Foundational Types

Foundational Types

Some of Lean's types are not defined in any Lean source files (even the prelude) since they come from its foundational type theory. This page provides basic documentation for these types.

For a more in-depth explanation of Lean's type theory, refer to TPiL.

Sort u

Sort uis the type of types in Lean, and Sort u : Sort (u + 1).

Instances For

Type u

Type uis notation for Sort (u + 1).

Instances For


Propis notation for Sort 0.

Instances For

Pi types, (a : α) → β a

The type of dependent functions is known as a pi type. Non-dependent functions and implications are a special case.

Note that these can also be written with the alternative notations:

Lean also permits ASCII-only spellings of the three variants:

Note that despite not itself being a function, (→)is available as infix notation for fun α β, α → β.