Abstract: 
This entry provides three lemmas to count the number of multisets
of a given size and finite carrier set. The first lemma provides a
cardinality formula assuming that the multiset's elements are chosen
from the given carrier set. The latter two lemmas provide formulas
assuming that the multiset's elements also cover the given carrier
set, i.e., each element of the carrier set occurs in the multiset at
least once. The proof of the first lemma uses the argument of
the recurrence relation for counting multisets. The proof of the
second lemma is straightforward, and the proof of the third lemma is
easily obtained using the first cardinality lemma. A challenge for the
formalization is the derivation of the required induction rule, which
is a special combination of the induction rules for finite sets and
natural numbers. The induction rule is derived by defining a suitable
inductive predicate and transforming the predicate's induction
rule. 
BibTeX: 
@article{Card_MultisetsAFP,
author = {Lukas Bulwahn},
title = {Cardinality of Multisets},
journal = {Archive of Formal Proofs},
month = jun,
year = 2016,
note = {\url{http://isaafp.org/entries/Card_Multisets.html},
Formal proof development},
ISSN = {2150914x},
}
