This paper presents natural synthesis, which generalizes the proof-theoretic synthesis technique to support very expressive logic theories. This approach leverages the natural proof methodology and reduces an intractable, unbounded-size synthesis problem to a tractable, bounded-size synthesis problem, which is amenable to be handled by modern inductive synthesis engines. The synthesized program admits a natural proof and is a provably-correct solution to the original synthesis problem. We explore the natural synthesis approach in the domain of imperative data-structure manipulations and present a novel syntax-guided synthesizer based on natural synthesis. The input to our system is a program template together with a rich functional specification that the synthesized program must meet. Our system automatically produces a program implementation along with necessary proof artifacts, namely loop invariants and ranking functions, and guarantees the total correctness with a natural proof.
Experiments show that our natural synthesizer can efficiently produce provably-correct implementations for sorted lists and binary search trees. To our knowledge, this is the first system that can automatically synthesize these programs, their functional correctness and their termination in tandem from bare-bones control flow skeletons.