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Propositional Equality

In the last chapter we learned, how dependent pairs and records can be used to calculate types from values only known at runtime by pattern matching on these values. We will now look at how we can describe relations - or contracts - between values as types, and how we can use values of these types as proofs that the contracts hold.

module Tutorial.Eq

import Data.Either
import Data.HList
import Data.Vect
import Data.String

%default total

Equality as a Type

Imagine, we'd like to concatenate the contents of two CSV files, both of which we stored on disk as tables together with their schemata as shown in our discussion about dependent pairs:

data ColType = I64 | Str | Boolean | Float

Schema : Type
Schema = List ColType

IdrisType : ColType -> Type
IdrisType I64     = Int64
IdrisType Str     = String
IdrisType Boolean = Bool
IdrisType Float   = Double

Row : Schema -> Type
Row = HList . map IdrisType

record Table where
  constructor MkTable
  schema : Schema
  size   : Nat
  rows   : Vect size (Row schema)

concatTables1 : Table -> Table -> Maybe Table

We will not be able to implement concatTables1 by appending the two row vectors, unless we can somehow verify that the two schemata are identical. "Well," I hear you say, "that shouldn't be a big issue! Just implement Eq for ColType". Let's give this a try:

Eq ColType where
  I64     == I64     = True
  Str     == Str     = True
  Boolean == Boolean = True
  Float   == Float   = True
  _       == _       = False

concatTables1 (MkTable s1 m rs1) (MkTable s2 n rs2) = case s1 == s2 of
  True  => ?what_now
  False => Nothing

Somehow, this doesn't seem to work. If we inspect the context of hole what_now, Idris still thinks that s1 and s2 are different, and if we go ahead and invoke Vect.(++) anyway in the True case, Idris will respond with a type error.

Tutorial.Relations> :t what_now
   m : Nat
   s1 : List ColType
   rs1 : Vect m (HList (map IdrisType s1))
   n : Nat
   s2 : List ColType
   rs2 : Vect n (HList (map IdrisType s2))
------------------------------
what_now : Maybe Table

The problem is, that there is no reason for Idris to unify the two values, even though (==) returned True because the result of (==) holds no other information than the type being a Bool. We think, if this is True the two values should be identical, but Idris is not convinced. In fact, the following implementation of Eq ColType would be perfectly fine as far as the type checker is concerned:

Eq ColType where
  _       == _       = True

So Idris is right in not trusting us. You might expect it to inspect the implementation of (==) and figure out on its own, what the True result means, but this is not how these things work in general, because most of the time the number of computational paths to check would be far too large. As a consequence, Idris is able to evaluate functions during unification, but it will not trace back information about function arguments from a function's result for us. We can do so manually, however, as we will see later.

A Type for equal Schemata

The problem described above is similar to what we saw when we talked about the benefit of singleton types: The types are not precise enough. What we are going to do now, is something we'll repeat time again for different use cases: We encode a contract between values in an indexed data type:

data SameSchema : (s1 : Schema) -> (s2 : Schema) -> Type where
  Same : SameSchema s s

First, note how SameSchema is a family of types indexed over two values of type Schema. But note also that the sole constructor restricts the values we allow for s1 and s2: The two indices must be identical.

Why is this useful? Well, imagine we had a function for checking the equality of two schemata, which would try and return a value of type SameSchema s1 s2:

sameSchema : (s1, s2 : Schema) -> Maybe (SameSchema s1 s2)

We could then use this function to implement concatTables:

concatTables : Table -> Table -> Maybe Table
concatTables (MkTable s1 m rs1) (MkTable s2 n rs2) = case sameSchema s1 s2 of
  Just Same => Just $ MkTable s1 _ (rs1 ++ rs2)
  Nothing   => Nothing

It worked! What's going on here? Well, let's inspect the types involved:

concatTables2 : Table -> Table -> Maybe Table
concatTables2 (MkTable s1 m rs1) (MkTable s2 n rs2) = case sameSchema s1 s2 of
  Just Same => ?almost_there
  Nothing   => Nothing

At the REPL, we get the following context for almost_there:

Tutorial.Relations> :t almost_there
   m : Nat
   s2 : List ColType
   rs1 : Vect m (HList (map IdrisType s2))
   n : Nat
   rs2 : Vect n (HList (map IdrisType s2))
   s1 : List ColType
------------------------------
almost_there : Maybe Table

See, how the types of rs1 and rs2 unify? Value Same, coming as the result of sameSchema s1 s2, is a witness that s1 and s2 are actually identical, because this is what we specified in the definition of Same.

All that remains to do is to implement sameSchema. For this, we will write another data type for specifying when two values of type ColType are identical:

data SameColType : (c1, c2 : ColType) -> Type where
  SameCT : SameColType c1 c1

We can now define several utility functions. First, one for figuring out if two column types are identical:

sameColType : (c1, c2 : ColType) -> Maybe (SameColType c1 c2)
sameColType I64     I64     = Just SameCT
sameColType Str     Str     = Just SameCT
sameColType Boolean Boolean = Just SameCT
sameColType Float   Float   = Just SameCT
sameColType _ _             = Nothing

This will convince Idris, because in each pattern match, the return type will be adjusted according to the values we matched on. For instance, on the first line, the output type is Maybe (SameColType I64 I64) as you can easily verify yourself by inserting a hole and checking its type at the REPL.

We will need two additional utilities: Functions for creating values of type SameSchema for the nil and cons cases. Please note, how the implementations are trivial. Still, we often have to quickly write such small proofs (I'll explain in the next section, why I call them proofs), which will then be used to convince the type checker about some fact we already take for granted but Idris does not.

sameNil : SameSchema [] []
sameNil = Same

sameCons :  SameColType c1 c2
         -> SameSchema s1 s2
         -> SameSchema (c1 :: s1) (c2 :: s2)
sameCons SameCT Same = Same

As usual, it can help understanding what's going on by replacing the right hand side of sameCons with a hole an check out its type and context at the REPL. The presence of values SameCT and Same on the left hand side forces Idris to unify c1 and c2 as well as s1 and s2, from which the unification of c1 :: s1 and c2 :: s2 immediately follows. With these, we can finally implement sameSchema:

sameSchema []        []        = Just sameNil
sameSchema (x :: xs) (y :: ys) =
  [| sameCons (sameColType x y) (sameSchema xs ys) |]
sameSchema (x :: xs) []        = Nothing
sameSchema []        (x :: xs) = Nothing

What we described here is a far stronger form of equality than what is provided by interface Eq and the (==) operator: Equality of values that is accepted by the type checker when trying to unify type level indices. This is also called propositional equality: We will see below, that we can view types as mathematical propositions, and values of these types a proofs that these propositions hold.

Type Equal

Propositional equality is such a fundamental concept, that the Prelude exports a general data type for this already: Equal, with its only data constructor Refl. In addition, there is a built-in operator for expressing propositional equality, which gets desugared to Equal: (=). This can sometimes lead to some confusion, because the equals symbol is also used for definitional equality: Describing in function implementations that the left-hand side and right-hand side are defined to be equal. If you want to disambiguate propositional from definitional equality, you can also use operator (===) for the former.

Here is another implementation of concatTables:

eqColType : (c1,c2 : ColType) -> Maybe (c1 = c2)
eqColType I64     I64     = Just Refl
eqColType Str     Str     = Just Refl
eqColType Boolean Boolean = Just Refl
eqColType Float   Float   = Just Refl
eqColType _ _             = Nothing

eqCons :  {0 c1,c2 : a}
       -> {0 s1,s2 : List a}
       -> c1 = c2 -> s1 = s2 ->  c1 :: s1 = c2 :: s2
eqCons Refl Refl = Refl

eqSchema : (s1,s2 : Schema) -> Maybe (s1 = s2)
eqSchema []        []        = Just Refl
eqSchema (x :: xs) (y :: ys) = [| eqCons (eqColType x y) (eqSchema xs ys) |]
eqSchema (x :: xs) []        = Nothing
eqSchema []        (x :: xs) = Nothing

concatTables3 : Table -> Table -> Maybe Table
concatTables3 (MkTable s1 m rs1) (MkTable s2 n rs2) = case eqSchema s1 s2 of
  Just Refl => Just $ MkTable _ _ (rs1 ++ rs2)
  Nothing   => Nothing

Exercises part 1

In the following exercises, you are going to implement some very basic properties of equality proofs. You'll have to come up with the types of the functions yourself, as the implementations will be incredibly simple.

Note: If you can't remember what the terms "reflexive", "symmetric", and "transitive" mean, quickly read about equivalence relations here.

  1. Show that SameColType is a reflexive relation.

  2. Show that SameColType is a symmetric relation.

  3. Show that SameColType is a transitive relation.

  4. Let f be a function of type ColType -> a for an arbitrary type a. Show that from a value of type SameColType c1 c2 follows that f c1 and f c2 are equal.

For (=) the above properties are available from the Prelude as functions sym, trans, and cong. Reflexivity comes from the data constructor Refl itself.

  1. Implement a function for verifying that two natural numbers are identical. Try using cong in your implementation.

  2. Use the function from exercise 5 for zipping two Tables if they have the same number of rows.

    Hint: Use Vect.zipWith. You will need to implement custom function appRows for this, since Idris will not automatically figure out that the types unify when using HList.(++):

    appRows : {ts1 : _} -> Row ts1 -> Row ts2 -> Row (ts1 ++ ts2)

We will later learn how to use rewrite rules to circumvent the need of writing custom functions like appRows and use (++) in zipWith directly.

Programs as Proofs

A famous observation by mathematician Haskell Curry and logician William Alvin Howard leads to the conclusion, that we can view a type in a programming language with a sufficiently rich type system as a mathematical proposition and a total program calculating a value of this type as a proof that the proposition holds. This is also known as the Curry-Howard isomorphism.

For instance, here is a simple proof that one plus one equals two:

onePlusOne : the Nat 1 + 1 = 2
onePlusOne = Refl

The above proof is trivial, as Idris solves this by unification. But we already stated some more interesting things in the exercises. For instance, the symmetry and transitivity of SameColType:

sctSymmetric : SameColType c1 c2 -> SameColType c2 c1
sctSymmetric SameCT = SameCT

sctTransitive : SameColType c1 c2 -> SameColType c2 c3 -> SameColType c1 c3
sctTransitive SameCT SameCT = SameCT

Note, that a type alone is not a proof. For instance, we are free to state that one plus one equals three:

onePlusOneWrong : the Nat 1 + 1 = 3

We will, however, have a hard time implementing this in a provably total way. We say: "The type the Nat 1 + 1 = 3 is uninhabited", meaning, that there is no value of this type.

When Proofs replace Tests

We will see several different use cases for compile time proofs, a very straight forward one being to show that our functions behave as they should by proofing some properties about them. For instance, here is a proposition that map on list does not change the number of elements in the list:

mapListLength : (f : a -> b) -> (as : List a) -> length as = length (map f as)

Read this as a universally quantified statement: For all functions f from a to b and for all lists as holding values of type a, the length of map f as is the same the as the length of the original list.

We can implement mapListLength by pattern matching on as. The Nil case will be trivial: Idris solves this by unification. It knows the value of the input list (Nil), and since map is implemented by pattern matching on the input as well, it follows immediately that the result will be Nil as well:

mapListLength f []        = Refl

The cons case is more involved, and we will do this stepwise. First, note that we can proof that the length of a map over the tail will stay the same by means of recursion:

mapListLength f (x :: xs) = case mapListLength f xs of
  prf => ?mll1

Let's inspect the types and context we have here:

 0 b : Type
 0 a : Type
   xs : List a
   f : a -> b
   x : a
   prf : length xs = length (map f xs)
------------------------------
mll1 : S (length xs) = S (length (map f xs))

So, we have a proof of type length xs = length (map f xs), and from the implementation of map Idris concludes that what we are actually looking for is a result of type S (length xs) = S (length (map f xs)). This is exactly what function cong from the Prelude is for ("cong" is an abbreviation for congruence). We can thus implement the cons case concisely like so:

mapListLength f (x :: xs) = cong S $ mapListLength f xs

Please take a moment to appreciate what we achieved here: A proof in the mathematical sense that our function will not affect the length of our list. We no longer need a unit test or similar program to verify this.

Before we continue, please note an important thing: In our case expression, we used a variable for the result from the recursive call:

mapListLength f (x :: xs) = case mapListLength f xs of
  prf => cong S prf

Here, we did not want the two lengths to unify, because we needed the distinction in our call to cong. Therefore: If you need a proof of type x = y in order for two variables to unify, use the Refl data constructor in the pattern match. If, on the other hand, you need to run further computations on such a proof, use a variable and the left and right-hand sides will remain distinct.

Here is another example from the last chapter: We want to show that parsing and printing column types behaves correctly. Writing proofs about parsers can be very hard in general, but here it can be done with a mere pattern match:

showColType : ColType -> String
showColType I64      = "i64"
showColType Str      = "str"
showColType Boolean  = "boolean"
showColType Float    = "float"

readColType : String -> Maybe ColType
readColType "i64"      = Just I64
readColType "str"      = Just Str
readColType "boolean"  = Just Boolean
readColType "float"    = Just Float
readColType s          = Nothing

showReadColType : (c : ColType) -> readColType (showColType c) = Just c
showReadColType I64     = Refl
showReadColType Str     = Refl
showReadColType Boolean = Refl
showReadColType Float   = Refl

Such simple proofs give us quick but strong guarantees that we did not make any stupid mistakes.

The examples we saw so far were very easy to implement. In general, this is not the case, and we will have to learn about several additional techniques in order to proof interesting things about our programs. However, when we use Idris as a general purpose programming language and not as a proof assistant, we are free to choose whether some aspect of our code needs such strong guarantees or not.

A Note of Caution: Lowercase Identifiers in Function Types

When writing down the types of proofs as we did above, one has to be very careful not to fall into the following trap: In general, Idris will treat lowercase identifiers in function types as type parameters (erased implicit arguments). For instance, here is a try at proofing the identity functor law for Maybe:

mapMaybeId1 : (ma : Maybe a) -> map id ma = ma
mapMaybeId1 Nothing  = Refl
mapMaybeId1 (Just x) = ?mapMaybeId1_rhs

You will not be able to implement the Just case, because Idris treats id as an implicit argument as can easily be seen when inspecting the context of mapMaybeId1_rhs:

Tutorial.Relations> :t mapMaybeId1_rhs
 0 a : Type
 0 id : a -> a
   x : a
------------------------------
mapMaybeId1_rhs : Just (id x) = Just x

As you can see, id is an erased argument of type a -> a. And in fact, when type-checking this module, Idris will issue a warning that parameter id is shadowing an existing function:

Warning: We are about to implicitly bind the following lowercase names.
You may be unintentionally shadowing the associated global definitions:
  id is shadowing Prelude.Basics.id

The same is not true for map: Since we explicitly pass arguments to map, Idris treats this as a function name and not as an implicit argument.

You have several options here. For instance, you could use an uppercase identifier, as these will never be treated as implicit arguments:

Id : a -> a
Id = id

mapMaybeId2 : (ma : Maybe a) -> map Id ma = ma
mapMaybeId2 Nothing  = Refl
mapMaybeId2 (Just x) = Refl

As an alternative - and this is the preferred way to handle this case - you can prefix id with part of its namespace, which will immediately resolve the issue:

mapMaybeId : (ma : Maybe a) -> map Prelude.id ma = ma
mapMaybeId Nothing  = Refl
mapMaybeId (Just x) = Refl

Note: If you have semantic highlighting turned on in your editor (for instance, by using the idris2-lsp plugin), you will note that map and id in mapMaybeId1 get highlighted differently: map as a function name, id as a bound variable.

Exercises part 2

In these exercises, you are going to proof several simple properties of small functions. When writing proofs, it is even more important to use holes to figure out what Idris expects from you next. Use the tools given to you, instead of trying to find your way in the dark!

  1. Proof that map id on an Either e returns the value unmodified.

  2. Proof that map id on a list returns the list unmodified.

  3. Proof that complementing a strand of a nucleobase (see the previous chapter) twice leads to the original strand.

    Hint: Proof this for single bases first, and use cong2 from the Prelude in your implementation for sequences of nucleic acids.

  4. Implement function replaceVect:

    replaceVect : (ix : Fin n) -> a -> Vect n a -> Vect n a

    Now proof, that after replacing an element in a vector using replaceAt accessing the same element using index will return the value we just added.

  5. Implement function insertVect:

    insertVect : (ix : Fin (S n)) -> a -> Vect n a -> Vect (S n) a

    Use a similar proof as in exercise 4 to show that this behaves correctly.

Note: Functions replaceVect and insertVect are available from Data.Vect as replaceAt and insertAt.

Into the Void

Remember function onePlusOneWrong from above? This was definitely a wrong statement: One plus one does not equal three. Sometimes, we want to express exactly this: That a certain statement is false and does not hold. Consider for a moment what it means to proof a statement in Idris: Such a statement (or proposition) is a type, and a proof of the statement is a value or expression of this type: The type is said to be inhabited. If a statement is not true, there can be no value of the given type. We say, the given type is uninhabited. If we still manage to get our hands on a value of an uninhabited type, that is a logical contradiction and from this, anything follows (remember ex falso quodlibet).

So this is how to express that a proposition does not hold: We state that if it would hold, this would lead to a contradiction. The most natural way to express a contradiction in Idris is to return a value of type Void:

onePlusOneWrongProvably : the Nat 1 + 1 = 3 -> Void
onePlusOneWrongProvably Refl impossible

See how this is a provably total implementation of the given type: A function from 1 + 1 = 3 to Void. We implement this by pattern matching, and there is only one constructor to match on, which leads to an impossible case.

We can also use contradictory statements to proof other such statements. For instance, here is a proof that if the lengths of two lists are not the same, then the two list can't be the same either:

notSameLength1 : (List.length as = length bs -> Void) -> as = bs -> Void
notSameLength1 f prf = f (cong length prf)

This is cumbersome to write and pretty hard to read, so there is function Not in the prelude to express the same thing more naturally:

notSameLength : Not (List.length as = length bs) -> Not (as = bs)
notSameLength f prf = f (cong length prf)

Actually, this is just a specialized version of the contraposition of cong: If from a = b follows f a = f b, then from not (f a = f b) follows not (a = b):

contraCong : {0 f : _} -> Not (f a = f b) -> Not (a = b)
contraCong fun x = fun $ cong f x

Interface Uninhabited

There is an interface in the Prelude for uninhabited types: Uninhabited with its sole function uninhabited. Have a look at its documentation at the REPL. You will see, that there is already an impressive number of implementations available, many of which involve data type Equal.

We can use Uninhabited, to for instance express that the empty schema is not equal to a non-empty schema:

Uninhabited (SameSchema [] (h :: t)) where
  uninhabited Same impossible

Uninhabited (SameSchema (h :: t) []) where
  uninhabited Same impossible

There is a related function you need to know about: absurd, which combines uninhabited with void:

Tutorial.Eq> :printdef absurd
Prelude.absurd : Uninhabited t => t -> a
absurd h = void (uninhabited h)

Decidable Equality

When we implemented sameColType, we got a proof that two column types are indeed the same, from which we could figure out, whether two schemata are identical. The types guarantee we do not generate any false positives: If we generate a value of type SameSchema s1 s2, we have a proof that s1 and s2 are indeed identical. However, sameColType and thus sameSchema could theoretically still produce false negatives by returning Nothing although the two values are identical. For instance, we could implement sameColType in such a way that it always returns Nothing. This would be in agreement with the types, but definitely not what we want. So, here is what we'd like to do in order to get yet stronger guarantees: We'd either want to return a proof that the two schemata are the same, or return a proof that the two schemata are not the same. (Remember that Not a is an alias for a -> Void).

We call a property, which either holds or leads to a contradiction a decidable property, and the Prelude exports data type Dec prop, which encapsulates this distinction.

Here is a way to encode this for ColType:

decSameColType :  (c1,c2 : ColType) -> Dec (SameColType c1 c2)
decSameColType I64 I64         = Yes SameCT
decSameColType I64 Str         = No $ \case SameCT impossible
decSameColType I64 Boolean     = No $ \case SameCT impossible
decSameColType I64 Float       = No $ \case SameCT impossible

decSameColType Str I64         = No $ \case SameCT impossible
decSameColType Str Str         = Yes SameCT
decSameColType Str Boolean     = No $ \case SameCT impossible
decSameColType Str Float       = No $ \case SameCT impossible

decSameColType Boolean I64     = No $ \case SameCT impossible
decSameColType Boolean Str     = No $ \case SameCT impossible
decSameColType Boolean Boolean = Yes SameCT
decSameColType Boolean Float   = No $ \case SameCT impossible

decSameColType Float I64       = No $ \case SameCT impossible
decSameColType Float Str       = No $ \case SameCT impossible
decSameColType Float Boolean   = No $ \case SameCT impossible
decSameColType Float Float     = Yes SameCT

First, note how we could use a pattern match in a single argument lambda directly. This is sometimes called the lambda case style, named after an extension of the Haskell programming language. If we use the SameCT constructor in the pattern match, Idris is forced to try and unify for instance Float with I64. This is not possible, so the case as a whole is impossible.

Yet, this was pretty cumbersome to implement. In order to convince Idris we did not miss a case, there is no way around treating every possible pairing of constructors explicitly. However, we get much stronger guarantees out of this: We can no longer create false positives or false negatives, and therefore, decSameColType is provably correct.

Doing the same thing for schemata requires some utility functions, the types of which we can figure out by placing some holes:

decSameSchema' :  (s1, s2 : Schema) -> Dec (SameSchema s1 s2)
decSameSchema' []        []        = Yes Same
decSameSchema' []        (y :: ys) = No ?decss1
decSameSchema' (x :: xs) []        = No ?decss2
decSameSchema' (x :: xs) (y :: ys) = case decSameColType x y of
  Yes SameCT => case decSameSchema' xs ys of
    Yes Same => Yes Same
    No  contra => No $ \prf => ?decss3
  No  contra => No $ \prf => ?decss4

The first two cases are not too hard. The type of decss1 is SameSchema [] (y :: ys) -> Void, which you can easily verify at the REPL. But that's just uninhabited, specialized to SameSchema [] (y :: ys), and this we already implemented further above. The same goes for decss2.

The other two cases are harder, so I already filled in as much stuff as possible. We know that we want to return a No, if either the heads or tails are provably distinct. The No holds a function, so I already added a lambda, leaving a hole only for the return value. Here are the type and - more important - context of decss3:

Tutorial.Relations> :t decss3
   y : ColType
   xs : List ColType
   ys : List ColType
   x : ColType
   contra : SameSchema xs ys -> Void
   prf : SameSchema (y :: xs) (y :: ys)
------------------------------
decss3 : Void

The types of contra and prf are what we need here: If xs and ys are distinct, then y :: xs and y :: ys must be distinct as well. This is the contraposition of the following statement: If x :: xs is the same as y :: ys, then xs and ys are the same as well. We must therefore implement a lemma, which proves that the cons constructor is injective:

consInjective :  SameSchema (c1 :: cs1) (c2 :: cs2)
              -> (SameColType c1 c2, SameSchema cs1 cs2)
consInjective Same = (SameCT, Same)

We can now pass prf to consInjective to extract a value of type SameSchema xs ys, which we then pass to contra in order to get the desired value of type Void. With these observations and utilities, we can now implement decSameSchema:

decSameSchema :  (s1, s2 : Schema) -> Dec (SameSchema s1 s2)
decSameSchema []        []        = Yes Same
decSameSchema []        (y :: ys) = No absurd
decSameSchema (x :: xs) []        = No absurd
decSameSchema (x :: xs) (y :: ys) = case decSameColType x y of
  Yes SameCT => case decSameSchema xs ys of
    Yes Same   => Yes Same
    No  contra => No $ contra . snd . consInjective
  No  contra => No $ contra . fst . consInjective

There is an interface called DecEq exported by module Decidable.Equality for types for which we can implement a decision procedure for propositional equality. We can implement this to figure out if two values are equal or not.

Exercises part 3

  1. Show that there can be no non-empty vector of Void by writing a corresponding implementation of uninhabited

  2. Generalize exercise 1 for all uninhabited element types.

  3. Show that if a = b cannot hold, then b = a cannot hold either.

  4. Show that if a = b holds, and b = c cannot hold, then a = c cannot hold either.

  5. Implement Uninhabited for Crud i a. Try to be as general as possible.

    data Crud : (i : Type) -> (a : Type) -> Type where
  Create : (value : a) -> Crud i a
  Update : (id : i) -> (value : a) -> Crud i a
  Read   : (id : i) -> Crud i a
  Delete : (id : i) -> Crud i a

  6. Implement DecEq for ColType.

  7. Implementations such as the one from exercise 6 are cumbersome to write as they require a quadratic number of pattern matches with relation to the number of data constructors. Here is a trick how to make this more bearable.

    1. Implement a function ctNat, which assigns every value of type ColType a unique natural number.

    2. Proof that ctNat is injective. Hint: You will need to pattern match on the ColType values, but four matches should be enough to satisfy the coverage checker.

    3. In your implementation of DecEq for ColType, use decEq on the result of applying both column types to ctNat, thus reducing it to only two lines of code.

    We will later talk about with rules: Special forms of dependent pattern matches, that allow us to learn something about the shape of function arguments by performing computations on them. These will allow us to use a similar technique as shown here to implement DecEq requiring only n pattern matches for arbitrary sum types with n data constructors.

Rewrite Rules

One of the most important use cases of propositional equality is to replace or rewrite existing types, which Idris can't unify automatically otherwise. For instance, the following is no problem: Idris know that 0 + n equals n, because plus on natural numbers is implemented by pattern matching on the first argument. The two vector lengths therefore unify just fine.

leftZero :  List (Vect n Nat)
         -> List (Vect (0 + n) Nat)
         -> List (Vect n Nat)
leftZero = (++)

However, the example below can't be implemented as easily (try it!), because Idris can't figure out on its own that the two lengths unify.

rightZero' :  List (Vect n Nat)
           -> List (Vect (n + 0) Nat)
           -> List (Vect n Nat)

Probably for the first time we realize, just how little Idris knows about the laws of arithmetics. Idris is able to unify values when

  • all values in a computation are known at compile time
  • one expression follows directly from the other due to the pattern matches used in a function's implementation.

In expression n + 0, not all values are known (n is a variable), and (+) is implemented by pattern matching on the first argument, about which we know nothing here.

However, we can teach Idris. If we can proof that the two expressions are equivalent, we can replace one expression for the other, so that the two unify again. Here is a lemma and its proof, that n + 0 equals n, for all natural numbers n.

addZeroRight : (n : Nat) -> n + 0 = n
addZeroRight 0     = Refl
addZeroRight (S k) = cong S $ addZeroRight k

Note, how the base case is trivial: Since there are no variables left, Idris can immediately figure out that 0 + 0 = 0. In the recursive case, it can be instructive to replace cong S with a hole and look at its type and context to figure out how to proceed.

The Prelude exports function replace for substituting one variable in a term by another, based on a proof of equality. Make sure to inspect its type first before looking at the example below:

replaceVect : Vect (n + 0) a -> Vect n a
replaceVect as = replace {p = \k => Vect k a} (addZeroRight n) as

As you can see, we replace a value of type p x with a value of type p y based on a proof that x = y, where p is a function from some type t to Type, and x and y are values of type t. In our replaceVect example, t equals Nat, x equals n + 0, y equals n, and p equals \k => Vect k a.

Using replace directly is not very convenient, because Idris can often not infer the value of p on its own. Indeed, we had to give its type explicitly in replaceVect. Idris therefore provides special syntax for such rewrite rules, which will get desugared to calls to replace with all the details filled in for us. Here is an implementation of replaceVect with a rewrite rule:

rewriteVect : Vect (n + 0) a -> Vect n a
rewriteVect as = rewrite sym (addZeroRight n) in as

One source of confusion is that rewrite uses proofs of equality the other way round: Given an y = x it replaces p x with p y. Hence the need to call sym in our implementation above.

Use Case: Reversing Vectors

Rewrite rules are often required when we perform interesting type-level computations. For instance, we have already seen many interesting examples of functions operating on Vect, which allowed us to keep track of the exact lengths of the vectors involved, but one key functionality has been missing from our discussions so far, and for good reasons: Function reverse. Here is a possible implementation, which is how reverse is implemented for lists:

revOnto' : Vect m a -> Vect n a -> Vect (m + n) a
revOnto' xs []        = xs
revOnto' xs (x :: ys) = revOnto' (x :: xs) ys


reverseVect' : Vect n a -> Vect n a
reverseVect' = revOnto' []

As you might have guessed, this will not compile as the length indices in the two clauses of revOnto' do not unify.

The nil case is a case we've already seen above: Here n is zero, because the second vector is empty, so we have to convince Idris once again that m + 0 = m:

revOnto : Vect m a -> Vect n a -> Vect (m + n) a
revOnto xs [] = rewrite addZeroRight m in xs

The second case is more complex. Here, Idris fails to unify S (m + len) with m + S len, where len is the length of ys, the tail of the second vector. Module Data.Nat provides many proofs about arithmetic operations on natural numbers, one of which is plusSuccRightSucc. Here's its type:

Tutorial.Eq> :t plusSuccRightSucc
Data.Nat.plusSuccRightSucc :  (left : Nat)
                           -> (right : Nat)
                           -> S (left + right) = left + S right

In our case, we want to replace S (m + len) with m + S len, so we will need the version with arguments flipped. However, there is one more obstacle: We need to invoke plusSuccRightSucc with the length of ys, which is not given as an implicit function argument of revOnto. We therefore need to pattern match on n (the length of the second vector), in order to bind the length of the tail to a variable. Remember, that we are allowed to pattern match on an erased argument only if the constructor used follows from a match on another, unerased, argument (ys in this case). Here's the implementation of the second case:

revOnto {n = S len} xs (x :: ys) =
  rewrite sym (plusSuccRightSucc m len) in revOnto (x :: xs) ys

I know from my own experience that this can be highly confusing at first. If you use Idris as a general purpose programming language and not as a proof assistant, you probably will not have to use rewrite rules too often. Still, it is important to know that they exist, as they allow us to teach complex equivalences to Idris.

A Note on Erasure

Single value data types like Unit, Equal, or SameSchema have not runtime relevance, as values of these types are always identical. We can therefore always use them as erased function arguments while still being able to pattern match on these values. For instance, when you look at the type of replace, you will see that the equality proof is an erased argument. This allows us to run arbitrarily complex computations to produce such values without fear of these computations slowing down the compiled Idris program.

Exercises part 4

  1. Implement plusSuccRightSucc yourself.

  2. Proof that minus n n equals zero for all natural numbers n.

  3. Proof that minus n 0 equals n for all natural numbers n

  4. Proof that n * 1 = n and 1 * n = n for all natural numbers n.

  5. Proof that addition of natural numbers is commutative.

  6. Implement a tail-recursive version of map for vectors.

  7. Proof the following proposition:

    mapAppend :  (f : a -> b)
          -> (xs : List a)
          -> (ys : List a)
          -> map f (xs ++ ys) = map f xs ++ map f ys

  8. Use the proof from exercise 7 to implement again a function for zipping two Tables, this time using a rewrite rule plus Data.HList.(++) instead of custom function appRows.

Conclusion

The concept of types as propositions, values as proofs is a very powerful tool for writing provably correct programs. We will therefore spend some more time defining data types for describing contracts between values, and values of these types as proofs that the contracts hold. This will allow us to describe necessary pre- and postconditions for our functions, thus reducing the need to return a Maybe or other failure type, because due to the restricted input, our functions can no longer fail.

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