On Characterizing Nat in Agda

Oct 6, 2020

"If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither ‘number’ nor ‘size,’ but always form." - Grothendieck

The 19th century brought about an unprecedented revolution in the foundations of mathematics culminating in the Zermelo–Frankel (ZF) axioms. The ZF axioms gave a logical basis to Cantor’s set theory as a foundational theory, in which the main branches of mathematics (analysis, topology, geometry and number theory) could be encoded and thusly understood. It was through this newly-systematized approach that the axiomatic method was born. This is of historical and philosophical interest as it marks a decisive structuralist turn in mathematical thought. It is at this time that mathematical objects begin to be characterized not by any particular construction, but by the collection of axioms (or laws) an object satisfies. \(\def\N{\mathsf{N}}\)

Consider a paradigmatic case: characterizing the natural numbers. In this post we will explore two different formulations of the natural numbers, the first of which is given by Peano in the late 19th century, and the second of which is a more categorical approach inspired by Lawvere. This will provide us a springboard upon which to explore how we can formulate and reason about mathematical structures in Agda. Since Agda is based on dependent type theory, our formulations will be constructive. This means our constructions will be proof relevant. In this setting, proofs are first-class entities and as such our algebraic structures will encode both the operations a structure has along with proofs of the properties that it satisfies.

We will assume the reader has some familiarity with Agda (with relevant materials noted throughout) and has an interest in formalizing mathematics.

Let us begin with Peano’s axioms. A set \(\N\) satisfies Peano’s axioms if the following properties hold:

How can we prove that this uniquely determines the natural numbers? Our strategy would go roughly as follows:

Let’s try to formalise this constructively in Agda. We start with a few imports we will need later:

module Peano where

open import Relation.Binary.PropositionalEquality
  using (__; refl; __; cong; trans; sym)
open import Function
  using (__)

Typically, we define the (unary) natural numbers as follows:

data: Set where
  Zero :
  Succ :->

-- This allows us to use numeric literals.

Our first port of call is to formulate equivalence relations. In Agda we usually encode algebraic structures as records. As mentioned in the introduction, since proofs are first-class, we carry around not just the structure of the binary relation but also the properties it satisfies:

record EqRel (A : Set) : Set₁ where
    __ : A  A  Set
    reflexivity  :  {a : A}      a ≃ a
    symmetry     :  {a b : A}    a ≃ b  b ≃ a
    transitivity :  {a b c : A}  a ≃ b  b ≃ c  a ≃ c

We note that this definition is slightly different from what is typical in mathematics. Rather than a subset of the diagonal, we make use of a two-argument dependent type \(\_ \simeq \_\). Given \(\mathrm{a}, \mathrm{b} : \mathrm{A}\), the type \(\mathrm{a}\;\simeq\;\mathrm{b}\) gives the collection of evidence that \(\mathrm{a}\) and \(\mathrm{b}\) are equal. The axioms this satisfies are reflexivity (that any element is equal to itself) symmetry (that we can freely reverse equalities) and transitivity (that we can compose equalities).

Let’s show that Agda’s built-in equality type, \(\equiv\), is an equivalence relation on \(\mathbb{N}\). As a brief reminder, here is how the equality type is defined, if we ignore level polymorphism:

data __ {A : Set} (x : A) : A  Set where
  refl : x ≡ x

That is to say, we can give a term \(\mathrm{refl}\) of type \(\mathrm{a} \equiv \mathrm{b}\) so long as Agda can directly compute that \(\mathrm{a}\) and \(\mathrm{b}\) are equal within the particular context. For example, if we define addition:

_+_ :->->
_+_ Zero     m = m
_+_ (Succ n) m = Succ (n + m)

then we can give the following (unnamed) definition:

_ : 2 + 24
_ = refl

as Agda can compute that both sides are equal.

Coming back to \(\mathbb{N}\), let’s show how \(\equiv\) satisfies the properties of an equivalence relation:

ℕ-refl :  {n :}  n ≡ n
ℕ-refl = refl

ℕ-symm :  {n m :}  n ≡ m  m ≡ n
ℕ-symm n≡m rewrite n≡m = refl

ℕ-trans :  {m n r :}  m ≡ n  n ≡ r  m ≡ r
ℕ-trans  m≡n n≡r rewrite m≡n | n≡r = refl

Here we make use of Agda’s rewrite construction. By providing an equality proof of the form \(\mathrm{a} \equiv \mathrm{b}\), the rewrite construction will replace subexpressions in the goal of the form \(\mathrm{a}\) with \(\mathrm{b}\). For example, in \(\mathbb{N}\mathrm{-symm}\), we use the equality term we are given, \(\mathrm{n} \equiv \mathrm{m}\), as an argument to rewrite so that each appearance of \(\mathrm{n}\) is replaced with \(\mathrm{m}\). At this point we may fill the hole with \(\mathrm{refl} : \mathrm{m} \equiv \mathrm{m}\).

It is worth noting that we haven’t used anything special about \(\mathbb{N}\) and these same definitions would work to prove that any set forms an equivalence relation under \(\equiv\).

Now we can write an instance of EqRel for \(\mathbb{N}\):

open EqRel {{...}} public

  ≡-Nat : EqRel ℕ
  ≡-Nat =
    { __          = __
    ; reflexivity  = ℕ-refl
    ; symmetry     = ℕ-symm
    ; transitivity = ℕ-trans

Here we use Agda’s instance arguments mechanism, the analog to Haskell’s typeclass instances. We start by bringing the fields of EqRel into scope for those instances which can be resolved. This is essentially equivalent to us defining top-level functions of the form:

__         :  {A : Set} {{_ : EqRel A}}  A  A  Set
reflexivity :  {A : Set} {{_ : EqRel A}} {a b : A}  (a ≃ a)
-- etc.

Implicit arguments \(\{\{\_ : \mathrm{EqRel}\; \mathrm{A}\}\}\) are resolved by searching for instances we have available in scope. In particular, we define an instance of EqRel for \(\mathbb{N}\) which means that we may use these methods on \(\mathbb{N}\) and Agda will infer the instance we have provided.

Now we are in a position to formalise the Peano axioms. In much the same way as we have done with equivalence relations, we again use records to encode algebraic structure:

record Peano (N : Set) {{rel : EqRel N}} : Set₁ where
    zero        : N
    succ        : N  N
    s-injective :  {a b : N}  (succ a)(succ b)  a ≃ b
    zero≠succ   :  (a : N)  zero ≃ (succ a) 
    induction   :
        (P : N  Set) (a : N)
        (z : P zero)
        (s :  {b : N}  P b  P (succ b))
        P a
    induction-zero :
        (P : N  Set)
        (z : P zero)
        (s :  {b : N}  P b  P (succ b))
        (induction P zero z s ≡ z)
    induction-succ :
        (P : N  Set) (a : N)
        (z : (P zero))
        (s :  {b : N}  P b  P (succ b))
        (induction P (succ a) z s ≡ s (induction P a z s))

Several things are worth noting about this defintion:

    class Eq a => Ord a

Let’s now prove that \(\mathbb{N}\) satisfies induction and injectivity of \(\mathrm{Succ}\):

ℕ-induction :
   (P : Set) (a :)
   (P Zero)
   (∀ {b :}  (P b)  (P (Succ b)))
   (P a)
ℕ-induction P Zero p[zero] p[succ] = p[zero]
ℕ-induction P (Succ n) p[zero] p[succ]
  = p[succ] (ℕ-induction P n p[zero] p[succ])

Succ-inj :  {n m :}  (Succ n)(Succ m)  n ≡ m
Succ-inj refl = refl

This is much as we might expect, induction is identical to the recursion principle for \(\mathbb{N}\) and \(\mathrm{Succ}\)-\(\mathrm{inj}\) follows definitionally after we case match on equality. We can now make \(\mathbb{N}\) an instance of \(\mathrm{Peano}\):

  ℕ-Peano : Peano ℕ
  ℕ-Peano =
      { zero           = Zero
      ; succ           = Succ
      ; s-injective    = Succ-inj
      ; zero≠succ      = λ n ()
      ; induction      = ℕ-induction
      ; induction-zero = λ P z s   → refl
      ; induction-succ = λ P a z s → refl

In the last two cases, the \(\mathrm{induction}\) laws hold definitionally from how we have defined \(\mathbb{N}\)-\(\mathrm{induction}\).

Now, suppose \(\mathsf{N}\) is a set satisfying the Peano axioms, we want to then define functions to and from \(\mathbb{N}\):

from-ℕ :  {N : Set} {{_ : EqRel N}}  {{ _ : Peano N}}  N
from-ℕ {N} n = induction  _ -> N) n zero succ

to-ℕ :  {N : Set} {{_ : EqRel N}}  {{_ : Peano N}}  N 
to-ℕ n = induction  _ ) n zero succ

Pleasantly both definitions are essentially identical, using instance resolution to determine the relevant induction principle and values to use. Here we see the power of constructive induction. We don’t use induction to prove a property per se, but to compute. Since the dependent types in question are constant, induction simply is recursion!

Now we can show these maps form equivalences. To get a flavour let us step through the development for the first equality:

to∘from :
   {N : Set} {{ _ : EqRel N }}  {{peano : Peano N}}
   (n :)  to-ℕ {N} (from-ℕ n) ≡ n
to∘from n = {!!}

Asking Agda for the goal gives us:

Goal: Peano.induction peano (λ _ → ℕ)
      (ℕ-induction (λ _ → N) n (Peano.zero peano) (Peano.succ peano)) 0
      ≡ n
n     : ℕ
peano : Peano N  (not in scope, instance)
_     : EqRel N    (instance)
N     : Set      (not in scope)

We can see that in order for \(\mathbb{N}\)-\(\mathrm{induction}\) to make progress we need to split on \(\mathrm{n}\):

to∘from :
   {N : Set} {{ _ : EqRel N }}  {{peano : Peano N}}
   (n :)  to-ℕ {N} (from-ℕ n) ≡ n
to∘from Zero     = {!!}
to∘from (Succ n) = {!!}

The new goal for \(\mathrm{Zero}\) is:

Goal: Peano.induction peano (λ _ → ℕ) (Peano.zero peano) 0 Succ ≡ 0

But this is precisely our \(\mathrm{induction}\)-\(\mathrm{zero}\) principle! Similarly we can now use the \(\mathrm{induction}\)-\(\mathrm{succ}\) principle in the second case and then recurse giving us:

to∘from :
   {N : Set} {{ _ : EqRel N }}  {{peano : Peano N}}
   (n :)  to-ℕ {N} (from-ℕ n) ≡ n
to∘from Zero =  (induction-zero  _ ) Zero Succ)
to∘from {N} {{_}} {{peano}} (Succ n)
       _ )
      (ℕ-induction  _  N) n (Peano.zero peano) (Peano.succ peano))
  | to∘from {N} n
  = refl

The slightly gnarly explicit arguments in the second case are to help along the rewrite construction as it didn’t seem to cooperate with a less verbose alternative.

The other proof is similarly a case of following our nose (or rather following the typechecker). We first remind ourselves of some equality principles we have imported above (again simplifying away level polymorphism):

-- Funtions preserve equality:
  : {A : Set} {B : Set} (f : A  B) {x y : A}
   x ≡ y  f x ≡ f y

-- The transitivity principle for ≡
trans :  {a b c : A}  a ≡ b  b ≡ c  a ≡ c

We also will need the fact that we can lift any propositional equality into an equivalence relation:

liftEq :  {A : Set}  {{r : EqRel A}}  {a b : A}  a ≡ b  (a ≃ b)
liftEq refl = reflexivity

With these can now give the proof:

from∘to :
   {N : Set}{{ rel : EqRel N}}  {{peano : Peano N}}  (n : N) 
  from-ℕ (to-ℕ n) ≃ n
from∘to {N} n = liftEq (prop-eq {N})
             -- We make use of liftEq as we prove the
             -- stronger claim that from-ℕ (to-ℕ n) ≡ n
  -- In the zero case we apply induction-zero underneath from-ℕ
  -- and then use the definition of from-ℕ.
    :  {N} {{_ : EqRel N}} {{peano : Peano N}}
     from-ℕ {N} (induction {N}  _ ) zero Zero Succ) ≡ zero
  zero-lem {N} =
      pf1 : from-ℕ (induction {N}  _ ) zero Zero Succ) ≡ from-ℕ Zero
      pf1 = cong from-ℕ (induction-zero  _ ) Zero Succ)

      pf2 : from-ℕ Zero ≡ zero
      pf2 = refl
      trans pf1 pf2
  -- In the successor case we similarly apply induction-succ
  -- underneath from-ℕ and then recurse on the previous proof.
    :  {N} {{_ : EqRel N}} {{peano : Peano N}} (prev : N)
     from-ℕ (induction  _ ) prev Zero Succ) ≡ prev
     from-ℕ (induction  _ ) (succ prev) Zero Succ) ≡ succ prev
  succ-lem prev pf =
      pf1 : from-ℕ (induction  _ ) (succ prev) Zero Succ)
          ≡ from-ℕ (Succ (induction  _ ) prev Zero Succ))
      pf1 = cong from-ℕ (induction-succ  _ ) prev Zero Succ)

      pf2 : from-ℕ (Succ (induction  _ ) prev Zero Succ))(succ prev)
      pf2 = cong succ pf
      trans pf1 pf2

    :  {N} {{_ : EqRel N}} {{peano : Peano N}}
     from-ℕ (to-ℕ n) ≡ n
  prop-eq =
     -- we use induction on the principle
     -- we are trying to show with the above
     -- lemmas.
         n  from-ℕ (to-ℕ  n) ≡ n)
         {prev}  succ-lem prev)

This shows that any two types which satisfy the Peano axioms are equivalent in the sense that there are maps between them which form an isomorphism up to equivalence.

This is quite interesting as it stands, but we might wonder if there is a more direct characterization of the natural numbers. After all, our original definition as a recursive algebraic data type seems to give a perfectly good specification of what the natural numbers are. Let us turn to a characterization of \(\mathbb{N}\) given by Lawvere.

We define the category of discrete dynamical systems, whose:

Lawvere then observed that the natural numbers are the initial object in the category of discrete dynamical systems. In other words, every other dynamical system receives a unique map from the discrete dynamical system \(\left( \mathbb{N}\; , \; 0 : \mathbb{N}\; ,\; \mathrm{s}\; : \; \mathbb{N} \rightarrow \mathbb{N}\right)\).

Let us phrase this idea in terms of language more familiar to functional programmers. First define a “pattern functor” for \(\mathbb{N}\):

data NatP (r : Set) : Set where
  ZeroP : NatP r
  SuccP : r  NatP r

This has the same shape as \(\mathbb{N}\) but we leave the recursion open (this same pattern of open recursion is the animating idea behind recursion schemes). We can show that this is a functor (we don’t worry about the functor laws):

record Functor (F : Set  Set) : Set₁ where
  constructor Func
    Arr :  {A B : Set}  (f : A  B)  F A  F B

open Functor {{...}} public

  NatP-Functor : Functor NatP
  NatP-Functor = Func map
    map :  {A} {B}  (A  B)  NatP A  NatP B
    map f ZeroP     = ZeroP
    map f (SuccP x) = SuccP (f x)

Functional programmers might recognise that the discrete dynamical systems discussed above are in fact \(\mathrm{F}\)-algebras for this pattern functor, which we define as follows:

record Alg (F : Set  Set) (A : Set) : Set where
    μ : F A  A
open Alg {{...}} public

In particular if we define discrete dynamical systems:

record Dyn (A : Set) : Set where
  constructor D
    basepoint : A
    self-map  : A  A

then we can show then any dynamical system gives rise to an algebra:

from-Dyn :  {A : Set}  Dyn A  Alg NatP A
from-Dyn {A} (D basepoint self-map) = record { μ = alg }
  alg : NatP A  A
  alg ZeroP     = basepoint
  alg (SuccP a) = self-map a

and we leave, as an exercise, to show that there is an isomorphism between \(\mathrm{NatP}\) algebra structures on \(\mathrm{A}\) and discrete dynamical system structures (an observation we can upgrade to an isomorphism between the respective categories).

Let’s give the algebra structure for \(\mathbb{N}\):

  ℕ-Alg : Alg NatP ℕ
  ℕ-Alg = record { μ = alg }
    alg : NatP ℕ 
    alg ZeroP     = Zero
    alg (SuccP x) = Succ x

Just as in Lawvere’s characterization, we now wish to show that this algebra is initial. For that, we first have to define maps between algebras. Suppose \(\mathrm{A}\) and \(\mathrm{B}\) are \(\mathrm{F}\)-algebras. We then say a map \(m : \mathrm{A} \rightarrow \mathrm{B}\) is an \(\mathrm{F}\)-algebra homomorphism when the following diagram commutes (where the downward arrows are the algebra maps):

\[ \begin{array}{lll} F A & \xrightarrow{F m} & F B \\ \downarrow & & \downarrow \\ A & \xrightarrow{m} & B \\ \end{array} \]

In other words, the algebra map commutes with the map in question, or in equations: \[ f \circ \mu_{A} \equiv \mu_{B} \circ (F f) \]

record Alg-Homo (A B : Set) {F : Set  Set} {{f : Functor F}}
  (FA : Alg F A) (FB : Alg F B) : Set₁ where
  constructor AlgHom
    →-fun  : A  B
      :  (fa : F A)
       →-fun (Alg.μ FA fa)(Alg.μ FB) (Arr →-fun fa)

Now we can try to prove that the algebra structure on \(\mathbb{N}\) is initial. We first show there is an induced map to every other \(\mathrm{F}\)-\(\mathrm{algebra}\):

  :  {A : Set}
   {{FA : Alg NatP A}}
   Alg-Homo ℕ A ℕ-Alg FA
ℕ-weakly-initial {A} = AlgH init-ℕ law
  init-ℕ : A
  init-ℕ Zero     = μ ZeroP
  init-ℕ (Succ n) = μ (SuccP (init-ℕ n))

  law : (nP : NatP ℕ)  init-ℕ (μ nP) ≡ μ (Arr init-ℕ nP)
  law ZeroP     = refl
  law (SuccP x) = refl

We define the map via the structure of the algebra. \(\mathrm{Zero}\) maps to the basepoint of \(\mathrm{A}\) and for the successor we apply the self-map and then recurse. For the \(\mu\)-\(\mathrm{law}\) we first case split as this is how \(\mathrm{init}\)-\(\mathbb{N}\) is defined. At this point we can see that the laws hold definitionally from how we have defined the map.

We can then show uniqueness:

ℕ-init-uniq :
   {A : Set}
   {{FA : Alg NatP A}}
   (alg-hom : Alg-Homo ℕ A ℕ-Alg FA)
    (n :)
   (Alg-Homo.→-fun alg-hom n)
(Alg-Homo.→-fun (ℕ-weakly-initial {{FA}}) n)
ℕ-init-uniq alg-hom Zero = μ-comm ZeroP
    open Alg-Homo alg-hom public
ℕ-init-uniq {{FA}} alg-hom (Succ n) =
    pf1 :  →-fun (Succ n) ≡ μ (SuccP (→-fun n))
    pf1 = μ-comm (SuccP n)

    pf2 :  μ (SuccP (→-fun n)) ≡ μ (SuccP (Alg-Homo.→-fun ℕ-weakly-initial n))
    pf2 = cong (μ ∘ SuccP) (ℕ-init-uniq alg-hom n)
    trans pf1 pf2
    open Alg-Homo alg-hom public

We don’t infer algebra homomorphisms and so we need to pass them as arguments and open the various records to bring their fields into scope. In the first case we have the following goal:

→-fun 0 ≡ μ ZeroP

We note that this is definitionally equivalent to showing:

→-fun (μ ZeroP) ≡ μ (Arr →-fun ZeroP)

and this is the \(\mu\)-\(\mathrm{comm}\) law!

In the successor case we have to prove:

→-fun (Succ n) ≡ μ (SuccP (Alg-Homo.→-fun ℕ-weakly-initial n))

We use a similar observation as above to first rewrite the left-hand-side using the \(\mu\)-\(\mathrm{comm}\) law. At that point (as before) we can rewrite the inner part by recursion.

One nice thing about this characterization (beyond being much simpler to reason about!) is that the same idea of initial algebra semantics works just as well for any algebraic data type. For example we can do exactly the same for lists as we have for \(\mathbb{N}\):

data ListP (A : Set) (r : Set) : Set where
  NilP  : ListP A r
  ConsP : A  r  ListP A r

data List (A : Set) : Set where
  Nil  : List A
  Cons : A  List A  List A

  ListP-Functor :  {A}   Functor (ListP A)
  ListP-Functor {A} = Func map-L
    map-L :  {B} {C}  (B  C)  ListP A B  ListP A C
    map-L f NilP = NilP
    map-L f (ConsP a b) = ConsP a (f b)

record HasFold (A : Set) (B : Set) : Set₁ where
  constructor F
    initial  : B
    operator : A  B  B

fromHasFold :  {A B : Set}   HasFold A B  Alg (ListP A) B
fromHasFold {A} {B} (F initial operator) = record { μ = alg }
  alg : ListP A B  B
  alg NilP = initial
  alg (ConsP a b) = operator a b

toHasFold :  {A B : Set}  Alg (ListP A) B  HasFold A B
toHasFold {A} {B} record { μ = μ } = F init op
  init : B
  init = μ NilP

  op : A  B  B
  op a b = μ (ConsP a b)

  List-Alg :  {A}  Alg (ListP A) (List A)
  List-Alg {A} = record { μ = alg }
    alg : ListP A (List A)  List A
    alg NilP = Nil
    alg (ConsP a as) = Cons a as

foldr :  {A B : Set}  (A  B  B)  B  List A  B
foldr op init Nil         = init
foldr op init (Cons a as) = op a (foldr op init as)

foldr-Alg :  {A B : Set}
   {{FA : Alg (ListP A) B}}
   List A  B
foldr-Alg = foldr  a b  μ (ConsP a b)) (μ NilP)

  :  {A B : Set}
   {{FA : Alg (ListP A) B}}
   Alg-Homo (List A) B (List-Alg {A}) FA
List-weakly-initial {A} {B} = AlgHom foldr-Alg law

  law : (fa : ListP A (List A)) 
    foldr  a b  μ (ConsP a b)) (μ NilP) (μ fa)

    μ (Arr (foldr  a b  μ (ConsP a b)) (μ NilP)) fa)
  law NilP         = refl
  law (ConsP a as) = refl

and we see that out of initiality naturally comes \(\mathrm{foldr}\)! \(\mathrm{F}\)-algebra semantics give rise to recursion principles! The essence of \(\mathbb{N}\), from this point of view, lies in its recursive structure. After all, in a dependently-typed setting recursion and induction are really two sides of the same coin.

Thank you for reading! The full code for these examples is available here. Hopefully this has given some ideas for how we can explore mathematical ideas in Agda leveraging the typechecker to guide our proofs. Feel free to contact me here with questions, thoughts, ideas, or all of the above.

With warmest thanks to Sam Derbyshire, Reed Mullanix, Alixandra Prybyla and Shon Feder for their valuable feedback.