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

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

L\(\exists\forall\)N

• 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
  • Types really are first-class objects: link
  • We can define functions: link

Propositions and Proofs

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

Propositions and Proofs

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

link

Tactics

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

Inductive types

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)

• Very strong and generic type system
• Many different ways of writing the same thing (Steep learning curve!)
• Many language versions, not clearly marked
• Fancy symbol usage in syntax

Lessons to l\(\exists\forall\)rn:

Resources:

Lean book on github

natural number game