Spivey's Generalized Recurrence for Bell Numbers

This entry defines the Bell numbers [1] as the cardinality of set partitions for a carrier set of given size, and derives Spivey’s generalized recurrence relation for Bell numbers [2] following his elegant and intuitive combinatorial proof. As the set construction for the combinatorial proof requires construction of three intermediate structures, the main difficulty of the formalization is handling the overall combinatorial argument in a structured way. The introduced proof structure allows us to compose the combinatorial argument from its subparts, and supports to keep track how the detailed proof steps are related to the overall argument. To obtain this structure, this entry uses set monad notation for the set construction’s definition, introduces suitable predicates and rules, and follows a repeating structure in its Isar proof.