Proof Trick: Small Inversions

International audienceWe show how an inductive hypothesis can be inverted with small proof terms, using just dependent elimination with a diagonal predicate. The technique works without any auxiliary type such as True, False, eq. It can also be used to discriminate, in some sense, the constructors of an inductive type of sort Prop in Coq

Builtin types viewed as inductive families

This research was funded by the Engineering and Physical Sciences Research Council (grant number EP/T007265/1).State of the art optimisation passes for dependently typed languages can help erase the redundant information typical of invariant-rich data structures and programs. These automated processes do not dramatically change the structure of the data, even though more efficient representations could be available. Using Quantitative Type Theory, we demonstrate how to define an invariant-rich, typechecking time data structure packing an efficient runtime representation together with runtime irrelevant invariants. The compiler can then aggressively erase all such invariants during compilation. Unlike other approaches, the complexity of the resulting representation is entirely predictable, we do not require both representations to have the same structure, and yet we are able to seamlessly program as if we were using the high-level structure.Publisher PD

Builtin types viewed as inductive families

State of the art optimisation passes for dependently typed languages can help erase the redundant information typical of invariant-rich data structures and programs. These automated processes do not dramatically change the structure of the data, even though more efficient representations could be available. Using Quantitative Type Theory as implemented in Idris 2, we demonstrate how to define an invariant-rich, typechecking-time data structure packing an efficient runtime representation together with runtime irrelevant invariants. The compiler can then aggressively erase all such invariants during compilation. Unlike other approaches, the complexity of the resulting representation is entirely predictable, we do not require both representations to have the same structure, and yet we are able to seamlessly program as if we were using the high-level structure