Safe Haskell  SafeInferred 

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 nonrecursive 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 nonrecursive 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 typechecked and evaluated. You cannot directly typecheck 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 userfriendly (and I apologize for that!). For example, Annah reuses Morte's typechecker which means that error messages are in terms of lowlevel 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/MorteTutorial.html
Annah is a superset of Morte that implements many of the higherlevel constructs mentioned in the Morte tutorial, which is why you should not skip reading the Morte tutorial.
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 lowlevel 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 <CtrlD> λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
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) →
Annah is a superset of Morte, so any Morte expression is also a valid Annah expression:
$ annah \(a : *) > \(x : a) > x <CtrlD> λ(a : *) → λ(x : a)
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 <CtrlD> λ(a : *) → λ(x : a)
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 <CtrlD> λ(a : *) → λ(x : a)
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) → → λ(f1 : ∀(x10 : _A10) → ∀(x11 : _A11) → ... → ∀(x1j : _A1j) → → ... → λ(fi : ∀(xi0 : _Ai0) → ∀(xi1 : _Ai1) → ... → ∀(xij : _Aij) → → e ) (λ(x00 : _A00) → λ(x01 : _A01) → ... → λ(x0j : _A0j) (λ(x10 : _A10) → λ(x11 : _A11) → ... → λ(x1j : _A1j) ... (λ(xi0 : _Ai0) → λ(xi1 : _Ai1) → ... → λ(xij : _Aij)
The above 'e'
is the "body" of the let expression and f0
through
fi
are the "letbound terms". Due to the above translation, each
"letbound" term is only in scope for the "body" of the letexpression
and the types of subsequent "letbound" 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) → → λ(const : ∀(a : *) → ∀(b : *) → ∀(x : a) → ∀(y : b → id ) (λ(a : *) → λ(x : a) → (λ(a : *) → λ(b : *) → λ(x : a) → λ(y :
... which normalizes to:
λ(a : *) → λ(x : a)
The definition of const
is dead code that is optimized away by βreduction because the letbound const
term is never used within the body of the let
expression.
Data types
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) → Bo let True : Bool = λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → let False : Bool = λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → let if : Bool → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Boo λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) in e
... which in turn desugars to:
( λ(Bool : *) → λ(True : Bool → λ(False : Bool → λ(if : Bool → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → → e ) (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Boo (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → F (λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
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)) <CtrlD> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ
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
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 Na
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 autogenerate 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 righthand 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 ./Tru
Annah actually does resolve the imports for the purposes of typechecking 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) 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(Fa
... and you can also expand derived expressions, too:
$ morte ./not ./True <CtrlD> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
... desugaring them with Annah if necessary:
$ annah  morte let doubleNegate (b : ./Bool ) : /Bool = ./not (./not b) in doubleNegate ./True <CtrlD> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
Autogenerate 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 autogenerating 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 <CtrlD>
... 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) →
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) → Bo
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 <CtrlD> * ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo
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 <CtrlD> ∀(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
... and we can use imports to simplify the type to:
∀(x : ./Bool ) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → B
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)
.. which is the same as:
λ(x : ./Bool ) →
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) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → $ morte < ./Bool/False ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
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 <CtrlD> annah: Context: Bool : * True : Bool False : Bool if : ∀(x : Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → B 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
typechecker 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) → B
... 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)
... 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) → → if ) (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Boo (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → F (λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
That then normalizes to;
$ annah desugar < ./Bool/if.annah  morte ∀(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) λ(x : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
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 <CtrlD>
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 types
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)) <CtrlD> λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ
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 <CtrlD>
... and now we can build natural numbers using these files:
$ morte ./Nat/Succ (./Nat/Succ (./Nat/Succ ./Nat/Zero )) <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ
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 : Na λ(x : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat
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 )) <CtrlD> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
It works! The result is identical to ./Bool/True
:
$ morte ./Bool/True <CtrlD> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
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 )  βreduce = ./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 (S
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 <CtrlD>
... and now these files can be used to build .Even@ or @.Odd
values:
$ morte ./Even/SuccE (./Odd/SuccO ./Even/ZeroE ) <CtrlD> ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) λ(Even : *) → λ(Odd : *) → λ(SuccE : ∀(predE : Odd) → Even) → λ(ZeroE : Even) → λ(SuccO : ∀(predO : Even) → Odd) → SuccE (S
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 <CtrlD> ∀(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) λ(x : ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) → E
$ 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 λ(x : ∀(Even : *) → ∀(Odd : *) → ∀(SuccE : ∀(predE : Odd) → Even) → ∀(ZeroE : Even) → ∀(SuccO : ∀(predO : Even) → Odd) →
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) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ
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 ) ) <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ
... 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) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ
... or you can "import" the entire Prelude into your current directory
using wget
:
$ wget np r cutdirs=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 toplevel 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, thelength
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 theMonoid.annah
file which was the Annah expression used to create theMonoid
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 : λ(a : *) → λ(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → List) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → xs Nat (λ
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(_ :
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'sBool
type *Cmd
corresponds to the operational monad (i.e. Control.Monad.Operational.Program
) *Defer
corresponds to Data.Functor.Coyoneda.Coyoneda
*List
corresponds to Haskell lists except that AnnahList
s are always finite because they are encoded recursively *Maybe
corresponds to Haskell'sMaybe
type constructor *Nat
corresponds to Haskell'sNatural
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'sIdentity
type constructor. Mnemonic: "Product type with one field" *Prod2
corresponds to Haskell's 2tuple type constructor. Mnemonic: "Product type with two fields" *Sum0
corresponds to Haskell'sVoid
type. Mnemonic: "Sum type with zero fields" *Sum1
also corresponds to Haskell'sIdentity
type constructor. Mnemonic: "Sum type with one field" *Sum2
corresponds to Haskell'sEither
type constructor. Mnemonic: "Sum type with two fields" *IO
corresponds to a very simpleIO
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'sFunctor
class *Monoid
corresponds to Haskell'sMonoid
class *Monad
corresponds to Haskell'sMonad
class *Category
corresponds to Haskell'sCategory
class
However, the specification of each type class radically differs from how Haskell encodes things. We'll revisit this in a later section.
Natural numbers
The Prelude provides addition and multiplication for natural numbers:
$ cat > three ./Nat/Succ (./Nat/Succ (./Nat/Succ ./Nat/Zero )) <CtrlD>
$ morte ./Nat/(+) ./three ./three <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zer
$ morte ./Nat/(*) ./three ./three ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : 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 <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zer
$ annah  morte ./Nat/(*) 3 3 ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : 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 <CtrlD> ∀(List : *) → ∀(Cons : ∀(head : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(tail : List) → List) → ∀(Nil : 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
Annah also provides support for list literals:
$ annah > bools [nil ./Bool , ./Bool/True , ./Bool/False , ./Bool/True ] <CtrlD>
$ cat bools λ(List : *) → λ(Cons : ∀(head : ./Bool ) → ∀(tail : List) → List) → λ(Nil : List) → Cons ./Bool/True (Cons ./Bool/False (Cons ./Bool/True
The general format for lists is:
[nil elementType, element0, element1, ..., elementN]
Here are some examples of operations on lists:
$ morte ./List/null ./Bool ./bools <CtrlD> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
$ morte ./List/length ./Bool (./List/(++) ./Bool ./bools ./bools ) <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Nil : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Nil : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Ni
$ annah  morte \(a : *) > \(xs : ./List a) > ./List/(++) a xs [nil a] <CtrlD> ∀(a : *) → ∀(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → List) → ∀(List : *) → ∀(Cons : a → List → List) → ∀(Nil : L λ(a : *) → λ(xs : ∀(List : *) → ∀(Cons : ∀(head : a) → ∀(tail : List) → List) → ∀(Nil : List) → Li
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 <CtrlD> ./Nat/sum [nil ./Nat , 1, 2, 3, 4] ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))
$ annah  morte <CtrlD> ./Bool/and [nil ./Bool , ./Bool/True , ./Bool/False , ./Bool/True ] ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
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 <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat)
$ annah  morte ./Monoid/mappend ./Nat ./Nat/sum 4 5 <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : 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 ] <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat)
$ annah  morte ./Nat/sum [nil ./Nat , 4, 5] <CtrlD> ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : 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); }) <CtrlD> ∀(List : *) → ∀(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → → ∀(Nil : List) → List λ(List : *) → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → λ(Nil : List → Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ Zer ( 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 thedo
block in the./Monad
instance for your type constructor followed by thedo
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); } <CtrlD> ∀(Cmd : *) → ∀(Bind : ∀(b : *) → (∀(List : *) → ∀(Cons : ∀(head : b) → ∀(tail : List) → List) → ∀(Nil : List) → List) → (b → Cm → ∀(Pure : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → → Cmd λ(Cmd : *) → λ(Bind : ∀(b : *) → (∀(List : *) → ∀(Cons : ∀(head : b) → ∀(tail : List) → List) → ∀(Nil : List) → List) → (b → C → λ(Pure : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → Bind (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → N ( λ(List : *) → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → λ(Nil : List → Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → S ( Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ ( Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Z Nil ) ) ) ( λ(x : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → → Bind (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → N ( λ(List : *) → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → λ(Nil : List → Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ Zer ( 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) → → Bind (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → N ( λ(List : *) → λ(Cons : ∀(head : ∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → ∀(tail : List) → Cons (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → x Nat Succ (y Nat Succ ) Pure ) )
... which is equivalent to:
./Cmd ./List./Nat λ(Cmd : *) → λ(Bind : ∀(b : *) → ./List b → (b → Cmd) → → λ(Pure : ./Nat → Cm → 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
bonafide ./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; }) <CtrlD> ./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/pu
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) → ∀(Put_ : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO → ∀(Pure_ : (∀(Prod0 : *) → ∀(Make : Prod0) → Prod0) → → IO λ(IO : *) → λ(Get_ : ((∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → IO → λ(Put_ : (∀(Nat : *) → ∀(Succ : ∀(pred : Nat) → Nat) → ∀(Zero : Nat) → Nat) → I → λ(Pure_ : (∀(Prod0 : *) → ∀(Make : Prod0) → Prod0) → → Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze ( Put_ (λ(Nat : *) → λ(Succ : ∀(pred : Nat) → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ Ze (Pure_ (λ(Prod0 : *) → λ(Make : Prod0) → Mak ) ) ) ) ) ) ) ) )
If you clean that up a bit you get a syntax tree for printing 4
10 times:
λ(IO : *) → λ(Get_ : (./Nat → IO) → → λ(Put_ : ./Nat → IO → → λ(Pure_ : ./Prod0 → I → 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) <CtrlD> ∀(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) → 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 : N
... which if we simplify we get:
λ(IO : *) → λ(Get_ : (./Nat → IO) → → λ(Put_ : ./Nat → IO → → λ(Pure_ : ./Prod0 → I → 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 <CtrlD> ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bo λ(Bool : *) → λ(True : Bool) → λ(False : Bool) →
The above code creates a composition chain of three functions, reading from left to right:

./Natsum@, 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
withid
* Theid
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 outofthe 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/HaskellAnnahLibrary/tree/master/Prelude