Domain Theory: An Introduction

ion Let τ = λx.σ. By the induction hypothesis, σ defines a function of its free variables. Let those free variables be of types D0, . . . ,Dn where Dn is the type of x. Then σ defines an approximable function g : D0 × . . .×Dn → D′ where D′ is the type of σ. Using Theorem 3.12, the function curry(g) : D0 × . . .×Dn−1 → (Dn → D′) yields an approximable function, but this is just the function defined by τ . The reader can generalize this proof for multiple bound variables in τ . 2 Given this, the equation τ = σ states that the two terms define the same approximable function of their free variables. As an example,