Morte.Tutorial

Morte is a minimalist implementation of the calculus of constructions that comes with a parser, type-checker, optimizer, and pretty-printer.

You can think of Morte as a very low-level intermediate language for functional languages. This virtual machine was designed with the following design principles, in descending order of importance:

• Be super-optimizable - by disabling unrestricted recursion
• Be portable - so you can transmit code between different languages
• Be efficient - so that Morte can scale to large code bases
• Be simple - so people can reason about Morte's soundness

This library does not provide any front-end or back-end language for Morte. These will be provided as separate libraries in the future.

The "Introduction" section walks through basic usage of the compiler and library.

The "Desugaring" section explains how to desugar complex abstractions to Morte's core calculus.

The "Optimization" section explains how Morte optimizes programs, providing several long-form example programs and their optimized output.

Introduction

You can test out your first Morte program using the morte executable provided by this library. This executable reads a Morte program from stdin, outputs the type of the program to stderr, and outputs the optimized program to stdout.

We'll begin by translating Haskell's identity function to Morte. For reference, id is defined in Haskell as:

id :: a -> a
id x = x

We will enter the equivalent Morte program at the command line:

$ morte
\(a : *) -> \(x : a) -> x <Enter>
<Ctrl-D>
∀(a : *) → ∀(x : a) → a

λ(a : *) → λ(x : a) → x
$

The compiler outputs two lines. The first line is the type, which is output to stderr. The second line is the optimized program, which is output to stdout.

Compare the type output by the compiler with the equivalent Haskell type:

-- Haskell
id :: a -> a

-- Morte
∀(a : *) → ∀(x : a) → a

The first thing you'll notice is that Morte explicitly quantifies all types. In Haskell, you can do this, too, using the ExplicitForAll extension:

id :: forall a . a -> a

The Morte compiler uses a Unicode forall symbol to sweeten the output, but Morte also accepts other equivalents, too, such as:

-- Ascii '∀'
\/(a : *) -> a -> a

-- English
forall (a : *) -> a -> a

-- Unicode Capital Pi
Π(a : *) -> a -> a

-- ASCII 'Π'
|~|(a : *) -> a -> a

Also, Morte accepts both Unicode and ASCII arrow symbols.

The compiler's last output line is the optimized program, which in this case is identical to our original program (except sweetened with Unicode). Compare to the equivalent Haskell code:

-- Haskell code, desugared to a lambda expression
id = \x -> x

λ(a : *) → λ(x : a) → x

Notice that Morte explicitly binds the type 'a' as an additional parameter. We use this to assign a type to the bound variable x. In Morte, all bound variables must be explicitly annotated with a type because Morte does not perform any type inference.

Now let's modify our program to accept an external type, such as String and then we can specialize our identity function to that type. Remember that the type is just another argument to our function, so we specialize our identity function by just applying it to String.

We'll use a file this time instead of entering the program at the command line:

-- id.mt

-- Morte accepts comments

-- Also, whitespace is not significant
\(String : *) ->
    (\(a : *) -> \(x : a) -> x) String

Then we'll type-check and optimize this program:

$ morte < id.mt
∀(String : *) → ∀(x : String) → String

λ(String : *) → λ(x : String) → x

Morte optimizes our program to the identity function on Strings, but if you notice carefully this is indistinguishable from our original identity function because we still take the String type as parameter. The only difference is that we've renamed 'a' to String.

In fact, Morte knows this and can detect when two expressions are equal up to renaming of bound variables (a.k.a. "alpha-equivalence"). The compiler does not support testing programs for equality, but the library does:

$ ghci
Prelude> import Morte.Core
Prelude Morte.Core> :set -XOverloadedStrings
Prelude Morte.Core> let id = Lam "a" (Const Star) (Lam "x" "a" "x")
Prelude Morte.Core> let id' = Lam "String" (Const Star) (App id "String")
Prelude Morte.Core> id == id'
True

In fact, Morte's equality operator also detects "beta-equivalence" and "eta-equivalence", too, which you can think of as equivalence of normal forms.

We can even use this equality operator to prove the equivalence of many (but not all) complex programs, but first we need to learn how to define more complex abstractions using Morte's restrictive language, as outlined in the next section.

Desugaring

The Expr type defines Morte's syntax, which is very simple:

data Expr
    = Const Const        -- Type system constants
    | Var Var            -- Bound variables
    | Lam Var Expr Expr  -- Lambda
    | Pi  Var Expr Expr  -- "forall"
    | App Expr Expr      -- Function application

For example, you can see what id' from the previous section expands out to by using the Show instance for Expr:

Lam (V "String" 0) (Const Star) (
    App (Lam (V "a" 0) (Const Star) (
             Lam (V "x" 0) (Var (V "a" 0)) (Var (V "x" 0))))
        (Var (V "String" 0)))

... although Morte provides syntactic sugar for building expressions by hand using the OverloadedStrings instance, so you could instead write:

Lam "String" (Const Star) (
    App (Lam "a" (Const Star)( Lam "x" "a" "a")) "String" )

Note that Morte's syntax does not include:

• let expressions
• case expressions
• Built-in values other than functions
• Built-in types other than function types
• newtypes
• Support for multiple expressions/statements
• Modules or imports
• Recursion / Corecursion

Future front-ends to Morte will support these higher-level abstractions, but for now you must desugar all of these to lambda calculus before Morte can type-check and optimize your program. The following sections explain how to desugar these abstractions from a Haskell-like language.

Let

Given a non-recursive let statement of the form:

let var1 :: type1
    var1 = expr1

    var2 :: type2
    var2 = expr2

    ...

    varN :: typeN
    varN = exprN

in  result

You can desugar that to:

(\(var1 : type1) -> \(var2 : type2) -> ... -> \(varN : typeN) -> result) expr1 expr2 ... exprN

Remember that whitespace is not significant, so you can also write that as:

(   \(var1 : type1)
->  \(var2 : type2)
->  ...
->  \(varN : typeN)
->  result
)
expr1
expr2
...
exprN

The Morte compiler does not mistake expr1 through exprN for additional top-level expresions, because a Morte program only consists of a single expression.

Carefully note that the following expression:

let var1 : type1
    var1 = expr1

    var2 : type2
    var2 = expr2

in  result

... is not the same as:

let var1 : type1
    var1 = expr1

in  let var2 : type2
        var2 = expr2

    in  result

They desugar to different Morte code and sometimes the distinction between the two is significant.

Using let, you can desugar this:

let id : forall (a : *) -> a -> a
    id = \(a : *) -> \(x : *) -> x

in  id (forall (a : *) -> a -> a) id

... into this:

-- id2.mt

(   \(id : forall (a : *) -> a -> a)
->  id (forall (a : *) -> a -> a) id  -- Apply the identity function to itself
)

-- id
(\(a : *) -> \(x : a) -> x)

... and the compiler will type-check and optimize that to:

$ morte < id2.mt
∀(a : *) → a → a

λ(a : *) → λ(x : a) → x

Simple types

The following sections use a technique known as Boehm-Berarducci encoding to convert recursive data types to lambda terms. If you already know what Boehm-Berarducci encoding is then you can skip these sections. You might already recognize this technique by the names of overlapping techniques such as CPS-encoding, Church-encoding, or F-algebras.

I'll first explain how to desugar a somewhat complicated non-recursive type and then show how this trick specializes to simpler types. The first example is quite long, but you'll see that it gets much more compact in the simpler examples.

Given the following non-recursive type:

let data T a b c = A | B a | C b c

in  result

You can desugar that to the following Morte code:

    -- The type constructor
(   \(T : * -> * -> * -> *)

    -- The value constructors
->  \(A : forall (a : *) -> forall (b : *) -> forall (c : *)           -> T a b c)
->  \(B : forall (a : *) -> forall (b : *) -> forall (c : *) -> a      -> T a b c)
->  \(C : forall (a : *) -> forall (b : *) -> forall (c : *) -> b -> c -> T a b c)

    -- Pattern match on T
->  \(  matchT
    :   forall (a : *) -> forall (b : *) -> forall (c : *)
    ->  T a b c
    ->  forall (r : *)
    ->  r              -- `A` branch of the pattern match
    ->  (a -> r)       -- `B` branch of the pattern match
    ->  (b -> c -> r)  -- `C` branch of the pattern match
    ->  r
    )
-> result
)

-- A value of type `T a b c` is just a preformed pattern match
(   \(a : *) -> \(b : *) -> \(c : *)
->  forall (r : *)
->  r              -- A branch of the pattern match
->  (a -> r)       -- B branch of the pattern match
->  (b -> c -> r)  -- C branch of the pattern match
->  r
)

-- Constructor for A
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(r : *)
->  \(A : r)
->  \(B : a -> r)
->  \(C : b -> c -> r)
->  A
)

-- Constructor for B
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(va : a)
->  \(r : *)
->  \(A : r)
->  \(B : a -> r)
->  \(C : b -> c -> r)
->  B va
)

-- Constructor for C
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(vb : b)
->  \(vc : c)
->  \(r : *)
->  \(A : r)
->  \(B : a -> r)
->  \(C : b -> c -> r)
->  C vb vc
)

-- matchT is just the identity function
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(t : forall (r : *) -> r -> (a -> r) -> (b -> c -> r) -> r)
->  t
)

Within the result expression, you could assemble values of type 'T' using the constructors:

Context:
String : *
Int    : *
Bool   : *
s      : String
i      : Int
b      : Bool

A String Int Bool     : T String Int Bool
B String Int Bool s   : T String Int Bool
C String Int Bool i b : T String Int Bool

... and you could pattern match on any value of type 'T' using matchT:

Context:
String : *
Int    : *
Bool   : *
r      : *  -- The result type of all three case branches
t      : T String Int Bool

matchT String Int Bool r t
    (                                ...)  -- Branch if you match `A`
    (\(s : String) ->                ...)  -- Branch if you match `B`
    (\(i : Int   ) -> \(b : Bool) -> ...)  -- Branch if you match `C`

Now let's see how this specializes to a simpler example: Haskell's Bool type.

-- let data Bool = True | False
--
-- in  result

(   \(Bool : *)
->  \(True  : Bool)
->  \(False : Bool)
->  \(if : Bool -> forall (r : *) -> r -> r -> r)
->  result
)

-- Bool
(forall (r : *) -> r -> r -> r)

-- True
(\(r : *) -> \(x : r) -> \(_ : r) -> x)

-- False
(\(r : *) -> \(_ : r) -> \(x : r) -> x)

-- if
(\(b : forall (r : *) -> r -> r -> r) -> b)

Notice that if is our function to pattern match on a Bool. The two branches of the if correspond to the then and else clauses.

Using this definition of Bool we can define a simple program:

-- bool.mt

-- let data Bool = True | False
--
-- in  if True then One else Zero

(   \(Bool : *)
->  \(True  : Bool)
->  \(False : Bool)
->  \(if : Bool -> forall (r : *) -> r -> r -> r)
->  \(Int  : *)
->  \(Zero : Int)
 -> \(One  : Int)
->  if True Int One Zero
)

-- Bool
(forall (r : *) -> r -> r -> r)

-- True
(\(r : *) -> \(x : r) -> \(_ : r) -> x)

-- False
(\(r : *) -> \(_ : r) -> \(x : r) -> x)

-- if
(\(b : forall (r : *) -> r -> r -> r) -> b)

If you type-check and optimize this, you get:

$ morte < bool.mt
∀(Int : *) → ∀(Zero : Int) → ∀(One : Int) → Int

λ(Int : *) → λ(Zero : Int) → λ(One : Int) → One

The compiler reduces the program to One. All the dead code has been eliminated. Also, if you study the output program closely, you'll notice that it's equivalent to False and the program's type is equivalent to the Bool type. Try flipping the Zero and One arguments to if and see what happens.

Now let's implement Haskell's binary tuple type, except using a named type and constructor since Morte does not support tuple syntax:

-- let Pair a b = P a b
--
-- in  result

(   \(Pair : * -> * -> *)
->  \(P    : forall (a : *) -> forall (b : *) -> a -> b -> Pair a b)
->  \(fst  : forall (a : *) -> forall (b : *) -> Pair a b -> a)
->  \(snd  : forall (a : *) -> forall (b : *) -> Pair a b -> b)
->  result
)

-- Pair
(\(a : *) -> \(b : *) -> forall (r : *) -> (a -> b -> r) -> r)

-- P
(   \(a : *)
->  \(b : *)
->  \(va : a)
->  \(vb : b)
->  \(r : *)
->  \(Pair : a -> b -> r)
->  Pair va vb
)

-- fst
(   \(a : *)
->  \(b : *)
->  \(p : forall (r : *) -> (a -> b -> r) -> r)
->  p a (\(x : a) -> \(_ : b) -> x)
)

-- snd
(   \(a : *)
->  \(b : *)
->  \(p : forall (r : *) -> (a -> b -> r) -> r)
->  p b (\(_ : a) -> \(x : b) -> x)
)

Here we provide fst and snd functions instead of matchPair.

Let's write a simple program that uses this Pair type:

-- pair.mt

-- let Pair a b = P a b
--
-- in  \x y -> snd (P x y)

(   \(Pair : * -> * -> *)
->  \(P    : forall (a : *) -> forall (b : *) -> a -> b -> Pair a b)
->  \(fst  : forall (a : *) -> forall (b : *) -> Pair a b -> a)
->  \(snd  : forall (a : *) -> forall (b : *) -> Pair a b -> b)
->  \(a : *) -> \(x : a) -> \(y : a) -> snd a a (P a a x y)
)

-- Pair
(\(a : *) -> \(b : *) -> forall (r : *) -> (a -> b -> r) -> r)

-- P
(   \(a : *)
->  \(b : *)
->  \(va : a)
->  \(vb : b)
->  \(r : *)
->  \(Pair : a -> b -> r)
->  Pair va vb
)

-- fst
(   \(a : *)
->  \(b : *)
->  \(p : forall (r : *) -> (a -> b -> r) -> r)
->  p a (\(x : a) -> \(_ : b) -> x)
)

-- snd
(   \(a : *)
->  \(b : *)
->  \(p : forall (r : *) -> (a -> b -> r) -> r)
->  p b (\(_ : a) -> \(x : b) -> x)
)

If you compile and type-check that you get:

$ morte < pair.mt
∀(a : *) → ∀(x : a) → ∀(y : a) → a

λ(a : *) → λ(x : a) → λ(y : a) → y

This is also equal to our previous program. Just rename 'a' to Int, rename 'x' to Zero and rename 'y' to One.

You can also import data types from whatever backend you use by accepting those types and functions on those types as explicit arguments to your program. For example, if you want to use machine integers, hardware arithmetic and integer literals, then you can just parametrize your program on the type, operations, and literal values:

    \(Int  : *)                  -- Foreign type
->  \((+)  : Int -> Int -> Int)  -- Foreign function
->  \((*)  : Int -> Int -> Int)  -- Foreign function
->  \(litA : Int)                -- Foreign integer literal
->  \(litB : Int)                -- Foreign integer literal
->  \(litC : Int)                -- Foreign integer literal
...

However, the more types and operations you encode natively within Morte the more the optimizer can simplify your program. This is because there is no runtime performance penalty from using natively encoded data types. Morte will optimize these all away at compile time because they are just ordinary functions under the hood and Morte optimizes away all function calls.

Newtypes

Defining a newtype is no different than defining a data type with a single constructor with one field:

-- let newtype Name = MkName { getName :: String }
--
-- in  result

(   \(Name    : *)
->  \(MkName  : String -> Name  )
->  \(getName : Name   -> String)
->  result
)

-- Name
String

-- MkName
(\(str : String) -> str)

-- getName
(\(str : String) -> str)

Within the expression result, Name is actually a new type, meaning that a value of type Name will not type-check as a String and, vice versa, a value of type String will not type-check as a Name. You would have to explicitly convert back and forth between Name and String using the MkName and getName functions.

We can prove this using the following example program:

-- newtype.mt

-- let newtype Name = MkName { getName :: String }
--
-- in  (f :: Name -> Name) (x :: String)

(   \(Name    : *)
->  \(MkName  : String -> Name  )
->  \(getName : Name   -> String)
->  \(f : Name -> Name) -> \(x : String) -> f x
)

-- Name
String

-- MkName
(\(str : String) -> str)

-- getName
(\(str : String) -> str)

That program fails to type-check, giving the following error message:

$ morte < newtype.mt
Context:
Name : *
MkName : String → Name
getName : Name → String
f : Name → Name
x : String

Expression: f x

Error: Function applied to argument of the wrong type

Expected type: Name
Argument type: String

There is never a performance penalty for using newtypes, but this is just a special case of the fact that there is no performance penalty for using any natively encoded data types in Morte.

Recursion

Defining a recursive data type is very similar to defining a non-recursive type. Let's use lists as an example:

let data List a = Cons a (List a) | Nil

in  result

The equivalent Morte code is:

-- let data List a = Cons a (List a) | Nil
--
-- in  result

(   \(List : * -> *)
->  \(Cons : forall (a : *) -> a -> List a -> List a)
->  \(Nil  : forall (a : *)                -> List a)
->  \(  foldr
    :   forall (a : *) -> List a -> forall (r : *) -> (a -> r -> r) -> r -> r
    )
->  result
)

-- List
(   \(a : *)
->  forall (list : *)
->  (a -> list -> list)  -- Cons
->  list                 -- Nil
->  list
)

-- Cons
(   \(a : *)
->  \(va  : a)
->  \(vas : forall (list : *) -> (a -> list -> list) -> list -> list)
->  \(list : *)
->  \(Cons : a -> list -> list)
->  \(Nil  : list)
->  Cons va (vas list Cons Nil)
)

-- Nil
(   \(a : *)
->  \(list : *)
->  \(Cons : a -> list -> list)
->  \(Nil  : list)
->  Nil
)

-- foldr
(   \(a : *)
->  \(vas : forall (list : *) -> (a -> list -> list) -> list -> list)
->  vas
)

Here I use the list type variable where previous examples would use 'r' to emphasize that the continuations that a List consumes both have the same shape as the list constructors. You just replace all recursive references to the data type with the type of the final result, pretending that the final result is a list.

Let's extend the List example with the Bool code to implement Haskell's all function and use it on an actual List of Bools:

-- all.mt

-- let data Bool = True | False
--
--     data List a = Cons a (List a) | Nil
--
-- in  let (&&) :: Bool -> Bool -> Bool
--         (&&) b1 b2 = if b1 then b2 else False
--
--         bools :: List Bool
--         bools = Cons True (Cons True (Cons True Nil))
--
--     in  foldr bools (&&) True

(   \(Bool : *)
->  \(True  : Bool)
->  \(False : Bool)
->  \(if : Bool -> forall (r : *) -> r -> r -> r)
->  \(List : * -> *)
->  \(Cons : forall (a : *) -> a -> List a -> List a)
->  \(Nil  : forall (a : *)                -> List a)
->  \(  foldr
    :   forall (a : *) -> List a -> forall (r : *) -> (a -> r -> r) -> r -> r
    )
->  (   \((&&) : Bool -> Bool -> Bool)
    ->  \(bools : List Bool)
    ->  foldr Bool bools Bool (&&) True
    )

    -- (&&)
    (\(x : Bool) -> \(y : Bool) -> if x Bool y False)

    -- bools
    (Cons Bool True (Cons Bool True (Cons Bool True (Nil Bool))))
)

-- Bool
(forall (r : *) -> r -> r -> r)

-- True
(\(r : *) -> \(x : r) -> \(_ : r) -> x)

-- False
(\(r : *) -> \(_ : r) -> \(x : r) -> x)

-- if
(\(b : forall (r : *) -> r -> r -> r) -> b)

-- List
(   \(a : *)
->  forall (list : *)
->  (a -> list -> list)  -- Cons
->  list                 -- Nil
->  list
)

-- Cons
(   \(a : *)
->  \(va  : a)
->  \(vas : forall (list : *) -> (a -> list -> list) -> list -> list)
->  \(list : *)
->  \(Cons : a -> list -> list)
->  \(Nil  : list)
->  Cons va (vas list Cons Nil)
)

-- Nil
(   \(a : *)
->  \(list : *)
->  \(Cons : a -> list -> list)
->  \(Nil  : list)
->  Nil
)

-- foldr
(   \(a : *)
->  \(vas : forall (list : *) -> (a -> list -> list) -> list -> list)
->  vas
)

If you type-check and optimize the program, the compiler will statically evaluate the entire computation, reducing the program to True:

$ morte < all.mt
∀(r : *) → r → r → r

λ(r : *) → λ(x : r) → λ(_ : r) → x

Here's another example of encoding a recursive type, using natural numbers:

-- let data Nat = Succ Nat | Zero
--
-- in  result

(   \(Nat : *)
->  \(Succ : Nat -> Nat)
->  \(Zero : Nat)
->  \(foldNat : Nat -> forall (r : *) -> (r -> r) -> r -> r)
->  result
)

-- Nat
(   forall (nat : *)
->  (nat -> nat)  -- Succ
->  nat           -- Zero
->  nat
)

(   \(n : forall (nat : *) -> (nat -> nat) -> nat -> nat)
->  \(nat : *)
->  \(Succ : nat -> nat)
->  \(Zero : nat)
->  Succ (n nat Succ Zero)
)

(   \(nat : *)
->  \(Succ : nat -> nat)
->  \(Zero : nat)
->  Zero
)

(   \(n : forall (nat : *) -> (nat -> nat) -> nat -> nat)
->  n
)

As an exercise, try implementing (+) for the Nat type, then implementing Haskell's sum, then using sum on a List of Nats. Verify that the compiler statically computes the sum as a Church-encoded numeral.

The encoding outlined in this section is equivalent to an F-algebra encoding of a recursive type, which is any encoding of the following shape:

forall (x : *) -> (F x -> x) -> x

.. where F is a strictly-positive functor.

Our List a encoding is isomorphic to an F-algebra encoding where:

F x = Maybe (a, x)

... and our Nat encoding is isomorphic to an F-algebra encoding where:

F x = Maybe x

Existential Quantification

You can translate existential quantified types to use universal quantification. For example, consider the following existentially quantified Haskell type:

let data Example = forall s . MkExample s (s -> String)

in  result

The equivalent Morte program is:

-- let data Example = forall s . Example s (s -> String)
--
-- in  result

\(String : *) ->
(   \(Example : *)
->  \(MkExample : forall (s : *) -> s -> (s -> String) -> Example)
->  \(  matchExample
    :   Example
    ->  forall (x : *)
    ->  (forall (s : *) -> s -> (s -> String) -> x)
    ->  x
    )
->  result
)

-- Example
(   forall (x : *)
->  (forall (s : *) -> s -> (s -> String) -> x)  -- MkExample
->  x
)

-- MkExample
(   \(s : *)
->  \(vs : s)
->  \(fs : s -> String)
->  \(x : *)
->  \(MkExample : forall (s : *) -> s -> (s -> String) -> x)
->  MkExample s vs fs
)

-- matchExample
(   \(e : forall (x : *) -> (forall (s : *) -> s -> (s -> String) -> x) -> x)
->  e
)

More generally, for every constructor that you existentially quantify with a type variable 's' you just add a (forall (s : *) -> ...) prefix to that constructor's continuation. If you "pattern match" against the constructor corresponding to that continuation you will bind the existentially quantified type.

For example, we can pattern match against the MkExample constructor like this:

\(e : Example) -> matchExample e
      (\(s : *) -> (x : s) -> (f : s -> String) -> expr) 

The type 's' will be in scope for expr and we can safely apply the bound function to the bound value if we so chose to extract a String, despite not knowing which type 's' we bound:

\(e : Example) -> matchExample e
      (\(s : *) -> (x : s) -> (f : s -> String) -> f x) 

The two universal quantifiers in the definition of the Example type statically forbid the type 's' from leaking from the pattern match.

Corecursion

Recursive types can only encode finite data types. If you want a potentially infinite data type (such as an infinite list), you must encode the type in a different way.

For example, consider the following infinite stream type:

codata Stream a = Cons a (Stream a)

If you tried to encode that as a recursive type, you would end up with this Morte type:

\(a : *) -> forall (x : *) -> (a -> x -> x) -> x

However, this type is uninhabited, meaning that you cannot create a value of the above type for any choice of 'a'. Try it, if you don't believe me.

Potentially infinite types must be encoded using a dual trick, where we store them as an existentially quantified seed and a generating step function that emits one layer alongside a new seed.

For example, the above Stream type would translate to the following non-recursive representation. The StreamF constructor represents one layer and the Stream type lets us generate an infinite number of layers by providing an initial seed of type s and a generation function of type (s -> StreamF a s):

-- Replace the corecursive occurrence of `Stream` with `s`
data StreamF a s = Cons a s

data Stream a = forall s . MkStream s (s -> StreamF a s)

The above type will work for any type 's' as the 's' is existentially quantified. The end user of the Stream will never be able to detect what the original type of s was, because the MkStream constructor closes over that information permanently.

An example Stream is the following lazy stream of natural numbers:

nats :: Stream Int
nats = MkStream 0 (\n -> Cons n (n + 1))

Internally, the above Stream uses an Int as its internal state, but that is completely invisible to all downstream code, which cannot access the concrete type of the internal state any longer.

In fact, this trick of using a seed and a generating step function is a special case of a F-coalgebra encoding of a corecursive type, which is anything of the form:

exists s . (s, s -> F s)

... where F is a strictly-positive functor.

Once you F-coalgebra encode the Stream type you can translate the type to Morte using the rules for existential quantification given in the previous section:

(forall (x : *) -> (forall (s : *) -> s -> (s -> StreamF a s) -> x) -> x

See the next section for some example Stream code.

Optimization

You might wonder why Morte forbids recursion, forcing us to encode data types as F-algebras or F-coalgebras. Morte imposes this restriction in order to super-optimize your program. For example, consider the following program which maps the identity function over a list:

-- mapid1.mt

(    \(List : * -> *)
->   \(map  : forall (a : *) -> forall (b : *) -> (a -> b) -> List a -> List b)
->   \(id   : forall (a : *) -> a -> a)
    ->   \(a : *) -> map a a (id a)
)

-- List
(\(a : *) -> forall (x : *) -> (a -> x -> x) -> x -> x)

-- map
(   \(a : *)
->  \(b : *)
->  \(f : a -> b)
->  \(l : forall (x : *) -> (a -> x -> x) -> x -> x)
->  \(x : *)
->  \(Cons : b -> x -> x)
->  \(Nil: x)
->  l x (\(va : a) -> \(vx : x) -> Cons (f va) vx) Nil
)

-- id
(\(a : *) -> \(va : a) -> va)

If we examine the compiler output, we'll see that the compiler fuses away the map, leaving behind the identity function on lists:

$ morte < mapid1.mt
∀(a : *) → (∀(x : *) → (a → x → x) → x → x) → ∀(x : *) → (a → x → x) → x → x

λ(a : *) → λ(l : ∀(x : *) → (a → x → x) → x → x) → l

We can prove this by replacing our map with the identity function on lists:

-- mapid2.mt

(    \(List : * -> *)
->   \(map  : forall (a : *) -> forall (b : *) -> (a -> b) -> List a -> List b)
->   \(id   : forall (a : *) -> a -> a)
    ->   \(a : *) -> id (List a)
)

-- List
(\(a : *) -> forall (x : *) -> (a -> x -> x) -> x -> x)

-- map
(   \(a : *)
->  \(b : *)
->  \(f : a -> b)
->  \(l : forall (x : *) -> (a -> x -> x) -> x -> x)
->  \(x : *)
->  \(Cons : b -> x -> x)
->  \(Nil: x)
->  l x (\(va : a) -> \(vx : x) -> Cons (f va) vx) Nil
)

-- id
(\(a : *) -> \(va : a) -> va)

The compiler output for this is alpha-equivalent:

$ morte < mapid2.mt
∀(a : *) → (∀(x : *) → (a → x → x) → x → x) → ∀(x : *) → (a → x → x) → x → x

λ(a : *) → λ(va : ∀(x : *) → (a → x → x) → x → x) → va

However, we don't have to trust our fallible eyes. We can enlist the morte library to mechanically check that the two programs are equal:

$ ghci
Prelude> import qualified Data.Text.Lazy.IO as Text
Prelude Text> txt1 <- Text.readFile "mapid1.mt"
Prelude Text> txt2 <- Text.readFile "mapid2.mt"
Prelude Text> import Morte.Parser
Prelude Text Morte.Parser> let e1 = exprFromText txt1
Prelude Text Morte.Parser> let e2 = exprFromText txt2
Prelude Text Morte.Parser> import Control.Applicative
Prelude Text Morte.Parser Control.Applicative> liftA2 (==) e1 e2
Right True

We just mechanically proved that map id == id. When we transform our code to a non-recursive form we've done most of the work. The compiler can then check that the two programs are equal by just optimizing both programs and verifying that they produce identical optimized code.

Using this same trick we can also prove the other map fusion law:

map (f . g) = map f . map g

Here is the first program, corresponding to the left-hand side of the equation:

-- mapcomp1.mt

-- map (f . g)

(   \(List : * -> *)
->  \(map  : forall (a : *) -> forall (b : *) -> (a -> b) -> List a -> List b)
->  \(  (.)
    :   forall (a : *)
    ->  forall (b : *)
    ->  forall (c : *)
    ->  (b -> c)
    ->  (a -> b)
    ->  (a -> c)
    )
    ->  \(a : *)
    ->  \(b : *)
    ->  \(c : *)
    ->  \(f : b -> c)
    ->  \(g : a -> b)
    ->  map a c ((.) a b c f g)
)

-- List
(\(a : *) -> forall (x : *) -> (a -> x -> x) -> x -> x)

-- map
(   \(a : *)
->  \(b : *)
->  \(f : a -> b)
->  \(l : forall (x : *) -> (a -> x -> x) -> x -> x)
->  \(x : *)
->  \(Cons : b -> x -> x)
->  \(Nil: x)
->  l x (\(va : a) -> \(vx : x) -> Cons (f va) vx) Nil
)

-- (.)
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(f : b -> c)
->  \(g : a -> b)
->  \(va : a)
->  f (g va)
)

... and here is the second program, corresponding to the right-hand side:

-- mapcomp2.mt

(   \(List : * -> *)
->  \(map  : forall (a : *) -> forall (b : *) -> (a -> b) -> List a -> List b)
->  \(  (.)
    :   forall (a : *)
    ->  forall (b : *)
    ->  forall (c : *)
    ->  (b -> c)
    ->  (a -> b)
    ->  (a -> c)
    )
    ->  \(a : *)
    ->  \(b : *)
    ->  \(c : *)
    ->  \(f : b -> c)
    ->  \(g : a -> b)
    ->  (.) (List a) (List b) (List c) (map b c f) (map a b g)
)

-- List
(\(a : *) -> forall (x : *) -> (a -> x -> x) -> x -> x)

-- map
(   \(a : *)
->  \(b : *)
->  \(f : a -> b)
->  \(l : forall (x : *) -> (a -> x -> x) -> x -> x)
->  \(x : *)
->  \(Cons : b -> x -> x)
->  \(Nil: x)
->  l x (\(va : a) -> \(vx : x) -> Cons (f va) vx) Nil
)

-- (.)
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(f : b -> c)
->  \(g : a -> b)
->  \(va : a)
->  f (g va)
)

Verify using the morte library that those produce identical expressions. For reference, they both generate the following optimized program that loops over the list just once, applying 'f' and 'g' to every value:

$ morte < mapcomp1.mt
∀(a : *) → ∀(b : *) → ∀(c : *) → ∀(f : b → c) → ∀(g : a → b) → (∀(x : *) → (a 
→ x → x) → x → x) → ∀(x : *) → (c → x → x) → x → x

λ(a : *) → λ(b : *) → λ(c : *) → λ(f : b → c) → λ(g : a → b) → λ(l : ∀(x : *) 
→ (a → x → x) → x → x) → λ(x : *) → λ(Cons : c → x → x) → l x (λ(va : a) → Con
s (f (g va)))

We can also prove map fusion for corecursive streams as well. Just use the following program:

-- corecursive.mt

-- first :: (a -> b) -> (a, c) -> (b, c)
-- first f (va, vb) = (f va, vb) 
-- 
-- data Stream a = Cons (a, Stream a)
-- 
-- map :: (a -> b) -> Stream a -> Stream b
-- map f (Cons (va, s)) = Cons (first f (va, map f s))
-- 
-- -- exampleA = exampleB
-- 
-- exampleA :: Stream a -> Stream a
-- exampleA = map id
-- 
-- exampleB :: Stream a -> Stream a
-- exampleB = id
-- 
-- -- exampleC = exampleD
-- 
-- exampleC :: (b -> c) -> (a -> b) -> Stream a -> Stream c
-- exampleC f g = map (f . g)
-- 
-- exampleD :: (b -> c) -> (a -> b) -> Stream a -> Stream c
-- exampleD f g = map f . map g

(   \(id : forall (a : *) -> a -> a)
->  \(  (.)
    :   forall (a : *)
    ->  forall (b : *)
    ->  forall (c : *)
    ->  (b -> c)
    ->  (a -> b)
    ->  (a -> c)
    )
->  \(Pair : * -> * -> *)
->  \(P : forall (a : *) -> forall (b : *) -> a -> b -> Pair a b)
->  \(  first
    :   forall (a : *)
    ->  forall (b : *)
    ->  forall (c : *)
    ->  (a -> b)
    ->  Pair a c
    ->  Pair b c
    )

->  (   \(Stream : * -> *)
    ->  \(  map
        :   forall (a : *)
        ->  forall (b : *)
        ->  (a -> b)
        ->  Stream a
        ->  Stream b
        )

        -- exampleA = exampleB
    ->  (   \(exampleA : forall (a : *) -> Stream a -> Stream a)
        ->  \(exampleB : forall (a : *) -> Stream a -> Stream a)

        -- exampleC = exampleD
        ->  \(  exampleC
            :   forall (a : *)
            ->  forall (b : *)
            ->  forall (c : *)
            ->  (b -> c)
            ->  (a -> b)
            ->  Stream a
            ->  Stream c
            )

        ->  \(  exampleD
            :   forall (a : *)
            ->  forall (b : *)
            ->  forall (c : *)
            ->  (b -> c)
            ->  (a -> b)
            ->  Stream a
            ->  Stream c
            )

        -- Uncomment the example you want to test
        ->  exampleA
--      ->  exampleB
--      ->  exampleC
--      ->  exampleD
        )

        -- exampleA
        (\(a : *) -> map a a (id a))
  
        -- exampleB
        (\(a : *) -> id (Stream a))

        -- exampleC
        (   \(a : *)
        ->  \(b : *)
        ->  \(c : *)
        ->  \(f : b -> c)
        ->  \(g : a -> b)
        ->  map a c ((.) a b c f g)
        )

        --  exampleD
        (   \(a : *)
        ->  \(b : *)
        ->  \(c : *)
        ->  \(f : b -> c)
        ->  \(g : a -> b)
        ->  (.) (Stream a) (Stream b) (Stream c) (map b c f) (map a b g)
        )
    )

    -- Stream
    (   \(a : *)
    ->  forall (x : *)
    ->  (forall (s : *) -> s -> (s -> Pair a s) -> x)
    ->  x
    )

    -- map
    (   \(a : *)
    ->  \(b : *)
    ->  \(f : a -> b)
    ->  \(  st
        :   forall (x : *) -> (forall (s : *) -> s -> (s -> Pair a s) -> x) -> x
        )
    ->  \(x : *)
    ->  \(S : forall (s : *) -> s -> (s -> Pair b s) -> x)
    ->  st
        x
        (   \(s : *)
        ->  \(seed : s)
        ->  \(step : s -> Pair a s)
        ->  S
            s
            seed
            (\(seed : s) -> first a b s f (step seed))
        )
    )
)

-- id
(\(a : *) -> \(va : a) -> va)

-- (.)
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(f : b -> c)
->  \(g : a -> b)
->  \(va : a)
->  f (g va)
)

-- Pair
(\(a : *) -> \(b : *) -> forall (x : *) -> (a -> b -> x) -> x)

-- P
(   \(a : *)
->  \(b : *)
->  \(va : a)
->  \(vb : b)
->  \(x : *)
->  \(P : a -> b -> x)
->  P va vb
)

-- first
(   \(a : *)
->  \(b : *)
->  \(c : *)
->  \(f : a -> b)
->  \(p : forall (x : *) -> (a -> c -> x) -> x)
->  \(x : *)
->  \(Pair : b -> c -> x)
->  p x (\(va : a) -> \(vc : c) -> Pair (f va) vc)
)

Both exampleA and exampleB generate identical optimized expressions, corresponding to the identity function on Stream:

$ morte < corecursive.mt
∀(a : *) → (∀(x : *) → (∀(s : *) → s → (s → ∀(x : *) → (a → s → x) → x) → x) →
x) → ∀(x : *) → (∀(s : *) → s → (s → ∀(x : *) → (a → s → x) → x) → x) → x

λ(a : *) → λ(st : ∀(x : *) → (∀(s : *) → s → (s → ∀(x : *) → (a → s → x) → x) 
→ x) → x) → st

Similarly, both exampleC and exampleD generate identical optimized expressions, corresponding to applying f and g to every value emitted by the generating step function:

$ morte < corecursive.mt
∀(a : *) → ∀(b : *) → ∀(c : *) → (b → c) → (a → b) → (∀(x : *) → (∀(s : *) → s
 → (s → ∀(x : *) → (a → s → x) → x) → x) → x) → ∀(x : *) → (∀(s : *) → s → (s 
→ ∀(x : *) → (c → s → x) → x) → x) → x

λ(a : *) → λ(b : *) → λ(c : *) → λ(f : b → c) → λ(g : a → b) → λ(st : ∀(x : *)
 → (∀(s : *) → s → (s → ∀(x : *) → (a → s → x) → x) → x) → x) → λ(x : *) → λ(S
 : ∀(s : *) → s → (s → ∀(x : *) → (c → s → x) → x) → x) → st x (λ(s : *) → λ(s
eed : s) → λ(step : s → ∀(x : *) → (a → s → x) → x) → S s seed (λ(seed : s) → 
λ(x : *) → λ(Pair : c → s → x) → step seed x (λ(va : a) → Pair (f (g va)))))

Normalization

Morte has a very simple optimization scheme. The only thing that Morte does to optimize programs is beta-reduce them and eta-reduce them to their normal form. Since Morte's core calculus is non-recursive, this reduction is guaranteed to terminate.

The way Morte compares expressions for equality is just to compare their normal forms. Note that this definition of equality does not detect all equal programs. Here's an example of an equality that Morte does not currently detect (but might detect in the future):

k : forall (x : *) -> (a -> x) -> x

k (f . g) = f (k g)

This is an example of a free theorem: an equality that can be deduced purely from the type of k. Morte may eventually use free theorems to further normalize expression, but for now it does not.

Normalization leads to certain emergent properties when optimizing recursive code or corecursive code. If you optimize a corecursive loop you will produce code equivalent to a while loop where the seed is the initial state of the loop and the generating step function unfolds one iteration of the loop. If you optimize a recursive loop you will generate an unrolled loop. See the next section for an example of Morte generating a very large unrolled loop.

Normalization confers one very useful property: the runtime performance of a Morte program is completely impervious to abstraction. Adding additional abstraction layers may increase compile time, but runtime performance will remain constant. The runtime performance of a program is solely a function of the program's normal form, and adding additional abstraction layers never changes the normal form your program.

Effects

Morte uses the Haskell approach to effects, where effects are represented as terms within the language and evaluation order has no impact on order of effects. This is by necessity: if evaluation triggered side effects then Morte would be unable to optimize expressions by normalizing them.

The following example encodes IO within Morte as an abstract syntax tree of effects (a.k.a. a "free monad"). Encoding IO as a free monad is not strictly necessary, but doing so makes Morte aware of the monad laws, which allows it to greatly simplify the program:

-- recursive.mt

-- The Haskell code we will translate to Morte:
--
--     import Prelude hiding (
--         (+), (*), IO, putStrLn, getLine, (>>=), (>>), return )
-- 
--     -- Simple prelude
--
--     data Nat = Succ Nat | Zero
--
--     zero :: Nat
--     zero = Zero
--
--     one :: Nat
--     one = Succ Zero
--
--     (+) :: Nat -> Nat -> Nat
--     Zero   + n = n
--     Succ m + n = m + Succ n
--
--     (*) :: Nat -> Nat -> Nat
--     Zero   * n = Zero
--     Succ m * n = n + (m * n)
--
--     foldNat :: Nat -> (a -> a) -> a -> a
--     foldNat  Zero    f x = x
--     foldNat (Succ m) f x = f (foldNat m f x)
--
--     data IO r = PutStrLn String (IO r) | GetLine (String -> IO r) | Return r
--
--     putStrLn :: String -> IO U
--     putStrLn str = PutStrLn str (Return Unit)
--
--     getLine :: IO String
--     getLine = GetLine Return
--
--     return :: a -> IO a
--     return = Return
--
--     (>>=) :: IO a -> (a -> IO b) -> IO b
--     PutStrLn str io >>= f = PutStrLn str (io >>= f)
--     GetLine k       >>= f = GetLine (\str -> k str >>= f)
--     Return r        >>= f = f r
--
--     -- Derived functions
--
--     (>>) :: IO U -> IO U -> IO U
--     m >> n = m >>= \_ -> n
--
--     two :: Nat
--     two = one + one
--
--     three :: Nat
--     three = one + one + one
--
--     four :: Nat
--     four = one + one + one + one
--
--     five :: Nat
--     five = one + one + one + one + one
--
--     six :: Nat
--     six = one + one + one + one + one + one
--
--     seven :: Nat
--     seven = one + one + one + one + one + one + one
--
--     eight :: Nat
--     eight = one + one + one + one + one + one + one + one
--
--     nine :: Nat
--     nine = one + one + one + one + one + one + one + one + one
--
--     ten :: Nat
--     ten = one + one + one + one + one + one + one + one + one + one
--
--     replicateM_ :: Nat -> IO U -> IO U
--     replicateM_ n io = foldNat n (io >>) (return Unit)
--
--     ninetynine :: Nat
--     ninetynine = nine * ten + nine
--
--     main_ :: IO U
--     main_ = getLine >>= putStrLn

-- "Free" variables
(   \(String : *   )
->  \(U : *)
->  \(Unit : U)

    -- Simple prelude
->  (   \(Nat : *)
    ->  \(zero : Nat)
    ->  \(one : Nat)
    ->  \((+) : Nat -> Nat -> Nat)
    ->  \((*) : Nat -> Nat -> Nat)
    ->  \(foldNat : Nat -> forall (a : *) -> (a -> a) -> a -> a)
    ->  \(IO : * -> *)
    ->  \(return : forall (a : *) -> a -> IO a)
    ->  \((>>=)
        :   forall (a : *)
        ->  forall (b : *)
        ->  IO a
        ->  (a -> IO b)
        ->  IO b
        )
    ->  \(putStrLn : String -> IO U)
    ->  \(getLine : IO String)

        -- Derived functions
    ->  (   \((>>) : IO U -> IO U -> IO U)
        ->  \(two   : Nat)
        ->  \(three : Nat)
        ->  \(four  : Nat)
        ->  \(five  : Nat)
        ->  \(six   : Nat)
        ->  \(seven : Nat)
        ->  \(eight : Nat)
        ->  \(nine  : Nat)
        ->  \(ten   : Nat)
        ->  (   \(replicateM_ : Nat -> IO U -> IO U)
            ->  \(ninetynine : Nat)
            ->  replicateM_ ninetynine ((>>=) String U getLine putStrLn)
            )

            -- replicateM_
            (   \(n : Nat)
            ->  \(io : IO U)
            ->  foldNat n (IO U) ((>>) io) (return U Unit)
            )

            -- ninetynine
            ((+) ((*) nine ten) nine)
        )

        -- (>>)
        (   \(m : IO U)
        ->  \(n : IO U)
        ->  (>>=) U U m (\(_ : U) -> n)
        )

        -- two
        ((+) one one)

        -- three
        ((+) one ((+) one one))

        -- four
        ((+) one ((+) one ((+) one one)))

        -- five
        ((+) one ((+) one ((+) one ((+) one one))))

        -- six
        ((+) one ((+) one ((+) one ((+) one ((+) one one)))))

        -- seven
        ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one one))))))

        -- eight
        ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one one)))))))
        -- nine
        ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one one))))))))

        -- ten
        ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one ((+) one one)))))))))
    )

    -- Nat
    (   forall (a : *)
    ->  (a -> a)
    ->  a
    ->  a
    )

    -- zero
    (   \(a : *)
    ->  \(Succ : a -> a)
    ->  \(Zero : a)
    ->  Zero
    )

    -- one
    (   \(a : *)
    ->  \(Succ : a -> a)
    ->  \(Zero : a)
    ->  Succ Zero
    )

    -- (+)
    (   \(m : forall (a : *) -> (a -> a) -> a -> a)
    ->  \(n : forall (a : *) -> (a -> a) -> a -> a)
    ->  \(a : *)
    ->  \(Succ : a -> a)
    ->  \(Zero : a)
    ->  m a Succ (n a Succ Zero)
    )

    -- (*)
    (   \(m : forall (a : *) -> (a -> a) -> a -> a)
    ->  \(n : forall (a : *) -> (a -> a) -> a -> a)
    ->  \(a : *)
    ->  \(Succ : a -> a)
    ->  \(Zero : a)
    ->  m a (n a Succ) Zero
    )

    -- foldNat
    (   \(n : forall (a : *) -> (a -> a) -> a -> a)
    ->  n
    )

    -- IO
    (   \(r : *)
    ->  forall (x : *)
    ->  (String -> x -> x)
    ->  ((String -> x) -> x)
    ->  (r -> x)
    ->  x
    )

    -- return
    (   \(a : *)
    ->  \(va : a)
    ->  \(x : *)
    ->  \(PutStrLn : String -> x -> x)
    ->  \(GetLine : (String -> x) -> x)
    ->  \(Return : a -> x)
    ->  Return va
    )

    -- (>>=)
    (   \(a : *)
    ->  \(b : *)
    ->  \(m : forall (x : *)
        ->  (String -> x -> x)
        ->  ((String -> x) -> x)
        ->  (a -> x)
        ->  x
        )
    ->  \(f : a
        ->  forall (x : *)
        -> (String -> x -> x)
        -> ((String -> x) -> x)
        -> (b -> x)
        -> x
        )
    ->  \(x : *)
    ->  \(PutStrLn : String -> x -> x)
    ->  \(GetLine : (String -> x) -> x)
    ->  \(Return : b -> x)
    ->  m x PutStrLn GetLine (\(va : a) -> f va x PutStrLn GetLine Return)
    )

    -- putStrLn
    (   \(str : String)
    ->  \(x : *)
    ->  \(PutStrLn : String -> x -> x  )
    ->  \(GetLine  : (String -> x) -> x)
    ->  \(Return   : U -> x)
    ->  PutStrLn str (Return Unit)
    )

    -- getLine
    (   \(x : *)
    ->  \(PutStrLn : String -> x -> x  )
    ->  \(GetLine  : (String -> x) -> x)
    ->  \(Return   : String -> x)
    -> GetLine Return
    )
)

If you type-check and normalize this program, the compiler will produce an unrolled syntax tree representing a program that echoes 99 lines from standard input to standard output:

$ morte < recursive.mt
∀(String : *) → ∀(U : *) → ∀(Unit : U) → ∀(x : *) → (String → x → x) → ((Strin
g → x) → x) → (U → x) → x

λ(String : *) → λ(U : *) → λ(Unit : U) → λ(x : *) → λ(PutStrLn : String → x → 
x) → λ(GetLine : (String → x) → x) → λ(Return : U → x) → GetLine (λ(va : Strin
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (GetLine (λ(va : Strin
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (GetLine (λ(va : Strin
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (GetLine (λ(va : Strin
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (GetLine (λ(va : Strin
...
<snip>
...
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (GetLine (λ(va : Strin
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (GetLine (λ(va : Strin
g) → PutStrLn va (GetLine (λ(va : String) → PutStrLn va (Return Unit))))))))))
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
))))))))))))))))))))))))))))))))

This program can then be passed to a backend language which interprets the syntax tree, translating GetLine and PutStrLn to read and write commands.

Notice that although our program is built using the high-level replicateM_ function, you'd never be able to tell by looking at the optimized program. By encoding effects as a free monad, we expose the monad laws to Morte, which allows the normalizer to optimize away monadic abstractions like replicateM_.

You can also build corecursive programs with effects. Here is an example of a corecursive IO syntax tree and a program that infinitely echoes standard input to standard output:

-- corecursive.mt

-- data IOF r s = PutStrLn String s | GetLine (String -> s) | Return r
--
-- data IO r = forall s . MkIO s (s -> IOF r s)
--
-- main = MkIO Nothing (maybe (\str -> PutStrLn str Nothing) (GetLine Just))

(   \(String : *)
->  (   \(Maybe : * -> *)
    ->  \(Just : forall (a : *) -> a -> Maybe a)
    ->  \(Nothing : forall (a : *) -> Maybe a)
    ->  \(  maybe
        :   forall (a : *) -> Maybe a -> forall (x : *) -> (a -> x) -> x -> x
        )
    ->  \(IOF : * -> * -> *)
    ->  \(  PutStrLn
        :   forall (r : *)
        ->  forall (s : *)
        ->  String
        ->  s
        ->  IOF r s
        )
    ->  \(  GetLine
        :   forall (r : *)
        ->  forall (s : *)
        ->  (String -> s)
        ->  IOF r s
        )
    ->  \(  Return
        :   forall (r : *)
        ->  forall (s : *)
        ->  r
        ->  IOF r s
        )
    ->  (   \(IO : * -> *)
        ->  \(  MkIO
            :   forall (r : *) -> forall (s : *) -> s -> (s -> IOF r s) -> IO r
            )
        ->  (   \(main : forall (r : *) -> IO r)
            ->  main
            )

            -- main
            (   \(r : *)
            ->  MkIO
                r
                (Maybe String)
                (Nothing String)
                (   \(m : Maybe String)
                ->  maybe
                        String
                        m
                        (IOF r (Maybe String))
                        (\(str : String) ->
                            PutStrLn r (Maybe String) str (Nothing String)
                        )
                        (GetLine r (Maybe String) (Just String))
                )
            )
        )

        -- IO
        (   \(r : *)
        ->  forall (x : *)
        ->  (forall (s : *) -> s -> (s -> IOF r s) -> x)
        ->  x
        )

        -- MkIO
        (   \(r : *)
        ->  \(s : *)
        ->  \(seed : s)
        ->  \(step : s -> IOF r s)
        ->  \(x : *)
        ->  \(k : forall (s : *) -> s -> (s -> IOF r s) -> x)
        ->  k s seed step
        )
    )

    -- Maybe
    (\(a : *) -> forall (x : *) -> (a -> x) -> x -> x)

    -- Just
    (   \(a : *)
    ->  \(va : a)
    ->  \(x : *)
    ->  \(Just : a -> x)
    ->  \(Nothing : x)
    ->  Just va
    )

    -- Nothing
    (   \(a : *)
    ->  \(x : *)
    ->  \(Just : a -> x)
    ->  \(Nothing : x)
    ->  Nothing
    )

    -- maybe
    (\(a : *) -> \(m : forall (x : *) -> (a -> x) -> x -> x) -> m)

    -- IOF
    (   \(r : *)
    ->  \(s : *)
    ->  forall (x : *)
    ->  (String -> s -> x)
    ->  ((String -> s) -> x)
    ->  (r -> x)
    ->  x
    )

    -- PutStrLn
    (   \(r : *)
    ->  \(s : *)
    ->  \(str : String)
    ->  \(vs : s)
    ->  \(x : *)
    ->  \(PutStrLn : String -> s -> x)
    ->  \(GetLine : (String -> s) -> x)
    ->  \(Return : r -> x)
    ->  PutStrLn str vs
    )

    -- GetLine
    (   \(r : *)
    ->  \(s : *)
    ->  \(k : String -> s)
    ->  \(x : *)
    ->  \(PutStrLn : String -> s -> x)
    ->  \(GetLine : (String -> s) -> x)
    ->  \(Return : r -> x)
    ->  GetLine k
    )

    -- Return
    (   \(r : *)
    ->  \(s : *)
    ->  \(vr : r)
    ->  \(x : *)
    ->  \(PutStrLn : String -> s -> x)
    ->  \(GetLine : (String -> s) -> x)
    ->  \(Return : r -> x)
    ->  Return vr
    )

)

If you compile this corecursive program you will get a state machine which can then be passed to a backend to step the state machine indefinitely:

$ morte < corecursive.mt
∀(String : *) → ∀(r : *) → ∀(x : *) → (∀(s : *) → s → (s → ∀(x : *) → (String 
→ s → x) → ((String → s) → x) → (r → x) → x) → x) → x

λ(String : *) → λ(r : *) → λ(x : *) → λ(k : ∀(s : *) → s → (s → ∀(x : *) → (St
ring → s → x) → ((String → s) → x) → (r → x) → x) → x) → k (∀(x : *) → (String
 → x) → x → x) (λ(x : *) → λ(Just : String → x) → λ(Nothing : x) → Nothing) (λ
(m : ∀(x : *) → (String → x) → x → x) → m (∀(x : *) → (String → (∀(x : *) → (S
tring → x) → x → x) → x) → ((String → ∀(x : *) → (String → x) → x → x) → x) → 
(r → x) → x) (λ(str : String) → λ(x : *) → λ(PutStrLn : String → (∀(x : *) → (
String → x) → x → x) → x) → λ(GetLine : (String → ∀(x : *) → (String → x) → x 
→ x) → x) → λ(Return : r → x) → PutStrLn str (λ(x : *) → λ(Just : String → x) 
→ λ(Nothing : x) → Nothing)) (λ(x : *) → λ(PutStrLn : String → (∀(x : *) → (St
ring → x) → x → x) → x) → λ(GetLine : (String → ∀(x : *) → (String → x) → x → 
x) → x) → λ(Return : r → x) → GetLine (λ(va : String) → λ(x : *) → λ(Just : St
ring → x) → λ(Nothing : x) → Just va)))

Any manipulations of this corecursive syntax tree within Morte will compile to efficient state transitions.

Imports

In Morte, expressions are the unit of code distribution. For example, suppose that you wanted to distribute the Nat type with its two constructors: Succ and Nat. You could use Boehm-Berarducci encoding to separately encode each term and type as its own independent lambda expression and save each of them in their own file:

$ cat Nat
    forall (Nat : *)
->  forall (Succ : Nat -> Nat)
->  forall (Zero : Nat)
->  Nat

$ cat Zero
    \(Nat : *)
->  \(Succ : Nat -> Nat)
->  \(Zero : Nat)
->  Zero

$ cat Succ
    \(   n
    :   forall (Nat : *)
    ->  forall (Succ : Nat -> Nat)
    ->  forall (Zero : Nat)
    -> Nat
    )
->  \(Nat : *)
->  \(Succ : Nat -> Nat)
->  \(Zero : Nat)
->  Succ (n Nat Succ Zero)

You can then import any of these expressions by their file name (as long as you prefix relative paths with ./). For example:

$ morte
./Succ (./Succ (./Succ ./Zero ))
<Ctrl-D>
∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat

λ(Nat : *) → λ(Succ : Nat → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))

Take care that you must have whitespace after the hashtag import. If you were to write:

$ morte
./Succ (./Succ (./Succ ./Zero))

... then you would get this parsing error:

Line:   2
Column: 1

Parsing: EOF

Error: Parsing failed

This is because morte permits parentheses in file names, so in the above program morte thinks that you are trying to import a file named Zero)). This causes a parsing failure because morte can no longer find the program's closing parentheses.

When you use imports, each hashtag is replaced with the expression from that file, so it would be as if you wrote the following really long expression:

$ morte
-- ./Succ
(   \(   n
    :   forall (Nat : *)
    ->  forall (Succ : Nat -> Nat)
    ->  forall (Zero : Nat)
    -> Nat
    )
->  \(Nat : *)
->  \(Succ : Nat -> Nat)
->  \(Zero : Nat)
->  Succ (n Nat Succ Zero)
)
(   -- ./Succ
    (   \(   n
        :   forall (Nat : *)
        ->  forall (Succ : Nat -> Nat)
        ->  forall (Zero : Nat)
        -> Nat
        )
    ->  \(Nat : *)
    ->  \(Succ : Nat -> Nat)
    ->  \(Zero : Nat)
    ->  Succ (n Nat Succ Zero)
    )
    (   -- ./Succ
        (   \(   n
            :   forall (Nat : *)
            ->  forall (Succ : Nat -> Nat)
            ->  forall (Zero : Nat)
            -> Nat
            )
        ->  \(Nat : *)
        ->  \(Succ : Nat -> Nat)
        ->  \(Zero : Nat)
        ->  Succ (n Nat Succ Zero)
        )
        -- ./Zero
        (   \(Nat : *)
        ->  \(Succ : Nat -> Nat)
        ->  \(Zero : Nat)
        ->  Zero
        )
    )
)
<Ctrl-D>
∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat

λ(Nat : *) → λ(Succ : Nat → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))

With Boehm-Beraducci encoding, the implementations of Succ, Zero, and Nat work out in such a way that when you import them they behave just like the constructors they are supposed to represent. This was an intentional design feature of Boehm-Berarducci encoding known as "representability".

The Morte repository provides a simple Prelude of utilities that you can try out. For example, if you cd to the local directory of the repository, you can run expressions like:

$ morte
./id ./Bool ./Bool/True
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool

λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
$ morte > exampleNumber  # Save an example number to the file `exampleNumber`
./Nat/Succ (./Nat/Succ (./Nat/Succ ./Nat/Zero ))
<Ctrl-D>
∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat

$ morte  # Now we can import our saved number
./Nat/(+) ./exampleNumber ./exampleNumber
<Ctrl-D>
∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat

λ(Nat : *) → λ(Succ : Nat → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero)))))
$ morte
./even ./exampleNumber
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool

λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
$ morte > exampleList  # Save an example list to the file `exampleList`
./List/Cons ./Bool ./Bool/True (./List/Cons ./Bool ./Bool/True (./List/Cons ./Bool ./Bool/False (./List/Nil ./Bool )))
<Ctrl-D>
∀(List : *) → ∀(Cons : ∀(head : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → ∀(tail : List) → List) → ∀(Nil : List) → List

$ morte
./length ./Bool ./exampleList
<Ctrl-D>
∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat

λ(Nat : *) → λ(Succ : Nat → Nat) → λ(Zero : Nat) → Succ (Succ (Succ Zero))
$ morte
./Bool/and ./exampleList
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool

λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
$ morte > double  # We can even save functions
\(n : ./Nat ) -> ./Nat/(+) n n
<Ctrl-D>
∀(n : ∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat) → ∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat
$ morte  # ... then reuse those saved functions
./double ./exampleNumber
<Ctrl-D>
∀(Nat : *) → ∀(Succ : Nat → Nat) → ∀(Zero : Nat) → Nat

λ(Nat : *) → λ(Succ : Nat → Nat) → λ(Zero : Nat) → Succ (Succ (Succ (Succ (Succ (Succ Zero)))))

Notice that some paths (like ./Bool) actually point to directories. If you provide a directory then Morte will look for a file named '@' located underneath that directory and use that instead. So, for example, if we specify ./Bool that actually translates to the file located at Bool/@.

You can also import expressions hosted on network endpoints. For example, there are several example expressions hosted at:

http://sigil.place/tutorial/morte/1.2

You can browse the above link and see that it's just an ordinary static web server hosting a directory full of source code. We can use curl to see that the hosted files are just ordinary UTF8-encoded text expressions:

$ curl http://sigil.place/tutorial/morte/1.2/id
λ(a : *) → λ(x : a) → x

We can either import these expressions directly by referencing their URLs:

$ morte
http://sigil.place/tutorial/morte/1.2/id       
    http://sigil.place/tutorial/morte/1.2/Bool
    http://sigil.place/tutorial/morte/1.2/Bool/True
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool

λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True

... or we could use local files to create short aliases for these URLs:

$ echo "http://sigil.place/tutorial/morte/1.2/id" > id
$ mkdir Bool
$ echo "http://sigil.place/tutorial/morte/1.2/Bool" > Bool/@
$ echo "http://sigil.place/tutorial/morte/1.2/Bool/True" > Bool/True

Now whenever we reference these local files they will in turn download the expressions hosted on the URL that they point to:

$ morte
./id ./Bool ./Bool/True  -- Exact same result, except now using remote code
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool

λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True

In fact, the remote directory in the morte repository does exactly this, mirroring the structure of the local directory except all the files are just references to network-hosted expressions. If you cd to the remote directory all of the above examples will still work, except this time when you compile the examples they will download the expressions from the server.

Remote expressions can reference other remote expressions, and the compiler will track down and resolve all of these references, taking care to avoid cycles. This means that you can host code on the network that references other people's code on the network just by supplying the appropriate URL wherever you want to embed their expression.

For example, suppose that we wished to take our contrived example and host it on the network. We'd simply host this text on any URL that we own:

http://sigil.place/tutorial/morte/1.2/id
    http://sigil.place/tutorial/morte/1.2/Bool
    http://sigil.place/tutorial/morte/1.2/Bool/True

... and then other people would be able to import our code within their programs by referencing our URL. Then when they downloaded our expression the compiler would further resolve the URLs embedded within our expression.

Note that when we host an expression on the network we can no longer use local imports within our code. For example, you cannot host code like this:

./id ./Bool ./Bool/True

... because client compilers that download your code would have no way of retrieving your local files. Fortunately, the compiler enforces that all code downloaded from the network may only contain URL imports and will fail with a ReferentiallyOpaque exception at compile time if any downloaded code contains a local file import.

Portability

You can use Morte as a standard format for transmitting code between functional languages. This requires you to encode the source language to Morte and decode the Morte into the destination language.

If every functional language has a Morte encoder/decoder, then eventually there can be a code utility analogous to pandoc that converts code written in any of these languages to code written in any other.

Additionally, Morte provides a standard Binary interface that you can use for serializing and deserializing code. You may find this useful for transmitting code between distributed services, even within the same language.

Conclusion

The primary purpose of Morte is a proof-of-concept that a non-recursive calculus of constructions is the ideal system for the super-optimization of functional programs. Morte uses a simple, yet powerful, optimization scheme that consists entirely of normalizing terms using the ordinary reduction rules of lambda calculus. Morte emphasizes pushing optimization complexity out of the virtual machine and into the translation of abstractions to the calculus of constructions. However, that means that the hard work has only just begun and Morte still needs front-end compilers to translate from high-level functional languages to the calculus of constructions.

The secondary purpose of Morte is to serve as a standardized format for encoding and transmission of functional code between distributed services or different functional languages. Morte restricts itself to lambda calculus in order to reuse the large body of research for translating programming abstractions to and from the polymorphic lambda calculus.

Finally, you can use Morte as an equational reasoning engine to learn how high-level abstractions reduce to low-level abstractions. If you are teaching lambda calculus you can use Morte as a teaching tool for how to encode abstractions within lambda calculus.

If you have problems, questions, or feature requests, you can open an issue on the issue tracker on Github:

https://github.com/Gabriel439/Haskell-Morte-Library/issues