The Lean Standard Library ========================= The Lean standard library is contained in the following files and directories: * [init](init/init.md) : constants and theorems needed for low-level system operations * [logic](logic/logic.md) : logical constructs and axioms * [data](data/data.md) : concrete datatypes and type constructors * [algebra](algebra/algebra.md) : algebraic structures * [tools](tools/tools.md) : additional tools Modules can be loaded individually, but they are also be loaded by importing the following packages: * [standard](standard.lean) : constructive logic and datatypes * [classical](classical.lean) : 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 * quotient 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](../hott/hott.md), a library for homotopy type theory based on a predicative foundation.