Algebraic Foundations of Proof Refinement

We contribute a general apparatus for dependenttactic-based proof refinement in the LCF tradition, in which thestatements of subgoals may express a dependency on the proofs ofother subgoals; this form of dependency is extremely useful andcan serve as an algorithmic alternative to extensions of LCF basedon non-local instantiation of schematic variables. Additionally,we introduce a novel behavioral distinction between refinementrules and tactics based on naturality. Our framework, calledDependent LCF, is already deployed in the nascent RedPRLproof assistant for computational cubical type theory.