ConditionalProbability.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 9 curated AI-written annotations and 485 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Polynomial Freiman-Ruzsa proof paper, arXiv v2

1 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

lemma ProbabilityTheory.sum_meas_smul_cond_fiber' Documented declaration: sum_meas_smul_cond_fiber' line 18 source
Source documentation
The law of total probability for a random variable taking finitely many values: a measure μ can be expressed as a linear combination of its conditional measures μ[|X ← x] on fibers of a random variable X valued in a fintype.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, Measurable, FiniteRange, MeasureTheory.Measure, MeasureTheory.IsFiniteMeasure, Eq, Finset.sum, FiniteRange.toFinset, HSMul.hSMul, ENNReal, Set, Set.preimage, Singleton.singleton, ProbabilityTheory.cond
Proof uses: MeasureTheory.Measure.ext, MeasurableSet, Trans.trans, Inter.inter, Eq.mpr, HMul.hMul, id, congrFun', congrArg, Eq.trans, congrFun, MeasureTheory.Measure.coe_finsetSum, Finset.sum_congr, Eq.refl, Finset, Membership.mem, Finset.sum_apply, mul_comm, of_eq_true, True and 29 more
57 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma sum_meas_smul_cond_fiber' {X : Ω → α} (hX : Measurable X) [finX : FiniteRange X]
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    ∑ x ∈ finX.toFinset, μ (X ⁻¹' {x}) • μ[|X ← x] = μ := by
…
```
Mentions: FiniteRange