annah-1.0.0: src/Annah/Tutorial.hs
{-| Annah is a tiny language that serves to illustrate how various programming
constructs can be desugared to lambda calculus. The most sophisticated
feature that Annah supports is desugaring mutually recursive datatypes
to non-recursive lambda expressions.
Annah is not intended to be used as a production language. Rather, Annah is
a step along the way towards a production language that I factored out as
a reusable library that others can learn from and possibly fork for their
own use cases.
Under the hood, all Annah expressions are translated to a minimalist
implementation of the calculus of constructions called Morte, which only
supports non-recursive lambda expressions and their types. You can find
the Morte compiler and library here:
<http://hackage.haskell.org/package/morte>
Annah piggybacks on Morte meaning all Annah expressions are translated to
Morte expressions and then those Morte expressions are type-checked and
evaluated. You cannot directly type-check or evaluate Annah expressions;
you have to desugar them to Morte expressions first before you can do
anything else with them.
Annah is not very user-friendly (and I apologize for that!). For example,
Annah reuses Morte's type-checker which means that error messages are in
terms of low-level lambda calculus expressions and not in terms of the
original Annah source code.
Most notably, Annah does not provide support for text, due to the gross
inefficiency of encoding even basic ASCII text in lambda calculus. Text
handling would be better served by a backend with primitive support for
text literals and operations on text.
This tutorial assumes that you have first read the Morte tutorial, which
you can find here:
<http://hackage.haskell.org/package/morte/docs/Morte-Tutorial.html>
Annah is a superset of Morte that implements many of the higher-level
constructs mentioned in the Morte tutorial, which is why you should not skip
reading the Morte tutorial.
-}
module Annah.Tutorial (
-- * Introduction
-- $introduction
-- * Let
-- $let
-- * Data types
-- $datatypes
-- * Imports
-- $imports
-- * Autogenerate Types
-- $types
-- * Folds
-- $folds
-- * Recursive types
-- $recursive
-- * Prelude
-- $prelude
-- * Natural numbers
-- $nats
-- * Lists
-- $lists
-- * Monoids
-- $monoids
-- * Commands
-- $commands
-- * IO
-- $io
-- * Paths
-- $paths
-- * Conclusion
-- $conclusion
) where
{- $introduction
This library comes with a binary executable that you can use to compile
Annah expressions to Morte expressions. This executable can be used in two
separate ways.
First, you can read an Annah expression from standard input and the program
will output the equivalent low-level Morte expression to standard output:
> $ annah
> type Bool
> data True
> data False
> fold if
> in
>
> let not (b : Bool) : Bool = if b Bool False True
> in not False
> <Ctrl-D>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
Second, you can read an Annah expression in from a file if you provide the
file name on the command line using the @compile@ subcommand:
> $ cat example.annah
> type Bool
> data True
> data False
> fold if
> in
>
> let not (b : Bool) : Bool = if b Bool False True
> in not False
> $ annah compile example.annah
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
Annah is a superset of Morte, so any Morte expression is also a valid Annah
expression:
> $ annah
> \(a : *) -> \(x : a) -> x
> <Ctrl-D>
> λ(a : *) → λ(x : a) → x
Like Morte, Annah is an explicitly typed language (i.e. no type inference).
-}
{- $let
Annah supports let expressions which can be used to introduce functions and
values. For example, this is how you can define the polymorphic identity
function in Annah:
> $ annah
> let id (a : *) (x : a) : a = x
> in id
> <Ctrl-D>
> λ(a : *) → λ(x : a) → x
You can define more than one thing in a let expression as long as you
prefix each definition with @let@:
> $ annah
> let id (a : *) (x : a) : a = x
> let const (a : *) (b : *) (x : a) (y : b) : a = x
> in id
> <Ctrl-D>
> λ(a : *) → λ(x : a) → x
The general form of a @let@ expression is:
> let f0 (x00 : _A00) (x01 : _A01) ... (x0j : _A0j) _B0 = b0
> let f1 (x10 : _A10) (x11 : _A11) ... (x1j : _A1j) _B1 = b1
> ...
> let fi (xi0 : _Ai0) (xi1 : _Ai1) ... (xij : _Aij) _Bi = bi
> in e
The above let expression desugars to the following lambda expression:
> ( λ(f0 : ∀(x00 : _A00) → ∀(x01 : _A01) → ... → ∀(x0j : _A0j) → _B0)
> → λ(f1 : ∀(x10 : _A10) → ∀(x11 : _A11) → ... → ∀(x1j : _A1j) → _B1)
> → ...
> → λ(fi : ∀(xi0 : _Ai0) → ∀(xi1 : _Ai1) → ... → ∀(xij : _Aij) → _Bi)
> → e
> )
>
> (λ(x00 : _A00) → λ(x01 : _A01) → ... → λ(x0j : _A0j) → b0)
> (λ(x10 : _A10) → λ(x11 : _A11) → ... → λ(x1j : _A1j) → b1)
> ...
> (λ(xi0 : _Ai0) → λ(xi1 : _Ai1) → ... → λ(xij : _Aij) → bi)
The above @\'e\'@ is the \"body\" of the let expression and @f0@ through
@fi@ are the \"let-bound terms\". Due to the above translation, each
\"let-bound\" term is only in scope for the \"body\" of the let-expression
and the types of subsequent \"let-bound\" terms.
To give a concrete example, our original @id@+@const@ let expression:
> let id (a : *) (x : a) : a = x
> let const (a : *) (b : *) (x : a) (y : b) : a = x
> in id
... was equivalent to:
> ( λ(id : ∀(a : *) → ∀(x : a) → a)
> → λ(const : ∀(a : *) → ∀(b : *) → ∀(x : a) → ∀(y : b) → a
> → id
> )
>
> (λ(a : *) → λ(x : a) → x)
> (λ(a : *) → λ(b : *) → λ(x : a) → λ(y : b) → x)
... which normalizes to:
> λ(a : *) → λ(x : a) → x
The definition of @const@ is dead code that is optimized away by β-reduction
because the let-bound @const@ term is never used within the body of the let
expression.
-}
{- $datatypes
Annah lets you define datatypes that scope over an expression. For example,
if you write:
> type Bool
> data True
> data False
> fold if
> in e
... then within the expression @\'e\'@ you will be able to use the @Bool@
type, the @True@ and @False@ values, and the @if@ fold.
The above definition of @Bool@ desugars to the following @let@ expression:
> let Bool : * = ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
> let True : Bool = λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
> let False : Bool = λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
> let if : Bool → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool =
> λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x
> in e
... which in turn desugars to:
> ( λ(Bool : *)
> → λ(True : Bool)
> → λ(False : Bool)
> → λ(if : Bool → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
> → e
> )
>
> (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
> (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
> (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
> (λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x)
Annah also supports recursive datatypes. For example, you can define
natural numbers like this:
> $ annah
> type Nat
> data Succ (pred : Nat)
> data Zero
> in Succ (Succ (Succ Zero))
> <Ctrl-D>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
Notice how we can omit the @fold@ line, which is optional.
You can also omit field names, too, and this code is also valid:
> $ annah
> type Nat
> data Succ Nat
> data Zero
> in Succ (Succ (Succ Zero))
> λ(Nat : *) → λ(Succ : Nat → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
Field names are just used to give nicer names to bound variables in the
desugared datatype definition and field names default to @\'_\'@ if you omit
the name.
You can find out how any given type or constructor is encoded by just
returning the constructor as the result of the let expression:
> $ annah
> type Nat
> data Succ (pred : Nat)
> data Zero
> in Succ
> λ(pred : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (pred Nat Succ Zero)
-}
{- $imports
Annah supports imports using the same syntax as Morte but you may only
import Morte expressions (/not/ Annah expressions). You can embed a file
path or http URL anywhere within an expression and Annah will substitute in
the Morte expression (encoded as plain text) located at that path or URL.
The reason Annah does not support importing Annah expressions is that Annah
does not actually resolve the imports. Annah piggybacks off of Morte's
support for imports, and Morte only supports importing Morte expressions.
Imports are extremely useful when combined with datatypes because you can
create a separate file for each type and constructor of a datatype. To
illustrate this we'll manually encode @Bool@, @True@, @False@, and @if@ as
separate Annah files (and later we will see how we can auto-generate these
files):
> $ cat Bool.annah
> type Bool
> data True
> data False
> fold if
> in Bool
> $ cat True.annah
> type Bool
> data True
> data False
> fold if
> in True
> $ cat False.annah
> type Bool
> data True
> data False
> fold if
> in False
> $ cat if.annah
> type Bool
> data True
> data False
> fold if
> in if
Then we will translate each of them to a file encoding the equivalent Morte
expression without the @\".annah\"@ file suffix:
> $ annah compile Bool.annah > Bool
> $ annah compile True.annah > True
> $ annah compile False.annah > False
> $ annah compile if.annah > if
Now that we've created a file for each type and term we can import them
within other expressions. For example, now we can define the @not@ function
in terms of imported types and values:
> $ cat not.annah
> let not (b : ./Bool ) : ./Bool = ./if b ./Bool ./False ./True
> in not
Don't worry if you don't understand what the above expression means just
yet. This tutorial will explain what the right-hand side means in the
section on \"Folds\".
We can run this file through Annah, which will desugar and normalize the
expression, but will preserve the original imports:
> $ annah compile not.annah > not
> $ cat not
> λ(b : ./Bool ) → ./if b ./Bool ./False ./True
Annah actually does resolve the imports for the purposes of type-checking
the expression, but deliberately does not resolve the imports for the final
normalized expression. Annah does this to keep the expression \"dynamically
linked\" so that the expression can continue to reflect changes to
dependencies.
If you prefer to statically link the expression then you can use Morte:
> $ echo "./not" | morte
> ∀(b : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(b : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → b (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False) (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
... and you can also expand derived expressions, too:
> $ morte
> ./not ./True
> <Ctrl-D>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
... desugaring them with Annah if necessary:
> $ annah | morte
> let doubleNegate (b : ./Bool ) : /Bool = ./not (./not b)
> in doubleNegate ./True
> <Ctrl-D>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
-}
{- $types
Creating one file per type, fold, and data constructor gets tedious pretty
quickly, so the @annah@ executable provides a convenient subcommand named
@types@ for auto-generating these files.
Just run the @annah types@ command and provide a datatype definition on
standard input:
> $ annah types
> type Bool
> data True
> data False
> fold if
> <Ctrl-D>
... and @annah@ will create one directory for each type in the datatype
definition:
> $ ls
> Bool/ Bool.annah
Each type's directory will have two files per data constructor associated
with the type and two files for the @fold@, too:
> $ ls Bool
> @ False False.annah if if.annah True True.annah
Everything comes in two flavors: the original Annah code and the equivalent
Morte code:
> $ cat Bool/True.annah
> type Bool
> data True
> data False
> fold if
> in True
> $ cat Bool/True
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
The Morte code for the type is located as a file named @\@@ underneath the
type's directory:
> $ cat Bool.annah
> type Bool
> data True
> data False
> fold if
> in Bool
> $ cat Bool/@
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
This is because Morte supports importing the directory by name if there is a
file named @\@@ underneath the directory. So, for example if you import
@./Bool@ and it's a directory then Morte will import @.\/Bool\/\@@ instead:
> $ morte
> ./Bool
> <Ctrl-D>
> *
>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
-}
{- $folds
Every datatype definition comes with an optional @fold@ which you can use to
pattern match on a value of that type. You can see what arguments the
pattern match expects just by querying the type of the fold:
> $ morte
> ./Bool/if
> <Ctrl-D>
> ∀(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x
... and we can use imports to simplify the type to:
> ∀(x : ./Bool ) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
This type says that @if@ expects the following arguments:
* A value named @x@ of type @./Bool@ to pattern match on (like
@.\/Bool\/True@ or @.\/Bool\/False@)
* The type of the result for each branch of the pattern match
* The result to return if our value equals @.\/Bool\/True@
* The result to return if our value equals @.\/Bool\/False@
Carefully note that the second argument is named @Bool@ but can actually be
any type. Similarly, the third and fourth arguments are named after the
@True@ and @False@ constructors but they actually represent how to handle
each branch of the pattern match.
So, for example, when we write:
> let not (b : ./Bool ) : ./Bool = ./if b ./Bool ./False ./True
> in not
... it's as if we wrote the following Haskell code using pattern matching:
> let not :: Bool -> Bool
> not b = case b of
> True -> False
> False -> True
> in not
We could even format our code to parallel the layout of a Haskell pattern
match:
> let not (b : ./Bool ) : ./Bool =
> ./if b ./Bool
> ./False
> ./True
> in not
The only difference is that in the Annah code we have to explicitly supply
the expected type of the result after the value that we pattern match on
(i.e. the @./Bool@ immediately after the @./if b@).
Our @./not@ function technically did not need to use the @./if@ @fold@. For
example, we could instead write:
> $ cat not.annah
> let not (b : ./Bool ) : ./Bool = b ./Bool ./False ./True
> in not
The @./if@ was unnecessary because it was just the identity function on
@./Bool@s:
> $ cat if
> λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x
.. which is the same as:
> λ(x : ./Bool ) → x
The reason we can omit the @if@ is that all values of type @./Bool@ are
already preformed pattern matches. We can prove this to ourselves by
consulting the definitions of @.\/Bool\/True@ and @.\/Bool\/False@:
> $ morte < ./Bool/True
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
> $ morte < ./Bool/False
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
In other words, @.\/Bool\/True@ is just a preformed pattern match that
always returns the first branch that you supply. Vice versa,
@.\/Bool\/False@ is just a preformed pattern match that always returns the
second branch that you supply.
In fact, all @fold@s are optional when you save a type and associated data
constructors as separate files. The only time we truly require the @fold@
is when we pattern match on the type within the "body" of a datatype
expression, like in our very first example:
> type Bool
> data True
> data False
> fold if
> in -- Everything below here is the "body" of the `Bool` datatype definition
>
> let not (b : Bool) : Bool = if b Bool False True
> in not False
@Bool@ and @./Bool@ are not the same type within the "body" of the @Bool@
datatype definition. If you omit the @if@ then you will get the following
type error:
> $ annah
> type Bool
> data True
> data False
> fold if
> in
>
> let not (b : Bool) : Bool = b Bool False True
> in not False
> <Ctrl-D>
> annah:
> Context:
> Bool : *
> True : Bool
> False : Bool
> if : ∀(x : Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
> b : Bool
>
> Expression: b Bool
>
> Error: Only functions may be applied to values
The @Context@ the compiler prints in the error message shows that the
type-checker views the @Bool@ type as abstract and not the type of a
pattern match. However, the same @Context@ says that @if@ has the correct
type to convert between the abstract @Bool@ type and the type we expect for
a pattern match:
> if : ∀(x : Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
... which we can simplify to just:
> -- The type of the bound variable named `if`
> if : ∀(x : Bool) → ./Bool
In other words, @Bool@ and @./Bool@ are different types from the type
checker's point of view. That is why you must explicitly convert from
@Bool@ to @./Bool@ using @if@ while inside that context.
However, once you save @./Bool@, @./True@, @./False@ and @./if@ to separate
files the distinction between @Bool@ and @./Bool@ vanishes. The type of
@./if@ (the file) is not the same as the type of @if@ (the bound variable):
> -- The type of the file named `./if`
> ./if : ∀(x : ./Bool ) → ./Bool
You can deduce why the distinction disappears when you save things to
separate files if you desugar the datatype definitions. For example our
@if.annah@ file was defined as:
> type Bool
> data True
> data False
> fold if
> in if
We can use the @annah desugar@ subcommand to see what that code desugars to
before normalization:
> $ annah desugar < ./Bool/if.annah
> (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → λ(if : ∀(x : Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → if) (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True) (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False) (λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x)
... which we can clean up a bit to get:
> ( λ(Bool : *)
> → λ(True : Bool)
> → λ(False : Bool)
> → λ(if : Bool → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
> → if
> )
>
> (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
> (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
> (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
> (λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x)
That then normalizes to;
> $ annah desugar < ./Bool/if.annah | morte
> ∀(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → x
There is also another use for storing @fold@s as files and using them, even
if they are not immediately necessary. Saving a @fold@ to a file lets you
provide a stable interface for pattern matching on a value if you ever
want to change the internal implementation of a type without breaking
backwards compatibility.
For example, suppose that a user writes the following @not@ function using
@./if@:
> let not (b : ./Bool ) : ./Bool = ./if b ./Bool ./False ./True
> in not
... but we later decide we want to flip the order of of the @True@ and
@False@ constructors in our datatype definition:
> $ annah types
> type Bool
> data False
> data True
The above changes would break the user's code unless we change @./if@ to
export the pattern match order that the user expects:
> $ cat > if
> \(b : ./Bool )
> -> \(Bool : *)
> -> \(True : Bool)
> -> \(False : Bool)
> -> b Bool False True
> <Ctrl-D>
Now the user's code continues to work as if nothing ever happened.
So saving @fold@s to files and using them to pattern match is not strictly
necessary, but if you do use them then you can change the underlying
implementation of the type without breaking backwards compatibility.
There's no way that you can force users to use the @fold@ that you provide
since all saved expressions are encoded in lambda calculus, which does not
provide any support for implementation hiding or encapsulation. The best
you can do is to simply warn users that you might break their code some
day if they perform a \"raw pattern match\" (i.e. a pattern match without
the use of a saved @fold@).
-}
{- $recursive
Annah supports recursive and mutually recursive types. We saw an example
of recursive types with natural numbers:
> $ annah
> type Nat
> data Succ (pred : Nat)
> data Zero
> fold foldNat
>
> in Succ (Succ (Succ Zero))
> <Ctrl-D>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
What might not be obvious is that if you save each type and constructor to
a separate file then you can build a natural number just from the files.
To illustrate this, we will compile our datatype definition to separate
files:
> $ annah types
> type Nat
> data Succ (pred : Nat)
> data Zero
> fold foldNat
> <Ctrl-D>
... and now we can build natural numbers using these files:
> $ morte
> ./Nat/Succ (./Nat/Succ (./Nat/Succ ./Nat/Zero ))
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
This gives the exact same result as before, but now we are programming
directly at the "top level" using files instead of programming inside the
body of a datatype definition.
We can also fold natural numbers using our @.\/Nat\/foldNat@ function.
Let's consult the type of the function:
> ∀(x : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(x : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → x
If we clean up that type we get:
> ∀(x : ./Nat )
> → ∀(Nat : *)
> → ∀(Succ : ∀(pred : Nat) → Nat)
> → ∀(Zero : Nat)
> → Nat
Conceptually, when we fold a @./Nat@ value using @.\/Nat\/foldNat@ we just
replace each @.\/Nat\/Succ@ constructor with the argument of the fold
labeled @Succ@ (i.e. the third argument). Similarly, we substitute each
@.\/Nat\/Zero@ constructor with the fourth argument labeled @Zero@.
We also supply a type parameter named @Nat@ as the second argument. This
type parameter must match the input and output of whatever we use to replace
the @.\/Nat\/Succ@ and @.\/Nat\/Zero@.
For example, suppose that we wanted to write a function to test if a @./Nat@
was an even number. We would just substitute every @Zero@ constructor with
@.\/Bool\/True@ and substitute every @.\/Nat\/Succ@ constructor with
@./not@. The code for that would be:
> $ cat not.annah # Update `not.annah` to use our new file layout
> let not (b : ./Bool ) : ./Bool =
> ./Bool/if b ./Bool
> ./Bool/False
> ./Bool/True
> in not
> $ cat isEven.annah
> let isEven (n : ./Nat ) : ./Bool =
> ./Nat/foldNat n ./Bool
> ./not -- Replace every `./Nat/Succ` with `./not`
> ./Bool/True -- Replace every `./Nat/Zero` with `./Bool/True`
> in isEven
The let definitions are not strictly necessary since we could just write:
> $ cat not.annah
> \(b : ./Bool ) ->
> ./Bool/if b ./Bool
> ./Bool/False
> ./Bool/True
> $ cat isEven.annah
> \(n : ./Nat ) ->
> ./Nat/foldNat n ./Bool
> ./not
> ./Bool/True
... but the let definitions help the readability of the code by naming the
functions and documenting their expected return types.
Then we can compile our Annah expression to Morte code:
> $ annah compile not.annah > not
> $ annah compile isEven.annah > isEven
... and test that @./isEven@ works:
> $ morte
> ./isEven (./Nat/Succ (./Nat/Succ ./Nat/Zero ))
> <Ctrl-D>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
It works! The result is identical to @.\/Bool\/True@:
> $ morte
> ./Bool/True
> <Ctrl-D>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
Conceptually, what happened was that @./isEven@ just performed the
desired substitutions, replacing every @.\/Nat\/Succ@ with @./not@ and
replacing every @.\/Nat\/Zero@ with @.\/Bool\/True@:
> ./isEven (./Nat/Succ (./Nat/Succ ./Nat/Zero ))
>
> -- Constructor substitution
> = ./not (./not ./Bool/True )
>
> -- β-reduce
> = ./Bool/True
Note that this is not really the path the compiler takes under the hood, but
it's equivalent.
We can also encode mutually recursive types such as the following type
declaration for even and odd numbers:
> $ annah
> type Even
> data SuccE (predE : Odd)
> data ZeroE
> fold foldEven
>
> type Odd
> data SuccO (predO : Even)
> fold foldOdd
>
> in SuccE (SuccO ZeroE)
> λ(Even : *) → λ(Odd : *) → λ(SuccE : ∀(predE : Odd) → Even) → λ(ZeroE : Even) → λ(SuccO : ∀(predO : Even) → Odd) → SuccE (SuccO ZeroE)
Like before, we can encode each type and term separately as files and the
files:
> annah types
> type Even
> data SuccE (predE : Odd)
> data ZeroE
> fold foldEven
>
> type Odd
> data SuccO (predO : Even)
> fold foldOdd
> <Ctrl-D>
... and now these files can be used to build @./Even@ or @./Odd@ values:
> $ morte
> ./Even/SuccE (./Odd/SuccO ./Even/ZeroE )
> <Ctrl-D>
> ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Even
>
> λ(Even : *) → λ(Odd : *) → λ(SuccE : ∀(predE : Odd) → Even) → λ(ZeroE : Even) → λ(SuccO : ∀(predO : Even) → Odd) → SuccE (SuccO ZeroE)
We can also consume mutually recursive types just by folding them. Each
type is already a preformed fold and we can consult each type's respective
@fold@ function to see what arguments the @fold@ expects:
> $ morte
> ./Even/foldEven
> <Ctrl-D>
> ∀(x : ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Even) → ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Even
>
> λ(x : ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Even) → x
> $ morte
> ./Odd/foldOdd
> ∀(x : ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Odd) → ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Odd
>
> λ(x : ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → Odd) → x
If we clean up the type of the @.\/Even\/foldEven@ function we get this:
> ∀(x : ./Even )
> → ∀(Even : *)
> → ∀(Odd : *)
> → ∀(SuccE : ∀(predE : Odd) → Even)
> → ∀(ZeroE : Even)
> → ∀(SuccO : ∀(predO : Even) → Odd)
> → Even
Conceptually, when we fold an @./Even@ value using @.\/Even\/foldEven@ we
just replace each @.\/Even\/SuccE@ constructor with the argument of the fold
labeled @SuccE@ (i.e. the fourth argument). Similarly, we substitute each
@.\/Even\/ZeroE@ constructor with the fifth argument named @ZeroE@ and
substitute each @.\/Odd\/SuccO@ constructor with the sixth argument named
@SuccO@.
We also supply two type parameters named @Even@ and @Odd@. These type
parameters must match the input and output of whatever we use to replace
the @SuccE@, @ZeroE@ and @SuccO@ constructors.
For example, suppose that we wanted to write a function that converts an
@./Even@ value to a @./Nat@. We would just replace every @.\/Even\/SuccE@
and @.\/Odd\/SuccO@ constructor with @Succ@ and replace every
@.\/Even\/ZeroE@ constructor with @Zero@, like this:
> $ cat evenToNat.annah
> let evenToNat (e : ./Even ) : ./Nat =
> ./Even/foldEven e ./Nat ./Nat
> ./Nat/Succ -- Replace every `./Even/SuccE` with `Succ`
> ./Nat/Zero -- Replace every `./Even/ZeroE` with `Zero`
> ./Nat/Succ -- Replace every `./Odd/SuccO` with `Succ`
> in evenToNat
Now we can \"compile\" our @evenToNat@ function to Morte code:
> annah evenToNat.annah > evenToNat
... and test that it correctly converts @./Even@ values to their
equivalent @./Nat@ values:
> $ morte
> ./evenToNat (./Even/SuccE (./Odd/SuccO ./Even/ZeroE ))
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ Zero)
It works! We can began with the number two, encoded as an @./Even@ number
and ended with two encoded as a @./Nat@.
As before, the @./evenTonat@ function was just performing the desired
substitution, replacing each @.\/Even\/SuccE@ and @.\/Odd\/SuccO@ with
@.\/Nat\/Succ@ and replacing @.\/Odd\/ZeroE@ with @.\/Nat\/Zero@:
> ./evenToNat (./Even/SuccE (./Odd/SuccO ./Even/ZeroE ))
>
> -- Constructor substitution
> = ./Nat/Succ (./Nat/Succ ./Nat/Zero )
Again, this is not the path the compiler takes under the hood, but it's
equivalent.
-}
{- $prelude
Annah also comes with a Prelude of utility types and terms. This Prelude is
hosted remotely here:
<http://sigil.place/prelude/annah/1.0/>
You can visit the above link to browse the Prelude and see what is
available.
There are several ways that you can use the Prelude. The most direct
approach is to use expressions from the Prelude directly by referencing
their URLs, like this:
> $ morte
> http://sigil.place/prelude/annah/1.0/Nat/Succ
> ( http://sigil.place/prelude/annah/1.0/Nat/Succ
> ( http://sigil.place/prelude/annah/1.0/Nat/Succ
> http://sigil.place/prelude/annah/1.0/Nat/Zero
> )
> )
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
... or you can selectively \"alias\" remote references locally by creating
local files that refer to the remote URLs:
> $ echo "http://sigil.place/prelude/annah/1.0/Nat/Succ" > Succ
> $ echo "http://sigil.place/prelude/annah/1.0/Nat/Zero" > Zero
> $ morte
> ./Succ (./Succ (./Succ ./Zero ))
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
... or you can \"import\" the entire Prelude into your current directory
using @wget@:
> $ wget -np -r --cut-dirs=3 http://sigil.place/prelude/annah/1.0/
> $ cd sigil.place
> $ ls
> (->) Defer.annah List.annah Path Sum0.annah
> (->).annah Eq Maybe Path.annah Sum1
> Bool Eq.annah Maybe.annah Prod0 Sum1.annah
> Bool.annah Functor Monad Prod0.annah Sum2
> Category Functor.annah Monad.annah Prod1 Sum2.annah
> Category.annah index.html Monoid Prod1.annah
> Cmd IO Monoid.annah Prod2
> Cmd.annah IO.annah Nat Prod2.annah
> Defer List Nat.annah Sum0
This tutorial will assume that you have imported the Prelude locally.
The Prelude is organized according to the following rules:
* Each type (like @./Bool@ or @./Nat@) is a top-level directory. You can
reference that type in your code by its directory
* Each constructor of that type lives underneath the type's directory. For
example, @True@ is located underneath the @./Bool@ directory
* Functions associated with each type are also located underneath the type's
directory. For example, the @length@ function is located underneath the
@./List@ directory.
* Every expression is provided as both the original Annah code (with a
@*.annah@ suffix) and Morte code (with no suffix). For example, you
will find the @Monoid.annah@ file which was the Annah expression used to
create the @Monoid@ file which is a Morte expression.
In order to use an expression within Morte you must explicitly import the
expression within the Morte code, like this:
> $ echo "./List/length" | morte # Good
> ∀(a : *) → ∀(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → List) → ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Nil : Nat) → Nat
>
> λ(a : *) → λ(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → List) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → xs Nat (λ(_ : a) → Succ)
Reading the expression through standard input will (usually) not work:
> $ morte < List/length # Bad
> ../List: openFile: does not exist (No such file or directory)
The reason why is that everything in the Prelude uses relative imports to
reference each other. This is what allows the Prelude to correctly
function both when you reference the Prelude remotely and when you download
the Prelude locally. If you read the expression through standard input
then Morte incorrectly concludes that any further imports are relative to
your current directory. However, if you explicitly import the expression
within the code then Morte correctly concludes that transitive imports are
relative to the imported file's path.
For example, the @List/length@ file has the following contents:
> cat List/length
> λ(a : *) → λ(xs : ../List a) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → xs Nat (λ(_ : a) → Succ)
There is one relative reference within that file to @../List@. That
reference is relative to the current file's directory (i.e. relative to
@List/@) which means that it still points to the same directory: @List@. We
could have also used just @.@ to refer to the current directory but that
would be less readable.
However, if you read in @List/length@ from standard input, then @morte@
looks for @../List@ expression relative to your present working directory
and fails.
Annah's Prelude has some similarities to Haskell's standard libraries and
some differences. The rough correspondences are:
* @(->)@ corresponds to Haskell's @(->)@ type constructor
* @Bool@ corresponds to Haskell's `Bool` type
* @Cmd@ corresponds to the operational monad (i.e.
"Control.Monad.Operational".`Control.Monad.Operational.Program`)
* @Defer@ corresponds to
"Data.Functor.Coyoneda".`Data.Functor.Coyoneda.Coyoneda`
* @List@ corresponds to Haskell lists except that Annah @List@s are always
finite because they are encoded recursively
* @Maybe@ corresponds to Haskell's `Maybe` type constructor
* @Nat@ corresponds to Haskell's `Numeric.Natural.Natural` type, except
much less efficient than its Haskell counterpart
* @Path@ corresponds to a free category. As far as I know there is no
standard Haskell implementation for free categories to reference
* @Prod0@ corresponds to Haskell's @()@ type. Mnemonic: \"Product type with
zero fields\"
* @Prod1@ corresponds to Haskell's `Data.Functor.Identity` type constructor.
Mnemonic: \"Product type with one field\"
* @Prod2@ corresponds to Haskell's 2-tuple type constructor. Mnemonic:
\"Product type with two fields\"
* @Sum0@ corresponds to Haskell's `Data.Void.Void` type. Mnemonic: \"Sum
type with zero fields\"
* @Sum1@ also corresponds to Haskell's `Data.Functor.Identity` type
constructor. Mnemonic: \"Sum type with one field\"
* @Sum2@ corresponds to Haskell's `Either` type constructor. Mnemonic:
\"Sum type with two fields\"
* @IO@ corresponds to a very simple `IO` type constructor that only supports
two operations:
> ./IO/get : ./IO ./Nat
> ./IO/put : ./Nat -> ./IO ./Prod0
In addition to those types, Annah also encodes several of Haskell's type
classes as values. Neither Annah nor Morte supports type classes /per se/.
Instead, each class is encoded as a type constructor and each instance is
a term of the corresponding type:
* @Functor@ corresponds to Haskell's `Functor` class
* @Monoid@ corresponds to Haskell's `Data.Monoid.Monoid` class
* @Monad@ corresponds to Haskell's `Monad` class
* @Category@ corresponds to Haskell's `Control.Category.Category` class
However, the specification of each type class radically differs from how
Haskell encodes things. We'll revisit this in a later section.
-}
{- $nats
The Prelude provides addition and multiplication for natural numbers:
> $ cat > three
> ./Nat/Succ (./Nat/Succ (./Nat/Succ ./Nat/Zero ))
> <Ctrl-D>
> $ morte
> ./Nat/(+) ./three ./three
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero)))))
> $ morte
> ./Nat/(*) ./three ./three
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))
Also, Annah provides basic syntactic support for natural number literals:
> $ annah | morte
> ./Nat/(+) 3 3
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero)))))
> $ annah | morte
> ./Nat/(*) 3 3
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))
-}
{- $lists
The Prelude provides operations on lists, too:
> $ annah | morte
> ./List/replicate ./Bool 3 ./Bool/True
> <Ctrl-D>
> ∀(List : *) → ∀(Cons : ∀(head : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(tail : List) → List) → ∀(Nil : List) → List
>
> λ(List : *) → λ(Cons : ∀(head : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(tail : List) → List) → λ(Nil : List) → Cons (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True) (Cons (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True) (Cons (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True) Nil))
Annah also provides support for list literals:
> $ annah > bools
> [nil ./Bool , ./Bool/True , ./Bool/False , ./Bool/True ]
> <Ctrl-D>
> $ cat bools
> λ(List : *) → λ(Cons : ∀(head : ./Bool ) → ∀(tail : List) → List) → λ(Nil : List) → Cons ./Bool/True (Cons ./Bool/False (Cons ./Bool/True Nil))
The general format for lists is:
> [nil elementType, element0, element1, ..., elementN]
Here are some examples of operations on lists:
> $ morte
> ./List/null ./Bool ./bools
> <Ctrl-D>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
> $ morte
> ./List/length ./Bool (./List/(++) ./Bool ./bools ./bools )
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Nil : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Nil : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Nil)))))
> $ annah | morte
> \(a : *) -> \(xs : ./List a) -> ./List/(++) a xs [nil a]
> <Ctrl-D>
> ∀(a : *) → ∀(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → List) → ∀(List : *) → ∀(Cons : a → List → List) → ∀(Nil : List) → List
>
> λ(a : *) → λ(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → List) → xs
The last example shows how @morte@ can optimized away @xs ++ []@ to just
@xs@.
-}
{- $monoids
Annah also provides several folds on lists, like @sum@ or @and@:
> $ annah | morte
> <Ctrl-D>
> ./Nat/sum [nil ./Nat , 1, 2, 3, 4]
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))
> $ annah | morte
> <Ctrl-D>
> ./Bool/and [nil ./Bool , ./Bool/True , ./Bool/False , ./Bool/True ]
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
What's interesting about these folds is their type:
> $ cat Nat/sum.annah
> let sum : ../Monoid ../Nat = \(xs : ../List ../Nat ) -> xs ../Nat ./(+) 0
> in sum
> $ cat Bool/and.annah
> let and : ../Monoid ../Bool =
> \(xs : ../List ../Bool ) -> xs ../Bool ./(&&) ./True
> in and
You might have been expecting their types to be something like this:
> sum : ../List ../Nat -> ../Nat
> and : ../List ../Bool -> ../Bool
... and you would have been right because that is actually what their types
are! This is because of how @./Monoid.annah@ is defined:
> $ cat Monoid.annah
> let Monoid (m : *) : * = ./List m -> m
> in Monoid
In other words, a `Monoid` \"instance\" for a type @m@ is just a function
that folds a @./List@ of @m@s into a single @m@. The @./sum@ and @./and@
functions that fold lists also double as @./Monoid@ instances.
You can recover the traditional Haskell `Monoid` operations like `mempty`
and `mappend` from the above @./Monoid@ definition:
> $ cat Monoid/mempty.annah
> let mempty (m : *) (monoid : ./Monoid m) : m =
> monoid [nil m]
> in mempty
> $ cat Monoid/mappend.annah
> let mappend (m : *) (monoid : ./Monoid m) (l : m) (r : m) : m =
> monoid [nil m, l, r]
> in mappend
For example:
> $ morte
> ./Monoid/mempty ./Nat ./Nat/sum
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Zero
> $ annah | morte
> ./Monoid/mappend ./Nat ./Nat/sum 4 5
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))
However, in practice it's easier to just use the folds directly instead of
using @.\/Monoid\/mempty@ or @.\/Monoid\/mappend@:
> $ annah | morte
> ./Nat/sum [nil ./Nat ]
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Zero
> $ annah | morte
> ./Nat/sum [nil ./Nat , 4, 5]
> <Ctrl-D>
> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
>
> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))
-}
{- $commands
Annah also provides syntactic support for chaining commands using @do@
notation, in a style very similar to Haskell. The following examples will
all give very large outputs so I will tidy the output results, although
there is not a good way to tidy the output in general:
For example, here is how you write a list comprehension in Annah.
> $ annah | morte # Output cleaned up by hand
> ./List/Monad ./Nat (do ./List {
> x : ./Nat <- [nil ./Nat , 1, 2, 3];
> y : ./Nat <- [nil ./Nat , 4, 5, 6];
> _ : ./Nat <- ./List/pure ./Nat (./Nat/(+) x y);
> })
> <Ctrl-D>
> ∀(List : *)
> → ∀(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → List)
> → ∀(Nil : List)
> → List
>
> λ(List : *)
> → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → List)
> → λ(Nil : List)
> → Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ Zero)))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))
> Nil
> )
> )
> )
> )
> )
> )
> )
> )
... which is equivalent to:
> ./List ./Nat
>
> [nil ./Nat , 5, 6, 7, 6, 7, 8, 7, 8, 9]
Annah @do@ notation has a few important differences from Haskell's @do@
notation:
* Every command's return type must be annotated; even the final command
* Braces are required and semicolons are required on all lines
* You must annotate the monad's type constructor right after the @do@
* You (usually) wrap the @do@ block in the @./Monad@ instance for your
type constructor followed by the @do@ block's return value
Here is an example diagram to illustrate the last rule:
> +-- Monad instance for ./List
> |
> | +-- The return value of block ...
> | |
> v v
> ./List/Monad ./Nat (do ./List {
> x : ./Nat <- [nil ./Nat , 1, 2, 3];
> y : ./Nat <- [nil ./Nat , 4, 5, 6];
> _ : ./Nat <- ./List/pure ./Nat (./Nat/(+) x y);
> }) ^
> |
> +-- ... which must match this return value
You actually don't have to wrap the @do@ block in a @./Monad@ instance, but
you will get a different result. Let's see what happens if we omit the
@./Monad@ instance:
> $ annah | morte # Output cleaned up by hand
> do ./List {
> x : ./Nat <- [nil ./Nat , 1, 2, 3];
> y : ./Nat <- [nil ./Nat , 4, 5, 6];
> _ : ./Nat <- ./List/pure ./Nat (./Nat/(+) x y);
> }
> <Ctrl-D>
> ∀(Cmd : *)
> → ∀(Bind : ∀(b : *) → (∀(List : *) → ∀(Cons : ∀(head : b) → ∀(tail : List) → List) → ∀(Nil : List) → List) → (b → Cmd) → Cmd)
> → ∀(Pure : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Cmd)
> → Cmd
>
> λ(Cmd : *)
> → λ(Bind : ∀(b : *) → (∀(List : *) → ∀(Cons : ∀(head : b) → ∀(tail : List) → List) → ∀(Nil : List) → List) → (b → Cmd) → Cmd)
> → λ(Pure : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Cmd)
> → Bind
> (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat)
> ( λ(List : *)
> → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → List)
> → λ(Nil : List)
> → Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → Succ)
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ Zero))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero)))
> Nil
> )
> )
> )
> ( λ(x : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat)
> → Bind
> (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat)
> ( λ(List : *)
> → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → List)
> → λ(Nil : List)
> → Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ Zero)))))
> ( Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero))))))
> Nil
> )
> )
> )
> ( λ(y : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat)
> → Bind
> (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat)
> ( λ(List : *)
> → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → List)
> → Cons
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → x Nat Succ (y Nat Succ Zero))
> )
> Pure
> )
> )
... which is equivalent to:
> ./Cmd ./List./Nat
>
> λ(Cmd : *)
> → λ(Bind : ∀(b : *) → ./List b → (b → Cmd) → Cmd)
> → λ(Pure : ./Nat → Cmd)
> → Bind
> ./Nat
> [nil ./Nat , 1, 2, 3]
> ( λ(x : ./Nat )
> → Bind
> ./Nat
> [nil ./Nat 4, 5, 6]
> ( λ(y : ./Nat )
> → Bind
> ./Nat
> [nil ./Nat (./Nat/(+) x y)]
> Pure
> )
> )
The @do@ notation is desugaring to a data type named @./Cmd@ that inserts
placeholders for each @<-@ (pronounced: \"bind\"). In the Haskell world
this datatype is commonly known as the \"operational\" monad.
So why did we wrap the @do@ block in @.\/List\/Monad@? Well, let's check
out the type of the @.\/List\/Monad@ function:
> $ cat ./List/Monad.annah
> let Monad: ../Monad ../List
> = \(a : *)
> -> \(m : ../Cmd ../List a)
> -> m (../List a) (\(b : *) -> ./(>>=) b a) (./pure a)
> in Monad
Hmmm, that's weird. Wasn't it supposed to be a function? Actually, it is!
To see why, let's check out how @./Monad@ is defined:
> let Monad (m : * -> *) : * = forall (a : *) -> ./Cmd m a -> m a
> in Monad
A @./Monad m@ is a function that transforms a @./Cmd m a@ into an @m a@ by
replacing each @Bind@ with the correct \"bind\" operation for that `Monad`
and replaces each @Pure@ with the correct \"pure\" operation for that
`Monad`. Therefore a @./Monad ./List@ is a function that transforms a
@.\/Cmd .\/List a@ into a @./List a@.
That's why we wrap the @do@ block in @.\/List\/Monad@ because the @do@
block starts out with this type:
> do ./List { ... } : ./Cmd ./List ./Nat
... and then when we apply the @.\/List\/Monad function we get back a
bona-fide @./List@:
> ./List/Monad ./Nat (do ./List { ... }) ./List ./Nat
There are a couple of parallels between Annah's @./Monad@+@./Cmd@ and
Annah's @./Monoid@+@./List@:
* Both of them have syntactic support for building a placeholder of some
sort. List notation builds a @./List@ and @do@ notation builds a @./Cmd@
* Both of them have a way to fold the placeholder into a single value.
@./Monoid@s fold @./List@s and @./Monad@s fold @./Cmd@s.
-}
{- $io
Annah also supports a very simplistic @./IO@ type as a proof of concept for
how you would model a foreign function interface. For example, here is an
@./IO@ action that reads a @./Nat@ and writes out the same @./Nat@:
> $ annah
> ./IO/Monad ./Prod0 (do ./IO {
> n : ./Nat <- ./IO/get ;
> _ : ./Prod0 <- ./IO/put n;
> })
> <Ctrl-D>
> ./IO/Monad ./Prod0 (λ(Cmd : *) → λ(Bind : ∀(b : *) → ./IO b → (b → Cmd) → Cmd) → λ(Pure : ./Prod0 → Cmd) → Bind ./Nat ./IO/get (λ(n : ./Nat ) → Bind ./Prod0 (./IO/put n) Pure))
Annah also provides utilities similar to Haskell for chaining commands, such
as @.\/Monad\/replicateM_.annah@ which lets you repeat a command a fixed
number of times:
> $ cat Monad/replicateM_.annah
> let replicateM_ (m : * -> *) (n : ../Nat ) (cmd : m ../Prod0 )
> : ../Cmd m ../Prod0
> = ./sequence_ m (../List/replicate (m ../Prod0 ) n cmd)
> in replicateM_
Notice that @.\/Monad\/replicateM_@ does not take a @./Monad@ instance as
an argument. Instead, @.\/Monad\/replicateM_@ returns a @./Cmd@ which
you can fold with the appropriate @./Monad@ instance:
For example:
> $ annah | morte # Output cleaned up by hand
> ./IO/Monad ./Prod0 (./Monad/replicateM_ ./IO 10 (./IO/put 4))
> ∀(IO : *)
> → ∀(Get_ : ((∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO) → IO)
> → ∀(Put_ : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO → IO)
> → ∀(Pure_ : (∀(Prod0 : *) → ∀(Make : Prod0) → Prod0) → IO)
> → IO
>
> λ(IO : *)
> → λ(Get_ : ((∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO) → IO)
> → λ(Put_ : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO → IO)
> → λ(Pure_ : (∀(Prod0 : *) → ∀(Make : Prod0) → Prod0) → IO)
> → Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> ( Put_
> (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Zero))))
> (Pure_ (λ(Prod0 : *) → λ(Make : Prod0) → Make))
> )
> )
> )
> )
> )
> )
> )
> )
> )
If you clean that up a bit you get a syntax tree for printing @4@ 10 times:
> λ(IO : *)
> → λ(Get_ : (./Nat → IO) → IO)
> → λ(Put_ : ./Nat → IO → IO)
> → λ(Pure_ : ./Prod0 → IO)
> → Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Put_ 4 (Pure_ ./Prod0/Make ))))))))))
Let's try a more complicated program, that reads and writes integers 10
times:
> $ annah | morte
> let io : ./IO ./Prod0 = ./IO/Monad ./Prod0 (do ./IO {
> n : ./Nat <- ./IO/get ;
> _ : ./Prod0 <- ./IO/put n;
> })
> in ./IO/Monad ./Prod0 (./Monad/replicateM_ ./IO 10 io)
> <Ctrl-D>
> ∀(IO : *) → ∀(Get_ : ((∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO) → IO) → ∀(Put_ : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO → IO) → ∀(Pure_ : (∀(Prod0 : *) → ∀(Make : Prod0) → Prod0) → IO) → IO
>
> λ(IO : *) → λ(Get_ : ((∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO) → IO) → λ(Put_ : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO → IO) → λ(Pure_ : (∀(Prod0 : *) → ∀(Make : Prod0) → Prod0) → IO) → Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Get_ (λ(r : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Put_ r (Pure_ (λ(Prod0 : *) → λ(Make : Prod0) → Make)))))))))))))))))))))
... which if we simplify we get:
> λ(IO : *)
> → λ(Get_ : (./Nat → IO) → IO)
> → λ(Put_ : ./Nat → IO → IO)
> → λ(Pure_ : ./Prod0 → IO)
> → Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Get_ (λ(r : ./Nat ) →
> Put_ r (
> Pure_ ./Prod0/Make))))))))))))))))))))
In other words, we've built an abstract syntax tree representing ten
@Get_@ and @Put_@ nodes where each @Get_@ node threads its result to the
next @Put_@ node.
Annah cannot run this abstract syntax tree since Annah does not have a
backend to interpret this tree. The most Annah can do is model effects
without running them.
-}
{- $paths
Annah provides support for the `Category` type class, too, using an approach
very similar to the support for `Monoid` and `Monad`:
* Provide a placeholder type named @./Path@ (which is a \"free category\")
* Provide syntactic support for building @./Path@s
* Define a @./Category@ to be something that folds @./Path@s
> $ cat Category.annah
> let Category (cat : * -> * -> *) : * =
> forall (a : *) -> forall (b : *) -> ./Path cat a b -> cat a b
> in Category
Here is an example of composing several functions using the @./Category@
instance for functions:
> $ annah | morte
> let even (n : ./Nat ) : ./Bool = n ./Bool ./Bool/not ./Bool/True
>
> in let f : ./List ./Nat -> ./Bool =
> ./(->)/Category (./List ./Nat ) ./Bool
> [id ./(->) { ./List ./Nat } ./Nat/sum { ./Nat } even { ./Bool } ./Bool/not { ./Bool }]
>
> in f [nil ./Nat , 1, 2, 3, 4
> <Ctrl-D>
> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
>
> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
The above code creates a composition chain of three functions, reading from
left to right:
* @.\/Nat/sum@, which has type @.\/List .\/Nat -> .\/Nat@
* @even@, which has type @.\/Nat -> .\/Bool@
* @.\/Bool\/not@, which has type @.\/Bool -> .\/Bool@
Annah's path notation requires you to annotate the types along the way as
you compose each component. In the above example, you can find each
function's input type immediately to the left of that function and the
output type immediately to the right of each function. Types are surrounded
by braces to separate them from the things you compose.
Annah's path notation differs from lists in a couple of ways:
* You replace @nil@ with @id@
* The @id@ is followed by the type constructor that you are chaining
* You replace commas with intermediate types
You may find the notation easier to read if you put each composable
component on a separate line preceded by the corresponding input type:
> let even (n : ./Nat ) : ./Bool = n ./Bool ./Bool/not ./Bool/True
>
> in let f : ./List ./Nat -> ./Bool =
> ./(->)/Category (./List ./Nat ) ./Bool [id ./(->)
> { ./List ./Nat } ./Nat/sum
> { ./Nat } even
> { ./Bool } ./Bool/not
> { ./Bool }
> ]
>
> in f [nil ./Nat , 1, 2, 3, 4]
Annah's Prelude only provides support for one @./Category@ instance for
functions named @./(->)/Category@, so in practice the @./Category@ support
is not that handy out-of-the box and is mainly provided for completeness.
-}
{- $conclusion
Those are all the features that Annah supports! Annah is a very tiny
language and library that illustrates and implements basic idioms for
translating functional programming concepts into pure lambda calculus.
Hopefully you can use Annah to learn how to encode a subset of Haskell in a
completely total programming language. If you translate any Haskell
functions to Annah you can contribute them upstream to the Annah prelude by
submitting a pull request against the Annah repository:
<https://github.com/Gabriel439/Haskell-Annah-Library/tree/master/Prelude>
-}