From 4205dfebe1b890fe959f022c7f573be9a15928f7 Mon Sep 17 00:00:00 2001
From: Craig McLaughlin <thecraigmclaughlin@gmail.com>
Date: Wed, 22 Jan 2025 14:30:05 +1100
Subject: [PATCH] Added slide material for talk.

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+import Mathlib.Tactic.Tauto
+import Mathlib.Data.Fin.Fin2
+universe u
+-- # Lean by example
+
+-- I am going to present Lean4 features via a worked example:
+--
+--                  Hindley-Milner type inference!
+--
+-- The worked example is based on a presentation of Algorithm W due to:
+--
+--                  "Type Inference in Context"
+--                              by
+--             Gundry, McBride, and McKinna (MSFP '10).
+--
+-- It is an algorithm where the unification variables are explicitly
+-- tracked in the typing context as they are introduced during type
+-- inference.  The general idea is:
+--
+--   1. Introduce unification variables during type inference problems;
+--   2. Store these meta variables in the typing context;
+--   3. Upate their occurence in the context during unification.
+
+-- # Types
+--
+-- The types of Hindley-Milner system are RANGED OVER BY τ:
+--   * Arrow (or function) types
+--   * Metavariables (using de Bruijn indices!)
+--
+-- Omitted base types (Ints, Bools, etc.) for brevity.
+--
+-- We ensure well-scopedness of types by mandating metavariables
+-- (unification variables, flexible variables) are within the permitted
+-- bounds.  The representation is **de Bruijn indices**.
+inductive Ty : (n : Nat) → Type u where
+  | meta  : Fin n → Ty n
+  | arrow : Ty n → Ty n → Ty n
+
+-- # What's Fin?
+--
+-- `Fin n` for (n : Nat) represents the numbers i between 0 ≤ i < 1
+-- Can be expressed as the numbers between .. .. .
+inductive Scoped : Nat → Type u where
+  | zero : {n : Nat} → Scoped n
+  | succ : {n : Nat} → Scoped n → Scoped (n + 1)
+
+--------
+-- # Schemes
+--
+-- A scheme is a type where all the freely occurring type variables are
+-- bounded by forall-quantifiers.
+--
+-- In LaTeX you would write:
+--
+--      σ ::= τ | ∀α. σ
+--
+-- In Lean:
+inductive Sch : (n : Nat) → Type u where
+  | type : Ty n → Sch n
+  | all  : Ty (n + 1) → Sch n
+------
+-- # Entries in a Context
+--
+-- Our typing contexts are sequences of **entries** which can be:
+--
+--      * A **definition** for a unification variable, α := τ
+--      * A term variable binding a scheme, e.g. x:σ
+--      * An unknown unification variable α, β
+--
+-- Definitions for metavariables are obtain through the unification
+-- algorithm.
+
+-- # Entries in Lean
+--
+--      * An unknown unification variable α, β
+--      * A **definition** for a unification variable, α := τ
+--      * A term variable binding a scheme, e.g. x:σ
+--
+inductive Entry : (n : Nat) → Type u where
+  | hole : Entry n
+  | defn : Ty n → Entry n
+  | bind : Sch n → Entry n
+
+-- # Contexts
+--
+-- These are backward lists of entries, i.e. the most recent binding
+-- occurs on the **RIGHT**.
+inductive Ctx : (n : Nat) → Type u where
+  | ε : Ctx 0
+  | snoc  : Ctx n → Entry n → Ctx (n + 1)
+
+-- # A Digression On Notations
+--
+-- Lean syntax is fully extensible, exporting the same APIs and
+-- mechanisms used by builtin syntax to users.  The metaprogramming
+-- framework has several layers of increasing sophistication.
+--
+-- The most basic layer is operators of various fixity and notations.
+--
+-- Convenient operators for type formers:
+--
+prefix:65 "ν"   => Ty.meta
+--
+-- We can overload notations: `→` is also used for Lean function
+-- spaces!  Lean will try to work out what you meant, e.g.
+--
+infixl:60  " → " => Ty.arrow
+
+-- To construct a valid type, we must have a proof that every
+-- metavariable is "in scope", e.g.
+--
+#check (ν ⟨ 4 , Nat.lt_add_one 4 ⟩ : Ty 5)
+
+-- An abbreviation
+abbrev prf : 0 < 1 := Nat.lt_add_one 0
+#check (ν ⟨ 0 , prf ⟩ → ν ⟨ 0 , prf ⟩)
+
+-- {hide}
+open Ty
+open Sch
+open Entry
+open Ctx
+-- {show}
+--
+-- Notations for Context Extension
+--
+-- Backward lists associate to the left:
+--
+notation:80 Γ "•ₜ" σ => snoc Γ (bind σ)
+
+-- # Where are the variables?
+--
+-- In order to define the lambda terms of this calculus, we need to
+-- first define a means of mentioning **term** variables in the context.
+--
+inductive Var : {n : Nat} → (Γ : Ctx n) → Sch n → Type u where
+  | vz : {n : Nat} → {Γ : Ctx n} → {σ : Sch (n + 1)} → Var (Γ •ₜ σ) σ
+
+-- # Terms of the Hindley-Milner type system
+--
+--  t, s ::= x | λx.t | s t | let x = s in t
+--
+--
+inductive Term : (n : Nat) → Type u where
+  | var  : Term n
+  | lam  : Term n
+  | app  : Term n → Term n → Term n
+  | lett : Term n