On McMillan's theorem about uniquely decipherable codes

Let len : A −→ N be the length function on the free monoid of all strings over a given non-empty finite set A. Let con : A −→ A be the concatenation map which to every string of strings associates their concatenation. A uniquely decipherable code is a finite set C ⊆ A such that con is injective on the submonoid C of A. This submonoid is then isomorphic to the submonoid C = con[C] of A freely generated by C. Denoting by r the number of elements of the alphabet A, the Kraft sum K(C) of any finite C ⊆ A is defined as ∑