The Essence of Dependent Object Types

Abstract

Focusing on path-dependent types, the paper develops foundations for Scala from first principles. Starting from a simple calculus D$_{<:}$ of dependent functions, it adds records, intersections and recursion to arrive at DOT, a calculus for dependent object types. The paper shows an encoding of System F with subtyping in D$_{<:}$ and demonstrates the expressiveness of DOT by modeling a range of Scala constructs in it.

A Additional Rules and Detailed Proofs

1.1 A.1 Typing of Dependent Sums/$\varSigma$s

There are one subtyping rule and four typing rules (all admissible in D$_{<:}$) associated with the encoding of dependent sums given in Sect. 2.1.

The proofs of admissibility are left as an (easy) exercise to the reader.

1.2 A.2 Proof of Theorem 2

We want to show that the translation $-^*$ preserves typing, i.e. if $\varGamma \,\vdash _{F}\, t: T$ then $\varGamma ^* \,\vdash _{D}\, t^*: T^*$.

We start by stating and proving two helper lemmas. Write $[x.A:=U]T$ for the capture-avoiding substitution of $x.A$ for U in T.

Lemma 6

Substitution of type variables for types commutes with translation of types, i.e.

$$ ([X:=U]T)^* = [x_X.A:=U^*]T^* $$

Proof

The proof is by straight-forward induction on the structure of T.

Lemma 7

The following subtyping rules are admissible in D$_<$:

Proof

The proof is by induction on the structure of T.

Proof

(Theorem 2 ). Proof is by induction on (F$_{<:}$) typing derivations. The case for subsumption follows immediately from preservation of subtyping (Theorem 1). The only remaining non-trivial cases are type and term application.

Term Application Case. We need to show that

is admissible. First note that z is not in ${ fv}(T^*)$, hence $[z:=y]T^* = T^*$ for all y. In particular, using (Var) and (All-E), we have

where $\varGamma ' = \varGamma ^*, x: \forall (z\!:\!S^*)T^*, y: S^*$. Applying the induction hypothesis, context weakening and (Let), we obtain

The result follows by applying the induction hypothesis once more:

Type Application Case. By preservation of subtyping, we have $\varGamma ^* \,\vdash \,U^* <:S^*$, hence it suffices to show that

is admissible. By context weakening, (Typ-$<:$-Typ), (Bot) and (Sub), we have

where $\varGamma ' = \varGamma ^*, x: \forall (x_X\!:\!\{A: \bot ..S^*\})T^*, y_Y: \{A: U^*..U^*\}$. By (Var) and (All-E), we have

Next, we observe that, by Lemma 6

$$\begin{aligned} ([X:=U]T)^*&= [x_X.A:=U^*]T^* \\&= [y_Y.A:=U^*][x_X:=y_Y]T^* \end{aligned}$$

By Lemma 7, (Var) and (Sub), we thus have

The result follows from applying (Let) twice. Note that, being fresh, neither x nor $y_Y$ appear free in $([X:=U]T)^*$, hence the hygiene condition of (Let) holds.