2. Basics

This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures.

2.1. Calculating

We generally learn to carry out mathematical calculations without thinking of them as proofs. But when we justify each step in a calculation, as Lean requires us to do, the net result is a proof that the left-hand side of the calculation is equal to the right-hand side.

In Lean, stating a theorem is tantamount to stating a goal, namely, the goal of proving the theorem. Lean provides the rewriting tactic rw, to replace the left-hand side of an identity by the right-hand side in the goal. If a, b, and c are real numbers, mul_assoc a b c is the identity a * b * c = a * (b * c) and mul_comm a b is the identity a * b = b * a. Lean provides automation that generally eliminates the need to refer the facts like these explicitly, but they are useful for the purposes of illustration. In Lean, multiplication associates to the left, so the left-hand side of mul_assoc could also be written (a * b) * c. However, it is generally good style to be mindful of Lean’s notational conventions and leave out parentheses when Lean does as well.

Let’s try out rw.

example (a b c : ℝ) : a * b * c = b * (a * c) := by
  rw [mul_comm a b]
  rw [mul_assoc b a c]

The import lines at the beginning of the associated examples file import the theory of the real numbers from Mathlib, as well as useful automation. For the sake of brevity, we generally suppress information like this in the textbook.

You are welcome to make changes to see what happens. You can type the ℝ character as \R or \real in VS Code. The symbol doesn’t appear until you hit space or the tab key. If you hover over a symbol when reading a Lean file, VS Code will show you the syntax that can be used to enter it. If you are curious to see all available abbreviations, you can hit Ctrl-Shift-P and then type abbreviations to get access to the Lean 4: Show all abbreviations command. If your keyboard does not have an easily accessible backslash, you can change the leading character by changing the lean4.input.leader setting.

When a cursor is in the middle of a tactic proof, Lean reports on the current proof state in the Lean Infoview window. As you move your cursor past each step of the proof, you can see the state change. A typical proof state in Lean might look as follows:

1 goal
x y : ℕ,
h₁ : Prime x,
h₂ : ¬Even x,
h₃ : y > x
⊢ y ≥ 4

The lines before the one that begins with ⊢ denote the context: they are the objects and assumptions currently at play. In this example, these include two objects, x and y, each a natural number. They also include three assumptions, labelled h₁, h₂, and h₃. In Lean, everything in a context is labelled with an identifier. You can type these subscripted labels as h\1, h\2, and h\3, but any legal identifiers would do: you can use h1, h2, h3 instead, or foo, bar, and baz. The last line represents the goal, that is, the fact to be proved. Sometimes people use target for the fact to be proved, and goal for the combination of the context and the target. In practice, the intended meaning is usually clear.

Try proving these identities, in each case replacing sorry by a tactic proof. With the rw tactic, you can use a left arrow (\l) to reverse an identity. For example, rw [← mul_assoc a b c] replaces a * (b * c) by a * b * c in the current goal. Note that the left-pointing arrow refers to going from right to left in the identity provided by mul_assoc, it has nothing to do with the left or right side of the goal.

example (a b c : ℝ) : c * b * a = b * (a * c) := by
  sorry

example (a b c : ℝ) : a * (b * c) = b * (a * c) := by
  sorry

You can also use identities like mul_assoc and mul_comm without arguments. In this case, the rewrite tactic tries to match the left-hand side with an expression in the goal, using the first pattern it finds.

example (a b c : ℝ) : a * b * c = b * c * a := by
  rw [mul_assoc]
  rw [mul_comm]

You can also provide partial information. For example, mul_comm a matches any pattern of the form a * ? and rewrites it to ? * a. Try doing the first of these examples without providing any arguments at all, and the second with only one argument.

example (a b c : ℝ) : a * (b * c) = b * (c * a) := by
  sorry

example (a b c : ℝ) : a * (b * c) = b * (a * c) := by
  sorry

You can also use rw with facts from the local context.

example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
  rw [h']
  rw [← mul_assoc]
  rw [h]
  rw [mul_assoc]

Try these, using the theorem sub_self for the second one:

example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by
  sorry

example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by
  sorry

Multiple rewrite commands can be carried out with a single command, by listing the relevant identities separated by commas inside the square brackets.

example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
  rw [h', ← mul_assoc, h, mul_assoc]

You still see the incremental progress by placing the cursor after a comma in any list of rewrites.

Another trick is that we can declare variables once and for all outside an example or theorem. Lean then includes them automatically.

variable (a b c d e f : ℝ)

example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
  rw [h', ← mul_assoc, h, mul_assoc]

Inspection of the tactic state at the beginning of the above proof reveals that Lean indeed included all variables. We can delimit the scope of the declaration by putting it in a section ... end block. Finally, recall from the introduction that Lean provides us with a command to determine the type of an expression:

section
variable (a b c : ℝ)

#check a
#check a + b
#check (a : ℝ)
#check mul_comm a b
#check (mul_comm a b : a * b = b * a)
#check mul_assoc c a b
#check mul_comm a
#check mul_comm

end

The #check command works for both objects and facts. In response to the command #check a, Lean reports that a has type ℝ. In response to the command #check mul_comm a b, Lean reports that mul_comm a b is a proof of the fact a * b = b * a. The command #check (a : ℝ) states our expectation that the type of a is ℝ, and Lean will raise an error if that is not the case. We will explain the output of the last three #check commands later, but in the meanwhile, you can take a look at them, and experiment with some #check commands of your own.

Let’s try some more examples. The theorem two_mul a says that 2 * a = a + a. The theorems add_mul and mul_add express the distributivity of multiplication over addition, and the theorem add_assoc expresses the associativity of addition. Use the #check command to see the precise statements.

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by
  rw [mul_add, add_mul, add_mul]
  rw [← add_assoc, add_assoc (a * a)]
  rw [mul_comm b a, ← two_mul]

Whereas it is possible to figure out what it going on in this proof by stepping through it in the editor, it is hard to read on its own. Lean provides a more structured way of writing proofs like this using the calc keyword.

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b :=
  calc
    (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by
      rw [mul_add, add_mul, add_mul]
    _ = a * a + (b * a + a * b) + b * b := by
      rw [← add_assoc, add_assoc (a * a)]
    _ = a * a + 2 * (a * b) + b * b := by
      rw [mul_comm b a, ← two_mul]

Notice that the proof does not begin with by: an expression that begins with calc is a proof term. A calc expression can also be used inside a tactic proof, but Lean interprets it as the instruction to use the resulting proof term to solve the goal. The calc syntax is finicky: the underscores and justification have to be in the format indicated above. Lean uses indentation to determine things like where a block of tactics or a calc block begins and ends; try changing the indentation in the proof above to see what happens.

One way to write a calc proof is to outline it first using the sorry tactic for justification, make sure Lean accepts the expression modulo these, and then justify the individual steps using tactics.

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b :=
  calc
    (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by
      sorry
    _ = a * a + (b * a + a * b) + b * b := by
      sorry
    _ = a * a + 2 * (a * b) + b * b := by
      sorry

Try proving the following identity using both a pure rw proof and a more structured calc proof:

example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by
  sorry

The following exercise is a little more challenging. You can use the theorems listed underneath.

example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by
  sorry

#check pow_two a
#check mul_sub a b c
#check add_mul a b c
#check add_sub a b c
#check sub_sub a b c
#check add_zero a

We can also perform rewriting in an assumption in the context. For example, rw [mul_comm a b] at hyp replaces a * b by b * a in the assumption hyp.

example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by
  rw [hyp'] at hyp
  rw [mul_comm d a] at hyp
  rw [← two_mul (a * d)] at hyp
  rw [← mul_assoc 2 a d] at hyp
  exact hyp

In the last step, the exact tactic can use hyp to solve the goal because at that point hyp matches the goal exactly.

We close this section by noting that Mathlib provides a useful bit of automation with a ring tactic, which is designed to prove identities in any commutative ring as long as they follow purely from the ring axioms, without using any local assumption.

example : c * b * a = b * (a * c) := by
  ring

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by
  ring

example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by
  ring

example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by
  rw [hyp, hyp']
  ring

The ring tactic is imported indirectly when we import Mathlib.Data.Real.Basic, but we will see in the next section that it can be used for calculations on structures other than the real numbers. It can be imported explicitly with the command import Mathlib.Tactic. We will see there are similar tactics for other common kind of algebraic structures.

There is a variation of rw called nth_rw that allows you to replace only particular instances of an expression in the goal. Possible matches are enumerated starting with 1, so in the following example, nth_rw 2 [h] replaces the second occurrence of a + b with c.

example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by
  nth_rw 2 [h]
  rw [add_mul]

2.2. Proving Identities in Algebraic Structures

Mathematically, a ring consists of a collection of objects, \(R\), operations \(+\) \(\times\), and constants \(0\) and \(1\), and an operation \(x \mapsto -x\) such that:

  • \(R\) with \(+\) is an abelian group, with \(0\) as the additive identity and negation as inverse.

  • Multiplication is associative with identity \(1\), and multiplication distributes over addition.

In Lean, the collection of objects is represented as a type, R. The ring axioms are as follows:

variable (R : Type*) [Ring R]

#check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c))
#check (add_comm : ∀ a b : R, a + b = b + a)
#check (zero_add : ∀ a : R, 0 + a = a)
#check (neg_add_cancel : ∀ a : R, -a + a = 0)
#check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c))
#check (mul_one : ∀ a : R, a * 1 = a)
#check (one_mul : ∀ a : R, 1 * a = a)
#check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c)
#check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c)

You will learn more about the square brackets in the first line later, but for the time being, suffice it to say that the declaration gives us a type, R, and a ring structure on R. Lean then allows us to use generic ring notation with elements of R, and to make use of a library of theorems about rings.

The names of some of the theorems should look familiar: they are exactly the ones we used to calculate with the real numbers in the last section. Lean is good not only for proving things about concrete mathematical structures like the natural numbers and the integers, but also for proving things about abstract structures, characterized axiomatically, like rings. Moreover, Lean supports generic reasoning about both abstract and concrete structures, and can be trained to recognize appropriate instances. So any theorem about rings can be applied to concrete rings like the integers, ℤ, the rational numbers, ℚ, and the complex numbers ℂ. It can also be applied to any instance of an abstract structure that extends rings, such as any ordered ring or any field.

Not all important properties of the real numbers hold in an arbitrary ring, however. For example, multiplication on the real numbers is commutative, but that does not hold in general. If you have taken a course in linear algebra, you will recognize that, for every \(n\), the \(n\) by \(n\) matrices of real numbers form a ring in which commutativity usually fails. If we declare R to be a commutative ring, in fact, all the theorems in the last section continue to hold when we replace ℝ by R.

variable (R : Type*) [CommRing R]
variable (a b c d : R)

example : c * b * a = b * (a * c) := by ring

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring

example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring

example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by
  rw [hyp, hyp']
  ring

We leave it to you to check that all the other proofs go through unchanged. Notice that when a proof is short, like by ring or by linarith or by sorry, it is common (and permissible) to put it on the same line as the by. Good proof-writing style should strike a balance between concision and readability.

The goal of this section is to strengthen the skills you have developed in the last section and apply them to reasoning axiomatically about rings. We will start with the axioms listed above, and use them to derive other facts. Most of the facts we prove are already in Mathlib. We will give the versions we prove the same names to help you learn the contents of the library as well as the naming conventions.

Lean provides an organizational mechanism similar to those used in programming languages: when a definition or theorem foo is introduced in a namespace bar, its full name is bar.foo. The command open bar later opens the namespace, which allows us to use the shorter name foo. To avoid errors due to name clashes, in the next example we put our versions of the library theorems in a new namespace called MyRing.

The next example shows that we do not need add_zero or add_right_neg as ring axioms, because they follow from the other axioms.

namespace MyRing
variable {R : Type*} [Ring R]

theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add]

theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, neg_add_cancel]

#check MyRing.add_zero
#check add_zero

end MyRing

The net effect is that we can temporarily reprove a theorem in the library, and then go on using the library version after that. But don’t cheat! In the exercises that follow, take care to use only the general facts about rings that we have proved earlier in this section.

(If you are paying careful attention, you may have noticed that we changed the round brackets in (R : Type*) for curly brackets in {R : Type*}. This declares R to be an implicit argument. We will explain what this means in a moment, but don’t worry about it in the meanwhile.)

Here is a useful theorem:

theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by
  rw [← add_assoc, neg_add_cancel, zero_add]

Prove the companion version:

theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by
  sorry

Use these to prove the following:

theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by
  sorry

theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by
  sorry

With enough planning, you can do each of them with three rewrites.

We will now explain the use of the curly braces. Imagine you are in a situation where you have a, b, and c in your context, as well as a hypothesis h : a + b = a + c, and you would like to draw the conclusion b = c. In Lean, you can apply a theorem to hypotheses and facts just the same way that you can apply them to objects, so you might think that add_left_cancel a b c h is a proof of the fact b = c. But notice that explicitly writing a, b, and c is redundant, because the hypothesis h makes it clear that those are the objects we have in mind. In this case, typing a few extra characters is not onerous, but if we wanted to apply add_left_cancel to more complicated expressions, writing them would be tedious. In cases like these, Lean allows us to mark arguments as implicit, meaning that they are supposed to be left out and inferred by other means, such as later arguments and hypotheses. The curly brackets in {a b c : R} do exactly that. So, given the statement of the theorem above, the correct expression is simply add_left_cancel h.

To illustrate, let us show that a * 0 = 0 follows from the ring axioms.

theorem mul_zero (a : R) : a * 0 = 0 := by
  have h : a * 0 + a * 0 = a * 0 + 0 := by
    rw [← mul_add, add_zero, add_zero]
  rw [add_left_cancel h]

We have used a new trick! If you step through the proof, you can see what is going on. The have tactic introduces a new goal, a * 0 + a * 0 = a * 0 + 0, with the same context as the original goal. The fact that the next line is indented indicates that Lean is expecting a block of tactics that serves to prove this new goal. The indentation therefore promotes a modular style of proof: the indented subproof establishes the goal that was introduced by the have. After that, we are back to proving the original goal, except a new hypothesis h has been added: having proved it, we are now free to use it. At this point, the goal is exactly the result of add_left_cancel h.

We could equally well have closed the proof with apply add_left_cancel h or exact add_left_cancel h. The exact tactic takes as argument a proof term which completely proves the current goal, without creating any new goal. The apply tactic is a variant whose argument is not necessarily a complete proof. The missing pieces are either inferred automatically by Lean or become new goals to prove. While the exact tactic is technically redundant since it is strictly less powerful than apply, it makes proof scripts slightly clearer to human readers and easier to maintain when the library evolves.

Remember that multiplication is not assumed to be commutative, so the following theorem also requires some work.

theorem zero_mul (a : R) : 0 * a = 0 := by
  sorry

By now, you should also be able replace each sorry in the next exercise with a proof, still using only facts about rings that we have established in this section.

theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by
  sorry

theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by
  sorry

theorem neg_zero : (-0 : R) = 0 := by
  apply neg_eq_of_add_eq_zero
  rw [add_zero]

theorem neg_neg (a : R) : - -a = a := by
  sorry

We had to use the annotation (-0 : R) instead of 0 in the third theorem because without specifying R it is impossible for Lean to infer which 0 we have in mind, and by default it would be interpreted as a natural number.

In Lean, subtraction in a ring is provably equal to addition of the additive inverse.

example (a b : R) : a - b = a + -b :=
  sub_eq_add_neg a b

On the real numbers, it is defined that way:

example (a b : ℝ) : a - b = a + -b :=
  rfl

example (a b : ℝ) : a - b = a + -b := by
  rfl

The proof term rfl is short for “reflexivity”. Presenting it as a proof of a - b = a + -b forces Lean to unfold the definition and recognize both sides as being the same. The rfl tactic does the same. This is an instance of what is known as a definitional equality in Lean’s underlying logic. This means that not only can one rewrite with sub_eq_add_neg to replace a - b = a + -b, but in some contexts, when dealing with the real numbers, you can use the two sides of the equation interchangeably. For example, you now have enough information to prove the theorem self_sub from the last section:

theorem self_sub (a : R) : a - a = 0 := by
  sorry

Show that you can prove this using rw, but if you replace the arbitrary ring R by the real numbers, you can also prove it using either apply or exact.

Lean knows that 1 + 1 = 2 holds in any ring. With a bit of effort, you can use that to prove the theorem two_mul from the last section:

theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by
  norm_num

theorem two_mul (a : R) : 2 * a = a + a := by
  sorry

We close this section by noting that some of the facts about addition and negation that we established above do not need the full strength of the ring axioms, or even commutativity of addition. The weaker notion of a group can be axiomatized as follows:

variable (A : Type*) [AddGroup A]

#check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c))
#check (zero_add : ∀ a : A, 0 + a = a)
#check (neg_add_cancel : ∀ a : A, -a + a = 0)

It is conventional to use additive notation when the group operation is commutative, and multiplicative notation otherwise. So Lean defines a multiplicative version as well as the additive version (and also their abelian variants, AddCommGroup and CommGroup).

variable {G : Type*} [Group G]

#check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c))
#check (one_mul : ∀ a : G, 1 * a = a)
#check (inv_mul_cancel : ∀ a : G, a⁻¹ * a = 1)

If you are feeling cocky, try proving the following facts about groups, using only these axioms. You will need to prove a number of helper lemmas along the way. The proofs we have carried out in this section provide some hints.

theorem mul_inv_cancel (a : G) : a * a⁻¹ = 1 := by
  sorry

theorem mul_one (a : G) : a * 1 = a := by
  sorry

theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by
  sorry

Explicitly invoking those lemmas is tedious, so Mathlib provides tactics similar to ring in order to cover most uses: group is for non-commutative multiplicative groups, abel for abelian additive groups, and noncomm_ring for non-commutative rings. It may seem odd that the algebraic structures are called Ring and CommRing while the tactics are named noncomm_ring and ring. This is partly for historical reasons, but also for the convenience of using a shorter name for the tactic that deals with commutative rings, since it is used more often.

2.3. Using Theorems and Lemmas

Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality, like the fact that \(a + e^b \le a + e^c\) holds whenever \(b \le c\)? We have already seen that theorems can be applied to arguments and hypotheses, and that the apply and exact tactics can be used to solve goals. In this section, we will make good use of these tools.

Consider the library theorems le_refl and le_trans:

#check (le_refl : ∀ a : ℝ, a ≤ a)
#check (le_trans : a ≤ b → b ≤ c → a ≤ c)

As we explain in more detail in Section 3.1, the implicit parentheses in the statement of le_trans associate to the right, so it should be interpreted as a ≤ b → (b ≤ c → a ≤ c). The library designers have set the arguments a, b and c to le_trans implicit, so that Lean will not let you provide them explicitly (unless you really insist, as we will discuss later). Rather, it expects to infer them from the context in which they are used. For example, when hypotheses h : a ≤ b and h' : b ≤ c are in the context, all the following work:

variable (h : a ≤ b) (h' : b ≤ c)

#check (le_refl : ∀ a : Real, a ≤ a)
#check (le_refl a : a ≤ a)
#check (le_trans : a ≤ b → b ≤ c → a ≤ c)
#check (le_trans h : b ≤ c → a ≤ c)
#check (le_trans h h' : a ≤ c)

The apply tactic takes a proof of a general statement or implication, tries to match the conclusion with the current goal, and leaves the hypotheses, if any, as new goals. If the given proof matches the goal exactly (modulo definitional equality), you can use the exact tactic instead of apply. So, all of these work:

example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by
  apply le_trans
  · apply h₀
  · apply h₁

example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by
  apply le_trans h₀
  apply h₁

example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z :=
  le_trans h₀ h₁

example (x : ℝ) : x ≤ x := by
  apply le_refl

example (x : ℝ) : x ≤ x :=
  le_refl x

In the first example, applying le_trans creates two goals, and we use the dots to indicate where the proof of each begins. The dots are optional, but they serve to focus the goal: within the block introduced by the dot, only one goal is visible, and it must be completed before the end of the block. Here we end the first block by starting a new one with another dot. We could just as well have decreased the indentation. In the fourth example and in the last example, we avoid going into tactic mode entirely: le_trans h₀ h₁ and le_refl x are the proof terms we need.

Here are a few more library theorems:

#check (le_refl : ∀ a, a ≤ a)
#check (le_trans : a ≤ b → b ≤ c → a ≤ c)
#check (lt_of_le_of_lt : a ≤ b → b < c → a < c)
#check (lt_of_lt_of_le : a < b → b ≤ c → a < c)
#check (lt_trans : a < b → b < c → a < c)

Use them together with apply and exact to prove the following:

example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by
  sorry

In fact, Lean has a tactic that does this sort of thing automatically:

example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by
  linarith

The linarith tactic is designed to handle linear arithmetic.

example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by
  linarith

In addition to equations and inequalities in the context, linarith will use additional inequalities that you pass as arguments. In the next example, exp_le_exp.mpr h' is a proof of exp b ≤ exp c, as we will explain in a moment. Notice that, in Lean, we write f x to denote the application of a function f to the argument x, exactly the same way we write h x to denote the result of applying a fact or theorem h to the argument x. Parentheses are only needed for compound arguments, as in f (x + y). Without the parentheses, f x + y would be parsed as (f x) + y.

example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by
  linarith [exp_le_exp.mpr h']

Here are some more theorems in the library that can be used to establish inequalities on the real numbers.

#check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b)
#check (exp_lt_exp : exp a < exp b ↔ a < b)
#check (log_le_log : 0 < a → a ≤ b → log a ≤ log b)
#check (log_lt_log : 0 < a → a < b → log a < log b)
#check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d)
#check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b)
#check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c)
#check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d)
#check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d)
#check (add_lt_add_left : a < b → ∀ c, c + a < c + b)
#check (add_lt_add_right : a < b → ∀ c, a + c < b + c)
#check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b)
#check (add_pos : 0 < a → 0 < b → 0 < a + b)
#check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b)
#check (exp_pos : ∀ a, 0 < exp a)
#check add_le_add_left

Some of the theorems, exp_le_exp, exp_lt_exp use a bi-implication, which represents the phrase “if and only if.” (You can type it in VS Code with \lr of \iff). We will discuss this connective in greater detail in the next chapter. Such a theorem can be used with rw to rewrite a goal to an equivalent one:

example (h : a ≤ b) : exp a ≤ exp b := by
  rw [exp_le_exp]
  exact h

In this section, however, we will use the fact that if h : A ↔ B is such an equivalence, then h.mp establishes the forward direction, A → B, and h.mpr establishes the reverse direction, B → A. Here, mp stands for “modus ponens” and mpr stands for “modus ponens reverse.” You can also use h.1 and h.2 for h.mp and h.mpr, respectively, if you prefer. Thus the following proof works:

example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by
  apply add_lt_add_of_lt_of_le
  · apply add_lt_add_of_le_of_lt h₀
    apply exp_lt_exp.mpr h₁
  apply le_refl

The first line, apply add_lt_add_of_lt_of_le, creates two goals, and once again we use a dot to separate the proof of the first from the proof of the second.

Try the following examples on your own. The example in the middle shows you that the norm_num tactic can be used to solve concrete numeric goals.

example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry

example : (0 : ℝ) < 1 := by norm_num

example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by
  have h₀ : 0 < 1 + exp a := by sorry
  apply log_le_log h₀
  sorry

From these examples, it should be clear that being able to find the library theorems you need constitutes an important part of formalization. There are a number of strategies you can use:

  • You can browse Mathlib in its GitHub repository.

  • You can use the API documentation on the Mathlib web pages.

  • You can rely on Mathlib naming conventions and Ctrl-space completion in the editor to guess a theorem name (or Cmd-space on a Mac keyboard). In Lean, a theorem named A_of_B_of_C establishes something of the form A from hypotheses of the form B and C, where A, B, and C approximate the way we might read the goals out loud. So a theorem establishing something like x + y ≤ ... will probably start with add_le. Typing add_le and hitting Ctrl-space will give you some helpful choices. Note that hitting Ctrl-space twice displays more information about the available completions.

  • If you right-click on an existing theorem name in VS Code, the editor will show a menu with the option to jump to the file where the theorem is defined, and you can find similar theorems nearby.

  • You can use the apply? tactic, which tries to find the relevant theorem in the library.

example : 0 ≤ a ^ 2 := by
  -- apply?
  exact sq_nonneg a

To try out apply? in this example, delete the exact command and uncomment the previous line. Using these tricks, see if you can find what you need to do the next example:

example (h : a ≤ b) : c - exp b ≤ c - exp a := by
  sorry

Using the same tricks, confirm that linarith instead of apply? can also finish the job.

Here is another example of an inequality:

example : 2*a*b ≤ a^2 + b^2 := by
  have h : 0 ≤ a^2 - 2*a*b + b^2
  calc
    a^2 - 2*a*b + b^2 = (a - b)^2 := by ring
    _ ≥ 0 := by apply pow_two_nonneg

  calc
    2*a*b = 2*a*b + 0 := by ring
    _ ≤ 2*a*b + (a^2 - 2*a*b + b^2) := add_le_add (le_refl _) h
    _ = a^2 + b^2 := by ring

Mathlib tends to put spaces around binary operations like * and ^, but in this example, the more compressed format increases readability. There are a number of things worth noticing. First, an expression s ≥ t is definitionally equivalent to t ≤ s. In principle, this means one should be able to use them interchangeably. But some of Lean’s automation does not recognize the equivalence, so Mathlib tends to favor ≤ over ≥. Second, we have used the ring tactic extensively. It is a real timesaver! Finally, notice that in the second line of the second calc proof, instead of writing by exact add_le_add (le_refl _) h, we can simply write the proof term add_le_add (le_refl _) h.

In fact, the only cleverness in the proof above is figuring out the hypothesis h. Once we have it, the second calculation involves only linear arithmetic, and linarith can handle it:

example : 2*a*b ≤ a^2 + b^2 := by
  have h : 0 ≤ a^2 - 2*a*b + b^2
  calc
    a^2 - 2*a*b + b^2 = (a - b)^2 := by ring
    _ ≥ 0 := by apply pow_two_nonneg
  linarith

How nice! We challenge you to use these ideas to prove the following theorem. You can use the theorem abs_le'.mpr. You will also need the constructor tactic to split a conjunction to two goals; see Section 3.4.

example : |a*b| ≤ (a^2 + b^2)/2 := by
  sorry

#check abs_le'.mpr

If you managed to solve this, congratulations! You are well on your way to becoming a master formalizer.

2.4. More examples using apply and rw

The min function on the real numbers is uniquely characterized by the following three facts:

#check (min_le_left a b : min a b ≤ a)
#check (min_le_right a b : min a b ≤ b)
#check (le_min : c ≤ a → c ≤ b → c ≤ min a b)

Can you guess the names of the theorems that characterize max in a similar way?

Notice that we have to apply min to a pair of arguments a and b by writing min a b rather than min (a, b). Formally, min is a function of type ℝ → ℝ → ℝ. When we write a type like this with multiple arrows, the convention is that the implicit parentheses associate to the right, so the type is interpreted as ℝ → (ℝ → ℝ). The net effect is that if a and b have type ℝ then min a has type ℝ → ℝ and min a b has type ℝ, so min acts like a function of two arguments, as we expect. Handling multiple arguments in this way is known as currying, after the logician Haskell Curry.

The order of operations in Lean can also take some getting used to. Function application binds tighter than infix operations, so the expression min a b + c is interpreted as (min a b) + c. With time, these conventions will become second nature.

Using the theorem le_antisymm, we can show that two real numbers are equal if each is less than or equal to the other. Using this and the facts above, we can show that min is commutative:

example : min a b = min b a := by
  apply le_antisymm
  · show min a b ≤ min b a
    apply le_min
    · apply min_le_right
    apply min_le_left
  · show min b a ≤ min a b
    apply le_min
    · apply min_le_right
    apply min_le_left

Here we have used dots to separate proofs of different goals. Our usage is inconsistent: at the outer level, we use dots and indentation for both goals, whereas for the nested proofs, we use dots only until a single goal remains. Both conventions are reasonable and useful. We also use the show tactic to structure the proof and indicate what is being proved in each block. The proof still works without the show commands, but using them makes the proof easier to read and maintain.

It may bother you that the proof is repetitive. To foreshadow skills you will learn later on, we note that one way to avoid the repetition is to state a local lemma and then use it:

example : min a b = min b a := by
  have h : ∀ x y : ℝ, min x y ≤ min y x := by
    intro x y
    apply le_min
    apply min_le_right
    apply min_le_left
  apply le_antisymm
  apply h
  apply h

We will say more about the universal quantifier in Section 3.1, but suffice it to say here that the hypothesis h says that the desired inequality holds for any x and y, and the intro tactic introduces an arbitrary x and y to establish the conclusion. The first apply after le_antisymm implicitly uses h a b, whereas the second one uses h b a.

Another solution is to use the repeat tactic, which applies a tactic (or a block) as many times as it can.

example : min a b = min b a := by
  apply le_antisymm
  repeat
    apply le_min
    apply min_le_right
    apply min_le_left

We encourage you to prove the following as exercises. You can use either of the tricks just described to shorten the first.

example : max a b = max b a := by
  sorry
example : min (min a b) c = min a (min b c) := by
  sorry

Of course, you are welcome to prove the associativity of max as well.

It is an interesting fact that min distributes over max the way that multiplication distributes over addition, and vice-versa. In other words, on the real numbers, we have the identity min a (max b c) = max (min a b) (min a c) as well as the corresponding version with max and min switched. But in the next section we will see that this does not follow from the transitivity and reflexivity of ≤ and the characterizing properties of min and max enumerated above. We need to use the fact that ≤ on the real numbers is a total order, which is to say, it satisfies ∀ x y, x ≤ y ∨ y ≤ x. Here the disjunction symbol, ∨, represents “or”. In the first case, we have min x y = x, and in the second case, we have min x y = y. We will learn how to reason by cases in Section 3.5, but for now we will stick to examples that don’t require the case split.

Here is one such example:

theorem aux : min a b + c ≤ min (a + c) (b + c) := by
  sorry
example : min a b + c = min (a + c) (b + c) := by
  sorry

It is clear that aux provides one of the two inequalities needed to prove the equality, but applying it to suitable values yields the other direction as well. As a hint, you can use the theorem add_neg_cancel_right and the linarith tactic.

Lean’s naming convention is made manifest in the library’s name for the triangle inequality:

#check (abs_add : ∀ a b : ℝ, |a + b| ≤ |a| + |b|)

Use it to prove the following variant, using also add_sub_cancel_right:

example : |a| - |b| ≤ |a - b| :=
  sorry
end

See if you can do this in three lines or less. You can use the theorem sub_add_cancel.

Another important relation that we will make use of in the sections to come is the divisibility relation on the natural numbers, x ∣ y. Be careful: the divisibility symbol is not the ordinary bar on your keyboard. Rather, it is a unicode character obtained by typing \| in VS Code. By convention, Mathlib uses dvd to refer to it in theorem names.

example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z :=
  dvd_trans h₀ h₁

example : x ∣ y * x * z := by
  apply dvd_mul_of_dvd_left
  apply dvd_mul_left

example : x ∣ x ^ 2 := by
  apply dvd_mul_left

In the last example, the exponent is a natural number, and applying dvd_mul_left forces Lean to expand the definition of x^2 to x^1 * x. See if you can guess the names of the theorems you need to prove the following:

example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by
  sorry
end

With respect to divisibility, the greatest common divisor, gcd, and least common multiple, lcm, are analogous to min and max. Since every number divides 0, 0 is really the greatest element with respect to divisibility:

variable (m n : ℕ)

#check (Nat.gcd_zero_right n : Nat.gcd n 0 = n)
#check (Nat.gcd_zero_left n : Nat.gcd 0 n = n)
#check (Nat.lcm_zero_right n : Nat.lcm n 0 = 0)
#check (Nat.lcm_zero_left n : Nat.lcm 0 n = 0)

See if you can guess the names of the theorems you will need to prove the following:

example : Nat.gcd m n = Nat.gcd n m := by
  sorry

Hint: you can use dvd_antisymm, but if you do, Lean will complain that the expression is ambiguous between the generic theorem and the version Nat.dvd_antisymm, the one specifically for the natural numbers. You can use _root_.dvd_antisymm to specify the generic one; either one will work.

2.5. Proving Facts about Algebraic Structures

In Section 2.2, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures, such as commutative rings. We can use any axioms we want to describe an algebraic structure, not just equations. For example, a partial order consists of a set with a binary relation that is reflexive, transitive, and antisymmetric. like ≤ on the real numbers. Lean knows about partial orders:

variable {α : Type*} [PartialOrder α]
variable (x y z : α)

#check x ≤ y
#check (le_refl x : x ≤ x)
#check (le_trans : x ≤ y → y ≤ z → x ≤ z)
#check (le_antisymm : x ≤ y → y ≤ x → x = y)

Here we are adopting the Mathlib convention of using letters like α, β, and γ (entered as \a, \b, and \g) for arbitrary types. The library often uses letters like R and G for the carriers of algebraic structures like rings and groups, respectively, but in general Greek letters are used for types, especially when there is little or no structure associated with them.

Associated to any partial order, ≤, there is also a strict partial order, <, which acts somewhat like < on the real numbers. Saying that x is less than y in this order is equivalent to saying that it is less-than-or-equal to y and not equal to y.

#check x < y
#check (lt_irrefl x : ¬ (x < x))
#check (lt_trans : x < y → y < z → x < z)
#check (lt_of_le_of_lt : x ≤ y → y < z → x < z)
#check (lt_of_lt_of_le : x < y → y ≤ z → x < z)

example : x < y ↔ x ≤ y ∧ x ≠ y :=
  lt_iff_le_and_ne

In this example, the symbol ∧ stands for “and,” the symbol ¬ stands for “not,” and x ≠ y abbreviates ¬ (x = y). In Chapter 3, you will learn how to use these logical connectives to prove that < has the properties indicated.

A lattice is a structure that extends a partial order with operations ⊓ and ⊔ that are analogous to min and max on the real numbers:

variable {α : Type*} [Lattice α]
variable (x y z : α)

#check x ⊓ y
#check (inf_le_left : x ⊓ y ≤ x)
#check (inf_le_right : x ⊓ y ≤ y)
#check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y)
#check x ⊔ y
#check (le_sup_left : x ≤ x ⊔ y)
#check (le_sup_right : y ≤ x ⊔ y)
#check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z)

The characterizations of ⊓ and ⊔ justify calling them the greatest lower bound and least upper bound, respectively. You can type them in VS code using \glb and \lub. The symbols are also often called then infimum and the supremum, and Mathlib refers to them as inf and sup in theorem names. To further complicate matters, they are also often called meet and join. Therefore, if you work with lattices, you have to keep the following dictionary in mind:

  • ⊓ is the greatest lower bound, infimum, or meet.

  • ⊔ is the least upper bound, supremum, or join.

Some instances of lattices include:

  • min and max on any total order, such as the integers or real numbers with ≤

  • ∩ and ∪ on the collection of subsets of some domain, with the ordering ⊆

  • ∧ and ∨ on boolean truth values, with ordering x ≤ y if either x is false or y is true

  • gcd and lcm on the natural numbers (or positive natural numbers), with the divisibility ordering, ∣

  • the collection of linear subspaces of a vector space, where the greatest lower bound is given by the intersection, the least upper bound is given by the sum of the two spaces, and the ordering is inclusion

  • the collection of topologies on a set (or, in Lean, a type), where the greatest lower bound of two topologies consists of the topology that is generated by their union, the least upper bound is their intersection, and the ordering is reverse inclusion

You can check that, as with min / max and gcd / lcm, you can prove the commutativity and associativity of the infimum and supremum using only their characterizing axioms, together with le_refl and le_trans.

Using apply le_trans when seeing a goal x ≤ z is not a great idea. Indeed Lean has no way to guess which intermediate element y we want to use. So apply le_trans produces three goals that look like x ≤ ?a, ?a ≤ z and α where ?a (probably with a more complicated auto-generated name) stands for the mysterious y. The last goal, with type α, is to provide the value of y. It comes lasts because Lean hopes to automatically infer it from the proof of the first goal x ≤ ?a. In order to avoid this unappealing situation, you can use the calc tactic to explicitly provide y. Alternatively, you can use the trans tactic which takes y as an argument and produces the expected goals x ≤ y and y ≤ z. Of course you can also avoid this issue by providing directly a full proof such as exact le_trans inf_le_left inf_le_right, but this requires a lot more planning.

example : x ⊓ y = y ⊓ x := by
  sorry

example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by
  sorry

example : x ⊔ y = y ⊔ x := by
  sorry

example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by
  sorry

You can find these theorems in the Mathlib as inf_comm, inf_assoc, sup_comm, and sup_assoc, respectively.

Another good exercise is to prove the absorption laws using only those axioms:

theorem absorb1 : x ⊓ (x ⊔ y) = x := by
  sorry

theorem absorb2 : x ⊔ x ⊓ y = x := by
  sorry

These can be found in Mathlib with the names inf_sup_self and sup_inf_self.

A lattice that satisfies the additional identities x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z) and x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z) is called a distributive lattice. Lean knows about these too:

variable {α : Type*} [DistribLattice α]
variable (x y z : α)

#check (inf_sup_left x y z : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z)
#check (inf_sup_right x y z : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z)
#check (sup_inf_left x y z : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z))
#check (sup_inf_right x y z : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z))

The left and right versions are easily shown to be equivalent, given the commutativity of ⊓ and ⊔. It is a good exercise to show that not every lattice is distributive by providing an explicit description of a nondistributive lattice with finitely many elements. It is also a good exercise to show that in any lattice, either distributivity law implies the other:

variable {α : Type*} [Lattice α]
variable (a b c : α)

example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by
  sorry

example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by
  sorry

It is possible to combine axiomatic structures into larger ones. For example, a strict ordered ring consists of a ring together with a partial order on the carrier satisfying additional axioms that say that the ring operations are compatible with the order:

variable {R : Type*} [StrictOrderedRing R]
variable (a b c : R)

#check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b)
#check (mul_pos : 0 < a → 0 < b → 0 < a * b)

Chapter 3 will provide the means to derive the following from mul_pos and the definition of <:

#check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b)

It is then an extended exercise to show that many common facts used to reason about arithmetic and the ordering on the real numbers hold generically for any ordered ring. Here are a couple of examples you can try, using only properties of rings, partial orders, and the facts enumerated in the last two examples (beware that those rings are not assumed to be commutative, so the ring tactic is not available):

example (h : a ≤ b) : 0 ≤ b - a := by
  sorry

example (h: 0 ≤ b - a) : a ≤ b := by
  sorry

example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by
  sorry

Finally, here is one last example. A metric space consists of a set equipped with a notion of distance, dist x y, mapping any pair of elements to a real number. The distance function is assumed to satisfy the following axioms:

variable {X : Type*} [MetricSpace X]
variable (x y z : X)

#check (dist_self x : dist x x = 0)
#check (dist_comm x y : dist x y = dist y x)
#check (dist_triangle x y z : dist x z ≤ dist x y + dist y z)

Having mastered this section, you can show that it follows from these axioms that distances are always nonnegative:

example (x y : X) : 0 ≤ dist x y := by
  sorry

We recommend making use of the theorem nonneg_of_mul_nonneg_left. As you may have guessed, this theorem is called dist_nonneg in Mathlib.