Positively dependent types (original) (raw)
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Dependently Typed Functional Programs and their Proofs
1999
Research in dependent type theories [M-L71a] has, in the past, concentrated on its usein the presentation of theorems and theorem-proving. This thesis is concerned mainlywith the exploitation of the computational aspects of type theory for programming, ina context where the properties of programs may readily be specified and established.In particular, it develops technology for programming with dependent inductive familiesof datatypes
2009
In these lecture notes we give an introduction to functional programming with dependent types. We use the dependently typed programming language Agda which is an extension of Martin-Löf type theory. First we show how to do simply typed functional programming in the style of Haskell and ML. Some differences between Agda’s type system and the Hindley-Milner type system of Haskell and ML are also discussed. Then we show how to use dependent types for programming and we explain the basic ideas behind type-checking dependent types. We go on to explain the Curry-Howard identification of propositions and types. This is what makes Agda a programming logic and not only a programming language. According to Curry-Howard, we identify programs and proofs, something which is possible only by requiring that all program terminate. However, at the end of these notes we present a method for encoding partial and general recursive functions as total functions using dependent types.
Parsing Parses A Pearl of ( Dependently Typed ) Programming and Proof
2015
We present a functional parser for arbitrary context-free grammars, together with soundness and completeness proofs, all inside Coq. By exposing the parser in the right way with parametric polymorphism and dependent types, we are able to use the parser to prove its own soundness, and, with a little help from relational parametricity, prove its own completeness, too. Of particular interest is one strange instantiation of the type and value parameters: by parsing parse trees instead of strings, we convince the parser to generate its own completeness proof. We conclude with highlights of our experiences iterating through several versions of the Coq development, and some general lessons about dependently typed programming.
The Implicit Calculus of Constructions as a Programming Language with Dependent Types
2008
In this paper, we show how Miquel’s Implicit Calculus of Constructions (ICC) can be used as a programming language featuring dependent types. Since this system has an undecidable type-checking, we introduce a more verbose variant, called ICC* which fixes this issue. Datatypes and program specifications are enriched with logical assertions (such as preconditions, postconditions, invariants) and programs are decorated with proofs of those assertions. The point of using ICC* rather than the Calculus of Constructions (the core formalism of the Coq proof assistant) is that all of the static information (types and proof objects) is transparent, in the sense that it does not affect the computational behavior. This is concretized by a built-in extraction procedure that removes this static information. We also illustrate the main features of ICC* on classical examples of dependently typed programs.
Dependently Typed Programming with Finite Sets
2016
Definitions of many mathematical structures used in computer sci-ence are parametrized by finite sets. To work with such structures in proof assistants, we need to be able to explain what a finite set is. In constructive mathematics, a widely used definition is listability: a set is considered to be finite, if its elements can be listed com-pletely. In this paper, we formalize different variations of this def-inition in the Agda programming language. We develop a toolbox for boilerplate-free programming with finite sets that arise as sub-sets of some base set with decidable equality. Among other things we implement combinators for defining functions from finite sets and a prover for quantified formulas over decidable properties on finite sets.