# 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"
/-- info: success -/
#assertType 5 : Nat
/-- error: failure -/
#assertType [] : Nat
```

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.