Most interactive theorem provers provide support for some form of
user-customizable proof automation. In a number of popular systems,
such as Coq and Isabelle, this automation is achieved primarily
through tactics, which are programmed in a separate language
from that of the prover's base logic. While tactics are clearly
useful in practice, they can be difficult to maintain and compose
because, unlike lemmas, their behavior cannot be specified within the
expressive type system of the prover itself.
We propose a novel approach to proof automation in Coq that allows the
user to specify the behavior of custom automated routines in terms of
Coq's own type system. Our approach involves a sophisticated
application of Coq's canonical structures, which generalize
Haskell type classes and facilitate a flexible style of
dependently-typed logic programming. Specifically, just as Haskell
type classes are used to infer the canonical implementation of an
overloaded term at a given type, canonical structures can be used to
infer the canonical proof of an overloaded lemma for a
given instantiation of its parameters. We present a series of design
patterns for canonical structure programming that enable one to
carefully and predictably coax Coq's type inference engine into
triggering the execution of user-supplied algorithms during
unification, and we illustrate these patterns through several
realistic examples drawn from Hoare Type Theory. We assume no prior
knowledge of Coq and describe the relevant aspects of Coq type
inference from first principles.