positiveTimeChain (l : List (ℝ × ℝ≥0)) : List (ℝ × ℝ≥0) := l.filter (fun p => p.2≠0) |
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|
| @[simp] theorem chainDuration_positiveTimeChain (l : List (ℝ × ℝ≥0)) : |
| chainDuration (positiveTimeChain l)=chainDuration l := by |
| induction l with |
| | nil => rfl |
| | cons p l ih => |
| rcases p with ⟨m,d⟩ |
| by_cases hd : d=0 |
| · simpa [positiveTimeChain,hd,chainDuration_cons] using ih |
| · simp only [SK.Analytic.positiveTimeChain,List.filter_cons,hd,ne_eq,not_false_eq_true,decide_true,ite_true,chainDuration_cons] at * |
| rw [ih] |
|
|
| theorem positiveTimeChain_mem {l : List (ℝ × ℝ≥0)} {p : ℝ × ℝ≥0} (hp : p∈positiveTimeChain l) : |
| p∈l ∧ 0<p.2 := by |
| simpa only [positiveTimeChain,List.mem_filter,decide_eq_true_eq,ne_eq,pos_iff_ne_zero] using hp |
|
|
| theorem TimeChainModels.positiveTimeChain {α : ℝ → ℝ} {s : ℝ} {l : List (ℝ × ℝ≥0)} |
| (hl : TimeChainModels α s l) : TimeChainModels α s (positiveTimeChain l) := by |
| induction l generalizing s with |
| | nil => trivial |
| | cons p l ih => |
| rcases p with ⟨m,d⟩ |
| obtain ⟨hc,ht⟩ := hl |
| by_cases hd : d=0 |
| · simpa [SK.Analytic.positiveTimeChain,hd] using ih ht |
| · simp only [SK.Analytic.positiveTimeChain,List.filter_cons,hd,ne_eq,not_false_eq_true,decide_true,ite_true] |
| exact ⟨hc,ih ht⟩ |
|
|
| theorem TimeChainModels.sorted {α : ℝ → ℝ} (hα : Monotone α) {s : ℝ} {l : List (ℝ × ℝ≥0)} |
| (hl : TimeChainModels α s l) (hpos : ∀ p∈l,0<p.2) : l.Pairwise (fun p q => p.1≤q.1) := by |
| induction l generalizing s with |
| | nil => simp |
| | cons p l ih => |
| rcases p with ⟨m,d⟩ |
| obtain ⟨hc,ht⟩ := hl |
| have hd : 0<(d:ℝ) := hpos (m,d) (List.mem_cons_self ..) |
| have he : α s=m := hc s ⟨le_rfl,by linarith⟩ |
| apply List.pairwise_cons.mpr |
| constructor |
| · intro q hq |
| rw [← he] |
| exact (hα (le_add_of_nonneg_right d.coe_nonneg)).trans |
| (ht.mass_bounds hα hq (hpos q (List.mem_cons_of_mem _ hq))).1 |
| · exact ih ht (fun q hq => hpos q (List.mem_cons_of_mem _ hq)) |
|
|
| theorem quantile_interval_chain {k : ℕ} (β : ℝ) (Q : Fin (k+1) → ℝ) |
| (hQ : Q∈admissibleQuantiles k) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) (ht : t ≤ 1) : |
| ∃ l : List (ℝ × ℝ≥0), (chainDuration l:ℝ)=t-s ∧ |
| (∀ p∈l,p.1∈Icc (0:ℝ) 1) ∧ (∀ p∈l,0<p.2) ∧ TimeChainModels (quantileCDF k Q) s l := by |
| obtain ⟨l,hd,hm,hmodel⟩ := quantile_suffix_chain β Q hQ ⟨hs,hst.trans ht⟩ |
| have hsplit : chainDuration l=Real.toNNReal (t-s)+Real.toNNReal (1-t) := by |
| rw [hd] |
| apply NNReal.coe_injective |
| rw [NNReal.coe_add,Real.coe_toNNReal _ (by linarith),Real.coe_toNNReal _ (by linarith),Real.coe_toNNReal _ (by linarith)] |
| ring |
| obtain ⟨l₁,_,hd₁,_,hm₁,_,hmodel₁,_,_⟩ := scalarTimeChain_split_at β (quantileCDF k Q) l hm s hmodel _ _ hsplit |
| refine ⟨positiveTimeChain l₁,?_,fun p hp => hm₁ p (positiveTimeChain_mem hp).1, |
| fun p hp => (positiveTimeChain_mem hp).2,hmodel₁.positiveTimeChain⟩ |
| rw [chainDuration_positiveTimeChain,hd₁,Real.coe_toNNReal _ (sub_nonneg.mpr hst)] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
