Observational equality, now!

Abstract

This paper has something new and positive to say about propositional equality in programming and proof systems based on the Curry-Howard correspondence between propositions and types. We have found a way to present a propositional equality type• which is substitutive, allowing us to reason by replacing equal for equal in propositions; • which reflects the observable behaviour of values rather than their construction: in particular, we have extensionalityfunctions are equal if they take equal inputs to equal outputs; • which retains strong normalisation, decidable typechecking and canonicity-the property that closed normal forms inhabiting datatypes have canonical constructors; • which allows inductive data structures to be expressed in terms of a standard characterisation of well-founded trees; • which is presented syntactically-you can implement it directly, and we are doing so-this approach stands at the core of Epigram 2; • which you can play with now: we have simulated our system by a shallow embedding in Agda 2, shipping as part of the standard examples package for that system [20].Until now, it has always been necessary to sacrifice some of these aspects. The closest attempt in the literature is Altenkirch's construction of a setoid-model for a system with canonicity and extensionality on top of an intensional type theory with proof-irrelevant propositions [4]. Our new proposal simplifies Altenkirch's construction by adopting McBride's heterogeneous approach to equality [18].

Type theory in type theory using quotient inductive types

“…The theory poses a canonicity problem, i.e. can all closed term of type N be reduced to numerals, which can be adressed using techniques developed in the context of Observational Type Theory [5]. Recent work [8] suggest that also the harder problem of canonicity in the presence of univalence can be addressed.…”

“…Defining the logical predicate interpretation is tedious but feasible. Most of the work that needs to be done is coercing syntactic expressions using equality reasoning which can be simplified by using a heterogeneous equality [5] -two terms of different types are equal if there is an equality between the types and an equality between the terms up to this previous equality. …”

System FC with explicit kind equality

“…This rule, inspired by Observational Type Theory (Altenkirch et al 2007), provides a simple way of ensuring that proofs do not interfere with equality. Without it, we would need coercions analogous to the many "push" rules of the operational semantics.…”

Giving Haskell a promotion

“…Essentially, is serving as the single type in a "uni-typed" type system. 1 We are not prevented from writing nonsensical types such as Succ Bool or Vec Zero Int.…”

“…Our intention for the arguments of Vec is now clear, and writing Vec Zero Int will rightly yield a type error. 1 Well, not quite uni-typed, because we have arrow kinds, but close.…”