Welcome to the course

We start with a quick tour, where we:

• Use Lean as a calculator
• Define some functions and call them.
• Look at some types.
• Look at some proofs.

We will then see

• A glimpse of AI.
• A detailed example with programs and proofs.

Lean as a calculator.

We begin by using Lean as a calculator. We can use #eval to evaluate expressions.

#eval 1 + 2 -- 3

#eval "Hello " ++ "world!" -- "Hello world!"

def some_number : ℕ

An arbitrary number.

Equations

• some_number = 42

We next evaluate an expression involving a definition.

#eval some_number + 23 -- 65

Defining functions

We next define some functions. These are defined in terms of previously defined functions.

def add_two (n : ℕ) : ℕ

Add 2 to a natural number

Equations

• add_two n = n + 2

def cube (n : ℕ) : ℕ

Cube a natural number

Equations

• cube n = n * n * n

#eval cube (add_two 3) -- 125

def cube' (n : ℕ) : ℕ

Cube a natural number

Equations

• cube' n = n * n * n

def cube'' : ℕ → ℕ

Cube a natural number

Equations

• cube'' n = n * n * n

Types

Terms in Lean, including functions, have types, which can be seen using #check

#check 1 + 2 -- ℕ

#check "Hello " ++ "world!" -- String

#check add_two -- ℕ → ℕ
#check cube   -- ℕ → ℕ

#check ℕ -- Type
#check Type -- Type 1
#check ℕ → ℕ -- Type

We next define a function of two arguments, and look at its type. We see that this is defined as a function from ℕ to a function from ℕ to ℕ.

def sum_of_squares (x : ℕ) (y : ℕ) : ℕ

Sum of squares of natural numbers x and y

Equations

• sum_of_squares x y = x * x + y * y

#check sum_of_squares -- ℕ → ℕ → ℕ

#check sum_of_squares 3 -- ℕ → ℕ

We can also define this in a way that makes the type clearer.

def sum_of_squares' : ℕ → ℕ → ℕ

Sum of squares of natural numbers x and y

Equations

• sum_of_squares' x y = x * x + y * y