The fixed-point calculus is a toolbox of theorems for reasoning equationally about fixed points. However, the underlying concepts of the calculus are not defined equationally, including the central definition, that of least fixed point. Thus, although the key theorems of the fixed-point calculus are equational, their proofs are not. In this work, we give equational specifications for the main concepts of the fixed-point calculus for partially ordered sets, to allow more proofs to be written in an equational style. Since reasoning equationally can be cumbersome without appropriate abstractions, we employ the graphical language of string diagrams, based on a category of feasibility relations, to make reasoning more ergonomical. Our contributions culminate in graphical equational proofs of key results of the fixed-point calculus, including the rolling, diagonal, and square rules; fixed-point fusion; and the mutual recursion theorem.

Most code is executed more than once. If not entire programs then libraries remain unchanged from one run to the next. Just-in-time compilers expend considerable effort gathering insights about code they compiled many times, and often end up generating the same binary over and over again. We explore how to reuse compiled code across runs of different programs to reduce warm-up costs of dynamic languages. We propose to use speculative contextual dispatch to select versions of functions from an off-line curated code repository . That repository is a persistent database of previously compiled functions indexed by the context under which they were compiled. The repository is curated to remove redundant code and to optimize dispatch. We assess practicality by extending Ř, a compiler for the R language, and evaluating its performance. Our results suggest that the approach improves warmup times while preserving peak performance.