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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | 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)
@[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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