Is Cadlag not acc Pt large Left Jump Set
IsCadlag.not_accPt_largeLeftJumpSet
Plain-language statement
The set of large left jump times has no accumulation points. TODO: maybe to_dual can be extended to simplify this proof as the proof of the second part is very similar to the first part.
Exact Lean statement
lemma IsCadlag.not_accPt_largeLeftJumpSet [LinearOrder ι] [OrderTopology ι] [UniformSpace E]
{f : ι → E} (hf : IsCadlag f) {v : Set (E × E)} (hv : v ∈ uniformity E) (t : ι) :
¬ AccPt t (𝓟 (largeLeftJumpSet f v))Formal artifact
Lean source
lemma IsCadlag.not_accPt_largeLeftJumpSet [LinearOrder ι] [OrderTopology ι] [UniformSpace E] {f : ι → E} (hf : IsCadlag f) {v : Set (E × E)} (hv : v ∈ uniformity E) (t : ι) : ¬ AccPt t (𝓟 (largeLeftJumpSet f v)) := by obtain ⟨u, hu, husymm, huv⟩ := comp_comp_symm_mem_uniformity_sets hv intro ht have hsplit : (𝓝[<] t ⊓ 𝓟 (largeLeftJumpSet f v)).NeBot ∨ (𝓝[>] t ⊓ 𝓟 (largeLeftJumpSet f v)).NeBot := by simp_all [AccPt, ← nhdsLT_sup_nhdsGT t, inf_comm, inf_sup_left] refine hsplit.elim (fun h => ?_) (fun h => ?_) · have hnb : (𝓝[<] t).NeBot := h.mono inf_le_left rw [← frequently_mem_iff_neBot] at h contrapose! h have htt : ¬ IsBot t := by by_contra exact (not_neBot.2 <| nhdsLT_eq_bot_iff.2 (Or.inl this)) hnb obtain ⟨a, ha⟩ : ∃ a, a < t := by simpa [IsBot] using htt -- There exists `b < t` such that for any `x ∈ Ioo b t`, `f x` is close to `f.leftLim t`. obtain ⟨b, hb⟩ := (mem_nhdsLT_iff_exists_Ioo_subset' ha).1 <| (Uniform.tendsto_nhds_left.1 <| (tendsto_leftLim_of_tendsto (hf.left_limit t))).eventually_mem hu -- For any `x ∈ Ioo b t`, there exists `s ∈ Ioc b x` such that `f s` is closed `f.leftLim x`. have h2 (x) (hx : x ∈ Set.Ioo b t) : ∃ s, s ∈ Set.Ioc b x ∧ (f s, f.leftLim x) ∈ u := by by_cases! hn : (𝓝[<] x).NeBot · exact ((eventually_mem_set.2 (Ioc_mem_nhdsLT hx.1)).and <| (Uniform.tendsto_nhds_left.1 (tendsto_leftLim_of_tendsto (hf.left_limit x))).eventually_mem hu).exists · rw [leftLim_eq_of_eq_bot f hn] exact ⟨x, ⟨hx.1, refl x⟩, refl_mem_uniformity hu⟩ refine eventually_of_mem (Ioo_mem_nhdsLT hb.1) fun x hx => ?_ obtain ⟨s, hs⟩ := h2 x hx -- `(f x, f.leftLim t) ∈ u` and `(f s, f.leftLim t) ∈ u` imply that `(f x, s) ∈ u ∘ u`, which -- can be used with `(f s, f.leftLim x) ∈ u` to imply that `(f x, f.leftLim x) ∈ u ∘ u ∘ u`. simpa [largeLeftJumpSet] using huv <| SetRel.prodMk_mem_comp (SetRel.prodMk_mem_comp (hb.2 hx) (SetRel.symm _ (hb.2 ⟨hs.1.1, hs.1.2.trans_lt hx.2⟩))) hs.2 · -- We use a similar argument to prove this part. have hnb : (𝓝[>] t).NeBot := h.mono inf_le_left rw [← frequently_mem_iff_neBot] at h contrapose! h have htt : ¬ IsTop t := by by_contra exact (not_neBot.2 <| nhdsGT_eq_bot_iff.2 (Or.inl this)) hnb obtain ⟨a, ha⟩ : ∃ a, t < a := by simpa [IsTop] using htt -- Here one can also use the existence of a right limit instead of right continuity, so this -- theorem should also be true for functions with both left and right limits at each point. obtain ⟨b, hb⟩ := (mem_nhdsGT_iff_exists_Ioo_subset' ha).1 <| (Uniform.continuousWithinAt_iff'_left.1 <| hf.right_continuous t).eventually_mem hu have h2 (x) (hx : x ∈ Set.Ioo t b) : ∃ s, s ∈ Set.Ioc t x ∧ (f s, f.leftLim x) ∈ u := by by_cases! hn : (𝓝[<] x).NeBot · exact ((eventually_mem_set.2 (Ioc_mem_nhdsLT hx.1)).and <| (Uniform.tendsto_nhds_left.1 (tendsto_leftLim_of_tendsto (hf.left_limit x))).eventually_mem hu).exists · rw [leftLim_eq_of_eq_bot f hn] exact ⟨x, ⟨hx.1, refl x⟩, refl_mem_uniformity hu⟩ refine eventually_of_mem (Ioo_mem_nhdsGT hb.1) fun x hx => ?_ obtain ⟨s, hs⟩ := h2 x hx simpa [largeLeftJumpSet] using huv <| SetRel.prodMk_mem_comp (SetRel.prodMk_mem_comp (hb.2 hx) (SetRel.symm _ (hb.2 ⟨hs.1.1, hs.1.2.trans_lt hx.2⟩))) hs.2- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/Cadlag.lean:76-130
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Project documentation
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