Agda

Agda is a dependently typed functional programming language and interactive proof assistant. It is based on intuitionistic dependent type theory, primarily Martin-Löf type theory with a universe hierarchy, and serves both as a programming language and a formal proof environment.

In Agda, types express propositions and programs correspond to proofs. Terms inhabit types, and type checking amounts to verifying proofs or programs meet specifications. The core features include dependent function types, inductive data types, records, modules, and a rich type system with universes (Set0, Set1, etc.). The language supports implicit arguments, pattern matching, and constructs for defining functions by cases, with rewrite rules to support equational reasoning. Agda emphasizes totality checking and termination, helping ensure that all defined functions are well-behaved.

Agda programs are written in source files with the .agda extension. The environment provides interactive development through editor integration, typically with Emacs, and tooling for loading, editing, and checking definitions. The standard library supplies a large collection of data structures and utilities. Agda can compile to other languages, notably Haskell via the MAlonzo backend, or be used to extract verified code for further use.

The language is used in research and education to formalize mathematical proofs, verify algorithms, and experiment with type-theoretic ideas. It emphasizes constructive proofs and explicit witness terms, aligning with the Curry-Howard correspondence.