This paper presents an algorithm for synthesizing recursive functions that process algebraic datatypes. It is founded on proof-theoretic techniques that exploit both type information and input-output examples to prune the search space. The algorithm uses refinement trees, a data structure that succinctly represents constraints on the shape of generated code. We evaluate the algorithm by using a prototype implementation to synthesize more than 40 benchmarks and several non-trivial larger examples. Our results demonstrate that the approach meets or outperforms the state-of-the-art for this domain, in terms of synthesis time or attainable size of the generated programs.

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In this paper we study the problem of interoperability -combining constructs from two separate programming languages within one program -in the case where one of the two languages is dependently typed and the other is simply typed. We present a core calculus called SD, which combines dependentlyand simply-typed sub-languages and supports user-defined (dependent) datatypes, among other standard features. SD has "boundary terms" that mediate the interaction between the two sub-languages. The operational semantics of SD demonstrates how the necessary dynamic checks, which must be done when passing a value from the simply-typed world to the dependently typed world, can be extracted from the dependent type constructors themselves, modulo user-defined functions for marshaling values across the boundary. We establish type-safety and other meta-theoretic properties of SD, and contrast this approach to others in the literature. Disciplines Disciplines Engineering Comments CommentsAbstract In this paper we study the problem of interoperability-combining constructs from two separate programming languages within one program-in the case where one of the two languages is dependently typed and the other is simply typed. We present a core calculus called SD, which combines dependentlyand simply-typed sub-languages and supports user-defined (dependent) datatypes, among other standard features. SD has "boundary terms" that mediate the interaction between the two sub-languages. The operational semantics of SD demonstrates how the necessary dynamic checks, which must be done when passing a value from the simply-typed world to the dependently typed world, can be extracted from the dependent type constructors themselves, modulo user-defined functions for marshaling values across the boundary. We establish type-safety and other meta-theoretic properties of SD, and contrast this approach to others in the literature.Lemma 19 (Substitution through types). If Γ T 1 : K and Γ t 2 ∼ = t 2 and y 2 ∈ dom (Γ), then Γ [t 2 /y 2 ]T 1 ≡ [t 2 /y 2 ]T 1 .Proof. Induction on the derivation of Γ, y:T 2 T 1 : K . The cases are Case wf dty arr: The rule looks like Γ T 1 : * Γ, y:T 1 T 2 : * Γ (y : T 1 ) → T 2 : * wf dty arrSince y is a bound variable we can pick it to be different from y 2 and push the substitution in, so we must show Γ (y :By eq dty pi it suffices to show Γ 1 , y :T 2 [t 2 /y 2 ]T 1 ≡ [t 2 /y 2 ]T 1 and Γ 1 [t 2 /y 2 ]T 2 ≡ [t 2 /y 2 ]T 2 , both of which follow by IH (noting again that y 2 = y).Case wf dty pair: Similar to the previous case.Case wf dty data: We must show Γ [t 2 /y 2 ]B ≡ [t 2 /y 2 ]B , that is to say, Γ B ≡ B . This follows by eq dty trefl.Case wf dty unit: Similar to the previous case. 41Case wf dty app: The rule looks like Γ T : T 1 ⇒ * Γ t : T 1 Γ T t : * wf dty app By distributing the substitution and using eq dty app, it suffices to show Γ [t 2 /y 2 ]T ≡ [t 2 /y 2 ]T (which is direct by IH) and Γ [t 2 /y 2 ]t ∼ = [t 2 /y 2 ]t (which is by eq dtm subst).Lemma 20 (Context conversion (types)). Suppose Γ T 1 ≡ T 2 and...

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