lean2/library/library.md

The Lean Library

The Lean library is contained in the following modules and directories:

• init : constants and theorems needed for low-level system operations
• logic : logical constructs and axioms
• data : concrete datatypes and type constructors
• algebra : algebraic structures
• tools : additional tools

Modules can be loaded individually, but they are also be loaded by importing the following packages:

• standard : constructive logic and datatypes
• classical : classical mathematics

Lean's default logical framework is a version of the Calculus of Constructions with:

• an impredicative, proof-irrelevant type Prop of propositions
• universe polymorphism
• a non-cumulative hierarchy of universes, Type 1, Type 2, ... above Prop
• inductively defined types

The standard library does not rely on any axioms beyond this framework, and is hence constructive. It includes theories of the natural numbers, integers, lists, and so on.

The classical library imports the law of the excluded middle, choice functions, and propositional extensionality. See logic/axioms for details.

See also the hott library, a library for homotopy type theory based on a predicative foundation.