Axioms and Computation

We have seen that the version of the Calculus of Constructions that has been implemented in Lean includes dependent function types, inductive types, and a hierarchy of universes that starts with an impredicative, proof-irrelevant Prop at the bottom. In this chapter, we consider ways of extending the CIC with additional axioms and rules. Extending a foundational system in such a way is often convenient; it can make it possible to prove more theorems, as well as make it easier to prove theorems that could have been proved otherwise. But there can be negative consequences of adding additional axioms, consequences which may go beyond concerns about their correctness. In particular, the use of axioms bears on the computational content of definitions and theorems, in ways we will explore here.

Lean is designed to support both computational and classical reasoning. Users that are so inclined can stick to a "computationally pure" fragment, which guarantees that closed expressions in the system evaluate to canonical normal forms. In particular, any closed computationally pure expression of type Nat, for example, will reduce to a numeral.

Lean's standard library defines an additional axiom, propositional extensionality, and a quotient construction which in turn implies the principle of function extensionality. These extensions are used, for example, to develop theories of sets and finite sets. We will see below that using these theorems can block evaluation in Lean's kernel, so that closed terms of type Nat no longer evaluate to numerals. But Lean erases types and propositional information when compiling definitions to bytecode for its virtual machine evaluator, and since these axioms only add new propositions, they are compatible with that computational interpretation. Even computationally inclined users may wish to use the classical law of the excluded middle to reason about computation. This also blocks evaluation in the kernel, but it is compatible with compilation to bytecode.

The standard library also defines a choice principle that is entirely antithetical to a computational interpretation, since it magically produces "data" from a proposition asserting its existence. Its use is essential to some classical constructions, and users can import it when needed. But expressions that use this construction to produce data do not have computational content, and in Lean we are required to mark such definitions as noncomputable to flag that fact.

Using a clever trick (known as Diaconescu's theorem), one can use propositional extensionality, function extensionality, and choice to derive the law of the excluded middle. As noted above, however, use of the law of the excluded middle is still compatible with bytecode compilation and code extraction, as are other classical principles, as long as they are not used to manufacture data.

To summarize, then, on top of the underlying framework of universes, dependent function types, and inductive types, the standard library adds three additional components:

• the axiom of propositional extensionality
• a quotient construction, which implies function extensionality
• a choice principle, which produces data from an existential proposition.

The first two of these block normalization within Lean, but are compatible with bytecode evaluation, whereas the third is not amenable to computational interpretation. We will spell out the details more precisely below.

Historical and Philosophical Context

For most of its history, mathematics was essentially computational: geometry dealt with constructions of geometric objects, algebra was concerned with algorithmic solutions to systems of equations, and analysis provided means to compute the future behavior of systems evolving over time. From the proof of a theorem to the effect that "for every x, there is a y such that ...", it was generally straightforward to extract an algorithm to compute such a y given x.

In the nineteenth century, however, increases in the complexity of mathematical arguments pushed mathematicians to develop new styles of reasoning that suppress algorithmic information and invoke descriptions of mathematical objects that abstract away the details of how those objects are represented. The goal was to obtain a powerful "conceptual" understanding without getting bogged down in computational details, but this had the effect of admitting mathematical theorems that are simply false on a direct computational reading.

There is still fairly uniform agreement today that computation is important to mathematics. But there are different views as to how best to address computational concerns. From a constructive point of view, it is a mistake to separate mathematics from its computational roots; every meaningful mathematical theorem should have a direct computational interpretation. From a classical point of view, it is more fruitful to maintain a separation of concerns: we can use one language and body of methods to write computer programs, while maintaining the freedom to use nonconstructive theories and methods to reason about them. Lean is designed to support both of these approaches. Core parts of the library are developed constructively, but the system also provides support for carrying out classical mathematical reasoning.

Computationally, the purest part of dependent type theory avoids the use of Prop entirely. Inductive types and dependent function types can be viewed as data types, and terms of these types can be "evaluated" by applying reduction rules until no more rules can be applied. In principle, any closed term (that is, term with no free variables) of type Nat should evaluate to a numeral, succ (... (succ zero)...).

Introducing a proof-irrelevant Prop and marking theorems irreducible represents a first step towards separation of concerns. The intention is that elements of a type p : Prop should play no role in computation, and so the particular construction of a term t : p is "irrelevant" in that sense. One can still define computational objects that incorporate elements of type Prop; the point is that these elements can help us reason about the effects of the computation, but can be ignored when we extract "code" from the term. Elements of type Prop are not entirely innocuous, however. They include equations s = t : α for any type α, and such equations can be used as casts, to type check terms. Below, we will see examples of how such casts can block computation in the system. However, computation is still possible under an evaluation scheme that erases propositional content, ignores intermediate typing constraints, and reduces terms until they reach a normal form. This is precisely what Lean's virtual machine does.

Having adopted a proof-irrelevant Prop, one might consider it legitimate to use, for example, the law of the excluded middle, p ∨ ¬p, where p is any proposition. Of course, this, too, can block computation according to the rules of CIC, but it does not block bytecode evaluation, as described above. It is only the choice principles discussed in :numref:choice that completely erase the distinction between the proof-irrelevant and data-relevant parts of the theory.

Propositional Extensionality

Propositional extensionality is the following axiom:

namespace Hidden
axiom propext {a b : Prop} : (a ↔ b) → a = b
end Hidden

It asserts that when two propositions imply one another, they are actually equal. This is consistent with set-theoretic interpretations in which any element a : Prop is either empty or the singleton set {*}, for some distinguished element *. The axiom has the effect that equivalent propositions can be substituted for one another in any context:

theorem thm₁ (a b c d e : Prop) (h : a ↔ b) : (c ∧ a ∧ d → e) ↔ (c ∧ b ∧ d → e) :=
  propext h ▸ Iff.refl _

theorem thm₂ (a b : Prop) (p : Prop → Prop) (h : a ↔ b) (h₁ : p a) : p b :=
  propext h ▸ h₁

Function Extensionality

Similar to propositional extensionality, function extensionality asserts that any two functions of type (x : α) → β x that agree on all their inputs are equal.

universe u v
#check (@funext :
           {α : Type u}
         → {β : α → Type u}
         → {f g : (x : α) → β x}
         → (∀ (x : α), f x = g x)
         → f = g)

#print funext

From a classical, set-theoretic perspective, this is exactly what it means for two functions to be equal. This is known as an "extensional" view of functions. From a constructive perspective, however, it is sometimes more natural to think of functions as algorithms, or computer programs, that are presented in some explicit way. It is certainly the case that two computer programs can compute the same answer for every input despite the fact that they are syntactically quite different. In much the same way, you might want to maintain a view of functions that does not force you to identify two functions that have the same input / output behavior. This is known as an "intensional" view of functions.

In fact, function extensionality follows from the existence of quotients, which we describe in the next section. In the Lean standard library, therefore, funext is thus proved from the quotient construction.

Suppose that for α : Type we define the Set α := α → Prop to denote the type of subsets of α, essentially identifying subsets with predicates. By combining funext and propext, we obtain an extensional theory of such sets:

def Set (α : Type u) := α → Prop

namespace Set

def mem (x : α) (a : Set α) := a x

infix:50 (priority := high) "∈" => mem

theorem setext {a b : Set α} (h : ∀ x, x ∈ a ↔ x ∈ b) : a = b :=
  funext (fun x => propext (h x))

end Set

We can then proceed to define the empty set and set intersection, for example, and prove set identities:

def Set (α : Type u) := α → Prop
namespace Set
def mem (x : α) (a : Set α) := a x
infix:50 (priority := high) "∈" => mem
theorem setext {a b : Set α} (h : ∀ x, x ∈ a ↔ x ∈ b) : a = b :=
  funext (fun x => propext (h x))
def empty : Set α := fun x => False

notation (priority := high) "∅" => empty

def inter (a b : Set α) : Set α :=
  fun x => x ∈ a ∧ x ∈ b

infix:70 " ∩ " => inter

theorem inter_self (a : Set α) : a ∩ a = a :=
  setext fun x => Iff.intro
    (fun ⟨h, _⟩ => h)
    (fun h => ⟨h, h⟩)

theorem inter_empty (a : Set α) : a ∩ ∅ = ∅ :=
  setext fun x => Iff.intro
    (fun ⟨_, h⟩ => h)
    (fun h => False.elim h)

theorem empty_inter (a : Set α) : ∅ ∩ a = ∅ :=
  setext fun x => Iff.intro
    (fun ⟨h, _⟩ => h)
    (fun h => False.elim h)

theorem inter.comm (a b : Set α) : a ∩ b = b ∩ a :=
  setext fun x => Iff.intro
    (fun ⟨h₁, h₂⟩ => ⟨h₂, h₁⟩)
    (fun ⟨h₁, h₂⟩ => ⟨h₂, h₁⟩)
end Set

The following is an example of how function extensionality blocks computation inside the Lean kernel.

def f (x : Nat) := x
def g (x : Nat) := 0 + x

theorem f_eq_g : f = g :=
  funext fun x => (Nat.zero_add x).symm

def val : Nat :=
  Eq.recOn (motive := fun _ _ => Nat) f_eq_g 0

-- does not reduce to 0
#reduce val

-- evaluates to 0
#eval val

First, we show that the two functions f and g are equal using function extensionality, and then we cast 0 of type Nat by replacing f by g in the type. Of course, the cast is vacuous, because Nat does not depend on f. But that is enough to do the damage: under the computational rules of the system, we now have a closed term of Nat that does not reduce to a numeral. In this case, we may be tempted to reduce the expression to 0. But in nontrivial examples, eliminating cast changes the type of the term, which might make an ambient expression type incorrect. The virtual machine, however, has no trouble evaluating the expression to 0. Here is a similarly contrived example that shows how propext can get in the way.

theorem tteq : (True ∧ True) = True :=
  propext (Iff.intro (fun ⟨h, _⟩ => h) (fun h => ⟨h, h⟩))

def val : Nat :=
  Eq.recOn (motive := fun _ _ => Nat) tteq 0

-- does not reduce to 0
#reduce val

-- evaluates to 0
#eval val

Current research programs, including work on observational type theory and cubical type theory, aim to extend type theory in ways that permit reductions for casts involving function extensionality, quotients, and more. But the solutions are not so clear cut, and the rules of Lean's underlying calculus do not sanction such reductions.

In a sense, however, a cast does not change the meaning of an expression. Rather, it is a mechanism to reason about the expression's type. Given an appropriate semantics, it then makes sense to reduce terms in ways that preserve their meaning, ignoring the intermediate bookkeeping needed to make the reductions type correct. In that case, adding new axioms in Prop does not matter; by proof irrelevance, an expression in Prop carries no information, and can be safely ignored by the reduction procedures.

Quotients

Let α be any type, and let r be an equivalence relation on α. It is mathematically common to form the "quotient" α / r, that is, the type of elements of α "modulo" r. Set theoretically, one can view α / r as the set of equivalence classes of α modulo r. If f : α → β is any function that respects the equivalence relation in the sense that for every x y : α, r x y implies f x = f y, then f "lifts" to a function f' : α / r → β defined on each equivalence class ⟦x⟧ by f' ⟦x⟧ = f x. Lean's standard library extends the Calculus of Constructions with additional constants that perform exactly these constructions, and installs this last equation as a definitional reduction rule.

In its most basic form, the quotient construction does not even require r to be an equivalence relation. The following constants are built into Lean:

namespace Hidden
universe u v

axiom Quot : {α : Sort u} → (α → α → Prop) → Sort u

axiom Quot.mk : {α : Sort u} → (r : α → α → Prop) → α → Quot r

axiom Quot.ind :
    ∀ {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop},
      (∀ a, β (Quot.mk r a)) → (q : Quot r) → β q

axiom Quot.lift :
    {α : Sort u} → {r : α → α → Prop} → {β : Sort u} → (f : α → β)
    → (∀ a b, r a b → f a = f b) → Quot r → β
end Hidden

The first one forms a type Quot r given a type α by any binary relation r on α. The second maps α to Quot α, so that if r : α → α → Prop and a : α, then Quot.mk r a is an element of Quot r. The third principle, Quot.ind, says that every element of Quot.mk r a is of this form. As for Quot.lift, given a function f : α → β, if h is a proof that f respects the relation r, then Quot.lift f h is the corresponding function on Quot r. The idea is that for each element a in α, the function Quot.lift f h maps Quot.mk r a (the r-class containing a) to f a, wherein h shows that this function is well defined. In fact, the computation principle is declared as a reduction rule, as the proof below makes clear.

def mod7Rel (x y : Nat) : Prop :=
  x % 7 = y % 7

-- the quotient type
#check (Quot mod7Rel : Type)

-- the class of a
#check (Quot.mk mod7Rel 4 : Quot mod7Rel)

def f (x : Nat) : Bool :=
  x % 7 = 0

theorem f_respects (a b : Nat) (h : mod7Rel a b) : f a = f b := by
  simp [mod7Rel, f] at *
  rw [h]

#check (Quot.lift f f_respects : Quot mod7Rel → Bool)

-- the computation principle
example (a : Nat) : Quot.lift f f_respects (Quot.mk mod7Rel a) = f a :=
  rfl

The four constants, Quot, Quot.mk, Quot.ind, and Quot.lift in and of themselves are not very strong. You can check that the Quot.ind is satisfied if we take Quot r to be simply α, and take Quot.lift to be the identity function (ignoring h). For that reason, these four constants are not viewed as additional axioms.

They are, like inductively defined types and the associated constructors and recursors, viewed as part of the logical framework.

What makes the Quot construction into a bona fide quotient is the following additional axiom:

namespace Hidden
universe u v
axiom Quot.sound :
      ∀ {α : Type u} {r : α → α → Prop} {a b : α},
        r a b → Quot.mk r a = Quot.mk r b
end Hidden

This is the axiom that asserts that any two elements of α that are related by r become identified in the quotient. If a theorem or definition makes use of Quot.sound, it will show up in the #print axioms command.

Of course, the quotient construction is most commonly used in situations when r is an equivalence relation. Given r as above, if we define r' according to the rule r' a b iff Quot.mk r a = Quot.mk r b, then it's clear that r' is an equivalence relation. Indeed, r' is the kernel of the function a ↦ quot.mk r a. The axiom Quot.sound says that r a b implies r' a b. Using Quot.lift and Quot.ind, we can show that r' is the smallest equivalence relation containing r, in the sense that if r'' is any equivalence relation containing r, then r' a b implies r'' a b. In particular, if r was an equivalence relation to start with, then for all a and b we have r a b iff r' a b.

To support this common use case, the standard library defines the notion of a setoid, which is simply a type with an associated equivalence relation:

namespace Hidden
class Setoid (α : Sort u) where
  r : α → α → Prop
  iseqv : Equivalence r

instance {α : Sort u} [Setoid α] : HasEquiv α :=
  ⟨Setoid.r⟩

namespace Setoid

variable {α : Sort u} [Setoid α]

theorem refl (a : α) : a ≈ a :=
  iseqv.refl a

theorem symm {a b : α} (hab : a ≈ b) : b ≈ a :=
  iseqv.symm hab

theorem trans {a b c : α} (hab : a ≈ b) (hbc : b ≈ c) : a ≈ c :=
  iseqv.trans hab hbc

end Setoid
end Hidden

Given a type α, a relation r on α, and a proof p that r is an equivalence relation, we can define Setoid.mk r p as an instance of the setoid class.

namespace Hidden
def Quotient {α : Sort u} (s : Setoid α) :=
  @Quot α Setoid.r
end Hidden

The constants Quotient.mk, Quotient.ind, Quotient.lift, and Quotient.sound are nothing more than the specializations of the corresponding elements of Quot. The fact that type class inference can find the setoid associated to a type α brings a number of benefits. First, we can use the notation a ≈ b (entered with \approx) for Setoid.r a b, where the instance of Setoid is implicit in the notation Setoid.r. We can use the generic theorems Setoid.refl, Setoid.symm, Setoid.trans to reason about the relation. Specifically with quotients we can use the generic notation ⟦a⟧ for Quot.mk Setoid.r where the instance of Setoid is implicit in the notation Setoid.r, as well as the theorem Quotient.exact:

universe u
#check (@Quotient.exact :
         ∀ {α : Sort u} {s : Setoid α} {a b : α},
           Quotient.mk s a = Quotient.mk s b → a ≈ b)

Together with Quotient.sound, this implies that the elements of the quotient correspond exactly to the equivalence classes of elements in α.

Recall that in the standard library, α × β represents the Cartesian product of the types α and β. To illustrate the use of quotients, let us define the type of unordered pairs of elements of a type α as a quotient of the type α × α. First, we define the relevant equivalence relation:

private def eqv (p₁ p₂ : α × α) : Prop :=
  (p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)

infix:50 " ~ " => eqv

The next step is to prove that eqv is in fact an equivalence relation, which is to say, it is reflexive, symmetric and transitive. We can prove these three facts in a convenient and readable way by using dependent pattern matching to perform case-analysis and break the hypotheses into pieces that are then reassembled to produce the conclusion.

private def eqv (p₁ p₂ : α × α) : Prop :=
  (p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
  Or.inl ⟨rfl, rfl⟩

private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
  | (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
    Or.inr (by simp_all)

private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inl (by simp_all)

private theorem is_equivalence : Equivalence (@eqv α) :=
  { refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }

Now that we have proved that eqv is an equivalence relation, we can construct a Setoid (α × α), and use it to define the type UProd α of unordered pairs.

private def eqv (p₁ p₂ : α × α) : Prop :=
  (p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
  Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
  | (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
    Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
  { refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
instance uprodSetoid (α : Type u) : Setoid (α × α) where
  r     := eqv
  iseqv := is_equivalence

def UProd (α : Type u) : Type u :=
  Quotient (uprodSetoid α)

namespace UProd

def mk {α : Type} (a₁ a₂ : α) : UProd α :=
  Quotient.mk' (a₁, a₂)

notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂

end UProd

Notice that we locally define the notation {a₁, a₂} for unordered pairs as Quotient.mk (a₁, a₂). This is useful for illustrative purposes, but it is not a good idea in general, since the notation will shadow other uses of curly brackets, such as for records and sets.

We can easily prove that {a₁, a₂} = {a₂, a₁} using Quot.sound, since we have (a₁, a₂) ~ (a₂, a₁).

private def eqv (p₁ p₂ : α × α) : Prop :=
  (p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
  Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
  | (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
    Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
  { refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
instance uprodSetoid (α : Type u) : Setoid (α × α) where
  r     := eqv
  iseqv := is_equivalence
def UProd (α : Type u) : Type u :=
  Quotient (uprodSetoid α)
namespace UProd
def mk {α : Type} (a₁ a₂ : α) : UProd α :=
  Quotient.mk' (a₁, a₂)
notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂
theorem mk_eq_mk (a₁ a₂ : α) : {a₁, a₂} = {a₂, a₁} :=
  Quot.sound (Or.inr ⟨rfl, rfl⟩)
end UProd

To complete the example, given a : α and u : uprod α, we define the proposition a ∈ u which should hold if a is one of the elements of the unordered pair u. First, we define a similar proposition mem_fn a u on (ordered) pairs; then we show that mem_fn respects the equivalence relation eqv with the lemma mem_respects. This is an idiom that is used extensively in the Lean standard library.

private def eqv (p₁ p₂ : α × α) : Prop :=
  (p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
  Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
  | (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
    Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inl (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
    Or.inr (by simp_all)
  | (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
    Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
  { refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
instance uprodSetoid (α : Type u) : Setoid (α × α) where
  r     := eqv
  iseqv := is_equivalence
def UProd (α : Type u) : Type u :=
  Quotient (uprodSetoid α)
namespace UProd
def mk {α : Type} (a₁ a₂ : α) : UProd α :=
  Quotient.mk' (a₁, a₂)
notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂
theorem mk_eq_mk (a₁ a₂ : α) : {a₁, a₂} = {a₂, a₁} :=
  Quot.sound (Or.inr ⟨rfl, rfl⟩)
private def mem_fn (a : α) : α × α → Prop
  | (a₁, a₂) => a = a₁ ∨ a = a₂

-- auxiliary lemma for proving mem_respects
private theorem mem_swap {a : α} :
      ∀ {p : α × α}, mem_fn a p = mem_fn a (⟨p.2, p.1⟩)
  | (a₁, a₂) => by
    apply propext
    apply Iff.intro
    . intro
      | Or.inl h => exact Or.inr h
      | Or.inr h => exact Or.inl h
    . intro
      | Or.inl h => exact Or.inr h
      | Or.inr h => exact Or.inl h


private theorem mem_respects
      : {p₁ p₂ : α × α} → (a : α) → p₁ ~ p₂ → mem_fn a p₁ = mem_fn a p₂
  | (a₁, a₂), (b₁, b₂), a, Or.inl ⟨a₁b₁, a₂b₂⟩ => by simp_all
  | (a₁, a₂), (b₁, b₂), a, Or.inr ⟨a₁b₂, a₂b₁⟩ => by simp_all; apply mem_swap

def mem (a : α) (u : UProd α) : Prop :=
  Quot.liftOn u (fun p => mem_fn a p) (fun p₁ p₂ e => mem_respects a e)

infix:50 (priority := high) " ∈ " => mem

theorem mem_mk_left (a b : α) : a ∈ {a, b} :=
  Or.inl rfl

theorem mem_mk_right (a b : α) : b ∈ {a, b} :=
  Or.inr rfl

theorem mem_or_mem_of_mem_mk {a b c : α} : c ∈ {a, b} → c = a ∨ c = b :=
  fun h => h
end UProd

For convenience, the standard library also defines Quotient.lift₂ for lifting binary functions, and Quotient.ind₂ for induction on two variables.

We close this section with some hints as to why the quotient construction implies function extensionality. It is not hard to show that extensional equality on the (x : α) → β x is an equivalence relation, and so we can consider the type extfun α β of functions "up to equivalence." Of course, application respects that equivalence in the sense that if f₁ is equivalent to f₂, then f₁ a is equal to f₂ a. Thus application gives rise to a function extfun_app : extfun α β → (x : α) → β x. But for every f, extfun_app ⟦f⟧ is definitionally equal to fun x => f x, which is in turn definitionally equal to f. So, when f₁ and f₂ are extensionally equal, we have the following chain of equalities:

    f₁ = extfun_app ⟦f₁⟧ = extfun_app ⟦f₂⟧ = f₂

As a result, f₁ is equal to f₂.

Choice

To state the final axiom defined in the standard library, we need the Nonempty type, which is defined as follows:

namespace Hidden
class inductive Nonempty (α : Sort u) : Prop where
  | intro (val : α) : Nonempty α
end Hidden

Because Nonempty α has type Prop and its constructor contains data, it can only eliminate to Prop. In fact, Nonempty α is equivalent to ∃ x : α, True:

example (α : Type u) : Nonempty α ↔ ∃ x : α, True :=
  Iff.intro (fun ⟨a⟩ => ⟨a, trivial⟩) (fun ⟨a, h⟩ => ⟨a⟩)

Our axiom of choice is now expressed simply as follows:

namespace Hidden
universe u
axiom choice {α : Sort u} : Nonempty α → α
end Hidden

Given only the assertion h that α is nonempty, choice h magically produces an element of α. Of course, this blocks any meaningful computation: by the interpretation of Prop, h contains no information at all as to how to find such an element.

This is found in the Classical namespace, so the full name of the theorem is Classical.choice. The choice principle is equivalent to the principle of indefinite description, which can be expressed with subtypes as follows:

namespace Hidden
universe u
axiom choice {α : Sort u} : Nonempty α → α
noncomputable def indefiniteDescription {α : Sort u} (p : α → Prop)
                                        (h : ∃ x, p x) : {x // p x} :=
  choice <| let ⟨x, px⟩ := h; ⟨⟨x, px⟩⟩
end Hidden

Because it depends on choice, Lean cannot generate bytecode for indefiniteDescription, and so requires us to mark the definition as noncomputable. Also in the Classical namespace, the function choose and the property choose_spec decompose the two parts of the output of indefiniteDescription:

open Classical
namespace Hidden
noncomputable def choose {α : Sort u} {p : α → Prop} (h : ∃ x, p x) : α :=
  (indefiniteDescription p h).val

theorem choose_spec {α : Sort u} {p : α → Prop} (h : ∃ x, p x) : p (choose h) :=
  (indefiniteDescription p h).property
end Hidden

The choice principle also erases the distinction between the property of being Nonempty and the more constructive property of being Inhabited:

open Classical
theorem inhabited_of_nonempty : Nonempty α → Inhabited α :=
  fun h => choice (let ⟨a⟩ := h; ⟨⟨a⟩⟩)

In the next section, we will see that propext, funext, and choice, taken together, imply the law of the excluded middle and the decidability of all propositions. Using those, one can strengthen the principle of indefinite description as follows:

open Classical
universe u
#check (@strongIndefiniteDescription :
         {α : Sort u} → (p : α → Prop)
         → Nonempty α → {x // (∃ (y : α), p y) → p x})

Assuming the ambient type α is nonempty, strongIndefiniteDescription p produces an element of α satisfying p if there is one. The data component of this definition is conventionally known as Hilbert's epsilon function:

open Classical
universe u
#check (@epsilon :
         {α : Sort u} → [Nonempty α]
         → (α → Prop) → α)

#check (@epsilon_spec :
         ∀ {α : Sort u} {p : α → Prop} (hex : ∃ (y : α), p y),
           p (@epsilon _ (nonempty_of_exists hex) p))

The Law of the Excluded Middle

The law of the excluded middle is the following

open Classical

#check (@em : ∀ (p : Prop), p ∨ ¬p)

Diaconescu's theorem states that the axiom of choice is sufficient to derive the law of excluded middle. More precisely, it shows that the law of the excluded middle follows from Classical.choice, propext, and funext. We sketch the proof that is found in the standard library.

First, we import the necessary axioms, and define two predicates U and V:

namespace Hidden
open Classical
theorem em (p : Prop) : p ∨ ¬p :=
  let U (x : Prop) : Prop := x = True ∨ p
  let V (x : Prop) : Prop := x = False ∨ p

  have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
  have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
  sorry
end Hidden

If p is true, then every element of Prop is in both U and V. If p is false, then U is the singleton true, and V is the singleton false.

Next, we use some to choose an element from each of U and V:

namespace Hidden
open Classical
theorem em (p : Prop) : p ∨ ¬p :=
  let U (x : Prop) : Prop := x = True ∨ p
  let V (x : Prop) : Prop := x = False ∨ p
  have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
  have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
  let u : Prop := choose exU
  let v : Prop := choose exV

  have u_def : U u := choose_spec exU
  have v_def : V v := choose_spec exV
  sorry
end Hidden

Each of U and V is a disjunction, so u_def and v_def represent four cases. In one of these cases, u = True and v = False, and in all the other cases, p is true. Thus we have:

namespace Hidden
open Classical
theorem em (p : Prop) : p ∨ ¬p :=
  let U (x : Prop) : Prop := x = True ∨ p
  let V (x : Prop) : Prop := x = False ∨ p
  have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
  have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
  let u : Prop := choose exU
  let v : Prop := choose exV
  have u_def : U u := choose_spec exU
  have v_def : V v := choose_spec exV
  have not_uv_or_p : u ≠ v ∨ p :=
    match u_def, v_def with
    | Or.inr h, _ => Or.inr h
    | _, Or.inr h => Or.inr h
    | Or.inl hut, Or.inl hvf =>
      have hne : u ≠ v := by simp [hvf, hut, true_ne_false]
      Or.inl hne
  sorry
end Hidden

On the other hand, if p is true, then, by function extensionality and propositional extensionality, U and V are equal. By the definition of u and v, this implies that they are equal as well.

namespace Hidden
open Classical
theorem em (p : Prop) : p ∨ ¬p :=
  let U (x : Prop) : Prop := x = True ∨ p
  let V (x : Prop) : Prop := x = False ∨ p
  have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
  have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
  let u : Prop := choose exU
  let v : Prop := choose exV
  have u_def : U u := choose_spec exU
  have v_def : V v := choose_spec exV
  have not_uv_or_p : u ≠ v ∨ p :=
    match u_def, v_def with
    | Or.inr h, _ => Or.inr h
    | _, Or.inr h => Or.inr h
    | Or.inl hut, Or.inl hvf =>
      have hne : u ≠ v := by simp [hvf, hut, true_ne_false]
      Or.inl hne
  have p_implies_uv : p → u = v :=
    fun hp =>
    have hpred : U = V :=
      funext fun x =>
        have hl : (x = True ∨ p) → (x = False ∨ p) :=
          fun _ => Or.inr hp
        have hr : (x = False ∨ p) → (x = True ∨ p) :=
          fun _ => Or.inr hp
        show (x = True ∨ p) = (x = False ∨ p) from
          propext (Iff.intro hl hr)
    have h₀ : ∀ exU exV, @choose _ U exU = @choose _ V exV := by
      rw [hpred]; intros; rfl
    show u = v from h₀ _ _
  sorry
end Hidden

Putting these last two facts together yields the desired conclusion:

namespace Hidden
open Classical
theorem em (p : Prop) : p ∨ ¬p :=
  let U (x : Prop) : Prop := x = True ∨ p
  let V (x : Prop) : Prop := x = False ∨ p
  have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
  have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
  let u : Prop := choose exU
  let v : Prop := choose exV
  have u_def : U u := choose_spec exU
  have v_def : V v := choose_spec exV
  have not_uv_or_p : u ≠ v ∨ p :=
    match u_def, v_def with
    | Or.inr h, _ => Or.inr h
    | _, Or.inr h => Or.inr h
    | Or.inl hut, Or.inl hvf =>
      have hne : u ≠ v := by simp [hvf, hut, true_ne_false]
      Or.inl hne
  have p_implies_uv : p → u = v :=
    fun hp =>
    have hpred : U = V :=
      funext fun x =>
        have hl : (x = True ∨ p) → (x = False ∨ p) :=
          fun _ => Or.inr hp
        have hr : (x = False ∨ p) → (x = True ∨ p) :=
          fun _ => Or.inr hp
        show (x = True ∨ p) = (x = False ∨ p) from
          propext (Iff.intro hl hr)
    have h₀ : ∀ exU exV, @choose _ U exU = @choose _ V exV := by
      rw [hpred]; intros; rfl
    show u = v from h₀ _ _
  match not_uv_or_p with
  | Or.inl hne => Or.inr (mt p_implies_uv hne)
  | Or.inr h   => Or.inl h
end Hidden

Consequences of excluded middle include double-negation elimination, proof by cases, and proof by contradiction, all of which are described in the Section Classical Logic. The law of the excluded middle and propositional extensionality imply propositional completeness:

namespace Hidden
open Classical
theorem propComplete (a : Prop) : a = True ∨ a = False :=
  match em a with
  | Or.inl ha => Or.inl (propext (Iff.intro (fun _ => ⟨⟩) (fun _ => ha)))
  | Or.inr hn => Or.inr (propext (Iff.intro (fun h => hn h) (fun h => False.elim h)))
end Hidden

Together with choice, we also get the stronger principle that every proposition is decidable. Recall that the class of Decidable propositions is defined as follows:

namespace Hidden
class inductive Decidable (p : Prop) where
  | isFalse (h : ¬p) : Decidable p
  | isTrue  (h : p)  : Decidable p
end Hidden

In contrast to p ∨ ¬ p, which can only eliminate to Prop, the type Decidable p is equivalent to the sum type Sum p (¬ p), which can eliminate to any type. It is this data that is needed to write an if-then-else expression.

As an example of classical reasoning, we use choose to show that if f : α → β is injective and α is inhabited, then f has a left inverse. To define the left inverse linv, we use a dependent if-then-else expression. Recall that if h : c then t else e is notation for dite c (fun h : c => t) (fun h : ¬ c => e). In the definition of linv, choice is used twice: first, to show that (∃ a : A, f a = b) is "decidable," and then to choose an a such that f a = b. Notice that propDecidable is a scoped instance and is activated by the open Classical command. We use this instance to justify the if-then-else expression. (See also the discussion in Section Decidable Propositions).

open Classical

noncomputable def linv [Inhabited α] (f : α → β) : β → α :=
  fun b : β => if ex : (∃ a : α, f a = b) then choose ex else default

theorem linv_comp_self {f : α → β} [Inhabited α]
                       (inj : ∀ {a b}, f a = f b → a = b)
                       : linv f ∘ f = id :=
  funext fun a =>
    have ex  : ∃ a₁ : α, f a₁ = f a := ⟨a, rfl⟩
    have feq : f (choose ex) = f a  := choose_spec ex
    calc linv f (f a)
      _ = choose ex := dif_pos ex
      _ = a         := inj feq

From a classical point of view, linv is a function. From a constructive point of view, it is unacceptable; because there is no way to implement such a function in general, the construction is not informative.