“…The idea is that we start with the disjoint sum A + B, and for every element c of C we add a new path from inl( f (c)) to inr(g(c)). The induction principle states that, given a dependent type P : A ⊔ C B → Type, we can define a function h : We are using here the notion of dependent paths (see [5]): given a type X, a dependent type P : X → Type, a path p : x = x ′ in X and two points u : P(x) and v : P(x ′ ), the type u = P p v represents paths in P going from u to v and lying over p. Given h : (x : X) → Q(x) and q : x = X x ′ , the term apd h (q) is the application of h to q, which is a dependent path in Q, over q, and from h(x) to h(x ′ ).…”