We are interested in graph representation in the theorem prover Coq. We have chosen to represent graphs using coinductive types. We wanted to explore their use in Coq. Indeed, they make the graph representation succinct and elegant. Moreover, navigability is ensured by construction. We had to overcome the guardedness condition whose objective is to ensure validity of all operations made on coinductive objects. Its implementation in Coq is restrictive and sometimes forbids definitions, even semantically correct ones. A canonical formalization of graphs thus surmounts Coq's direct expressivity. We have designed a solution respecting these limitations. Then, we have defined a relation on graphs close to the notion of equivalence obtained on a classical representation, keeping however the advantages offered by coinduction. We show that this relation is equivalent to another one based on finite observations of the graphs.
