lean2/library/init/init.md

init

The files in this folder are required by low-level operations, and are always imported by default. You can suppress this behavior by beginning a file with the keyword "prelude".

Syntax declarations:

• reserved_notation
• tactic
• priority

Datatypes and logic:

• datatypes
• logic
• bool
• num
• nat

Support for well-founded recursion:

• relation
• wf
• wf_k
• measurable

Additional datatypes:

• prod
• sigma

The default import:

• default

Module init.logic defines "inhabited" and "nonempty" types. Constructively, inhabited types have a witness, while nonempty types are proof irrelevant. Classically (assuming the axioms in logic.axioms.hilbert) the two are equivalent. Type class inferences are set up to use "inhabited" however, so users should use that to declare that types have an element. Use "nonempty" in the hypothesis of a theorem when the theorem does not depend on the witness chosen.