3. Lean as a Programming Language

3.1. About Lean

Lean is a new programming language and interactive proof assistant being developed at Microsoft Research. It is currently in an experimental, development stage, which makes it a risky choice for this course. But in many ways it is an ideal system for working with logical syntax and putting logic to use. Lean is an exciting project, and the system fun to use. So please bear with us. Using Lean puts us out on the frontier, but if you adopt a pioneering attitude, you will be in a good position to enjoy all the cool things that Lean has to offer.

You can learn more about Lean on the Lean home page, on the Lean community home page, and by asking questions on the Lean Zulip chat, which you are heartily encouraged to join. To be more precise, there are currently two versions of Lean:

  • Lean 3 is reasonably stable, and primarily an interactive proof assistant. It has a very large mathematical library, known as mathlib.

  • Lean 4 is being designed as a performant programming language, and it is still under development. It can also be used as a proof assistant, though it does not yet have a substantial library. Its language and syntax are similar to that of Lean 3, but it is not backward compatible.

In this course, we will use Lean 4, even though it is still under development. It has the rough beginnings of a user manual and there is a tutorial on the underlying foundation. As we will see, Lean has a lot of features that make that worthwhile. In particular, Lean 4 is designed to be an ideal language for implementing powerful logic-based systems, as evidenced by the fact that most of Lean 4 is implemented in Lean 4 itself.

The goal of this section is to give you a better sense of what Lean is, how it can possibly be a programming language and proof assistant at the same time, and why that makes sense. The rest of the introduction will give you a quick tour of some of its features, and we will learn more about them as the course progresses.

At the core, Lean is an implementation of a formal logical foundation known as type theory. More specifically, it is an implementation of dependent type theory, and even more specifically than that, it implements a version of the Calculus of Inductive Constructions. Saying that it implements a formal logic foundation means that there is a precise grammar for writing expressions, and precise rules for using them. In Lean, every well-formed expression has a type.

#check 2 + 2
#check -5
#check [1, 2, 3]
#check #[1, 2, 3]
#check (1, 2, 3)
#check "hello world"
#check true
#check fun x => x + 1
#check fun x => if x = 1 then "yes" else "no"

You can find this example in the file using_lean_as_a_programming_language/examples1.lean in the LAMR/Examples folder of the course repository. We recommend copying that entire folder in the User folder, so you can edit the files and try examples of your own. You can always find the original file in the folder LAMR/Examples, which you should not edit.

If you hover over the #check statements or move your cursor to one of these lines and check the information window, Lean reports the result of the command. It tells you that 2 + 2 has type Nat, -5 has type Int, and so on. In fact, in the formal foundation, types are expressions as well. The types of all the expressions above are listed below:

#check Nat
#check Int
#check List Nat
#check Array Nat
#check Nat × Nat × Nat
#check String
#check Bool
#check Nat  Nat
#check Nat  String

Now Lean tells you each of these has type Type, indicating that they are all data types. If you know the type of an expression, you can ask Lean to confirm it:

#check (2 + 2 : Nat)
#check ([1, 2, 3] : List Nat)

Lean will report an error if it cannot construe the expression as having the indicated type.

In Lean, you can define new objects with the def command. The new definition becomes part of the environment: the defined expression is associated with the identifier that appears after the word def.

def four : Nat := 2 + 2

def isOne (x : Nat) : String := if x = 1 then "yes" else "no"

#check four
#print four

#check isOne
#print isOne

The type annotations indicate the intended types of the arguments and the result, but they can be omitted when Lean can infer them from the context:

def four' := 2 + 2

def isOne' x := if x = 1 then "yes" else "no"

So far, so good: in Lean, we can define expressions and check their types. What makes Lean into a programming language is that the logical foundation has a computational semantics, under which expressions can be evaluated.

#eval four
#eval isOne 3
#eval isOne 1

The #eval command evaluates the expression and then displays the return value. Evaluation can also have side effects, which are generally related to system IO. For example, displaying the string “Hello, world!” is a side effect of the following evaluation:

#eval IO.println "Hello, world!"

Theoretical computer scientists are used to thinking about programs as expressions and identifying the act of running the program with the act of evaluating the expression. In Lean, this view is made manifest, and the expressions are defined in a formal system with a precise specification.

But what makes Lean into a proof assistant? To start with, some expressions in the proof system express propositions:

#check 2 + 2 = 4
#check 2 + 2 < 5
#check isOne 3 = "no"
#check 2 + 2 < 5  isOne 3 = "no"

Lean confirms that each of these is a proposition by reporting that each of them has type Prop. Notice that they do not all express true propositions; theorem proving is about certifying the ones that are. But the language of Lean is flexible enough to express just about any meaningful mathematical statement at all. For example, here is the statement of Fermat’s last theorem:

def Fermat_statement : Prop :=
 a b c n : Nat, a * b * c  0  n > 2  a^n + b^n  c^n

In Lean’s formal system, data types are expressions of type Type, and if T is a type, an expression of type T denotes an object of that type. We have also seen that propositions are expressions of type Prop. In the formal system, if P is a proposition, a proof of P is just an expression of type P. This is the final piece of the puzzle: we use Lean as a proof assistant by writing down a proposition P, writing down an expression p, and asking Lean to confirm that p has type P. The fact that 2 + 2 = 4 has an easy proof, that we will explain later:

theorem two_plus_two_is_four : 2 + 2 = 4 := rfl

In contrast, proving Fermat’s last theorem is considerably harder.

theorem Fermat_last_theorem : Fermat_statement := sorry

Lean knows that sorry is not a real proof, and it flags a warning there. If you manage to replace sorry by a real Lean expression, please let us know. We will be very impressed.

So, in Lean, one can write programs and execute them, and one can state propositions and prove them. In fact, one can state propositions about programs and then prove those statements as well. This is known as software verification; it is a means of obtaining a strong guarantee that a computer program behaves as intended, something that is important, say, if you are using the software to control a nuclear reactor or fly an airplane.

This course is not about software verification. We will be using Lean 4 primarily as a programming language, one in which we can easily define logical expressions and manipulate them. To a small extent, we will also write some simple proofs in Lean. This will help us think about proof systems and rules, and understand how they work. Taken together, these two activities embody the general vision that animates this course: knowing how to work with formally specified expressions and rules opens up a world of opportunity. It is the key to unlocking the secrets of the universe.

3.2. Using Lean as a functional programming language

There is a preliminary user’s manual for Lean, still a work in progress, here. The fact that Lean is a functional programming language means that instead of presenting a program as a list of instructions, you simply define functions and ask Lean to evaluate them.

def foo n := 3 * n + 7

#eval foo 3
#eval foo (foo 3)

def bar n := foo (foo n) + 3

#eval bar 3
#eval bar (bar 3)

There is no global state: any value a function can act on is passed as an explicit argument and is never changed. For that reason, functional programming languages are amenable to parallelization.

Nonetheless, Lean can do handle system IO using the IO monad, and can accommodate an imperative style of programming using do notation.

def printExample : IO Unit:= do
  IO.println "hello"
  IO.println "world"

#eval printExample

Recursive definitions are built into Lean.

def factorial : Nat  Nat
  | 0       => 1
  | (n + 1) => (n + 1) * factorial n

#eval factorial 10
#eval factorial 100

Here is a solution to the Towers of Hanoi problem:

def hanoi (numPegs start finish aux : Nat) : IO Unit :=
  match numPegs with
  | 0     => pure ()
  | n + 1 => do
      hanoi n start aux finish
      IO.println s!"Move disk {n + 1} from peg {start} to peg {finish}"
      hanoi n aux finish start

#eval hanoi 7 1 2 3

You can also define things by recursion on lists:

def addNums : List Nat  Nat
  | []    => 0
  | a::as => a + addNums as

#eval addNums [0, 1, 2, 3, 4, 5, 6]

In fact, there are a number of useful functions built into Lean’s library. The function List.range n returns the list [0, 1, …, n-1], and the functions List.map and List.foldl and List.foldr implement the usual map and fold functions for lists. By opening the List namespace, we can refer to these as range, map, foldl, and foldr. In the examples below, the dollar sign has the same effect as putting parentheses around everything that appears afterward.

#eval List.range 7

section
open List

#eval range 7
#eval addNums $ range 7
#eval map (fun x => x + 3) $ range 7
#eval foldl (. + .) 0 $ range 7

end

The scope of the open command is limited to the section, and the cryptic inscription (. + .) is notation for the addition function. Lean also supports projection notation that is useful when the corresponding namespace is not open:

def myRange := List.range 7
#eval myRange.map fun x => x + 3

Because myRange has type List Nat, Lean interprets myrange.map fun x => x + 3 as List.map (fun x => x + 3) myrange. In other words, it automatically interprets map as being in the List namespace, and then it interprets myrange as the first List argument.

This course assumes you have some familiarity with functional programming. There is a free online textbook, Learn You a Haskell for Great Good that you might find helpful; porting some of the examples there to Lean is a good exercise. We will all suffer from the fact that documentation for Lean 4 barely exists at the moment, but we will do our best to provide you with enough examples for you to be able to figure out how to do what you need to do. One trick is to nose around the Lean code base itself. If you ctrl-click on the name of a function in the Lean library, the editor will jump to the definition, and you can look around and see what else is there. Another strategy is simply to ask us, ask each other, or ask questions on the Lean Zulip chat. We are all in this together.

When working with a functional programming language, there are often clever tricks for doing things that you may be more comfortable doing in an imperative programming language. For example, as explained in Section 2.3, here are Lean’s definitions of the reverse and append functions for lists:

namespace hidden

def reverseAux : List α  List α  List α
  | [],   r => r
  | a::l, r => reverseAux l (a::r)

def reverse (as : List α) :List α :=
  reverseAux as []

protected def append (as bs : List α) : List α :=
  reverseAux as.reverse bs

end hidden

The function reverseAux l r reverses the elements of list l and adds them to the front of r. When called from reverse l, the argument r acts as an accumulator, storing the partial result. Because reverseAux is tail recursive, Lean’s compiler can implement it efficiently as a loop rather than a recursive function. We have defined these functions in a namespace named hidden so that they don’t conflict with the ones in Lean’s library if you open the List namespace.

In Lean’s foundation, every function is totally defined. In particular, every function that Lean computes has to terminates (in principle) on every input. Lean 4 will eventually support arbitrary recursive definitions in which the arguments in a recursive call decrease by some measure, but some work is needed to justify these calls in the underlying foundation. In the meanwhile, we can always cheat by using the partial keyword, which will let us perform arbitrary recursive calls.

partial def gcd m n :=
  if n = 0 then m else gcd n (m % n)

#eval gcd 45 30
#eval gcd 37252 49824

Using partial takes us outside the formal foundation; Lean will not let us prove anything about gcd when we define it this way. Using partial also makes it easy for us to shoot ourselves in the foot:

partial def bad (n : Nat) : Nat := bad (n + 1)

On homework exercises, you should try to use structural recursion when you can, but don’t hesitate to use partial whenever Lean complains about a recursive definition. We will not penalize you for it.

The following definition of the Fibonacci numbers does not require the partial keyword:

def fib' : Nat  Nat
  | 0 => 0
  | 1 => 1
  | n + 2 => fib' (n + 1) + fib' n

But it is inefficient; you should convince yourself that the natural evaluation strategy requires exponential time. The following definition avoids that.

def fibAux : Nat  Nat × Nat
  | 0     => (0, 1)
  | n + 1 => let p := fibAux n
             (p.2, p.1 + p.2)

def fib n := (fibAux n).1

#eval (List.range 20).map fib

Producing a list of Fibonacci numbers, however, as we have done here is inefficient; you should convince yourself that the running time is quadratic. In the exercises, we ask you to define a function that computes a list of Fibonacci values with running time linear in the length of the list.

3.3. Inductive data types in Lean

One reason that computer scientists and logicians tend to like functional programming languages is that they often provide good support for defining inductive data types and then using structural recursion on such types. For example, here is a Lean definition of the extended binary trees that we defined in mathematical terms in Section 2.3:

import Init

inductive BinTree
  | empty : BinTree
  | node  : BinTree  BinTree  BinTree
  deriving Repr, DecidableEq, Inhabited

open BinTree

The command import Init imports a part of the initial library for us to use. The command open BinTree allows us to write empty and node instead of BinTree.empty and BinTree.node. Note the Lean convention of capitalizing the names of data types.

The last line of the definition, the one that begins with the word deriving, is boilerplate. It tells Lean to automatically generate a few additional functions that are useful. The directive deriving Repr tells Lean to define an internal function that can be used to represent any BinTree as a string. This is the string that is printed out by any #eval command whose argument evaluates to a BinTree. Adding DecidableEq defines a function that tests whether two BinTrees are equal, and adding Inhabited defines an arbitrary value of the data type to serve as a default value for function that need one. The following illustrates their use.

#eval node empty (node empty empty)

#eval empty == node empty empty  -- evaluates to false

#eval (arbitrary : BinTree)  -- BinTree.empty

We can now define the functions size and depth by structural recursion:

def size : BinTree  Nat
  | empty    => 0
  | node a b => 1 + size a + size b

def depth : BinTree  Nat
  | empty    => 0
  | node a b => 1 + Nat.max (depth a) (depth b)

def example_tree := node (node empty empty) (node empty (node empty empty))

#eval size example_tree
#eval depth example_tree

In fact, the List data type is also inductively defined.

#print List

You should try writing the inductive definition on your own. Call it MyList, and then try #print MyList to see how it compares.

3.4. Using Lean as an imperative programming language

The fact that Lean is a functional programming language means that there is no global notion of state. Functions take values as input and return values as output; there are no global or even local variables that are changed by the result of a function call.

But one of the interesting features of Lean is a functional programming language is that it incorporates features that make it feel like an imperative programming language. The following example shows how to print out, for each value \(i\) less than 100, the the sum of the numbers up to \(i\).

def showSums : IO Unit := do
  let mut sum := 0
  for i in [0:100] do
    sum := sum + i
    IO.println s!"i: {i}, sum: {sum}"

#eval showSums

You can use a loop not just to print values, but also to compute values. The following is a boolean test for primality:

def isPrime (n : Nat) : Bool := do
  if n < 2 then false else
    for i in [2:n] do
      if n % i = 0 then
        return false
      if i * i > n then
        return true
    true

You can use such a function with the list primitives to construct a list of the first 10,000 prime numbers.

#eval (List.range 10000).filter isPrime

But you can also use it with Lean’s support for arrays. Within the formal foundation these are modeled as lists, but the compiler implements them as dynamic arrays, and for efficiency it will modify values rather than copy them whenever the old value is not referred to by another part of an expression.

def primes (n : Nat) : Array Nat := do
  let mut result := #[]
  for i in [2:n] do
    if isPrime i then
      result := result.push i
  result

#eval (primes 10000).size

Notice the notation: #[] denotes a fresh array (Lean infers the type from context), and the Array.push function adds a new element at the end of the array.

The following example shows how to compute a two-dimensional array, a ten by ten multiplication table.

def mulTable : Array (Array Nat) := do
  let mut table := #[]
  for i in [:10] do
    let mut row := #[]
    for j in [:10] do
      row := row.push ((i + 1) * (j + 1))
    table := table.push row
  table

#eval mulTable

Alternatively, you can use the function Array.mkArray to initialize an array (in this case, to the values 0), and then use the Array.set! function to replace the elements later one.

def mulTable' : Array (Array Nat) := do
  let mut s : Array (Array Nat) := mkArray 10 (mkArray 10 0)
  for i in [:10] do
    for j in [:10] do
      s := s.set! i $ s[i].set! j ((i + 1) * (j + 1))
  s

Here we replace the ith row by the previous ith row, with the jth column updated.

The following snippet prints out the table. The idiom show T from t is a way of telling Lean that term t should have type T. Writing @id T t has a similar effect, as does writing (t : T). (A difference is that the first two expressions have type T exactly, whereas (t : T) ensures that t has a type that Lean recognizes as being equivalent to T.)

#eval show IO Unit from do
  for i in [:10] do
    for j in [:10] do
      let numstr := toString mulTable[i][j]
      -- print 1-3 spaces
      IO.print $ " ".pushn ' ' (3 - numstr.length)
      IO.print numstr
    IO.println ""

3.5. Exercises

  1. Using operations on List, write a Lean function that for every \(n\) returns the list of all the divisors of \(n\) that are less than \(n\).

  2. A natural number \(n\) is perfect if it is equal to the sum of the divisors less than \(n.\) Write a Lean function (with return type Bool) that determines whether a number \(n\) is perfect. Use it to find all the perfect numbers less than 1,000.

  3. Define a recursive function \(\fn{sublists}(\ell)\) that, for every list \(\ell\), returns a list of all the sublists of \(\ell\). For example, given the list \([1, 2, 3]\), it should compute the list

    \[[ [], [1], [2], [3], [1,2], [1,3], [2, 3], [1, 2, 3] ].\]

    The elements need not be listed in that same order.

  4. Prove in Lean that the length of \(\fn{sublists}(\ell)\) is \(2^{\fn{length}(\ell)}\).

  5. Define a function \(\fn{permutations}(\ell)\) that returns a list of all the permutations of \(\ell\).

  6. Prove in Lean that the length of \(\fn{permutations}(\ell)\) is \(\fn{factorial}(\fn{length}(\ell))\).

  7. Define in Lean a function that, assuming \(\ell\) is a list of lists representing an \(n \times n\) array, returns a list of lists representing the transpose of that array.

  8. Write a program that solves the Tower of Hanoi problem with \(n\) disks on the assumption that disks can only be moved to an adjacent peg. (See Section 2.5.)

  9. Write a program that solves the Tower of Hanoi problem with \(n\) disks on the assumption that disks can only be moved clockwise. (See Section 2.5.)

  10. Define a Lean data type of binary trees in which every node is numbered by a label. Define a Lean function to compute the sum of the nodes in such a tree. Also write functions to list the elements in a preorder, postorder, and inorder traversal.