  Recent work on homotopy type theory exploits an exciting new correspondence
between Martin-Lof's dependent type theory and the mathematical disciplines of
category theory and homotopy theory. The category theory and homotopy theory
suggest new principles to add to type theory, and type theory can be used in
novel ways to formalize these areas of mathematics. In this paper, we formalize
a basic result in algebraic topology, that the fundamental group of the circle
is the integers. Though simple, this example is interesting for several
reasons: it illustrates the new principles in homotopy type theory; it mixes
ideas from traditional homotopy-theoretic proofs of the result with
type-theoretic inductive reasoning; and it provides a context for understanding
an existing puzzle in type theory---that a universe (type of types) is
necessary to prove that the constructors of inductive types are disjoint and
injective.
