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Observational equality, now!
| Content Provider | ACM Digital Library |
|---|---|
| Author | Swierstra, Wouter Altenkirch, Thorsten McBride, Conor |
| 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 extensionality-- functions 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 [21]. 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 [19]. |
| Starting Page | 57 |
| Ending Page | 68 |
| Page Count | 12 |
| File Format | |
| ISBN | 9781595936776 |
| DOI | 10.1145/1292597.1292608 |
| Language | English |
| Publisher | Association for Computing Machinery (ACM) |
| Publisher Date | 2007-10-02 |
| Publisher Place | New York |
| Access Restriction | Subscribed |
| Subject Keyword | Type theory Equality |
| Content Type | Text |
| Resource Type | Article |
