All proofs
Project-declaredLean 4.33.0-rc1 · mathlib@0434c033

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

Canonical source
Full Lean sourceLean 4
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

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.33.0-rc1

Continuous Within At Iio indicator Ioc

continuousWithinAt_Iio_indicator_Ioc

Plain-language statement

The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.

probabilitystochastic processesmeasure theory

Source project: Brownian motion

Person-level attribution pending.

View proof record
Project-declaredLean 4.33.0-rc1

Dense comap val nhds Within Ioi ne Bot

Dense.comap_val_nhdsWithin_Ioi_neBot

Project documentation

This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.

probabilitystochastic processesmeasure theory

Source project: Brownian motion

Person-level attribution pending.

View proof record