lean4-htt/library/library.md

The Lean Core Library

The Lean core library is contained in the following files and directories:

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

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 0, Type 1, ... above Prop
• inductively defined types
• quotient types

Most of the core library does not rely on any axioms beyond this framework, and is hence fully constructive.

The following additional axioms are defined in init:

• quotients and propositional extensionality (in quot)
• Hilbert choice (in classical)

Function extensionality is derived from the quotient construction, and excluded middle is derived from Hilbert choice. For Hilbert choice and excluded middle, use open classical. The additional axioms are used sparingly. For example:

You can use #print axioms foo to see which axioms foo depends on.