Proof of `Collision and length bounds for independently sampled paths` (4th statement)
groundedproofs/Lax263307Proofs/PathSampling.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
Substitute the vertex-load bound into the two union bounds. The estimate and total cost at most control intersections; the edge-cost hypothesis controls adjacency. Summing the same load bound controls expected size.
Attribution
Bonnet–Chang, equations (8) and (9) and the length estimate in Lemma 11.