An attempt at explaining Decidable Equality

Introduction

If you’re just getting started with Idris, you might have heard that it is possible to prove things in Idris. But how does that make any sense? We can certainly write tests in programs, but how on earth do you “prove” something?? 4 years after I encountered Idris, I’m not sure I’d necessarily say I get it. But I do know a lot more than I started out with! And having tried to explain this to various people, both friends and family, hopefully I can write something at least semi-coherent which provides a nice explanation.

By the way, this post is also a Literate Idris file, which means you can load the markdown it’s based on at the REPL and try using the functions it defines!

What we’re aiming for

The goal at the end of this journey, is to be able to write the Dec interface (which models properties that either provably hold or provably cannot hold) and some instances of it. However, in order to do this we need to understand and implement certain other things like Equal and Void.

Worry not! This will all become clear in due course!

Hiding the built-in stuff

Since proofs and decidable equality are somewhat fundamental to Idris, there are a lot of definitions provided by default. But we want to understand these, not just use them, so let’s start by hiding these defaults in order to implement our own:

module DecEqIntro

%hide (==)
%hide (===)
%hide Void

%hide Prelude.Not
%hide Prelude.Dec
%hide Prelude.Uninhabited

%hide Builtin.Equal
%hide Builtin.Refl

%hide Builtin.sym
%hide Prelude.prim__void
%hide Prelude.void
%hide Prelude.absurd

%default total

Fundamentals

Rather than jumping in at the deep end and trying to implement DecEq immediately (alongside everything that we’ll need for it), let’s start with a couple of fundamental concepts. These can be difficult to wrap your head around in their own right, so it’s worth taking a moment to go through them. We’ll start with what is (hopefully) familiar ground: non-dependent types.

Non-dependent types promise nothing

If you’ve done some functional programming before, you might have seen something along these lines as the type of boolean equality:

(==) : Eq ty => ty -> ty -> Bool

This defines the == operator, which allows us to compare two instances of a type ty for boolean equality (i.e. True or False), provided that the type actually has a notion of “equals” (this what the interface constraint Eq ty => does).

There’s only one small problem with this: It doesn’t keep us from doing silly things:

namespace Nat
  export
  (==) : Nat -> Nat -> Bool
  (==) Z     Z     = False
  (==) Z     (S j) = True
  (==) (S k) Z     = True
  (==) (S k) (S j) = False

This clearly makes no sense in terms of what anyone would normally understand by (==), but as far as the type-checker is concerned, it is a valid implementation: We’ve taken 2 Nats and returned a Bool, just like we said we would in the type declaration.

Now, you could definitely argue that nobody in their right mind would implement (==) like this, and you’d definitely be right (or I really hope you would). And anyone who tried to use this (==) function would be thoroughly confused. But the point is that there is nothing in the type of the function which prevents us from doing silly things. In that sense, the type promises nothing about the return value of the function. This isn’t great…

Fortunately, with dependent types (types which can depend on values) we can define some stronger type-level guarantees.

The idea of proofs in Idris

Let’s start by trying to explain how we can have “proofs” in code. I’m going to try to explain this using a real-world parallel: bronze.

Bronze is an alloy made up of tin and copper, which revolutionised humanity’s tools around 3500 BCE (at the cost of stone chippers going the way of the saber-tooth tiger). Since we know that bronze requires access to tin and copper, if you find some old artefacts made of bronze, you know that whomever made the artefacts must have had access to tin and copper.

In that sense, the existence of bronze “proves” the existence of some tin and copper. Let’s relate that back to Idris by defining some datatypes:

public export
data Tin = Sn

public export
data Copper = Cu

public export
data Bronze : Type where
  MkBronze : Tin -> Copper -> Bronze

We can create something of type Tin with the Sn constructor, and something of type Copper with the Cu constructor. So far so good. The MkBronze constructor is where things get interesting!

The MkBronze constructor takes two arguments: one of type Tin and one of type Copper and returns something of type Bronze. If we think of constructors as special functions, we know that in order to call this MkBronze function we need to provide it with its required arguments (that’s just how functions work). So to create something of type Bronze, we’d first need to have some things of type Tin and type Copper which we could pass to the MkBronze function. Are you starting to see where I’m getting with this? : )

Proof by existence

Let’s take the leap to proofs! If we have something of type Bronze, we have a proof that we had something of type Tin and something of type Copper at some point. How is this a proof? Well, MkBronze is the only thing which can create something of type Bronze and MkBronze takes 2 very specific arguments: something of type Tin and something of type Copper. We have a thing of type Bronze, so clearly we were able to call MkBronze at some point. This in turn means we must have had something of type Tin and something of type Copper at that point, otherwise we wouldn’t have been able to pass them as arguments to MkBronze. So the existence of Bronze proves the existence of Tin and Copper!

Feel free to read over that previous bit a couple of times. The first time I realised this was how proofs in Idris worked, it took me a while to convince myself that I’d actually understood it correctly ^^

I get it now!

The reason we call this is a proof is because, according to all the rules of our functional programming “game”, it is a proof: Having Bronze means we could call MkBronze, which means we have proved that we had some Tin and Copper at some point; otherwise we’d never have been able to call MkBronze!

Awesome! With that in mind, let’s see how this idea can be used to define something slightly more useful: A proof of equality.

Proofs of equality

To define equality as a datatype, we’ll start by explicitly stating what equality is:

  • It is a relationship between 2 things,
  • which expresses that those things are, for all intents and purposes, equivalent; that they’re the same thing.

We can express this as an Idris datatype like so:

public export
data Equal : a -> b -> Type where
  Refl : {0 x : _} -> Equal x x

This is a tad more complicated than the Bronze data declaration, so let’s break it down:

  • data Equal : a -> b -> Type where — Defines a dependent type carrying two things of type a and b in the type.
  • Refl : — Declares a constructor for Equal. In this case we’re calling it Refl (short for “reflexive”: a mathematical concept meaning, roughly, the same as “is equal”).
  • Refl : {0 x : _} -> — Defines that the argument x to the Refl constructor is implicit (i.e. we don’t need to explicitly pass it when calling Refl) and that it is of a type that Idris can figure out by itself (_). We have also expressed that x is not needed at runtime (that’s what the 0 does), since we only actually care about the property that a and b are equal, and not the values themselves.
  • Refl : {0 x : _} -> Equal x x — Defines that if we give Refl some value x, then it returns an Equal x x, i.e. something of type Equal, where the a and b stored in Equal is the same thing: We put x in both places and there is only one argument x given, so it must be the same thing.

This Equal datatype with the Refl constructor can, based on the intuition I explained earlier, be considered a proof that 2 things are equal. If we have an Equal a b, then a must be the same as b or we wouldn’t have been able to call Refl to obtain that Equal.

We can construct some examples to illustrate this:

  1. If we pass Refl a number, e.g. 0, explicitly, it is equal to itself:
    public export
    ex1 : Equal 0 0
    ex1 = Refl {x=0}
    
  2. The unit type () is equal to itself:
    public export
    ex2 : Equal () ()
    ex2 = Refl {x=()}
    
  3. 0 is equal to 0 (this time, letting implicit arguments figure out that x=0):
    public export
    ex3 : Equal 0 0
    ex3 = Refl
    
  4. Unit is equal to unit (again, letting the implicit argument do its job):
    public export
    ex4 : Equal () ()
    ex4 = Refl
    

The double-edged sword of implicit arguments

The last 2 examples show why implicit arguments are useful, but also how they can lead to confusion in proof types: the Refl used in ex3 looks the same as the Refl used in ex4, but since the implicit arguments are different, they are actually different Refl calls (we see this in the type declaration).

On the flip side, when we know two things are equal, it is enough to express that in the type, and then as long as they actually are equal, the implicit argument takes care of the rest. That’s pretty handy!

Shorthand for Equal

Writing Equal all the time reads awkwardly (we don’t write verbatim “x equals y” all the time when writing equations), and it also quickly becomes tiring, so let’s finish the section on equality proofs by defining some infix syntax for Equal:

infixl 6 ===

public export
(===) : a -> b -> Type
(===) = Equal

This lets us use === in the middle of expressions where we would have used Equal instead, e.g.:

public export
ex5 : 3 === 3
ex5 = Refl

And just to show how equality proofs (and Idris) are actually quite powerful, here’s an example with computation:

public export
ex6 : (1 + 2 * 3) === 7
ex6 = Refl

Whooa, we’re halfway there!

We’ve had one proof, yes. But what about second proof?

I’m glad you asked! We’re trying to get to decidable equality. Which means we’ll need a proof of things being equal (we’ve just seen that) and a proof of things not being equal. Let’s try to do that second part next.

Proofs of False

To prove that something is false, we might try to write:

falseAttempt1 : (2 === 3) === False

But we cannot implement this! There is no way to call Refl to obtain the expression (2 === 3), which we need to show is False. (This is actually somewhat reassuring, since it suggests our approach to proofs is sound.)

Now we might be tempted to write:

falseAttempt2 : (2 == 3) === False

And while we might be able to implement this, it would actually not guarantee anything about our values! All it would prove, was that the function call 2 == 3 returned False. Which, as we saw at the beginning, guarantees nothing… Are we stuck? Only somewhat. We just need a different approach.

False cannot be

What we need is some datatype which captures the concept of “false” or “untrue”. In logic and type-theory, we often call this “bottom” or “void” and write it “⊥”. Now the question is “How on Earth do we express this as a datatype?” Especially in terms of constructors: What arguments would those even take??

Well, if we think slightly more maths-y (this may be a blessing or a curse, depending on how much you enjoy logic and proof-theory), a common approach is “proof by contradiction”: If we assume p to be true and we then arrive at a contradiction, it would be silly to keep going; our assumption was false.

A different way to say “our assumption was false” is: “we could never write (or, construct) that assumption according to our logic”. Now that’s beginning to sound more like something we can define in Idris!

The representation of false

We need a datatype which captures something we cannot construct; an untrue statement… Well, the easiest way to do that would just be to not provide any constructors, no? Then there’s nothing to call, so we can’t construct the datatype! Let’s try that!

public export
data Void : Type where

This is a perfectly valid datatype declaration, we can “forward-declare” data definitions just like we can for functions. Except if we never define any constructors (neither here nor later) then we can never create something of type Void but we can still talk about it.

Since we prove things by having instances of datatypes whose constructors can only be called if certain conditions are met (like Refl for Equal), we can never prove Void. This is great! How so? Well, in terms of logical reasoning and proofs, you can never prove something false to be true either! (This might sound obvious when put into English, but it’s important to realise nonetheless). There is nothing we can do to prove the statement “1 times 1 is 2” or that “the square root of 2 is 1”.

Since there are no constructors for Void, and we use constructors to prove things, Void represents an untrue statement for which no proof can exist. Pretty clever, no?

Interlude

Whew!… That took a lot longer than I initially expected! I know I said the fundamentals were challenging in their own right, but still…

If you’re still with me, let’s finally move on to what I actually promised this would be about: decidable equality!

Decidable Equality

We’ve finally arrived! Decidable equality! Let’s go!!!

What is decidable?

So, what does it mean for something to be decidable? It means we can decide whether that thing is true or not. Let’s try to express that in a datatype!

data Decide : Type -> Type where
  ItHolds : (prf : p) -> Decide p
  Unsound : (contra : Void) -> Decide Void

And let’s break it down to make sure we understand it:

  • data Decide : Type -> Type where — The Decide datatype depends on some Type (remember, we’re using datatypes as proofs).
  • ItHolds : (prf : p) -> Decide p — The ItHolds constructor takes a proof of type p, and constructs a Decide p (effectively proving p held since we could pass an instance of it to ItHolds).
  • Unsound : (contra : Void) -> Decide Void — Expresses something which cannot be (we would be passing some Void to it, but Void cannot exist so this statement is unable to be proven, i.e. ‘false’).

So far so good: We have a datatype for reasoning about decidable properties. Now let’s try to use it!

Trying to use Decide

Our Decide datatypes technically models any decidable property, so in order to arrive at decidable equality, we need to combine it with Equals. As we saw while covering the fundamentals, Equals describes a relationship between two things, so we’ll also need two things which can be equal. An easy example is Nats:

decideEquals : (n : Nat) -> (m : Nat) -> Decide (n === m)
decideEquals 0     0     = ItHolds Refl
decideEquals 0     (S j) = ?decide_rhs_2
decideEquals (S k) 0     = ?decide_rhs_3
decideEquals (S k) (S j) = ?decide_rhs_4

So far so good. We’ve case-split the function into the various options, and successfully proved the first case: if both arguments are 0, then we know that there is a sound proof of equality, which in this case is Refl.

However, if we try to implement any of the other holes, we run into some trouble. Consider the types of ?decide_rhs_2 and ?decide_rhs_3:

--- ?decide_rhs_2 : Decide (Equal 0     (S j))
--- ?decide_rhs_3 : Decide (Equal (S k) 0    )

We know these proofs cannot exist, and we have a concept for that: the Unsound constructor, right? Well yes but actually no. If we try to introduce Unsound on the right hand side of decide 0 (S j) Idris (rightfully) complains:

-- decideEquals 0     (S j) = Unsound ?decide_rhs_2

--- Error: While processing right hand side of decide. When unifying:
---     Decide Void
--- and:
---     Decide (0 === S j)
--- Mismatch between: Void and Equal 0 (S j).

When we defined Decide, we defined that Unsound takes an argument of type Void but we’re trying to construct an Equal 0 (S j). These are clearly not the same types (although they are both impossible). Thing is, as we went over when defining Void, there is no way we can pass Unsound something of type Void! Void cannot be constructed; that was kinda the whole idea… A similar issue pops up if we try using Unsound in the decide (S k) 0 case.

This is a problem… We’ve defined something which models the impossible (Void), but in order to use it we currently need to prove the impossible. This is contradictory. Are we stuck?

The solution: expressing contradictions

We are not stuck (at least, not completely). We just need to rethink a couple of things. Remember I talked about proof by contradiction? (It was a while back, I don’t blame you if you don’t). Those are our way out!

In case you need a refresher, the tl;dr for proofs by contradiction is:

  1. Make an assumption,
  2. show that from there you arrive at some contradiction,
  3. this means your assumption must be false,
  4. you’re done. Hurray!

What we’ve expressed with Void is something impossible. But we haven’t actually expressed how something might be impossible. There needs to be a link between impossible things and the Void datatype.

Sticking to our example of deciding if two Nats are provably equal, let’s define a case where Nats are provably not equal. But this time, via proof by contradiction.

We’ll start with (S k) ≠ Z, i.e. it is impossible for the successor of a number to be zero. We need to express this as a proof by contradiction… How about “If I can prove that (S k) is 0, then I have done the impossible”? That sounds pretty accurate, don’t you think? Let’s write that in Idris!

public export
succNotZero : (prf : (S k) === Z) -> Void

Now the question becomes “How do we implement this?”. We, the programmers, know that it is not possible. When we were coming up with Void, we saw that definitions based on unsound proofs of equality cannot be defined since we cannot match on the proof (it doesn’t exist). But how do we communicate this to Idris?

Impossible definitions

For things where there is no way to define them, Idris provides the impossible keyword. This tells the type-checker “Hey! Here, in this expression, there should be no way whatsoever to make this type-check”. So if we want to provide a definition for succNotZero, we can write:

succNotZero Refl impossible

And Idris is happy with this: There is no way to pass the implicit argument to Refl, which means we cannot pattern-match on the argument, which means we cannot construct Void. Well, that’s what we’ve been trying to say all along! Fantastic!

The trick to counter-proofs

The trick to counter-proofs is functions which return Void. We saw this when proving the successor of a Nat cannot be zero, and we can also use it to prove that zero cannot be the successor of a Nat:

public export
zeroNotSucc : (prf : Z === (S j)) -> Void
zeroNotSucc Refl impossible

Since Void cannot be constructed, neither can the argument(s) to a function which returns Void (or we’d be able to return something from it). Hopefully you agree that this is a proof by contradiction:

  1. We started out with an assumption (e.g. 0 === (S j)),
  2. expressed that this was contradictory (given the proof, we could construct Void — “construct Void” is a contradiction in terms),
  3. so our assumption must be false (Refl impossible).
  4. Idris agrees, since it cannot find a solution to the implicit argument for Refl. Hurray!

Decidable with proof-by-contradiction

With that cleared up, let’s make a better definition of something decidable. We previously had that Decide was either a proof or something impossible. Let’s refine that definition to:

Something decidable is either a proof, or something which should be impossible (i.e. a contradiction).

Let’s put this in a datatype, shall we?

public export
data Dec : Type -> Type where
  Yes : (prf : prop)            -> Dec prop
  No  : (contra : prop -> Void) -> Dec prop

The change here is that for the case where the property provably doesn’t hold, we provide a contra which explains how the property cannot hold. Oh and we’ve renamed the constructors to Yes and No. It’s a lot easier to read (and shorter to write).

Decidable equality (for real this time)

With a refined, and crucially more usable, datatype, let’s have a go at using it to define decidable equality again. As before, we’ll use Nats and start by case-splitting on the arguments:

decNat : (m : Nat) -> (n : Nat) -> Dec (m === n)
-- decNat 0     0     = ?decNat_rhs_1
-- decNat 0     (S j) = ?decNat_rhs_2
-- decNat (S k) 0     = ?decNat_rhs_3
-- decNat (S k) (S j) = ?decNat_rhs_4

The first case is trivial, 0 is equal to itself:

decNat 0     0     = Yes Refl

The second and third cases are false, so we start by prefixing them with No:

-- decNat 0     (S j) = No ?decNat_rhs_2
-- decNat (S k) 0     = No ?decNat_rhs_3

Now you’ll notice that rather than complaining, Idris provides the following types for the holes:

--- decNat_rhs_2 : Equal 0     (S j) -> Void
--- decNat_rhs_3 : Equal (S k) 0     -> Void

Now that’s something we can work with! With our refined Dec datatype, we don’t have to provide the impossible, we just have to provide a proof that the property arrives at a contradiction! Even better, we’ve already defined the two counter-proofs!

decNat 0     (S j) = No zeroNotSucc
decNat (S k) 0     = No succNotZero

The non-trivial case

For the final case, we run into a small problem: How do we deal with two non-zero numbers which might be equal?

decNat (S k) (S j) = ?decNat_rhs_4

We can’t just stick a Yes in there, since we don’t know if k equals j… In order to figure this out, we first need some more information about k and j, so let’s recurse:

-- decNat (S k) (S j) = case decNat k j of
--                           (Yes prf) -> ?decNat_rhs_5
--                           (No contra) -> ?decNat_rhs_6

This helps us a bit: If we know k === j (the Yes prf case), then (S k) === (S j); we’ve just proved that k and j are the same number, after all. Similarly, if we can prove that k and j being the same would lead to a contradiction, then using S shouldn’t change that.

Let’s put that in the case expression, and then inspect the holes:

-- decNat (S k) (S j) = case decNat k j of
--                           (Yes prf) -> Yes ?decNat_rhs_5
--                           (No contra) -> No ?decNat_rhs_6

For the first hole, we get:

---   k : Nat
---   j : Nat
---   prf : Equal k j
--- ------------------------------
--- decNat_rhs_5 : Equal (S k) (S j)

So we have a proof that k === j, but we want a proof that (S k) === (S j). Seems like we need some way to link prf and S, which we don’t at the moment. Let’s come back to that after looking at the No case:

---  k : Nat
---  j : Nat
---  contra : Equal k j -> Void
--- ------------------------------
--- decNat_rhs_6 : Equal (S k) (S j) -> Void

Hmm… Here we have a proof that if k === j, we’d have a contradiction. But we want a proof that if (S k) === (S j) we’d have a contradiction. This also seems like we need a link to S, which we don’t have at the moment…

We were on the verge of greatness

Unfortunately, missing the two links to S means we can’t quite complete the definition for decNat. Without them, there is no way to convince Idris that one proof (or counter-proof) leads to the next. This is a problem, but thankfully one which is not too difficult to solve.

Side note on rewrite

The builtin equality in Idris supports this thing called rewrite. It lets us use a proof to update the rhs, like so:

-- decNat (S k) (S j) = case decNat k j of
--                           (Yes prf) -> rewrite prf in Yes Refl
--                           (No contra) -> No ?decNat_rhs_6

Here, rewrite has used the proof prf (which shows that k and j are equal) to show that it is fine to construct the Refl for (S j). Unfortunately, rewrite is slightly magic and requires the builtin equality. So we can’t use it with our own equality (although the two are identical) and we have to do things by hand.

Well then. Let’s make the final push!

Linking k === j and S

First things first: The missing link between equality and the successor function. If k and j are the same, then applying S doesn’t change that. For the mathematically inclined, this is also known as a “congruence” relation:

public export
succCongruent : (prf : k === j) -> (S k) === (S j)

This definition gets interesting! If we start by defining:

-- succCongruent prf = ?succCongruent_rhs

Then the type of the hole becomes:

---  0 k : Nat
---  0 j : Nat
---    prf : Equal k j
--- ------------------------------
--- succCongruent_rhs : Equal (S k) (S j)

Which isn’t very helpful; it’s just re-told us what we’re trying to prove. However, if we now case-split on prf, the picture suddenly changes:

-- succCongruent Refl = ?succCongruent_rhs_0

---  0 k : Nat
---  0 j : Nat
--- ------------------------------
--- succCongruent_rhs_0 : Equal (S j) (S j)

What’s happened?? Suddenly we only care about j! This is because matching on Refl tells us something about k and j: Refl only takes a single argument, so by pattern-matching on Refl, Idris can deduce that k and j must have been the same (or we wouldn’t have been able to call Refl; just like with MkBronze!!). This is supremely useful, since Equal (S j) (S j) is trivial to prove:

succCongruent Refl = Refl

Once again, the implicit arguments confuse us here: The left hand side (lhs) and the right hand side (rhs) look identical, but the Refl on the lhs is actually Refl {x=j} and the Refl on the rhs is actually Refl {x=(S j)}. Idris can figure this out automatically which saves us some typing but admittedly looks a bit confusing. Trust me when I say it’s worth it. Especially for bigger proofs.

Linking k === j -> Void and S

With k === j -> (S k) === (S j) defined, can we take a similar approach to the counter-proof? I don’t see why not! If we have one contradiction, then mixing S doesn’t magically fix things, it just leads to another contradiction:

public export
succDiffers : (contra : k === j -> Void) -> (sPrf : (S k) === (S j)) -> Void

Another way to read this type is “If I know that k === j is nonsense, but at the same time someone has given me a proof that (S k) === (S j), then that proof must also be nonsense!”.

Like with succCongruent, our initial definition doesn’t match on anything but let’s us inspect the type of the rhs.

-- succDiffers contra sPrf = ?succDiffers_rhs

---  0 k : Nat
---  0 j : Nat
---    sPrf : Equal (S k) (S j)
---    contra : Equal k j -> Void
--- ------------------------------
--- succDiffers_rhs : Void

Ah, mince! We need to return Void here, but we know that that’s impossible! Is all lost? Not quite. If we try matching on sPrf (the proof that S magically fixed things) we get:

-- succDiffers contra Refl = ?succDiffers_rhs_0

---  0 k : Nat
---  0 j : Nat
---    contra : Equal j j -> Void
--- ------------------------------
--- succDiffers_rhs_0 : Void

Exactly how does this help us? Well, we now know that (S k) and (S j) were the same (that’s what sPrf was telling us) so k and j must have been the same. But we also have a counter-proof, a function, showing us that if that’s the case, then we have a contradiction. We can apply this to dismiss the proof as nonsense (or rather, show that the other proof also arrives at a contradiction):

succDiffers contra Refl = contra Refl

Why don’t we break the rules already?

Have we beaten the system? Have we returned Void? No! contra is impossible; there is no way to give its body a definition. What we have done is explained to Idris that “if I somehow give you a contra : j === j -> Void and a Refl {x=(S j)}, then I have broken the rules and we can use that contra to return Void”.

This is a very important thing to be able to show. It shows that breaking the rules in one place results in them being broken everywhere. If we take a false statement as true (i.e. have it exist as a function), then our entire system of reasoning becomes unsound and we can do silly things like construct Void from this statement.

Continuing to prove things after we have shown there is a contradiction is nonsensical.

Decidable equality (for real real this time)

With all the missing links now in place, we can finally, actually define decidable equality for natural numbers. No more tricks or princesses in other castles, I promise.

public export
decEqNat : (m : Nat) -> (n : Nat) -> Dec (m === n)
decEqNat 0     0     = Yes Refl
decEqNat 0     (S j) = No zeroNotSucc
decEqNat (S k) 0     = No succNotZero
{-
decEqNat (S k) (S j) = case decEqNat k j of
                            (Yes prf) = Yes ?decEqNat_rhs_5
                            (No contra) = No ?decEqNat_rhs_6
-}

That was where we left off last time. Except now that we have shown that there are links between proofs and counter-proofs and S, we can complete the definition by applying these as relevant:

decEqNat (S k) (S j) = case decEqNat k j of
                            (Yes prf) => Yes (succCongruent prf)
                            (No contra) => No (succDiffers contra)

This final case shows that if we have proved k and j to be equal, then a proof for “Are (S k) and (S j) equal?” is decidably Yes, and we obtain the relevant proof via succCongruent.

And vice-versa, if we have proved that k and j cannot be equal, then we can use that counter-proof to construct another one using succDiffers, which shows that (S k) and (S j) cannot be equal. And with that, the answer is decidably No.

We did it!

That’s decidable equality. Having Refl and Void allows us to write functions which can tell if two things are equal along with explaining how they are equal (or how they cannot possibly be equal). This is only possible thanks to dependent types, which lets us describe relationships between data in the type of things. Without it, we do not get any guarantees from the return types of functions.

Coming back to the very start where we were looking at ==. If we now compare the types of two functions for comparing Nats:

boolEqNat : (m : Nat) -> (n : Nat) -> Bool

decEqNat' : (m : Nat) -> (n : Nat) -> Dec (m === n)

(I’ve had to call the function decEqNat' since we already have decEqNat).

We can see that the type of the function now provides a relationship between the inputs. Where Bool doesn’t have any mention of whether it’s actually looking at m and n, Dec explicitly mentions m and n and states that the function will return a proof datatype relating them, if it returns a Yes. That’s pretty cool, no?

It is… but that sure took a while!…

Yeah, sorry about that. This took 4-5 times as long as I thought it would. For those bits where I went “ah but first, we actually also need […]”, that was genuinely me getting caught off-guard having forgotten a thing.

I hope that you still think it was worth it, and that you now understand both the concept and inner workings of decidable equality. Thanks for reading along! : )

Acknowledgements

  • Zoe Stafford (z-snails/z_snail) and Guillaume Allais (gallais) for help with understanding rewrite and void.
  • Tom Harley (magnostherobot) for reading this once I’d published it, and spotting that I’d confused “injective” for “congruent”.
  • the hacker known as “Alex” (mangopdf), for making me realise I was writing a textbook when I didn’t need to

Would you like to know more?

Really? You’ve not had enough?… Alrighty then!

(note: be aware that this will be a lot more crash-coursey than the previous stuff, since this blog entry is already extremely long)

Generalising stuff

You may not be surprised to hear that there are general ways of defining pretty much all of the above. Generalising is what logicians do best, after all. Without further ado, let’s get to it!

Generalising decidable equality

Decidable equality is a thing we can define for lots of types. So it would be nice to be able to just use a general function, e.g. decEq, for all these types. I got ya fam!

public export
interface DecEq ty where
  decEq : (a : ty) -> (b : ty) -> Dec (a === b)

Now anything that implements the DecEq interface has to specify an implementation of the decEq function, which will describe how to prove (and how to disprove) equality between two members of the type. And then you can just use decEq whenever you need a proof of equality for some type which implements it. Neat!

Generalising contradictions

We saw how to state that an expression cannot be made to type-check using the impossible keyword. When this applies to a datatype in general, we say that the type is “uninhabited”. Constructing an uninhabited type is a contradiction, so we define the following interface:

public export
interface Uninhabited ty where
  uninhabited : ty -> Void

This enforces that uninhabited types must be able to show that a contradiction occurs, e.g. using impossible:

public export
Uninhabited (True === False) where
  uninhabited Refl impossible

public export
Uninhabited (False === True) where
  uninhabited Refl impossible

Generalising counter-proofs

If we can generalise contradictions, can we also generalise counter-proofs? Why yes, we can!

We’ve discussed that once we have Void, anything goes (aka. it is pointless to continue after a contradiction has been found). The generalisation of this requires a tiny amount of trickery:

public export
void : (0 v : Void) -> a
void v = assert_total $ idris_crash "ERROR: Called 'void'"

The type of the function captures that from Void we can construct anything. However, the function definition crashes Idris with the given error message, whilst also asserting to the totality-checker that doing so is total. We are subverting the totality-checker here, but it is okay (necessary, even) because we should never be able to call void (we can only call it if we have a Void) and if we somehow do call void, the user should definitely know something has gone horribly wrong!

In the Idris source code, void is defined in terms of prim__void which is an external function. However, all external implementations of prim__void crash the runtime they target, so the definitions are effectively equivalent.

That’s absurd!

To finish generalising counter-proofs, we need one more thing: A link to generalised contradictions. Since a Void lets us construct anything and constructing Void is impossible, we call any type that lets us do this absurd:

public export
absurd : Uninhabited ty => ty -> a
absurd u = void (uninhabited u)

This absurd function expresses that if we have some instance of something Uninhabited, then that’s the same as having constructed Void (since uninhabited means the type cannot exist).

This is generally nice

The DecEq and Uninhabited interfaces, combined with the absurd function, allow us to write Dec things more nicely (if we can define the interfaces for the type):

public export
DecEq Bool where
  decEq False False = Yes Refl
  decEq False True  = No absurd
  decEq True  False = No absurd
  decEq True  True  = Yes Refl

Look at how nice and simple that implementation is! It’s so much easier to read than writing out custom proofs.

And this, is to go even further beyond!

Simple equality is not always the most interesting. Quite often, it is more fun to reason about more complicated properties of more complex structures. And that is totally possible! As long as you’re willing to do the work…

public export
data Repetition : a -> List a -> Type where
  Just : Repetition x [x]
  More : (x : a) -> Repetition x xs -> Repetition x (x :: xs)

Here we have a proof showing that a list xs consists purely of repetitions of x. It’s slightly a toy example, but it illustrates the point of being “willing to do the work” quite well.

We’d like to just write:

-- isRepetition : DecEq a => (x : a) -> (xs : List a) -> Dec (Repetition x xs)

But, as with natural numbers, we need some intermediary proofs before we can do that. Except, this time we’ve thrown a List into the mix! How is that a problem? Well, proofs over lists can fail in multiple places: either the head (the first element), or somewhere in the tail (the following elements). This means there’s a lot more to prove before we can write the final decidable thing.

public export
Uninhabited (Repetition x []) where
  uninhabited Just impossible
  uninhabited (More x ys) impossible

The easiest is to prove that we cannot have a repetition if the list is empty. There is no way to call either of the constructors, so we simply use Uninhabited and impossible. So far, so good.

Next, let’s try the positive cases. If we have a proof that x is the same as y, then we can put that in a singleton list for Repetition purposes:

public export
singletonRep : (prf : x === y) -> Repetition x [y]
singletonRep Refl = Just

Matching on Refl makes Idris recognise that the two are the same thing, so we can call Just without any problem; we only have one thing, which we can put in a list.

With that taken care of, let’s show that if x is the same as y, and we have an existing repetition of x, then we can extend the repetition to one more element:

public export
allGood :  {x : _}
        -> (another : x === y)
        -> (rep : Repetition x (z :: xs))
        -> Repetition x (y :: (z :: xs))
allGood Refl rep = More x rep

Again, matching on Refl makes Idris realise that x and y are the same thing, and we already know that (z :: xs) is just a repetition of x, so we can extend it with the extra x using More.

Another side note on rewrite

If we weren’t defining everything from scratch, we could use rewrite to prove the positive cases. We wouldn’t define intermediary proofs for each and every one of the positive cases. But unfortunately for us, we are defining everything from scratch, and so we need these extra positive proofs.

Ah well. It’s probably good for the soul or something.

Proving things impossible is hard

We’ve shown that we can prove things to be okay for a this more complicated property, but what about the cases where things are not okay? Well, those are pretty difficult: We don’t need to show that it is unlikely to succeed, we need to show that it is guaranteed to fail!

We’ll start with showing that if x cannot be the same as y, then a proof of repetition involving both (where y was the first element in the list) is a contradiction:

public export
headDiffers :  (contra : (x === y) -> Void)
            -> (prf : Repetition x (y :: _))
            -> Void
headDiffers contra Just = contra Refl
headDiffers contra (More x _) = contra Refl

This is already more confusing than the positive proofs. In the first line of the definition, we have a singleton list containing y as a repetition of x (meaning they are equal), but we also proved that x doesn’t equal y, so we can show that the Refl used for the repetition must be contradictory.

In the second line of the definition, we have the same thing, except there’s more than one element in the list involving y. However, the other elements are irrelevant since we know the first one forms a contradiction. So we use the first element to show that the Refl used for the repetition must be contradictory.

Now let’s do the tail case: If we have a proof that some repetition of x is actually a contradiction, then it doesn’t matter if we extend the repetition, it’ll still contain a contradiction!

public export
tailDiffers :  (contra : Repetition x (z :: xs) -> Void)
            -> (prf : Repetition x (y :: (z :: xs)))
            -> Void
tailDiffers contra (More x absurdTail) = contra absurdTail

We can only match on More since (y :: (z :: xs)) is not a singleton list (it has at least 2 elements). And when matching on More, we also gain access to the sublist that we know contains a contradiction! So we just refer to that contradiction.

Plus ça change…

Finally, let’s put together the isRepetition function. Note that we could not have written this definition if we’d missed any of the steps above…

public export
covering
isRepetition : DecEq a => (x : a) -> (xs : List a) -> Dec (Repetition x xs)

isRepetition x [] = No absurd

isRepetition x (y :: []) =
  case decEq x y of
       (Yes prf) => Yes (singletonRep prf)    -- rewrite prf in Yes Just
       (No contra) => No (headDiffers contra)

isRepetition x (y :: (z :: xs)) =
  case decEq x y of
       (Yes headOK) =>                        -- rewrite headOK in case...
          case isRepetition x (z :: xs) of
               (Yes tailOK) => Yes (allGood headOK tailOK)  -- rewrite tailOK...
               (No contra) => No (tailDiffers contra)

       (No contra) => No (headDiffers contra)

This may look familiar to the other Dec function we wrote, and that’s because it is! The general strategy for writing these is the same, but the intermediary steps and proofs/lemmas/functions are really where things get hairy. A good rule of thumb is “If it seems trivial, prove it!!” – chances are you’ll need that trivial (counter-)proof at some point in your more involved proofs.

That’s all folks!

Genuinely this time. If there’s more you want to learn, I strongly encourage you to experiment with datatypes for custom properties and proving these! You’ll likely get it wrong, both your datatypes and the properties involving them, but that’s just part of the “Type, define, refine” mantra of Idris.

gl hf!

Thomas Ekström Hansen
Thomas Ekström Hansen
PhD student in Computer Science

My research interests include low-level programming, type systems, and formal methods.

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