| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.TimeChainSplit |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| @[simp] theorem chainDuration_append (l r : List (ℝ × ℝ≥0)) : |
| chainDuration (l++r)=chainDuration l+chainDuration r := by |
| simp [chainDuration] |
|
|
| theorem TimeChainModels.append {α : ℝ → ℝ} {s : ℝ} {l r : List (ℝ × ℝ≥0)} |
| (hl : TimeChainModels α s l) (hr : TimeChainModels α (s+chainDuration l) r) : |
| TimeChainModels α s (l++r) := by |
| induction l generalizing s with |
| | nil => simpa only [chainDuration_nil,NNReal.coe_zero,add_zero,List.nil_append] using hr |
| | cons p l ih => |
| rcases p with ⟨m,t⟩ |
| rcases hl with ⟨hc,hl⟩ |
| exact ⟨hc,ih hl (by simpa only [chainDuration_cons,NNReal.coe_add,add_assoc] using hr)⟩ |
|
|
| def partitionTimeChain (n : ℕ) (q : Fin (n+1) → ℝ) (hq : Monotone q) |
| (m : Fin n → ℝ) : List (ℝ × ℝ≥0) := |
| List.ofFn (fun i : Fin n => (m i,NNReal.mk (q i.succ-q i.castSucc) (sub_nonneg.mpr (hq (Fin.castSucc_le_succ i))))) |
|
|
| theorem partitionTimeChain_cons (n : ℕ) (q : Fin (n+2) → ℝ) (hq : Monotone q) |
| (m : Fin (n+1) → ℝ) : |
| partitionTimeChain (n+1) q hq m = |
| (m 0,NNReal.mk (q 1-q 0) (sub_nonneg.mpr (hq (Fin.zero_le _)))):: |
| partitionTimeChain n (fun i => q i.succ) |
| (fun _ _ h => hq (Fin.succ_le_succ_iff.mpr h)) (fun i => m i.succ) := by |
| simp only [partitionTimeChain,List.ofFn_succ] |
| rfl |
|
|
| theorem partitionTimeChain_duration (n : ℕ) (q : Fin (n+1) → ℝ) (hq : Monotone q) |
| (m : Fin n → ℝ) : (chainDuration (partitionTimeChain n q hq m):ℝ)=q (Fin.last n)-q 0 := by |
| simp only [partitionTimeChain,chainDuration,List.map_ofFn,List.sum_ofFn, |
| NNReal.coe_sum,NNReal.coe_mk,Function.comp_apply,Finset.sum_sub_distrib] |
| have H₁ := Fin.sum_univ_succ q |
| have H₂ := Fin.sum_univ_castSucc q |
| linarith |
|
|
| theorem partitionTimeChain_models (n : ℕ) (q : Fin (n+1) → ℝ) (hq : Monotone q) |
| (m : Fin n → ℝ) {α : ℝ → ℝ} |
| (hc : ∀ i : Fin n, ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i) : |
| TimeChainModels α (q 0) (partitionTimeChain n q hq m) := by |
| induction n with |
| | zero => trivial |
| | succ n ih => |
| simp only [partitionTimeChain,List.ofFn_succ,TimeChainModels,NNReal.coe_mk] |
| constructor |
| · intro z hz |
| apply hc 0 z |
| simpa only [Fin.castSucc_zero,Fin.succ_zero_eq_one,add_sub_cancel] using hz |
| · have H := ih (fun i => q i.succ) (fun _ _ h => hq (Fin.succ_le_succ_iff.mpr h)) |
| (fun i => m i.succ) (fun i z hz => hc i.succ z hz) |
| simpa only [partitionTimeChain,Fin.succ_zero_eq_one,Fin.castSucc_zero,Fin.succ_castSucc,add_sub_cancel] using H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
