OAI.Problem336.CanonicalBrownianSampler.raw_innovation_hasLaw: intermediate lemma from OpenAI, The free uniform spanning forest is a factor of IID

Proved

OAI.Problem336.CanonicalBrownianSampler.raw_innovation_hasLaw

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

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Problem336.fusf_is_factor_iid, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The free uniform spanning forest is a factor of IID, OpenAI Math Release, September 25, 2026, result family 231, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Problem336.CanonicalBrownianSampler.raw_innovation_hasLaw of lean/OAI/Probability/SpanningForest/SpanningForest/CanonicalBrownianSampler.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Problem336.CanonicalBrownianSampler, with open MeasureTheory ProbabilityTheory, open scoped NNReal Topology, open CanonicalBrownianReference):

theorem raw_innovation_hasLaw (μ : ProbabilityMeasure (I → Bool))
    (S : I → ℕ → Finset I)
    (hS : ∀ i, Monotone (S i)) (hcover : ∀ i j, ∃ n, j ∈ S i n) :
    HasLaw (RawInnovationSampler.rawInnovation CanonicalBrownianReference.hidden CanonicalBrownianReference.noise (PosteriorDrift.drift μ S))
      ((noiseLaw I).map ContinuousPathLaw.raw) (law μ)

Definitions. The statement is written with OAI.Problem336.CanonicalBrownianReference.Space, OAI.Problem336.CanonicalBrownianReference.hidden, OAI.Problem336.CanonicalBrownianReference.instMeasurableSpacePath, OAI.Problem336.CanonicalBrownianReference.law, OAI.Problem336.CanonicalBrownianReference.noise, OAI.Problem336.CanonicalBrownianReference.noiseLaw, OAI.Problem336.ContinuousPathLaw.raw, OAI.Problem336.GlobalWiener.Path, OAI.Problem336.PosteriorDrift.drift and OAI.Problem336.RawInnovationSampler.rawInnovation (definitions bundle OAIS336000 of this split).

Used by. The proof of OAI.Problem336.FinalFactor.worksOnGraph of this split imports it.

Preamble

import Mathlib
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Data.List.Nodup
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic
import Definitions.Def_OAIS336000

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal Topology

namespace Problem336.CanonicalBrownianSampler

open CanonicalBrownianReference

variable {I : Type*} [countableIndices : Countable I]

attribute [local instance] OAI.Problem336.CanonicalBrownianSampler.instMeasurableSpacePath
theorem raw_innovation_hasLaw (μ : ProbabilityMeasure (I → Bool))
    (S : I → ℕ → Finset I)
    (hS : ∀ i, Monotone (S i)) (hcover : ∀ i j, ∃ n, j ∈ S i n) :
    HasLaw (RawInnovationSampler.rawInnovation CanonicalBrownianReference.hidden CanonicalBrownianReference.noise (PosteriorDrift.drift μ S))
      ((noiseLaw I).map ContinuousPathLaw.raw) (law μ) := by
  sorry

end Problem336.CanonicalBrownianSampler

end

end OAI
end

Source

OpenAI, The free uniform spanning forest is a factor of IID, OpenAI Math Release, September 25, 2026, https://github.com/openai/math (result family 231; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/SpanningForest/SpanningForest/CanonicalBrownianSampler.lean (declaration OAI.Problem336.CanonicalBrownianSampler.raw_innovation_hasLaw)

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