Lean 4 & Mathlib

Lean 4 is both a functional programming language and an interactive theorem prover. In Lean, a theorem is a type, and a proof is a program of that type. This is the propositions-as-types idea: the statement 'for every natural number $n$, $n + 0 = n$' is a type, and any term that inhabits it is a proof. The Lean kernel is a small piece of trusted code that type-checks every term. If the kernel accepts it, the theorem is established, however clever or messy the proof-writing process was.

You rarely write proof terms by hand. Instead you use tactics: short commands such as intro, rw (rewrite using an equation), simp (simplify), linarith (solve linear inequalities) and ring (prove polynomial identities). Tactics build the underlying term for you, and the kernel checks the result. The alternative is term mode, where you write the term directly. Tactics are usually faster for people; term mode is more compact when it works.

Mathlib is the community library on top of Lean. Its main design choice is that it is one library with one set of definitions. A group, a ring or a metric space is defined once, and everything else reuses it. This works because of the system: a statement such as 'in any , every nonzero element has an inverse' is proved once, and Lean automatically finds it whenever you work in a field such as the real numbers. The hierarchy (, , , ...) is organised so that more specialised structures inherit the results of more general ones.