Brutal [Meta]Introduction to Dependent Types in Agda

by Jan Malakhovski, v. 16.3, published , last updated Literate Source

Agda doesn’t lack tutorials and introductions, see below for a rather long list of recommendations. There’s a lot to read already, why another introduction? Because there is a gap. The Theory is huge and full of subtle details that are mostly ignored in tutorial implementations and hidden in language tutorials (so that unprepared are not scared away). Which is hardly surprising since the current state of art takes years to implement correctly, and even then some (considerable) problems remain.

Nevertheless, I think it is the hard parts that matter, and I always wanted a tutorial that at least mentioned their existence (well, obviously there is a set of dependently typed problems most people appreciate, e.g. undecidable type inference, but there is still a lot of issues that are not so well-understood). Moreover, after I stumbled upon some of these lesser known parts of dependently typed programming I started to suspect that hiding them behind the language goodnesses actually makes things harder to understand. “Dotted patterns” and “unification stuck” error in Agda are perfect examples. I claim that:

Having said that, this article serves somewhat controversial purposes:

Finally, before we start, a disclaimer: I verified my core thoughts about how all this stuff works by reading (parts of) Agda’s source code, but still, as Plato’s Socrates stated, “I know that I know nothing”.

Changelog

Table of Contents

Recommended reading

There is a whole page of Agda tutorials [3] on the Agda’s documentation page [4]. Personally, I recommend:

(This list is not an order, the best practice is to read them (and this page) simultaneously.23)

Same proposition holds for Coq [9], Idris [10] and, to a lesser extent, for Epigram [11].

For a general introduction to type theory field look no further than:

There’s also a number of theoretical books strongly related to the languages listed above:

And a number of tutorials which show how to implement a dependently typed language yourself:

Slow start

You want to use Emacs, trust me

There is agda2-mode for Emacs. It allows to:

Installation:

Don’t be scared away by this if you never used Emacs before, by default it looks and works like any conventional text editor (i.e. it’s not Vim; though, you can make Emacs to emulate Vim by installing and enabling evil-mode, and I do, but that’s unrelated).

I expect you to load the Literate Agda version of this document into Emacs and continue reading it there. You can, of course

How to open the Literate Agda file:

If Emacs appars to hang, or it asks if you want to do something you don’t, or you got yourself into some other kind of Emacs-mess somehow, you can always start fixing your problem by pressing C-g (which calls keyboard-quit Emacs LISP function, which is Emacs’ version of shell’s Control+C) repeatedly until it stops doing the thing you don’t it want doing.

The basics of syntax

In Agda a module definition always goes first:

{-# OPTIONS -WnoOpenPublicPrivate #-}
-- The above is needed because Agda 2.6.4.1 gives factually incorrect warnings.
-- Trying to fix the problem it points to the way it wants will break things.

module BrutalDepTypes where

Nested modules and modules with parameters are supported. One of the most common usages of nested modules is to hide some definitions from the top level namespace:

module ThrowAwayIntroduction where

Datatypes are written in GADTs-style:

  data Bool : Set where
    true false : Bool -- Note, we can list constructors of a same type
                      -- by interspersing them with spaces.

  -- input for ℕ is \bN,
  -- input for → is \to, but -> is fine too
  -- Naturals.
  data: Set where
    zero :
    succ :

  -- Identity container
  data Id (A : Set) : Set where
    pack : A  Id A

  -- input for ⊥ is \bot
  -- Empty type. Absurd. False proposition.
  data: Set where

Set here means the same thing as kind * in Haskell, i.e. a type of types (more on that below).

Agda is a total language. There is no Haskell-like undefined, all functions are guaranteed to terminate on all possible inputs (if not explicitly stated otherwise by a compiler flag or a function definition itself), which means that type is really empty.

Function declarations look very similar to those in Haskell:

  -- input for ₀ is \_0, ₁ is \_1 and so on
  idℕ₀ :
  idℕ₀ x = x

except function arguments have their names even in type expressions:

  -- Note, argument's name in a type might differ from a name used in pattern-matching
  idℕ₁ : (n :) 
  idℕ₁ x = x -- this `x` refers to the same argument as `n` in the type

with idℕ₀’s definition being a syntax sugar for:

  idℕ₂ : (_ :) 
  idℕ₂ x = x

where the underscore means “I don’t care about the name”, just like in Haskell.

Dependent types allow type expressions after an arrow to depend on expressions before the arrow, this is used to type polymorphic functions:

  id₀ : (A : Set)  A  A
  id₀ _ a = a

Note that this time A in the type cannot be changed into an underscore, but it’s fine to ignore this name in pattern matching.

Pattern matching looks as usual:

  not : Bool  Bool
  not true  = false
  not false = true

except if you make an error in a constructor name:

  not₀ : Bool  Bool
  not₀ true  = false
  not₀ fals  = true

Agda will say nothing. This might be critical sometimes:

  data Three : Set where
    COne CTwo CThree : Three

  three2ℕ : Three 
  three2ℕ COne = zero
  three2ℕ Ctwo = succ zero
  three2ℕ _    = succ (succ zero) -- intersects with the previous clause

Finally, Agda supports implicit arguments:

  id : {A : Set}  A  A
  id a = a

  idTest₀ :
  idTest₀ = id

Values of implicit arguments are derived from other arguments’ values and types by solving type equations (more on them below). You don’t have to apply them or pattern match on them explicitly, but you still can if you wish:

  -- positional:
  id₁ : {A : Set}  A  A
  id₁ {A} a = a

  idTest₁ :
  idTest₁ = id {}

  -- named:
  const₀ : {A : Set} {B : Set}  A  B  A
  const₀ {B = _} a _ = a

  constTest₀ :
  constTest₀ = const₀ {A =} {B =}

[It’s important to note that no proof search is ever done. Implicit arguments are completely orthogonal to computational aspect of a program, being implicit doesn’t imply anything else. Implicit variables are not treated any way special, they are not erased any way differently than others. They are just a kind of syntax sugar assisted by equation solving.]

It’s allowed to skip arrows between arguments in parentheses or braces:

  id₃ : {A : Set} (a : A)  A
  id₃ a = a

and to intersperse names of values of the same type by spaces inside parentheses and braces:

  const : {A B : Set}  A  B  A
  const a _ = a

What makes Agda’s syntax so confusing is the usage of underscore.

The first one, is the “I don’t care about the name” demonstrated by the const term above (which is the same syntax as Haskell uses).

The second one is “guess the value yourself”:

  idTest₃ :
  idTest₃ = id₀ _

which works exactly the same way as implicit arguments (this is different from Haskell where an underscore in a term body declares it to be a hole to be filled later, Agda has a similar mechanism called “goals”, which will be discussed below).

Or, to be more precise, it is the implicit arguments that work like arguments implicitly applied with underscores, except Agda does this once for each function definition, not for each call.

The second-and-a-half meaning is “guess the type yourself” (it shares the mechanism with the second meaning, its a dependently typed language, after all):

  unpack₀ : {A : _}  Id A  A
  unpack₀ (pack a) = a

which has a special syntax sugar:

  -- input for ∀ is \all or \forall
  unpack :  {A}  Id A  A
  unpack (pack a) = a

  -- explicit argument version:
  unpack₁ :  A  Id A  A
  unpack₁ _ (pack a) = a

extends to the right up to the first arrow:

  unpack₂ :  {A B}  Id A  Id B  A
  unpack₂ (pack a) _ = a

  unpack₃ :  {A} (_ : Id A) {B}  Id B  A
  unpack₃ (pack a) _ = a

Datatype syntax assumes implicit when there is no type specified:

  data ForAllId A (B : Id A) : Set where

It is important to note that Agda’s is quite different from Haskell’s (forall). When we say ∀ n in Agda it’s perfectly normal for n : ℕ to be inferred, but in Haskell ∀ n always means {n : Set}, [i.e. Haskell’s is an implicit (Hindley-Milner) version of second order universal quantifier while in Agda it’s just a syntax sugar].

Syntax misinterpreting becomes a huge problem when working with more than one universe level (more on that below). It is important to train yourself to desugar type expressions subconsciously (by doing in consciously at first). It will save hours of your time later. For instance, ∀ {A} → Id A → A means {A : _} → (_ : Id A) → A (where the last → A should be interpreted as → (_ : A)), i.e. the first A is a variable name, while the other expressions are types.

Finally, the third and the last meaning of an underscore is to mark arguments’ places in function names for the MixFix parser, i.e. an underscore in a function name marks the place where the arguments goes:

  if_then_else_ : {A : Set}  Bool  A  A  A
  if true then a else _ = a
  if false then _ else b = b

  -- Are two ℕs equal?
  _=ℕ?_ : Bool
  zero   =ℕ? zero   = true
  zero   =ℕ? succ m = false
  succ m =ℕ? zero   = false
  succ n =ℕ? succ m = n =ℕ? m

  -- Sum for ℕ.
  infix 6 _+_
  _+_ :
  zero   + n = n
  succ n + m = succ (n + m)

  ifthenelseTest₀ :
  ifthenelseTest₀ = if (zero + succ zero) =ℕ? zero
    then zero
    else succ (succ zero)

  -- Lists
  -- input for ∷ is \::
  data List (A : Set) : Set where
    []  : List A
    __ : A  List A  List A

  [_] : {A : Set}  A  List A
  [ a ] = a ∷ []

  listTest₁ : List ℕ
  listTest₁ = []

  listTest₂ : List ℕ
  listTest₂ = zero ∷ (zero ∷ (succ zero ∷ []))

Note the fixity declaration infix which has the same meaning as in Haskell. We didn’t write infixl for a reason. With declared associativity Agda would not print redundant parentheses, which is good in general, but would somewhat complicate explanation of a several things below.

There is a where construct, just like in Haskell:

  ifthenelseTest₁ :
  ifthenelseTest₁ = if (zero + succ zero) =ℕ? zero
    then zero
    else x
    where
    x = succ (succ zero)

While pattern matching, there is a special case when a type we are trying to pattern match on is obviously ([type inhabitance problem is undecidable in a general case]) empty. This special case is called an absurd pattern:

  -- ⊥ implies anything.
  ⊥-elim : {A : Set}  A
  ⊥-elim ()

which allows you to skip a right-hand side of a definition.

You can bind variables like that still:

  -- Absurd implies anything, take two.
  ⊥-elim₀ : {A : Set}  A
  ⊥-elim₀ x = ⊥-elim x

Agda has records, which work very much like newtype declarations in Haskell, i.e. they are datatypes with a single constructor [the tag for which is not stored in memory].

  record Pair (A B : Set) : Set where
    field
      first  : A
      second : B

  getFirst :  {A B}  Pair A B  A
  getFirst = Pair.first

Note, however, that to prevent name clashes record definition generates a module with field extractors inside.

There is a convention to define a type with one element as a record with no fields:

  -- input for ⊤ is \top
  -- One element type. Record without fields. True proposition.
  record: Set where

  tt :
  tt = record {}

A special thing about this convention is that an argument of an empty record type automatically gets the value record {} when applied implicitly or with underscore.

Lastly, Agda uses oversimplified lexer that splits tokens by spaces, parentheses, and braces. For instance (note the name of the variable binding):

  -- input for ‵ is \`
  -- input for ′ is \'
  ⊥-elim‵′ : {A : Set}  A
  ⊥-elim‵′ ∀x:⊥→-- = ⊥-elim ∀x:⊥→--

is totally fine. Also note that -- doesn’t generate a comment here.

The magic of dependent types

Let’s define the division by two:

  div2 :
  div2 zero = zero
  div2 (succ (succ n)) = succ (div2 n)

the problem with this definition is that Agda is total and we have to extend this function for odd numbers

  div2 (succ zero) = {!check me!}

by changing {!check me!} into some term, most common choice being zero.

Suppose now, we know that inputs to div2 are always even and we don’t want to extend div2 for the succ zero case. How do we constrain div2 to even naturals only? With a predicate! That is, even predicate:

  even : Set
  even zero =
  even (succ zero) =
  even (succ (succ n)) = even n

which returns with with a trivial proof tt when the argument is even and empty then the argument is odd.

Now the definition of div2e constrained to even naturals only becomes:

  div2e : (n :)  even n -- Note, we have to give a name `n` to the first argument here
  div2e zero p = zero
  div2e (succ zero) ()
  div2e (succ (succ y)) p = succ (div2e y p) -- Note, a proof of `even (succ (succ n))` translates
                                             -- to a proof of `even n` by the definition of `even`.

When programming with dependent types, a predicate on A becomes a function from A to types, i.e. A → Set. If a : A satisfies the predicate P : A → Set then the function P returns a type with each element being a proof of P a, in a case a doesn’t satisfy P it returns an empty type.

The magic of dependent types makes the type of the second argument of div2e change every time we pattern match on the first argument n. From the callee side, if the first argument is odd then the second argument would get type sometime (after a number of recursive calls) enabling the use of an absurd pattern. From the caller side, we are not able to call the function with an odd n, since we have no means to construct a value for the second argument in this case.

Type families and Unification

There is another way to define “even” predicate. This time with a datatype indexed by ℕ:

  data Even : Set where
    ezero  : Even zero
    e2succ : {n :}  Even n  Even (succ (succ n))

  twoIsEven : Even (succ (succ zero))
  twoIsEven = e2succ ezero

Even : ℕ → Set is a family of types indexed by ℕ and obeying the following rules:

Compare this to even : ℕ → Set definition translation:

In other words, the difference is that Even : ℕ → Set constructs a type whereas even : ℕ → Set returns a type when applied to an element of .

The proof that two is even even (succ (succ zero)) literally says “two is even because it has a trivial proof”, whereas the proof that two is even twoIsEven says “two is even because zero is even and two is the successor of the successor of zero”.

Even datatype allows us to define another non-extended division by two for :

  div2E : (n :)  Even n 
  div2E zero ezero = zero
  div2E (succ zero) ()
  div2E (succ (succ n)) (e2succ stilleven) = succ (div2E n stilleven) -- Compare this case to div2e.

Note, there is no case for div2E zero (e2succ x) since e2succ x has the wrong type, there is no such constructor in Even zero. For the succ zero case the type of the second argument is not , but is empty. How do we know that? Unification!

Unification is the most important (at least with pattern matching on inductive datatypes involved) and easily forgotten aspect of dependently typed programming. Given two terms M and N unification tries to find a substitution s such that using s on M gives the same result as using s on N. The precise algorithm definition is pretty long, but the idea is simple: to decide if two expressions could be unified we

For instance:

In the code above succ zero does not unify with any of the Even constructors’ indexes [zero, succ (succ n)], which means this type is obviously empty by the definition.

[Refer to “The view from the left” paper by McBride and McKinna [22] for more details on pattern matching with type families.]

More type families and less Unification

In datatype declaration things before the colon are called “parameters”, things between the colon and the Set are called “indexes”.

There is a famous datatype involving both of them:

  data Vec (A : Set) : Set where
    []  : Vec A zero
    __ :  {n}  A  Vec A n  Vec A (succ n)

Vec A n is a vector of values of type A and length n, Vec has a parameter of type Set and is indexed by values of type . Compare this definition to the definition of List and Even. Note also, that Agda tolerates different datatypes with constructors of the same name (see below for how this is resolved).

We can not omit the clause for an [] case in a function which takes a head of a List:

  head₀ :  {A}  List A  A
  head₀ []       = {!check me!}
  head₀ (a ∷ as) = a

but we have nothing to write in place of {!check me!} there (if we want to be total).

On the other hand, there is no [] constructor in a Vec A (succ n) type:

  head :  {A n}  Vec A (succ n)  A
  head (a ∷ as) = a

Note that there are no absurd patterns here, Vec A (succ n) is inhabited, it just happens that there is no [] in there.

The Vec type is famous for its concatenation function, which has a very nice type when compared to that for simple Lists:

  -- Concatenation for `List`s
  _++_ :  {A}  List A  List A  List A
  []       ++ bs = bs
  (a ∷ as) ++ bs = a ∷ (as ++ bs)

  -- Concatenation for `Vec`tors
  -- The length of a concatenation is the sum of lengths of arguments and is available in types.
  _++v_ :  {A n m}  Vec A n  Vec A m  Vec A (n + m)
  []       ++v bs = bs
  (a ∷ as) ++v bs = a ∷ (as ++v bs)

Compare _+_, _++_, and _++v_ definitions.

Why does the definition of _++v_ work? Because we defined _+_ this way!

Dotted patterns and Unification

Let’s define a substraction function:

  infix 6 _-_
  _-_ :
  zero   - _      = zero
  succ n - zero   = succ n
  succ n - succ m = n - m

Note that n - m = zero for m > n.

Let us get rid of this (succ n) - zero case with _≤_ relation:

  data __ : Set where
    z≤n :  {n}            zero ≤ n
    s≤s :  {n m}  n ≤ m  succ n ≤ succ m

We are now able to write a substraction that is not extended for m > n.

  sub₀ : (n m :)  m ≤ n 
  sub₀ n zero (z≤n .{n}) = n
  sub₀ .(succ n) .(succ m) (s≤s {m} {n} y) = sub₀ n m y

Note the dots. These are called “dotted patterns”. Now ignore them for a second.

Consider the first clause sub₀ n zero (z≤n {k}).

The second clause is sub₀ n m (s≤s {n'} {m'} y) (for fresh type variables n' and m').

Note that in the first clause for sub₀ we now have two pattern matches on n and in the second clause we now have two matches on n' and m' each. Which of those do we want to match on and bind to a variable? Dotted patterns allow us to make that choice by placing dots before the expressions we want to ignore.

In the above instance, to ask the compiler to match on the first n in the first clause of sub₀ we put a dot before the second occurrence of n. The second clause of sub₀ binds the second occurience of each variable instead. (Yes, from a computational point of view, this is stupid. Deliberately so, see below.)

In other words, dotted pattern say “do not match on this, it is the only possible value” to the compiler.

Rewritten with a case construct from Haskell (Agda doesn’t have case, see below) the code above becomes (in pseudo-Haskell):

  sub₀ n m less = case less of
    z≤n {k}     -> case m of -- [`m = zero`, `k = n`]
      zero    -> n
      succ m' -> __IMPOSSIBLE__  -- since `m = zero` doesn't merge with `m = succ m'`
    s≤s n' m' y -> sub₀ n' m' y  -- [`n = succ n'`, `m = succ n'`]

where __IMPOSSIBLE__ is “an undefined that is never executed”. [It is sometimes also called abort in the TT literature.] Absurd patterns translate to __IMPOSSIBLE__ too.

Note, that since we have [m = zero, k = n] in the first case, we can actually dot the first usage of zero too to optimize the match on m away completely:

  sub₁ : (n m :)  m ≤ n 
  sub₁ n .zero (z≤n .{n}) = n
  sub₁ .(succ n) .(succ m) (s≤s {m} {n} y) = sub₁ n m y

which translates to

sub₁ n m less = case less of
  z≤n {k}     -> n
  s≤s n' m' y -> sub₁ n' m' y

Finally, note that sub₀ and sub₁ extract m it from the proof of m ≤ n instead of just using the actual argument, which, of course, is a bit ridiculous. [I presented them first to show that dot patterns do not influence “terms are expression rewrite rules” semantics, but do influence computational behaviour of terms.]

A coventional definition for sub looks like this:

  sub : (n m :)  m ≤ n 
  sub n zero (z≤n .{n}) = n
  sub (succ n) (succ m) (s≤s .{m} .{n} y) = sub n m y

and translates into the following:

  sub n m less = case m of
    zero   -> case less of
        z≤n {k}       -> n
        s≤s {k} {l} y -> __IMPOSSIBLE__ -- since `zero` (`m`) can't be unified
                                        -- with `succ k`
    succ m' -> case n of
      zero   -> case less of
        z≤n {k}       -> __IMPOSSIBLE__ -- since `succ m'` (`m`) can't be unified
                                        -- with `zero`
        s≤s {k} {l} y -> __IMPOSSIBLE__ -- since `zero` (`n`) can't be unified
                                        -- with `succ l`
      succ n' -> case less of
        z≤n {k}       -> __IMPOSSIBLE__ -- since `succ n'` (`n`) can't be unified
                                        -- with `zero`
        s≤s {k} {l} y -> sub n' m' y

Exercise. Write out the unification constraints for the pseudo-Haskell translation above.

Also note, that for sub n zero the third argument is always z≤n {n}, so, in theory, we could have written

  sub₂ : (n m :)  m ≤ n 
  sub₂ n zero .(z≤n {n}) = n
  sub₂ (succ n) (succ m) (s≤s .{m} .{n} y) = sub₂ n m y

but Agda doesn’t allow this. Why? Because dotted patterns are inlined unification constraints and the unification algorithm does not generate any constraints that allow us to dot z≤n {n} in the first clause of sub₂ (a different unification algorithm could, though). [Actually, also note that in the second clause of sub₂ we, too, at least in theory, could have dotted the whole s≤s destructor subexpression while keeping the y bound (we don’t care about its value since nothing in sub₂ will actually use it for computation), but Agda doesn’t allow that either.]

This is also the reason why the first case of sub₀ has two possible implementations noted above (with a dot pattern on zero and without).

In the sub case, however, we can write

  sub₃ : (n m :)  m ≤ n 
  sub₃ n zero _ = n
  sub₃ (succ n) (succ m) (s≤s .{m} .{n} y) = sub₃ n m y

to sidestep the problem at least in the z≤n case.

Exercise. Translate the following definition into pseudo-Haskell with unification constraints:

  sub₄ : (n m :)  m ≤ n 
  sub₄ n zero (z≤n .{n}) = n
  sub₄ (succ .n) (succ .m) (s≤s {m} {n} y) = sub₄ n m y

Propositional equality and Unification

We shall now define the most useful type family, that is, Martin-Löf’s equivalence (values only version, though):

  -- ≡ is \==
  infix 4 __
  data __ {A : Set} (x : A) : A  Set where
    refl : x ≡ x

For x y : A the type x ≡ y has exactly one constructor refl if x and y are convertible, i.e. there exist such z that z →β✴ x and z →β✴ y, where →β✴ is “β-reduces in zero or more steps”. By a consequence from a Church-Rosser theorem and strong normalization convertibility can be solved by normalization. Which means that unification will both check convertibility and fill in any missing parts. In other words, x y : A the type x ≡ y has exactly one constructor refl if x and y unify with each other.

Let’s prove some of _≡_’s properties:

  -- _≡_ is symmetric
  sym : {A : Set} {a b : A}  a ≡ b  b ≡ a
  sym refl = refl

  -- transitive
  trans : {A : Set}{a b c : A}  a ≡ b  b ≡ c  a ≡ c
  trans refl refl = refl

  -- and congruent
  cong : {A B : Set} {a b : A}  (f : A  B)  a ≡ b  f a ≡ f b
  cong f refl = refl

Consider the case sym {A} {a} {b} (refl {x = a}). Matching on refl gives [b = a] equation, i.e. the clause actually is sym {A} {a} .{a} (refl {x = a}) which allows to write refl on the right-hand side. Other proofs are similar.

Note, we can prove sym the other way:

  sym′ : {A : Set}{a b : A}  a ≡ b  b ≡ a
  sym′ {A} .{b} {b} refl = refl

sym packs a into refl. sym′ packs b. “Are these two definitions equal?” is an interesting philosophical question. (From the Agda’s point of view they are.)

Since dotted patterns are just unification constraints, you don’t have to dot implicit arguments when you don’t bind or match on them.

_≡_ type family is called “propositional equality”. In Agda’s standard library it has a bit more general definition, see below.

Proving things interactively

With _≡_ we can finally prove something from basic number theory. Let’s do this interactively.

Our first victim is the associativity of _+_.

  +-assoc₀ :  a b c  (a + b) + c ≡ a + (b + c)
  +-assoc₀ a b c = {!!}

Note a mark {!!}.

Press C-c C-l (agda2-load in Emacs LISP and under M-x) to load and typecheck this buffer in Emacs agda2-mode (this file is quite large, give it a couple seconds). Your Emacs buffer should now be syntax-colored and the {!!} above should be transformed into a greenish “{ }2”.

Anything of the form {!expr!} with “expr” being any string (including empty) becomes a goal after a buffer gets loaded by agda2-mode. Typing {!!} is quite tedious, so there is a shortcut ?. All ? symbols are automatically transformed into {!!} when a buffer gets reloaded.

Goals are interactive “holes” in a buffer, pressing special key sequences (or calling agda2-* function via M-x) while inside a goal allows you to ask Agda questions about and perform actions on the code inside and around that goal. In this document “check me” in a goal means that that goal is not expected to be filled, it’s just an example.

Normally, you are going to spend most of your time editing with goals, but from time to time you might want to turn everything back to plain text (in case you screwed something up or something bugged out). Press C-c C-x C-q (agda2-quit) and most things should become black-and-white again. Press C-c C-l to continue.

Placing the cursor in the goal above and pressing C-c C-c a RET (agda2-make-case, make case by a) gives (ignore changes to the name of a function and “check me”s everywhere):

  +-assoc₁ :  a b c  (a + b) + c ≡ a + (b + c)
  +-assoc₁ zero b c = {!check me!}
  +-assoc₁ (succ a) b c = {!check me!}

Press C-c C-, (agda2-goal-and-context) to show goal type and the context while in the goal. Write refl in there and press C-c C-r (agda2-refine, refine), this would typecheck it and produce:

  +-assoc₂ :  a b c  (a + b) + c ≡ a + (b + c)
  +-assoc₂ zero b c = refl
  +-assoc₂ (succ a) b c = {!check me!}

C-c C-f (agda2-next-goal), write cong succ, refine, and you will get

  +-assoc₃ :  a b c  (a + b) + c ≡ a + (b + c)
  +-assoc₃ zero b c = refl
  +-assoc₃ (succ a) b c = cong succ {!check me!}

Next goal, goal type and context, press C-c C-a (agda2-auto proof search), and you will get this:

  +-assoc :  a b c  (a + b) + c ≡ a + (b + c)
  +-assoc zero b c = refl
  +-assoc (succ a) b c = cong succ (+-assoc a b c)

Done.

Similarly, we prove

  lemma-+zero :  a  a + zero ≡ a
  lemma-+zero zero = refl
  lemma-+zero (succ a) = cong succ (lemma-+zero a)

  lemma-+succ :  a b  succ a + b ≡ a + succ b
  lemma-+succ zero b = refl
  lemma-+succ (succ a) b = cong succ (lemma-+succ a b)

The commutativity for _+_ is not hard to follow too:

  -- A fun way to write transitivity
  infixr 5 _~_
  _~_ = trans

  +-comm :  a b  a + b ≡ b + a
  +-comm zero b = sym (lemma-+zero b)
  +-comm (succ a) b = cong succ (+-comm a b) ~ lemma-+succ b a

Nice way to “step” through a proof is to wrap some subexpression with {! !}, e.g.:

  +-comm₁ :  a b  a + b ≡ b + a
  +-comm₁ zero b = sym (lemma-+zero b)
  +-comm₁ (succ a) b = cong succ {!(+-comm a b)!} ~ lemma-+succ b a

reload, ask for a type, context and inferred type with C-c C-l followed by C-c C-., refine, wrap another subexpression, rinse and repeat. Sometimes I dream of a better interface for this.

Solving type equations

The second clause of +-comm is pretty fun example to infer implicit arguments by hand. Let’s do that. Algorithm is as follows:

Applying the first step of the algorithm to a term

trans (cong succ (+-comm1 a b)) (lemma-+succ b a)

gives:

trans {ma} {mb} {mc} {md} (cong {me} {mf} {mg} {mh} succ (+-comm a b)) (lemma-+succ b a)

with m* being metavariables.

a b : ℕ since _+_ : ℕ → ℕ → ℕ in the type of +comm. This gives the following system of equations (with duplicates and metavariable applications skipped):

trans (cong succ (+-comm a b)) (lemma-+succ b a) : __ {} (succ a + b) (b + succ a)
trans (cong succ (+-comm a b)) (lemma-+succ b a) : __ {} (succ (a + b)) (b + succ a) -- after normalization
ma =
mb = succ (a + b)
md = b + succ a
+-comm a b : __ {} (a + b) (b + a)
mg = (a + b)
me =
mh = (b + a)
mf =
cong succ (+-comm a b) : __ {} (succ (a + b)) (succ (b + a))
mc = succ (b + a)
lemma-+succ b a : __ {} (succ b + a) (b + succ a)
lemma-+succ b a : __ {} (succ (b + a)) (b + succ a) -- after normalization
trans (cong succ (+-comm a b)) (lemma-+succ b a) : __ {} (succ a + b) (b + succ a)

The most awesome thing about this is that from Agda’s point of view, a goal is just a metavariable of a special kind. When you ask for a type of a goal with C-c C-t or C-c C-, Agda prints everything it has for the corresponding metavariable. Funny things like ?0, ?1, and etc in agda2-mode outputs are references to these goal metavariables. For instance, in the following:

  metaVarTest : Vec ℕ (div2 (succ zero)) 
  metaVarTest = {!check me!}

the type of the goal mentions the name of the metavariable corresponding to the very first goal in this article (of the unimplementable case in div2).

By the way, to resolve datatype constructor overloading Agda infers the type for a constructor call expected at the call site, and unifies the inferred type with the types of all possible constructors of the same name. If there are no matches found, an error is reported. In case with more than one alternative available, an “unsolved meta” error for the corresponding return type metavariable is produced.

Termination checking, well-founded induction

Work in progress.

Propositional equality exercises

Exercise. Define multiplication by induction on the first argument:

  module Exercise where
    infix 7 _*_
    _*_ :
    n * m = {!!}

so that the following proof works:

    -- Distributivity.
    *+-dist :  a b c  (a + b) * c ≡ a * c + b * c
    *+-dist zero b c = refl
    -- λ is \lambda, \Gl
    *+-dist (succ a) b c = cong  x  c + x) (*+-dist a b c) ~ sym (+-assoc c (a * c) (b * c))

Now, fill in the following goals:

    *-assoc :  a b c  (a * b) * c ≡ a * (b * c)
    *-assoc zero b c = refl
    *-assoc (succ a) b c = *+-dist b (a * b) c ~ cong {!!} (*-assoc a b c)

    lemma-*zero :  a  a * zero ≡ zero
    lemma-*zero a = {!!}

    lemma-+swap :  a b c  a + (b + c) ≡ b + (a + c)
    lemma-+swap a b c = sym (+-assoc a b c) ~ {!!} ~ +-assoc b a c

    lemma-*succ :  a b  a + a * b ≡ a * succ b
    lemma-*succ a b = {!!}

    *-comm :  a b  a * b ≡ b * a
    *-comm a b = {!!}

Pressing C-c C-. while there is a term in a hole shows a goal type, context and the term’s inferred type. Incredibly useful key sequence for interactive proof editing.

Pattern matching with with

Consider the following implementation of a filter function in Haskell:

  filter :: (a  Bool)  [a]  [a]
  filter p [] = []
  filter p (a : as) = case p a of
    True  -> a : (filter p as)
    False -> filter p as

It could be rewritten into Agda like this:

  filter : {A : Set}  (A  Bool)  List A  List A
  filter p [] = []
  filter p (a ∷ as) with p a
  ... | true  = a ∷ (filter p as)
  ... | false = filter p as

which doesn’t look very different. But desugaring ... by the rules of Agda syntax makes it a bit less similar:

  filter₀ : {A : Set}  (A  Bool)  List A  List A
  filter₀ p [] = []
  filter₀ p (a ∷ as) with p a
  filter₀ p (a ∷ as) | true  = a ∷ (filter₀ p as)
  filter₀ p (a ∷ as) | false = filter₀ p as

There’s no direct analogue to case in Agda, with construction allows pattern matching on intermediate expressions (just like Haskell’s case), but (unlike case) on a top level only. Thus with effectively just adds a “derived” argument to a function. Just like with normal arguments, pattern matching on a derived argument might change some types in a context.

The top level restriction simplifies all the dependently typed stuff (mainly related to dotted patterns), but makes some things a little bit more awkward (in most cases you can emulate case with a subterm placed into a where block). Syntactically, vertical bars separate normal arguments from a derived ones and a derived ones from each other.

The with construct can be nested and multiple matches are allowed to be done in parallel, e.g. with can obfuscate the above definition as:

  filterN : {A : Set}  (A  Bool)  List A  List A
  filterN p [] = []
  filterN p (a ∷ as) with p a
  filterN p (a ∷ as) | true  with as
  filterN p (a ∷ as) | true | [] = a ∷ []
  filterN p (a ∷ as) | true | b ∷ bs with p b
  filterN p (a ∷ as) | true | b ∷ bs | true  = a ∷ (b ∷ filterN p bs)
  filterN p (a ∷ as) | true | b ∷ bs | false = a ∷ filterN p bs
  filterN p (a ∷ as) | false = filterN p as

  -- or alternatively
  filterP : {A : Set}  (A  Bool)  List A  List A
  filterP p [] = []
  filterP p (a ∷ []) with p a
  filterP p (a ∷ []) | true = a ∷ []
  filterP p (a ∷ []) | false = []
  filterP p (a ∷ (b ∷ bs)) with p a | p b
  filterP p (a ∷ (b ∷ bs)) | true  | true  = a ∷ (b ∷ filterP p bs)
  filterP p (a ∷ (b ∷ bs)) | true  | false = a ∷ filterP p bs
  filterP p (a ∷ (b ∷ bs)) | false | true  = b ∷ filterP p bs
  filterP p (a ∷ (b ∷ bs)) | false | false = filterP p bs

Let us prove that all these functions produce equal results when applied to equal arguments:

  filter≡filterN₀ : {A : Set}  (p : A  Bool)  (as : List A)  filter p as ≡ filterN p as
  filter≡filterN₀ p [] = refl
  filter≡filterN₀ p (a ∷ as) = {!check me!}

note the goal type (filter p (a ∷ as) | p a) ≡ (filterN p (a ∷ as) | p a) which shows p a as derived argument to filter function.

Remember that to reduce a + b we had to match on a in the proofs above, matching on b gave nothing interesting because _+_ was defined by induction on the first argument. Similarly, to finish the filter≡filterN proof we have to match on p a, as, and p b, essentially duplicating the form of filterN term:

  filter≡filterN : {A : Set}  (p : A  Bool)  (as : List A)  filter p as ≡ filterN p as
  filter≡filterN p [] = refl
  filter≡filterN p (a ∷ as) with p a
  filter≡filterN p (a ∷ as) | true with as
  filter≡filterN p (a ∷ as) | true | [] = refl
  filter≡filterN p (a ∷ as) | true | b ∷ bs with p b
  filter≡filterN p (a ∷ as) | true | b ∷ bs | true = cong  x  a ∷ (b ∷ x)) (filter≡filterN p bs)
  filter≡filterN p (a ∷ as) | true | b ∷ bs | false = cong (__ a) (filter≡filterN p bs)
  filter≡filterN p (a ∷ as) | false = filter≡filterN p as

Exercise. Guess the types for filter≡filterP and filterN≡filterP. Argue which of these is easier to prove? Do it (and get the other one almost for free by transitivity).

Rewriting with with and Unification

When playing with the proofs about filters you might have noticed that with does something interesting with a goal.

In the following hole

  filter≡filterN₁ : {A : Set}  (p : A  Bool)  (as : List A)  filter p as ≡ filterN p as
  filter≡filterN₁ p [] = refl
  filter≡filterN₁ p (a ∷ as) = {!check me!}

the type of the goal is (filter p (a ∷ as) | p a) ≡ (filterN p (a ∷ as) | p a). But after the following with

  filter≡filterN₂ : {A : Set}  (p : A  Bool)  (as : List A)  filter p as ≡ filterN p as
  filter≡filterN₂ p [] = refl
  filter≡filterN₂ p (a ∷ as) with p a | as
  ... | r | rs = {!check me!}

it becomes (filter p (a ∷ rs) | r) ≡ (filterN p (a ∷ rs) | r).

Same things might happen not only to a goal but to a context as a whole:

  strange-id : {A : Set} {B : A  Set}  (a : A)  (b : B a)  B a
  strange-id {A} {B} a ba with B a
  ... | r = {!check me!}

in the hole, both the type of ba and the goal’s type are r.

From these observations we conclude that with expr creates a new variable, say w, and “backwards-substitutes” expr to w in a context, changing all the occurrences of expr in the types of the context to w. Which means that in a resulting context every type that had expr as a subterm starts dependending on w.

This property allows using with for rewriting:

  lemma-+zero′ :  a  a + zero ≡ a
  lemma-+zero′ zero = refl
  lemma-+zero′ (succ a) with a + zero | lemma-+zero′ a
  lemma-+zero′ (succ a) | ._ | refl = refl

  -- same expression with expanded underscore:
  lemma-+zero′₀ :  a  a + zero ≡ a
  lemma-+zero′₀ zero = refl
  lemma-+zero′₀ (succ a) with a + zero | lemma-+zero′₀ a
  lemma-+zero′₀ (succ a) | .a | refl = refl

In second clauses these terms a + zero is replaced by a new variable, say w, which gives lemma-+zero′ a : w ≡ a. Pattern matching on refl gives [w = a] and so the dotted pattern appears. After that the goal type becomes succ a ≡ succ a.

This pattern

f ps with a | eqn
... | ._ | refl = rhs

is so common that it has its own shorthand:

f ps rewrite eqn = rhs

Exercise. Prove (on paper) that rewriting a goal type with with and propositional equality is a syntax sugar for expressions built from refl, sym, trans and cong.

Universes and postulates

When moving from Haskell to Agda expression “every type is of kind *, i.e. for any type X, X : *” transforms into “every ground type is of type Set, i.e. for any ground type X, X : Set”. If we are willing to be consistent, we can’t afford Set : Set because it leads to a number of paradoxes (more on them below). Still, we might want to construct things like “a list of types” and our current implementation of List can not express this.

To solve this problem Agda introduces an infinite tower of Sets, i.e. Set0 : Set1, Set1 : Set2, and so on with Set being an alias for Set0. Agda is also a predicative system which means that Set0 → Set0 : Set1, Set0 → Set1 : Set2, and so on, but not Set0 → Set1 : Set1. Note, however, that this tower is not cumulative, e.g. Set0 : Set2 and Set0 → Set1 : Set3 are false typing judgments.

[As far as I know, in theory nothing prevents us from making the tower cumulative, it’s just so happened that Agda selected this route and not another. Predicativity is a much more subtle matter (more on that below).]

A list of types now becomes:

  data List1 (A : Set1) : Set1 where
    []  : List1 A
    __ : A  List1 A  List1 A

which looks very much like the usual List definition.

To prevent a code duplication like that Agda allows universe polymorphic definitions by, essentially, defining a type for universe levels, a type very reminiscent of :

  data Level : ? where
    lzero : Level
    lsucc : Level  Level

  -- input for ⊔ is \sqcup
  -- maximum of two levels
  __ : Level  Level  Level
  lzero   ⊔ m = m
  lsucc n ⊔ lzero = lsucc n
  lsucc n ⊔ lsucc m = lsucc (n ⊔ m)

though, since there’s nothing we could possibly write in the place of ? there, older versions of Agda allowed us to define Level with postulate syntax instead:

  postulate Level : Set
  postulate lzero : Level
  postulate lsucc : Level  Level
  postulate __   : Level  Level  Level

followed by some BUILTIN pragmas:

  {-# BUILTIN LEVEL     Level #-}
  {-# BUILTIN LEVELZERO lzero #-}
  {-# BUILTIN LEVELSUC  lsucc #-}
  {-# BUILTIN LEVELMAX  _⊔_   #-}

Modern versions require the use of Agda.Primitive module instead:

  open import Agda.Primitive public

In practice, the difference between and Level is that we are not allowed to pattern match on elements of the latter.

Though, postulates still exist and allow you to define propositions without proofs, i.e. they say “trust me, I know this to be true”. Obviously, this can be exploited to infer contradictions

  postulate undefined : {A : Set}  A

and will produce errors when type-checking in safe mode and will abort the program on attempts at executing them.

Given the definition above, expression Set α for α : Level means “the Set of level α”.

We are now able to define universe polymorphic list in the following way:

  data PList₀ {α : Level} (A : Set α) : Set α where
    []  : PList₀ A
    __ : A  PList₀ A  PList₀ A

  -- or a bit nicer:
  data PList₁ {α} (A : Set α) : Set α where
    []  : PList₁ A
    __ : A  PList₁ A  PList₁ A

The Library

Modules and the end of throw away code

Note that we have been writing everything above inside a module called ThrowAwayIntroduction. From here on we are going to (mostly) forget about it and write a small standard library for Agda from scratch. The idea is to remove any module with a name prefixed by “ThrowAway” from this file to produce the library code. To make the implementation of this idea as simple as possible we place markers like:

{- end of ThrowAwayIntroduction -}

at the ends of throw away code. It allows to generate the library by a simple shell command:

cat BrutalDepTypes.lagda | sed '/^\\begin{code}/,/^\\end{code}/ ! d; /^\\begin{code}/ d; /^\\end{code}/ c \
' | sed '/^ *module ThrowAway/,/^ *.- end of ThrowAway/ d;'

We are now going to redefine everything useful from above in a universe polymorphic way (when applicable).

Each module in Agda has an export list. Everything defined in a module gets appended to it. To place things defined for export in another module into a current context there is an open construct:

open ModuleName

This doesn’t append ModuleName’s export list to current module’s export list. To do that we need to add public keyword at the end:

open import Agda.Primitive public

Common functions with types not for the faint of heart

Exercise. Understand what is going on in types of the following functions:

module Function where
  -- Dependent application
  infixl 0 _$_
  _$_ :  {α β}
       {A : Set α} {B : A  Set β}
       (f : (x : A)  B x)
       ((x : A)  B x)
  f $ x = f x

  -- Simple application
  infixl 0 _$′_
  _$′_ :  {α β}
       {A : Set α} {B : Set β}
       (A  B)  (A  B)
  f $′ x = f $ x

  -- input for ∘ is \o
  -- Dependent composition
  __ :  {α β γ}
       {A : Set α} {B : A  Set β} {C : {x : A}  B x  Set γ}
       (f : {x : A}  (y : B x)  C y)
       (g : (x : A)  B x)
       ((x : A)  C (g x))
  f ∘ g = λ x  f (g x)

  -- Simple composition
  _∘′_ :  {α β γ}
       {A : Set α} {B : Set β} {C : Set γ}
       (B  C)  (A  B)  (A  C)
  f ∘′ g = f ∘ g

  -- Flip
  flip :  {α β γ}
        {A : Set α} {B : Set β} {C : A  B  Set γ}
        ((x : A)  (y : B)  C x y)
        ((y : B)  (x : A)  C x y)
  flip f x y = f y x

  -- Identity
  id :  {α} {A : Set α}  A  A
  id x = x

  -- Constant function
  const :  {α β}
        {A : Set α} {B : Set β}
        (A  B  A)
  const x y = x

open Function public

Especially note the scopes of variable bindings in types.

Logic

Intuitionistic Logic module:

module Logic where
  -- input for ⊥ is \bot
  -- False proposition
  data: Set where

  -- input for ⊤ is \top
  -- True proposition
  record: Set where

  -- ⊥ implies anything at any universe level
  ⊥-elim :  {α} {A : Set α}  A
  ⊥-elim ()

Propositional negation is defined as follows:

  -- input for ¬ is \lnot
  ¬ :  {α}  Set α  Set α
  ¬ P = P 

The technical part of the idea of this definition is that the principle of explosion (“from a contradiction, anything follows”) gets a pretty straightforward proof.

Exercise. Prove the following propositions:

  module ThrowAwayExercise where
    contradiction :  {α β} {A : Set α} {B : Set β}  A  ¬ A  B
    contradiction = {!!}

    contraposition :  {α β} {A : Set α} {B : Set β}  (A  B)  (¬ B  ¬ A)
    contraposition = {!!}

    contraposition¬ :  {α β} {A : Set α} {B : Set β}  (A  ¬ B)  (B  ¬ A)
    contraposition¬ = {!!}

    →¬² :  {α} {A : Set α}  A  ¬ (¬ A)
    →¬² a = {!!}

    ¬³→¬ :  {α} {A : Set α}  ¬ (¬ (¬ A))  ¬ A
    ¬³→¬ = {!!}

Hint. Use C-c C-, here to see the goal type in its normal form.

From a more logical standpoint the idea of ¬ is that false proposition P should be isomorphic to (i.e. they should imply each other: ⊥ → P ∧ P → ⊥). Since ⊥ → P is true for all P there is only P → ⊥ left for us to prove.

From a computational point of view having a variable of type in a context means that there is no way execution of a program could reach this point. Which means we can match on the variable and use absurd pattern, ⊥-elim does exactly that.

Note that, being an intuitionistic system, Agda has no means to prove “double negation” rule. See for yourself:

    ¬²→ :  {α} {A : Set α}  ¬ (¬ A)  A
    ¬²→ ¬¬a = {!check me!}
  {- end of ThrowAwayExercise -}

Fun fact: proofs in the exercise above amounted to a scientific paper at the start of 20th century.

Solution for the exercise:

  private
   module DummyAB {α β} {A : Set α} {B : Set β} where
    contradiction : A  ¬ A  B
    contradiction a ¬a = ⊥-elim (¬a a)

    contraposition : (A  B)  (¬ B  ¬ A)
    contraposition = flip _∘′_

    contraposition¬ : (A  ¬ B)  (B  ¬ A)
    contraposition¬ = flip

  open DummyAB public

  private
   module DummyA {α} {A : Set α} where
    →¬² : A  ¬ (¬ A)
    →¬² = contradiction

    ¬³→¬ : ¬ (¬ (¬ A))  ¬ A
    ¬³→¬ ¬³a = ¬³a ∘′ →¬²

  open DummyA public

Exercise. Understand this solution.

Note clever module usage. Opening a module with parameters prefixes types of all the things defined there with these parameters. We will use this trick a lot.

Let us define conjunction, disjunction, and logical equivalence:

  -- input for ∧ is \and
  infixr 6 __ _,_
  record __ {α β} (A : Set α) (B : Set β) : Set (α ⊔ β) where
    constructor _,_
    field
      fst : A
      snd : B

  open __ public

  -- input for ∨ is \or
  data __ {α β} (A : Set α) (B : Set β) : Set (α ⊔ β) where
    inl : A  A ∨ B
    inr : B  A ∨ B

  un∨ :  {α} {A : Set α}  A ∨ A  A
  un∨ (inl a) = a
  un∨ (inr a) = a

  -- input for ↔ is \<->
  __ :  {α β} (A : Set α) (B : Set β)  Set (α ⊔ β)
  A ↔ B = (A  B)(B  A)

Make all this goodness available:

open Logic public

MLTT: types and properties

Some definitions from Per Martin-Löf’s type theory [14]:

module MLTT where
  -- input for ≡ is \==
  -- Propositional equality
  infix 4 __
  data __ {α} {A : Set α} (x : A) : A  Set α where
    refl : x ≡ x

  __ :  {α} {A : Set α} (x : A)  A  Set α
  x ≠ y = ¬ (x ≡ y)

  -- input for Σ is \Sigma
  -- Dependent pair
  infixr 6 _,_
  record Σ {α β} (A : Set α) (B : A  Set β) : Set (α ⊔ β) where
    constructor _,_
    field
      projl : A
      projr : B projl

  open Σ public

  -- Make rewrite syntax work
  {-# BUILTIN EQUALITY _≡_ #-}

The Σ type is a dependent version of _∧_ (the second field depends on the first), i.e. _∧_ is a specific case of Σ:

  -- input for × is \x
  _×_ :  {α β} (A : Set α) (B : Set β)  Set (α ⊔ β)
  A × B = Σ A  _  B)

  ×↔∧ :  {α β} {A : Set α} {B : Set β}  (A × B)(A ∧ B)
  ×↔∧ =  z  projl z , projr z) ,  z  fst z , snd z)

Personally, I use both _∧_ and _×_ occasionally since _×_ looks ugly in the normal form and makes goal types hard to read.

Some properties:

  module ≡-Prop where
   private
    module DummyA {α} {A : Set α} where
      -- _≡_ is symmetric
      sym : {x y : A}  x ≡ y  y ≡ x
      sym refl = refl

      -- _≡_ is transitive
      trans : {x y z : A}  x ≡ y  y ≡ z  x ≡ z
      trans refl refl = refl

      -- _≡_ is substitutive
      subst :  {γ} (P : A  Set γ) {x y}  x ≡ y  P x  P y
      subst P refl p = p

    private
     module DummyAB {α β} {A : Set α} {B : Set β} where
      -- _≡_ is congruent
      cong :  (f : A  B) {x y}  x ≡ y  f x ≡ f y
      cong f refl = refl

      subst₂ :  {} {P : A  B  Set} {x y u v}  x ≡ y  u ≡ v  P x u  P y v
      subst₂ refl refl p = p

    private
     module DummyABC {α β γ} {A : Set α} {B : Set β} {C : Set γ} where
      cong₂ :  (f : A  B  C) {x y u v}  x ≡ y  u ≡ v  f x u ≡ f y v
      cong₂ f refl refl = refl

    open DummyA public
    open DummyAB public
    open DummyABC public

Make all this goodness available:

open MLTT public

Decidable propositions

module Decidable where

Decidable proposition it is a proposition that has an explicit proof or disproval:

  data Dec {α} (A : Set α) : Set α where
    yes : ( a :   A)  Dec A
    no  : (¬a : ¬ A)  Dec A

This datatype is very much like Bool, except it also explains why the proposition holds or why it must not.

Decidable propositions are the glue that make your program work with the real world data.

Suppose we want to write a program that reads a natural number, say n, from stdin and divides it by two with div2E. To do that we need a proof that n is Even. The easiest way to do it is to define a function that decides if a given natural is Even:

  module ThrowAwayExample₁ where
    open ThrowAwayIntroduction

    ¬Even+2 :  {n}  ¬ (Even n)  ¬ (Even (succ (succ n)))
    ¬Even+2 ¬en (e2succ en) = contradiction en ¬en

    Even? :  n  Dec (Even n)
    Even? zero        = yes ezero
    Even? (succ zero) = no  ()) -- note an absurd pattern in
                                  -- an anonymous lambda expression
    Even? (succ (succ n)) with Even? n
    ... | yes a       = yes (e2succ a)
    ... | no  a¬      = no (¬Even+2 a¬)
  {- end of ThrowAwayExample₁ -}

then read n from stdin, feed it to Even?, match on the result and call div2E if n is Even.

Same idea applies to almost everything:

Using same idea we can define decidable dichotomous and trichotomous propositions:

  data Di {α β} (A : Set α) (B : Set β) : Set (α ⊔ β) where
    diyes : ( a :   A) (¬b : ¬ B)  Di A B
    dino  : (¬a : ¬ A) ( b :   B)  Di A B

  data Tri {α β γ} (A : Set α) (B : Set β) (C : Set γ) : Set (α ⊔ (β ⊔ γ)) where
    tri< : ( a :   A) (¬b : ¬ B) (¬c : ¬ C)  Tri A B C
    tri≈ : (¬a : ¬ A) ( b :   B) (¬c : ¬ C)  Tri A B C
    tri> : (¬a : ¬ A) (¬b : ¬ B) ( c :   C)  Tri A B C

Make all this goodness available:

open Decidable public

Natural numbers: operations, properties and relations

Consider this to be the answer (encrypted with rewrites) for the exercise way above:

module Data-ℕ where
  -- Natural numbers (positive integers)
  data: Set where
    zero :
    succ :

  module ℕ-Rel where
    infix 4 __ _<_ _>_

    data __ : Set where
      z≤n :  {n}            zero ≤ n
      s≤s :  {n m}  n ≤ m  succ n ≤ succ m

    _<_ : Set
    n < m = succ n ≤ m

    _>_ : Set
    n > m = m < n

    ≤-unsucc :  {n m}  succ n ≤ succ m  n ≤ m
    ≤-unsucc (s≤s a) = a

    <-¬refl :  n  ¬ (n < n)
    <-¬refl zero ()
    <-¬refl (succ n) (s≤s p) = <-¬refl n p

    ≡→≤ :  {n m}  n ≡ m  n ≤ m
    ≡→≤ {zero}   refl = z≤n
    ≡→≤ {succ n} refl = s≤s (≡→≤ {n} refl) -- Note this

    ≡→¬< :  {n m}  n ≡ m  ¬ (n < m)
    ≡→¬< refl = <-¬refl _

    ≡→¬> :  {n m}  n ≡ m  ¬ (n > m)
    ≡→¬> refl = <-¬refl _

    <→¬≡ :  {n m}  n < m  ¬ (n ≡ m)
    <→¬≡ = contraposition¬ ≡→¬<

    >→¬≡ :  {n m}  n > m  ¬ (n ≡ m)
    >→¬≡ = contraposition¬ ≡→¬>

    <→¬> :  {n m}  n < m  ¬ (n > m)
    <→¬> {zero} (s≤s z≤n) ()
    <→¬> {succ n} (s≤s p<) p> = <→¬> p< (≤-unsucc p>)

    >→¬< :  {n m}  n > m  ¬ (n < m)
    >→¬< = contraposition¬ <→¬>

  module ℕ-Op where
    open ≡-Prop

    pred :
    pred zero = zero
    pred (succ n) = n

    infixl 6 _+_
    _+_ :
    zero   + n = n
    succ n + m = succ (n + m)

    infixl 6 _-_
    _-_ :
    zero   - n      = zero
    n      - zero   = n
    succ n - succ m = n - m

    infixr 7 _*_
    _*_ :
    zero   * m = zero
    succ n * m = m + (n * m)

    private
     module Dummy₀ where
      lemma-+zero :  a  a + zero ≡ a
      lemma-+zero zero = refl
      lemma-+zero (succ a) rewrite lemma-+zero a = refl

      lemma-+succ :  a b  succ a + b ≡ a + succ b
      lemma-+succ zero b = refl
      lemma-+succ (succ a) b rewrite lemma-+succ a b = refl

    open Dummy₀

    -- + is associative
    +-assoc :  a b c  (a + b) + c ≡ a + (b + c)
    +-assoc zero b c = refl
    +-assoc (succ a) b c rewrite (+-assoc a b c) = refl

    -- + is commutative
    +-comm :  a b  a + b ≡ b + a
    +-comm zero b = sym $ lemma-+zero b
    +-comm (succ a) b rewrite +-comm a b | lemma-+succ b a = refl

    -- * is distributive by +
    *+-dist :  a b c  (a + b) * c ≡ a * c + b * c
    *+-dist zero b c = refl
    *+-dist (succ a) b c rewrite *+-dist a b c | +-assoc c (a * c) (b * c) = refl

    -- * is associative
    *-assoc :  a b c  (a * b) * c ≡ a * (b * c)
    *-assoc zero b c = refl
    *-assoc (succ a) b c rewrite *+-dist b (a * b) c | *-assoc a b c = refl

    private
     module Dummy₁ where
      lemma-*zero :  a  a * zero ≡ zero
      lemma-*zero zero = refl
      lemma-*zero (succ a) = lemma-*zero a

      lemma-+swap :  a b c  a + (b + c) ≡ b + (a + c)
      lemma-+swap a b c rewrite sym (+-assoc a b c) | +-comm a b | +-assoc b a c = refl

      lemma-*succ :  a b  a + a * b ≡ a * succ b
      lemma-*succ zero b = refl
      lemma-*succ (succ a) b rewrite lemma-+swap a b (a * b) | lemma-*succ a b = refl

    open Dummy₁

    -- * is commutative
    *-comm :  a b  a * b ≡ b * a
    *-comm zero b = sym $ lemma-*zero b
    *-comm (succ a) b rewrite *-comm a b | lemma-*succ b a = refl

  module ℕ-RelOp where
    open ℕ-Rel
    open ℕ-Op
    open ≡-Prop

    infix 4 _≡?_ _≤?_ _<?_

    _≡?_ : (n m :)  Dec (n ≡ m)
    zero    ≡? zero   = yes refl
    zero    ≡? succ m = no  ())
    succ n  ≡? zero   = no  ())
    succ n  ≡? succ m with n ≡? m
    succ .m ≡? succ m | yes refl = yes refl
    succ n  ≡? succ m | no ¬a    = no (¬a ∘ cong pred) -- Note this

    _≤?_ : (n m :)  Dec (n ≤ m)
    zero ≤? m = yes z≤n
    succ n ≤? zero = no  ())
    succ n ≤? succ m with n ≤? m
    ... | yes a = yes (s≤s a)
    ... | no ¬a = no (¬a ∘ ≤-unsucc)

    _<?_ : (n m :)  Dec (n < m)
    n <? m = succ n ≤? m

    cmp : (n m :)  Tri (n < m) (n ≡ m) (n > m)
    cmp zero zero     = tri≈  ()) refl  ())
    cmp zero (succ m) = tri< (s≤s z≤n)  ())  ())
    cmp (succ n) zero = tri>  ())  ()) (s≤s z≤n)
    cmp (succ n) (succ m) with cmp n m
    cmp (succ n) (succ m) | tri< a ¬b ¬c = tri< (s≤s a) (¬b ∘ cong pred) (¬c ∘ ≤-unsucc)
    cmp (succ n) (succ m) | tri≈ ¬a b ¬c = tri≈ (¬a ∘ ≤-unsucc) (cong succ b) (¬c ∘ ≤-unsucc)
    cmp (succ n) (succ m) | tri> ¬a ¬b c = tri> (¬a ∘ ≤-unsucc) (¬b ∘ cong pred) (s≤s c)

open Data-ℕ public

Exercise. Understand this. Now, remove all term bodies from ℕ-RelProp and ℕ-RelOp and reimplement everything yourself.

Lists and Vectors

module Data-List where
  -- List
  infixr 10 __
  data List {α} (A : Set α) : Set α where
    []  : List A
    __ : A  List A  List A

  module List-Op where
  private
   module DummyA {α} {A : Set α} where
    -- Singleton `List`
    [_] : A  List A
    [ a ] = a ∷ []

    -- Concatenation for `List`s
    infixr 10 _++_
    _++_ : List A  List A  List A
    []       ++ bs = bs
    (a ∷ as) ++ bs = a ∷ (as ++ bs)

    -- Filtering with decidable propositions
    filter :  {β} {P : A  Set β}  (∀ a  Dec (P a))  List A  List A
    filter p [] = []
    filter p (a ∷ as) with p a
    ... | yes _ = a ∷ (filter p as)
    ... | no  _ = filter p as

  open DummyA public

module Data-Vec where
  -- Vector
  infixr 5 __
  data Vec {α} (A : Set α) : Set α where
    []  : Vec A zero
    __ :  {n}  A  Vec A n  Vec A (succ n)

  module Vec-Op where
    open ℕ-Op

    private
     module DummyA {α} {A : Set α} where
      -- Singleton `Vec`
      [_] : A  Vec A (succ zero)
      [ a ] = a ∷ []

      -- Concatenation for `Vec`s
      infixr 5 _++_
      _++_ :  {n m}  Vec A n  Vec A m  Vec A (n + m)
      []       ++ bs = bs
      (a ∷ as) ++ bs = a ∷ (as ++ bs)

      head :  {n}  Vec A (succ n)  A
      head (a ∷ as) = a

      tail :  {n}  Vec A (succ n)  Vec A n
      tail (a ∷ as) = as

    open DummyA public
{-
Work in progress. TODO.

I find the following definition for List to be quite amusing in practice:

module VecLists where
  open Data-Vec

  private
   module DummyA {α} {A : Set α} where
     VecList = Σ ℕ (Vec A)
-}

Being in a List

Indexing allows to define pretty fun things:

module ThrowAwayMore₁ where
  open Data-List
  open List-Op

  -- input for ∈ is \in
  -- `a` is in `List`
  data __ {α} {A : Set α} (a : A) : List A  Set α where
    here  :  {as}    a ∈ (a ∷ as)
    there :  {b as}  a ∈ as  a ∈ (b ∷ as)

  -- input for ⊆ is \sub=
  -- `xs` is a subset of `ys`
  __ :  {α} {A : Set α}  List A  List A  Set α
  as ⊆ bs =  {x}  x ∈ as  x ∈ bs

The _∈_ relation says that “being in a List” for an element a : A means that a in the head of a List or in the tail of a List. For some a and as a value of type a ∈ as, that is “a is in a list as” is a position of an element a in as (there might be any number of elements in this type). Relation , that is “being a sublist”, carries a function that for each a in xs gives its position in as.

Examples:

  listTest₁ = zero ∷ zero ∷ succ zero ∷ []
  listTest₂ = zero ∷ succ zero ∷ []

  ∈Test₀ : zero ∈ listTest₁
  ∈Test₀ = here

  ∈Test₁ : zero ∈ listTest₁
  ∈Test₁ = there here

  ⊆Test : listTest₂ ⊆ listTest₁
  ⊆Test here = here
  ⊆Test (there here) = there (there here)
  ⊆Test (there (there ()))

Let us prove some properties for relation:

  ⊆-++-left :  {A : Set} (as bs : List A)  as ⊆ (bs ++ as)
  ⊆-++-left as [] n = n
  ⊆-++-left as (b ∷ bs) n = there (⊆-++-left as bs n)

  ⊆-++-right :  {A : Set} (as bs : List A)  as ⊆ (as ++ bs)
  ⊆-++-right [] bs ()
  ⊆-++-right (a ∷ as) bs here = here
  ⊆-++-right (a ∷ as) bs (there n) = there (⊆-++-right as bs n)
{- end of ThrowAwayMore₁ -}

Note how these proofs renumber elements of a given list.

Being in a List generalized: Any

By generalizing _∈_ relation from propositional equality (in x ∈ (x ∷ xs) both xs are propositionally equal) to arbitrary predicates we arrive at:

module Data-Any where
  open Data-List
  open List-Op

  -- Some element of a `List` satisfies `P`
  data Any {α γ} {A : Set α} (P : A  Set γ) : List A  Set (α ⊔ γ) where
    here  :  {a as}  (pa  : P a)       Any P (a ∷ as)
    there :  {a as}  (pas : Any P as)  Any P (a ∷ as)

  module Membership {α β γ} {A : Set α} {B : Set β} (P : B  A  Set γ) where
    -- input for ∈ is \in
    -- `P b a` holds for some element `a` from the `List`
    -- when P is `_≡_` this becomes the usual "is in" relation
    __ : B  List A  Set (α ⊔ γ)
    b ∈ as = Any (P b) as

    -- input for ∉ is \notin
    __ : B  List A  Set (α ⊔ γ)
    b ∉ as = ¬ (b ∈ as)

    -- input for ⊆ is \sub=
    __ : List A  List A  Set (α ⊔ β ⊔ γ)
    as ⊆ bs =  {x}  x ∈ as  x ∈ bs

    -- input for ⊈ is \sub=n
    __ : List A  List A  Set (α ⊔ β ⊔ γ)
    as ⊈ bs = ¬ (as ⊆ bs)

    -- input for ⊇ is \sup=
    _⊆⊇_ : List A  List A  Set (α ⊔ β ⊔ γ)
    as ⊆⊇ bs = (as ⊆ bs)(bs ⊆ as)

    ⊆-refl :  {as}  as ⊆ as
    ⊆-refl = id

    ⊆-trans :  {as bs cs}  as ⊆ bs  bs ⊆ cs  as ⊆ cs
    ⊆-trans f g = g ∘ f

    ⊆⊇-refl :  {as}  as ⊆⊇ as
    ⊆⊇-refl = id , id

    ⊆⊇-sym :  {as bs}  as ⊆⊇ bs  bs ⊆⊇ as
    ⊆⊇-sym (f , g) = g , f

    ⊆⊇-trans :  {as bs cs}  as ⊆⊇ bs  bs ⊆⊇ cs  as ⊆⊇ cs
    ⊆⊇-trans f g = (fst g ∘ fst f) , (snd f ∘ snd g)

    ∉[] :  {b}  b ∉ []
    ∉[]()

    -- When P is `_≡_` this becomes `b ∈ [ a ] → b ≡ a`
    ∈singleton→P :  {a b}  b ∈ [ a ]  P b a
    ∈singleton→P (here pba) = pba
    ∈singleton→P (there ())

    P→∈singleton :  {a b}  P b a  b ∈ [ a ]
    P→∈singleton pba = here pba

    ⊆-++-left :  as bs  as ⊆ (bs ++ as)
    ⊆-++-left as [] n = n
    ⊆-++-left as (b ∷ bs) n = there (⊆-++-left as bs n)

    ⊆-++-right :  as bs  as ⊆ (as ++ bs)
    ⊆-++-right [] bs ()
    ⊆-++-right (x ∷ as) bs (here pa) = here pa
    ⊆-++-right (x ∷ as) bs (there n) = there (⊆-++-right as bs n)

    ⊆-++-both-left :  {as bs} cs  as ⊆ bs  (cs ++ as)(cs ++ bs)
    ⊆-++-both-left [] as⊆bs n = as⊆bs n
    ⊆-++-both-left (x ∷ cs) as⊆bs (here pa) = here pa
    ⊆-++-both-left (x ∷ cs) as⊆bs (there n) = there (⊆-++-both-left cs as⊆bs n)

    ⊆-filter :  {σ} {Q : A  Set σ}  (q :  x  Dec (Q x))  (as : List A)  filter q as ⊆ as
    ⊆-filter q [] ()
    ⊆-filter q (a ∷ as) n with q a
    ⊆-filter q (a ∷ as) (here pa) | yes qa = here pa
    ⊆-filter q (a ∷ as) (there n) | yes qa = there (⊆-filter q as n)
    ⊆-filter q (a ∷ as) n         | no ¬qa = there (⊆-filter q as n)

Exercise. Note how general this code is. ⊆-filter covers a broad set of propositions, with “filtered list is a sublist (in the usual sense) of the original list” being a special case. Do C-c C-. in the following goal and explain the type:

module ThrowAwayMore₂ where
  goal = {!Data-Any.Membership.⊆-filter!}
{- end of ThrowAwayMore₂ -}

Explain the types of all the terms in Membership module.

Dual predicate: All

{-
Work in progress. TODO.

I didn't have a chance to use `All` yet (and I'm too lazy to implement this module right now),
but here is the definition:

module Data-All where
  open Data-List
  -- All elements of a `List` satisfy `P`
  data All {α β} {A : Set α} (P : A → Set β) : List A → Set (α ⊔ β) where
    []∀  : All P []
    _∷∀_ : ∀ {a as} → P a → All P as → All P (a ∷ as)
-}

Chained values

Work in progress. TODO.

module Data-Chain where
  open Data-List
  open List-Op

  data Chain {α γ} {A : Set α} (P : A  A  Set γ) : List A  Set (α ⊔ γ) where
    []c  : Chain P []
    [1]c :  {a}  Chain P (a ∷ [])
    _∷c_ :  {a b bs}  P a b  Chain P (b ∷ bs)  Chain P (a ∷ b ∷ bs)

Booleans

Are not that needed with Dec, actually, but lets define them for completeness:

module Data-Bool where
  -- Booleans
  data Bool : Set where
    true false : Bool

  module Bool-Op where
    if_then_else_ :  {α} {A : Set α}  Bool  A  A  A
    if true  then a else _ = a
    if false then _ else b = b

    not : Bool  Bool
    not true  = false
    not false = true

    and : Bool  Bool  Bool
    and true  x = x
    and false _ = false

    or : Bool  Bool  Bool
    or false x = x
    or true  x = true

open Data-Bool public

Other stuff

Work in progress. TODO. We need to prove something from A to Z. Quicksort maybe.

Pre-theoretical corner

This section discusses interesting things about Agda which are somewhere in between practice and pure theory.

module ThrowAwayPreTheory where
  open ≡-Prop
  open ℕ-Op

Equality and unification

Rewriting with equality hides a couple of catches.

Remember the term of lemma-+zero′ from above:

  lemma-+zero′ :  a  a + zero ≡ a
  lemma-+zero′ zero = refl
  lemma-+zero′ (succ a) with a + zero | lemma-+zero′ a
  lemma-+zero′ (succ a) | ._ | refl = refl

it typechecks, but the following proof doesn’t:

  lemma-+zero′′ :  a  a + zero ≡ a
  lemma-+zero′′ zero = refl
  lemma-+zero′′ (succ a) with a | lemma-+zero′′ a
  lemma-+zero′′ (succ a) | ._ | refl = refl

The problem here is that for arbitrary terms A and B to pattern match on refl : A ≡ B these A and B must unify. In lemma-+zero′ case we have a + zero backward-substituted into a new variable w, then we match on refl we get w ≡ a. On the other hand, in lemma-+zero′′ case we have a changed into w, an refl gets w + zero ≡ w type which is a malformed (recursive) unification constraint.

There is another catch. Our current definition of _≡_ allows to express equality on types, e.g. Bool ≡ ℕ.

This enables us to write the following term:

  lemma-unsafe-eq : (P : Bool ≡ ℕ)  Bool 
  lemma-unsafe-eq P b with Bool | P
  lemma-unsafe-eq P b | .| refl = b + succ zero

which type checks without errors.

Note, however, that lemma-unsafe-eq cannot be proven by simply pattern matching on P:

  lemma-unsafe-eq₀ : (P : Bool ≡ ℕ)  Bool 
  lemma-unsafe-eq₀ refl b = b
{- end of ThrowAwayPreTheory -}

Exercise. lemma-unsafe-eq is food for thought about computation safety under false assumptions.

Theoretical corner

In this section we shall talk about some theoretical stuff like datatype encodings and paradoxes. You might want to read some of the theoretical references like [12,14] first.

module ThrowAwayTheory where

In literature Agda’s arrow (x : X) → Y (where Y might have x free) is called dependent product type, or Π-type (“Pi-type”) for short. Dependent pair Σ is called dependent sum type, or Σ-type (“Sigma-type”) for short.

Finite types

Given , and Bool it is possible to define any finite type, that is a type with finite number of elements.

  module FiniteTypes where
    open Bool-Op

    _∨′_ : (A B : Set)  Set
    A ∨′ B = Σ Bool  x  if x then A else B)

    zero′  =
    one′   =
    two′   = Bool
    three′ = one′ ∨′ two′
    four′  = two′ ∨′ two′
    --- and so on

TODO. Say something about extensional setting and ⊤ = ⊥ → ⊥.

Simple datatypes

  module ΠΣ-Datatypes where

Given finite types, Π-types, and Σ-types it is possible to define non-inductive datatypes using the same scheme the definition of _∨′_ uses.

Non-inductive datatype without indexes has the following scheme:

data DataTypeName (Param1 : Param1Type) (Param2 : Param2Type) ... : Set whatever
  Cons1 : (Cons1Arg1 : Cons1Arg1Type) (Cons1Arg2 : Cons1Arg2Type) ...  DataTypeName Param1 Param2 ...
  Cons2 : (Cons2Arg1 : Cons2Arg1Type) ...  DataTypeName Param1 Param2 ...
  ...
  ConsN : (ConsNArg1 : ConsNArg1Type) ...  DataTypeName Param1 Param2 ...

Re-encoded into Π-types, Σ-types, and finite types it becomes:

DataTypeName : (Param1 : Param1Type) (Param2 : Param2Type) ...  Set whatever
DataTypeName Param1 Param2 ... = Σ FiniteTypeWithNElements choice where
  choice : FiniteTypeWithNElements  Set whatever
  choice element1 = Σ Cons1Arg1Type  Cons1Arg1  Σ Cons1Arg2Type  Cons1Arg2  ...))
  choice element2 = Σ Cons2Arg1Type  Cons2Arg1  ...)
  ...
  choice elementN = Σ ConsNArg1Type  ConsNArg1  ...)

For instance, Di type from above translates into:

    Di′ :  {α β} (A : Set α) (B : Set β)  Set (α ⊔ β)
    Di′ {α} {β} A B = Σ Bool choice where
      choice : Bool  Set (α ⊔ β)
      choice true  = A × ¬ B
      choice false = ¬ A × B

Datatypes with indices

Work in progress. TODO. The general idea: add them as parameters and place an equality proof inside.

Recursive datatypes

Work in progress. TODO. General ideas: W-types and μ.

Curry’s paradox

Negative occurrences make the system inconsistent.

Copy this to a separate file and typecheck:

{-# OPTIONS --no-positivity-check #-}
module CurrysParadox where
  data CS (C : Set) : Set where
    cs : (CS C  C)  CS C

  paradox :  {C}  CS C  C
  paradox (cs b) = b (cs b)

  loop :  {C}  C
  loop = paradox (cs paradox)

  contr :
  contr = loop

Universes and impredicativity

Work in progress. TODO.

{- end of ThrowAwayTheory -}

References

1. Malakhovski J. Agda-mode commands to show and use unification constraints between "Goal" and "Have". https://web.archive.org/web/20140326144257/http://code.google.com/p/agda/issues/detail?id=771.

2. Malakhovski J. Functional Programming and Proof Checking Course. https://oxij.org/teaching/itmo/fp/.

3. Agda Project Authors. Agda: Tutorials list. 2024. https://agda.readthedocs.io/en/latest/getting-started/tutorial-list.html.

4. Agda Project Authors. Agda: Documentation. 2024. https://agda.readthedocs.io/en/latest/index.html.

6. Bove A., Dybjer P. Dependent types at work. https://www.cse.chalmers.se/~peterd/papers/DependentTypesAtWork.pdf.

7. Norell U. Dependently typed programming in agda. 2008. https://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf.

8. Altenkirch T. Computer aided formal reasoning. https://www.cs.nott.ac.uk/~txa/g53cfr/.

9. Coq Project Authors. Coq: Documentation. 2015. https://coq.inria.fr/documentation.

10. Idris Project Authors. Idris: Documentation. 2015. https://www.idris-lang.org/pages/documentation.html.

11. Epigram. http://www.e-pig.org/.

12. Sørensen M.H.B., Urzyczyn P. Lectures on the Curry-Howard isomorphism. 1998. https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7385.

13. Thompson S. Type theory and functional programming. https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/.

14. Martin-Löf P. Intuitionistic type theory. Notes by Giovanni Sambin. 1980. https://web.archive.org/web/20230930164024/https://www.csie.ntu.edu.tw/~b94087/ITT.pdf.

15. Martin-Löf P. Intuitionistic type theory. https://intuitionistic.files.wordpress.com/2010/07/martin-lof-tt.pdf.

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18. Bauer A. How to implement dependent type theory I. https://math.andrej.com/2012/11/08/how-to-implement-dependent-type-theory-i/.

19. Bauer A. How to implement dependent type theory II. https://math.andrej.com/2012/11/11/how-to-implement-dependent-type-theory-ii/.

20. Bauer A. How to implement dependent type theory III. https://math.andrej.com/2012/11/29/how-to-implement-dependent-type-theory-iii/.

21. Nix: Purely functional package manager. https://nixos.org/nix/.

22. McBride C., McKinna J. The view from the left. http://strictlypositive.org/view.ps.gz.


  1. (As of writing of Version 15 of this document) Most of that proposal was implemented between 2013 and 2024: agda2-mode now has agda2-solve-maybe-all and agda2-elaborate-give commands. But multi-goal tracking still does not exists, unfortunately.↩︎

  2. By the way, this document is far from finished, but should be pretty useful in its current state.↩︎

  3. (As of Version 15) 11 years later and still unfinished…↩︎