Càdlàg path predicates

Definition

KLSModular_BrownianMotion_StochasticIntegral_Cadlag

by Lazyfill · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitiesklskls-conjecturekls-modularlog-concavemodule-contextvendored-brownianmotion

Vendored from the BrownianMotion library (https://github.com/RemyDegenne/brownian-motion, commit a7ad7af662044313bc80a7c0756360fb3745f931). IsRightContinuous means that a function is continuous within the strict right neighborhood at every point. IsCadlag is a structure requiring right continuity together with existence of a left limit at every point, expressed by a filter Tendsto statement.

Original module imports

public import Mathlib.Analysis.Normed.Group.Continuity
public import Mathlib.Topology.Algebra.MulAction
public import Mathlib.Topology.Instances.ENNReal.Lemmas
public import Mathlib.Topology.MetricSpace.Bounded
public import Mathlib.Topology.Order.Compact
public import Mathlib.Topology.Order.LeftRightLim

Original declaration excerpt

Shown in the original module context.

abbrev IsRightContinuous [TopologicalSpace E] [Preorder ι] (f : ι → E) :=
  ∀ a, ContinuousWithinAt f (Set.Ioi a) a

Module navigation

Project result: The KLS conjecture.

Original module · BrownianMotion.StochasticIntegral.Cadlag · Frozen source 408915f.

Definition code

import Mathlib.Analysis.Normed.Group.Continuity
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.Order.LeftRightLim

/-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
-- Vendored from https://github.com/RemyDegenne/brownian-motion at commit a7ad7af662044313bc80a7c0756360fb3745f931 (upstream path BrownianMotion/StochasticIntegral/Cadlag.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
section
open Filter TopologicalSpace Bornology
open scoped Topology ENNReal
variable {ι E : Type*} [TopologicalSpace ι]
/-- The predicate that a function is right continuous. -/
abbrev IsRightContinuous [TopologicalSpace E] [Preorder ι] (f : ι → E) :=
  ∀ a, ContinuousWithinAt f (Set.Ioi a) a
/-- A function is cadlag if it is right-continuous and has left limits. -/
structure IsCadlag [TopologicalSpace E] [Preorder ι] (f : ι → E) : Prop where
  right_continuous : IsRightContinuous f
  left_limit : ∀ x, ∃ l, Tendsto f (𝓝[<] x) (𝓝 l)

Source

Lean source: https://github.com/RemyDegenne/brownian-motion/blob/a7ad7af662044313bc80a7c0756360fb3745f931/BrownianMotion/StochasticIntegral/Cadlag.lean#L26-L78 (repository human-ai-mathematics/kls-lean, commit 408915f, module `BrownianMotion.StochasticIntegral.Cadlag`, definition bundle `KLSModular_BrownianMotion_StochasticIntegral_Cadlag`; the repository is private at upload time). Vendored from https://github.com/RemyDegenne/brownian-motion at commit a7ad7af662044313bc80a7c0756360fb3745f931 (Apache-2.0; upstream copyright notice retained in the Lean text).

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