Metamath

Metamath is a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems covering conventional results in logic, set theory, number theory, group theory, algebra, analysis, and topology, as well as topics in Hilbert spaces and quantum logic.

The Metamath language

While the large database of proved theorems follows conventional ZFC set theory, the Metamath language is a metalanguage, suitable for developing a wide variety of formal systems.

The set of symbols that can be used for constructing formulas is declared using $c and $v statements; for example:

The grammar for formulas is specified using a combination of $f and $a statements; for example:

Axioms and rules of inference are specified with $a statements along with ${ and $} for block scoping; for example:

The metamath program can convert statements to more conventional TeX notation; for example, the modus ponens axiom from set.mm: