L\(\exists\forall\)n Theorem Prover

The lean prover system is getting gaining a lot of popularity:

Quanta magazine called it one of the 5 most important progressions in math of 2020: original article
In use at many universities to teach topics in mathematics: e.g. lambda calculus

Integrates a theorem prover and a programming language
Based on dependent type theory
(Specifically, calculus of inductive constructions)
- Every expression has a type
- countable hierarchy of non-cumulative universes and inductive types: link

\(\alpha\)

Type

Type 1

Type

\(\alpha\)

Type

Type 1

Uses 'Propositions as Types' paradigm (Curry-Howard isomorphism)

then:

~p ≡ p -> false

p \(\Rightarrow\) q ≡ p -> q

p \(\land\) q ≡ (p, q)

p \(\lor\) q ≡ Either p q

p and q are propositions:
p : Prop q : Prop

Proving p \(\Rightarrow\) q \(\Rightarrow\) p is the same as providing an implementation for a function of type p \(\rightarrow\) q \(\rightarrow\) p

Tactics are instructions on how to build a proof. They offer a gateway to to L\(\exists\forall\)N automated reasoning capabilities.

An introduction to tactics is given in the natural number game.

We'll go through it if time allows.

inductive foo : Sort u
| constructor₁ : ... → foo
| constructor₂ : ... → foo
...
| constructorₙ : ... → foo

(Almost*) all types are a form of inductive types:

* All user-defined, and all built-in except for the universes and the dependent type constructor \(\Pi\), that I kept hidden.

Every type automatically comes with its own eliminator and based on that, we can recurse (of which induce is considered a special form)