The Lean Library ================ The Lean library is contained in the following modules and directories: * [general_notation](general_notation.lean) : commonly shared notation * [logic](logic/logic.md) : logical constructs and axioms * [data](data/data.md) : concrete datatypes and type constructors * [algebra](algebra/algebra.md) : algebraic structures * [hott](hott/hott.md) : homotopy type theory * [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 * [hott](hott/default.lean) : homotopy type theory Lean's default logical framework is a version of the Calculus of Constructions with: * an impredicative, proof-irrelevant type `Prop` of propositions * univerve 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. The `hott` library is compatible with the standard library, but favors "proof relevant" versions of the logical connectives, based on `Type` rather than `Prop`. See `hott` for details.