2014-08-25 16:11:46 +00:00
|
|
|
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
|
2014-09-16 18:58:54 +00:00
|
|
|
* [algebra](algebra/algebra.md) : algebraic structures
|
2014-08-25 16:11:46 +00:00
|
|
|
* [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
|
2014-08-26 06:00:50 +00:00
|
|
|
* univerve polymorphism
|
2014-08-25 16:11:46 +00:00
|
|
|
* 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.
|
