LeanMachineLearning

ProbabilityTheory.Kernel.traj_zero_map_eval_zero๐Ÿ”—

Lemma

No docstring.

๐Ÿ”—theorem
ProbabilityTheory.Kernel.traj_zero_map_eval_zero.{u_2}
  {X : โ„• โ†’ Type u_2} [(n : โ„•) โ†’ MeasurableSpace (X n)]
  {ฮบ : (n : โ„•) โ†’ Kernel ((i : โ†ฅ(Finset.Iic n)) โ†’ X โ†‘i) (X (n + 1))}
  [โˆ€ (n : โ„•), IsMarkovKernel (ฮบ n)] :
  (map (traj ฮบ 0) fun h => h โ†‘default) =
    deterministic โ‡‘(MeasurableEquiv.piUnique fun i => X โ†‘i) โ‹ฏ
ProbabilityTheory.Kernel.traj_zero_map_eval_zero.{u_2}
  {X : โ„• โ†’ Type u_2}
  [(n : โ„•) โ†’ MeasurableSpace (X n)]
  {ฮบ :
    (n : โ„•) โ†’
      Kernel
        ((i : โ†ฅ(Finset.Iic n)) โ†’ X โ†‘i)
        (X (n + 1))}
  [โˆ€ (n : โ„•), IsMarkovKernel (ฮบ n)] :
  (map (traj ฮบ 0) fun h => h โ†‘default) =
    deterministic
      โ‡‘(MeasurableEquiv.piUnique fun i =>
          X โ†‘i)
      โ‹ฏ

Code

lemma traj_zero_map_eval_zero :
    (Kernel.traj ฮบ 0).map (fun h โ†ฆ h (default : Iic 0))
      = Kernel.deterministic (MeasurableEquiv.piUnique (fun i : Iic 0 โ†ฆ X i))
        (MeasurableEquiv.piUnique _).measurable
Proof
by
  suffices (Kernel.traj ฮบ 0).map (fun h โ†ฆ h (default : Iic 0))
      = (Kernel.partialTraj ฮบ 0 0).map (MeasurableEquiv.piUnique (fun i : Iic 0 โ†ฆ X i)) by
    rwa [Kernel.partialTraj_zero, Kernel.deterministic_map] at this
    fun_prop
  rw [โ† Kernel.traj_map_frestrictLe, โ† Kernel.map_comp_right _ (by fun_prop) (by fun_prop)]
  rfl

Actions: Source ยท Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file ยท open it in the Lean web editor.

Dependency graph

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited โ€” so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 34 external constants

โœ“ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.