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Intermediate data structures are a common cause of inefficiency in functional programming.Fusionattempts to eliminate intermediate data structures by combining adjacent data traversals into one; existing fusion techniques, however, are based on predefined rewrite rules and hence are limited in expressiveness. In this work we explore a different approach to eliminating intermediate data structures, based on inductive program synthesis. We dub this approachsuperfusion(by analogy withsuperoptimization, which uses inductive synthesis for program optimization). Starting from a reference program annotated with data structures to be eliminated, superfusion first generates asketchwhere program fragments operating on those data structures are replaced with holes; it then fills the holes with constant-time expressions such that the resulting program is equivalent to the reference. The main technical challenge here is scalability because optimized programs are often complex, making the search space intractably large for naive enumeration. To address this challenge, our key insight is to first synthesize aghost functionthat describes the relationship between the original intermediate data structure and its compressed version; this function, although not used in the final program, serves to decompose the joint sketch filling problem into independent simpler problems for each hole. We implement superfusion in a tool called SuFu and evaluate it on a dataset of 290 tasks collected from prior work on deductive fusion and program restructuring. The results show that SuFu solves 264 out of 290 tasks, exceeding the capabilities of rewriting-based fusion systems and achieving comparable performance with specialized approaches to program restructuring on their respective domains. 

The problem of automatically proving the equality of terms over recursive functions and inductive data types is challenging, as such proofs often require auxiliary lemmas which must themselves be proven. Previous attempts at lemma discovery compromise on either efficiency or efficacy.Goal-directedapproaches are fast but limited in expressiveness, as they can only discover auxiliary lemmas which entail their goals.Theory explorationapproaches are expressive but inefficient, as they exhaustively enumerate candidate lemmas. We introducee-graph guided lemma discovery, a new approach to finding equational proofs that makes theory exploration goal-directed. We accomplish this by using e-graphs and equality saturation to efficiently construct and compactly represent the space ofallgoal-oriented proofs. This allows us to explore only those auxiliary lemmasguaranteedto help make progress on some of these proofs. We implemented our method in a new prover called CCLemma and compared it with three state-of-the-art provers across a variety of benchmarks. CCLemma performs consistently well on two standard benchmarks and additionally solves 50% more problems than the next best tool on a new challenging set.