Introduction

What's the goal of this book?

This book aims to build up enough knowledge about metaprogramming in Lean 4 so you can be comfortable enough to:

  • Start building your own meta helpers (defining new Lean notation such as , building new Lean commands such as #check, writing tactics such as use, etc.)
  • Read and discuss metaprogramming APIs like the ones in Lean 4 core and Mathlib4

We by no means intend to provide an exhaustive exploration/explanation of the entire Lean 4 metaprogramming API. We also don't cover the topic of monadic programming in Lean 4. However, we hope that the examples provided will be simple enough for the reader to follow and comprehend without a super deep understanding of monadic programming. The book Functional Programming in Lean is a highly recommended source on that subject.

Book structure

The book is organized in a way to build up enough context for the chapters that cover DSLs and tactics. Backtracking the pre-requisites for each chapter, the dependency structure is as follows:

  • "Tactics" builds on top of "Macros" and "Elaboration"
  • "DSLs" builds on top of "Elaboration"
  • "Macros" builds on top of "Syntax"
  • "Elaboration" builds on top of "Syntax" and "MetaM"
  • "MetaM" builds on top of "Expressions"

After the chapter on tactics, you find a cheat sheet containing a wrap-up of key concepts and functions. And after that, there are some chapters with extra content, showing other applications of metaprogramming in Lean 4. Most chapters contain exercises at the end of the chapter - and at the end of the book you will have full solutions to those exercises.

The rest of this chapter is a gentle introduction to what metaprogramming is, offering some small examples to serve as appetizers for what the book shall cover.

Note: the code snippets aren't self-contained. They are supposed to be run/read incrementally, starting from the beginning of each chapter.

What does it mean to be in meta?

When we write code in most programming languages such as Python, C, Java or Scala, we usually have to stick to a pre-defined syntax otherwise the compiler or the interpreter won't be able to figure out what we're trying to say. In Lean, that would be defining an inductive type, implementing a function, proving a theorem, etc. The compiler, then, has to parse the code, build an AST (abstract syntax tree) and elaborate its syntax nodes into terms that can be processed by the language kernel. We say that such activities performed by the compiler are done in the meta-level, which will be studied throughout the book. And we also say that the common usage of the language syntax is done in the object-level.

In most systems, the meta-level activities are done in a different language to the one that we use to write code. In Isabelle, the meta-level language is ML and Scala. In Coq, it's OCaml. In Agda, it's Haskell. In Lean 4, the meta code is mostly written in Lean itself, with a few components written in C++.

One cool thing about Lean, though, is that it allows us to define custom syntax nodes and implement meta-level routines to elaborate them in the very same development environment that we use to perform object-level activities. So for example, one can write notation to instantiate a term of a certain type and use it right away, in the same file! This concept is generally called reflection. We can say that, in Lean, the meta-level is reflected to the object-level.

If you have done some metaprogramming in languages such as Ruby, Python or Javascript, it probably took the form of making use of predefined metaprogramming methods to define something on the fly. For example, in Ruby you can use Class.new and define_method to define a new class and its new method (with new code inside!) on the fly, as your program is executing.

We don't usually need to define new commands or tactics "on the fly" in Lean, but spiritually similar feats are possible with Lean metaprogramming and are equally straightforward, e.g. you can define a new Lean command using a simple one-liner elab "#help" : command => do ...normal Lean code....

In Lean, however, we will frequently want to directly manipulate Lean's CST (Concrete Syntax Tree, Lean's Syntax type) and Lean's AST (Abstract Syntax Tree, Lean's Expr type) instead of using "normal Lean code", especially when we're writing tactics. So Lean metaprogramming is more challenging to master - a large chunk of this book is contributed to studying these types and how we can manipulate them.

Metaprogramming examples

Next, we introduce several examples of Lean metaprogramming, so that you start getting a taste for what metaprogramming in Lean is, and what it will enable you to achieve. These examples are meant as mere illustrations - do not worry if you don't understand the details for now.

Introducing notation (defining new syntax)

Often one wants to introduce new notation, for example one more suitable for (a branch of) mathematics. For instance, in mathematics one would write the function adding 2 to a natural number as x : Nat ↦ x + 2 or simply x ↦ x + 2 if the domain can be inferred to be the natural numbers. The corresponding lean definitions fun x : Nat => x + 2 and fun x => x + 2 use => which in mathematics means implication, so may be confusing to some.

We can introduce notation using a macro which transforms our syntax to Lean's syntax (or syntax we previously defined). Here we introduce the notation for functions.

import Lean

macro x:ident ":" t:term " ↦ " y:term : term => do
  `(fun $x : $t => $y)

#eval (x : Nat ↦ x + 2) 2 -- 4

macro x:ident " ↦ " y:term : term => do
  `(fun $x  => $y)

#eval (x ↦  x + 2) 2 -- 4

Building a command

Suppose we want to build a helper command #assertType which tells whether a given term is of a certain type. The usage will be:

#assertType <term> : <type>

Let's see the code:

elab "#assertType " termStx:term " : " typeStx:term : command =>
  open Lean Lean.Elab Command Term in
  liftTermElabM
    try
      let tp ← elabType typeStx
      discard $ elabTermEnsuringType termStx tp
      synthesizeSyntheticMVarsNoPostponing
      logInfo "success"
    catch | _ => throwError "failure"

#assertType 5  : Nat -- success
#assertType [] : Nat -- failure

We started by using elab to define a command syntax. When parsed by the compiler, it will trigger the incoming computation.

At this point, the code should be running in the CommandElabM monad. We then use liftTermElabM to access the TermElabM monad, which allows us to use elabType and elabTermEnsuringType to build expressions out of the syntax nodes typeStx and termStx.

First, we elaborate the expected type tp : Expr, then we use it to elaborate the term expression. The term should have the type tp otherwise an error will be thrown. We then discard the resulting term expression, since it doesn't matter to us here - we're calling elabTermEnsuringType as a sanity check.

We also add synthesizeSyntheticMVarsNoPostponing, which forces Lean to elaborate metavariables right away. Without that line, #assertType [] : ?_ would result in success.

If no error is thrown until now then the elaboration succeeded and we can use logInfo to output "success". If, instead, some error is caught, then we use throwError with the appropriate message.

Building a DSL and a syntax for it

Let's parse a classic grammar, the grammar of arithmetic expressions with addition, multiplication, naturals, and variables. We'll define an AST (Abstract Syntax Tree) to encode the data of our expressions, and use operators + and * to denote building an arithmetic AST. Here's the AST that we will be parsing:

inductive Arith : Type where
  | add : Arith → Arith → Arith -- e + f
  | mul : Arith → Arith → Arith -- e * f
  | nat : Nat → Arith           -- constant
  | var : String → Arith        -- variable

Now we declare a syntax category to describe the grammar that we will be parsing. Notice that we control the precedence of + and * by giving a lower precedence weight to the + syntax than to the * syntax indicating that multiplication binds tighter than addition (the higher the number, the tighter the binding). This allows us to declare precedence when defining new syntax.

declare_syntax_cat arith
syntax num                        : arith -- nat for Arith.nat
syntax str                        : arith -- strings for Arith.var
syntax:50 arith:50 " + " arith:51 : arith -- Arith.add
syntax:60 arith:60 " * " arith:61 : arith -- Arith.mul
syntax " ( " arith " ) "          : arith -- bracketed expressions

-- Auxiliary notation for translating `arith` into `term`
syntax " ⟪ " arith " ⟫ " : term

-- Our macro rules perform the "obvious" translation:
macro_rules
  | `(⟪ $s:str ⟫)              => `(Arith.var $s)
  | `(⟪ $num:num ⟫)            => `(Arith.nat $num)
  | `(⟪ $x:arith + $y:arith ⟫) => `(Arith.add ⟪ $x ⟫ ⟪ $y ⟫)
  | `(⟪ $x:arith * $y:arith ⟫) => `(Arith.mul ⟪ $x ⟫ ⟪ $y ⟫)
  | `(⟪ ( $x ) ⟫)              => `( ⟪ $x ⟫ )

#check ⟪ "x" * "y" ⟫
-- Arith.mul (Arith.var "x") (Arith.var "y") : Arith

#check ⟪ "x" + "y" ⟫
-- Arith.add (Arith.var "x") (Arith.var "y") : Arith

#check ⟪ "x" + 20 ⟫
-- Arith.add (Arith.var "x") (Arith.nat 20) : Arith

#check ⟪ "x" + "y" * "z" ⟫ -- precedence
-- Arith.add (Arith.var "x") (Arith.mul (Arith.var "y") (Arith.var "z")) : Arith

#check ⟪ "x" * "y" + "z" ⟫ -- precedence
-- Arith.add (Arith.mul (Arith.var "x") (Arith.var "y")) (Arith.var "z") : Arith

#check ⟪ ("x" + "y") * "z" ⟫ -- brackets
-- Arith.mul (Arith.add (Arith.var "x") (Arith.var "y")) (Arith.var "z")

Writing our own tactic

Let's create a tactic that adds a new hypothesis to the context with a given name and postpones the need for its proof to the very end. It's similar to the suffices tactic from Lean 3, except that we want to make sure that the new goal goes to the bottom of the goal list.

It's going to be called suppose and is used like this:

suppose <name> : <type>

So let's see the code:

open Lean Meta Elab Tactic Term in
elab "suppose " n:ident " : " t:term : tactic => do
  let n : Name := n.getId
  let mvarId ← getMainGoal
  mvarId.withContext do
    let t ← elabType t
    let p ← mkFreshExprMVar t MetavarKind.syntheticOpaque n
    let (_, mvarIdNew) ← MVarId.intro1P $ ← mvarId.assert n t p
    replaceMainGoal [p.mvarId!, mvarIdNew]
  evalTactic $ ← `(tactic|rotate_left)

example : 0 + a = a := by
  suppose add_comm : 0 + a = a + 0
  rw [add_comm]; rfl     -- closes the initial main goal
  rw [Nat.zero_add]; rfl -- proves `add_comm`

We start by storing the main goal in mvarId and using it as a parameter of withMVarContext to make sure that our elaborations will work with types that depend on other variables in the context.

This time we're using mkFreshExprMVar to create a metavariable expression for the proof of t, which we can introduce to the context using intro1P and assert.

To require the proof of the new hypothesis as a goal, we call replaceMainGoal passing a list with p.mvarId! in the head. And then we can use the rotate_left tactic to move the recently added top goal to the bottom.