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HoTTSQL: Proving Query Rewrites with Univalent SQL Semantics SQL is the lingua franca for retrieving structured data. Existing semantics for SQL, however, either do not model crucial features of the language (e.g., relational algebra lacks bag semantics, correlated subqueries, and aggregation), or make it hard to formally reason about SQL query rewrites (e.g., the SQL standard’s English is too…
(Dis)advantages of using Abstract Data Types in Proof Assistants This blog post explains the advantages and disadvantages of using Abstract Data Types (ADTs, see TaPL Chapter 24) in a Proof Assistant. On the plus side, ADTs promote data representation independence, code reuse, and clean code extraction; but they also do not support fix/match syntax, have to expose derived operations, and prohibit…
Machine Checkable Pictorial Mathematics for Category Theory Pictures are essential to conveying proofs and definitions (Pinto and Tall, 2002), and are thus common in traditional pen and paper mathematics. While proof assistants offer several benefits over pen and paper mathematics, proofs and definitions in proof assistants are entirely textual (i.e. lack images), and are thus often harder to…
3 Shocking Ways to Break Mathematics This post provides formal proofs of three paradoxes that occur in type systems with unrestricted recursion, negative data types, or type in type (Girard’s paradox). The title is a pun on BuzzFeedy headlines.
Abstract Data Types and Object Oriented Programming in Coq (Part I) This post describes how to use data abstractions, specifically Abstract Data Types (ADTs) and Object Oriented Programming (OOP), to hide a value’s representation in a functional programming language with dependent types.
Connections between Natural Numbers, Types, Sets, and Propositions This post shows connections between natural numbers, types, sets and propositions. This post draws ideas from the Curry Howard correspondence which shows a connection between functional programs and mathematical proofs, and Chris Taylor’s blog post which shows a connection between natural numbers and Haskell types.