Dependently Typed Records for Representing Mathematical Structure

This “unpackaged” approach can be improved by collecting all the parts of the meaning of group into a context, which need not be explicitly mentioned in every statement. A means of discharging some of the context is provided, so that statements made under that context can be instantiated with particular groups. However once the group context is discharged, all the parts of a group must be mentioned when using any general lemma about groups. Variations on this are supported by many proof tools, e.g. Coq’s Section mechanism [5], Lego’s Discharge [12], Automath contexts and Isabelle locales. A significant refinement is achieved by giving names to bits of context as in telescopes [7], or first-class contexts as in Martin-Löf’s framework with explicit substitutions [21]. With these, we need not discharge a context to instantiate definitions and lemmas. But contexts or telescopes are “flat”; they don’t show that structures are built from existing structures, sharing some parts, and inheriting some properties. We informally define “packaging” as any approach to collecting the parts of mathematical structures, supporting more abstract manipulation of structures. Packaging is a two-edged sword: once structures are packaged to gain abstraction, we need more tools for manipulating them.