Proof of `Joint distributions for the entropy-decrease argument`
groundedproofs/Lax263307Proofs/ConditioningDistribution.lean · lax-263307
What this proof establishes
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
The first and second conditioned moments define the probability distributions used by the entropy argument.
Proof strategy
The mass equation at normalizes the vertex distribution; the same equation at gives the marginal of the joint distribution. Symmetry is commutativity of insertion. A set containing the endpoints of an edge has zero moment. Clearing positive denominators gives the conditional-probability identity.
Attribution
Bonnet–Chang, proof of Lemma 12, equations (15) and (16).