Dafny and verification-aware programming, including proof by induction to verify properties of programs (for example, that an optimizer preserves semantics). Dafny Sketcher (https://github.com/namin/dafny-sketcher)
Multi-stage programming, a principled approach to writing programs that write programs, and its incarnation in multi-stage relational programming for faster synthesis of programs with holes—with the theoretical insight that a staged interpreter is a compiler, and a staged relational interpreter for a functional language can turn functions into relations running backwards for synthesis. multi-stage miniKanren (https://github.com/namin/staged-miniKanren)
Monte Carlo Tree Search, specifically the VerMCTS variant, and when this exploration-exploitation sweet spot is a good match for synthesis problems. VerMCTS (https://github.com/namin/llm-verified-with-monte-carlo-tree-...), and Holey (https://github.com/namin/holey).