morte-1.7.0: A bare-bones calculus of constructions

Morte.Core

Description

This module contains the core calculus for the Morte language. This language is a minimalist implementation of the calculus of constructions, which is in turn a specific kind of pure type system. If you are new to pure type systems you may wish to read "Henk: a typed intermediate language".

http://research.microsoft.com/en-us/um/people/simonpj/papers/henk.ps.gz

Morte is a strongly normalizing language, meaning that:

• Every expression has a unique normal form computed by normalize
• You test expressions for equality of their normal forms using ==
• Equational reasoning preserves normal forms

Strong normalization comes at a price: Morte forbids recursion. Instead, you must translate all recursion to F-algebras and translate all corecursion to F-coalgebras. If you are new to F-(co)algebras then you may wish to read Morte.Tutorial or read "Recursive types for free!":

Morte is designed to be a super-optimizing intermediate language with a simple optimization scheme. You optimize a Morte expression by just normalizing the expression. If you normalize a long-lived program encoded as an F-coalgebra you typically get a state machine, and if you normalize a long-lived program encoded as an F-algebra you typically get an unrolled loop.

Strong normalization guarantees that all abstractions encodable in Morte are "free", meaning that they may increase your program's compile times but they will never increase your program's run time because they will normalize to the same code.

Synopsis

# Syntax

data Var Source #

Label for a bound variable

The Text field is the variable's name (i.e. "x").

The Int field disambiguates variables with the same name if there are multiple bound variables of the same name in scope. Zero refers to the nearest bound variable and the index increases by one for each bound variable of the same name going outward. The following diagram may help:

                          +-refers to-+
|           |
v           |
\(x : *) -> \(y : *) -> \(x : *) -> x@0

+-------------refers to-------------+
|                                   |
v                                   |
\(x : *) -> \(y : *) -> \(x : *) -> x@1

This Int behaves like a De Bruijn index in the special case where all variables have the same name.

You can optionally omit the index if it is 0:

                          +refers to+
|         |
v         |
\(x : *) -> \(y : *) -> \(x : *) -> x

Zero indices are omitted when pretty-printing Vars and non-zero indices appear as a numeric suffix.

Constructors

 V Text Int
Instances
 Source # Instance detailsDefined in Morte.Core Methods(==) :: Var -> Var -> Bool #(/=) :: Var -> Var -> Bool # Source # Instance detailsDefined in Morte.Core MethodsshowsPrec :: Int -> Var -> ShowS #show :: Var -> String #showList :: [Var] -> ShowS # Source # Instance detailsDefined in Morte.Core Methods Source # Instance detailsDefined in Morte.Core Methodsput :: Var -> Put #putList :: [Var] -> Put # Source # Instance detailsDefined in Morte.Core Methodsrnf :: Var -> () # Source # Instance detailsDefined in Morte.Core Methodsbuild :: Var -> Builder #

data Const Source #

Constants for the calculus of constructions

The only axiom is:

⊦ * : □

... and all four rule pairs are valid:

⊦ * ↝ * : *
⊦ □ ↝ * : *
⊦ * ↝ □ : □
⊦ □ ↝ □ : □

Constructors

 Star Box
Instances
 Source # Instance detailsDefined in Morte.Core Methods Source # Instance detailsDefined in Morte.Core Methodssucc :: Const -> Const #pred :: Const -> Const #toEnum :: Int -> Const #enumFrom :: Const -> [Const] #enumFromThen :: Const -> Const -> [Const] #enumFromTo :: Const -> Const -> [Const] #enumFromThenTo :: Const -> Const -> Const -> [Const] # Source # Instance detailsDefined in Morte.Core Methods(==) :: Const -> Const -> Bool #(/=) :: Const -> Const -> Bool # Source # Instance detailsDefined in Morte.Core MethodsshowsPrec :: Int -> Const -> ShowS #show :: Const -> String #showList :: [Const] -> ShowS # Source # Instance detailsDefined in Morte.Core Methodsput :: Const -> Put #putList :: [Const] -> Put # Source # Instance detailsDefined in Morte.Core Methodsrnf :: Const -> () # Source # Instance detailsDefined in Morte.Core Methods

data Path Source #

Path to an external resource

Constructors

 File FilePath URL Text
Instances
 Source # Instance detailsDefined in Morte.Core Methods(==) :: Path -> Path -> Bool #(/=) :: Path -> Path -> Bool # Source # Instance detailsDefined in Morte.Core Methodscompare :: Path -> Path -> Ordering #(<) :: Path -> Path -> Bool #(<=) :: Path -> Path -> Bool #(>) :: Path -> Path -> Bool #(>=) :: Path -> Path -> Bool #max :: Path -> Path -> Path #min :: Path -> Path -> Path # Source # Instance detailsDefined in Morte.Core MethodsshowsPrec :: Int -> Path -> ShowS #show :: Path -> String #showList :: [Path] -> ShowS # Source # Instance detailsDefined in Morte.Core Methods

data Expr a Source #

Syntax tree for expressions

Constructors

 Const Const Const c ~ c Var Var Var (V x 0) ~ x Var (V x n) ~ x@n Lam Text (Expr a) (Expr a) Lam x A b ~ λ(x : A) → b Pi Text (Expr a) (Expr a) Pi x A B ~ ∀(x : A) → B Pi unused A B ~ A → B App (Expr a) (Expr a) App f a ~ f a Embed a Embed path ~ #path
Instances
 Source # Instance detailsDefined in Morte.Core Methods(>>=) :: Expr a -> (a -> Expr b) -> Expr b #(>>) :: Expr a -> Expr b -> Expr b #return :: a -> Expr a #fail :: String -> Expr a # Source # Instance detailsDefined in Morte.Core Methodsfmap :: (a -> b) -> Expr a -> Expr b #(<$) :: a -> Expr b -> Expr a # Source # Instance detailsDefined in Morte.Core Methodspure :: a -> Expr a #(<*>) :: Expr (a -> b) -> Expr a -> Expr b #liftA2 :: (a -> b -> c) -> Expr a -> Expr b -> Expr c #(*>) :: Expr a -> Expr b -> Expr b #(<*) :: Expr a -> Expr b -> Expr a # Source # Instance detailsDefined in Morte.Core Methodsfold :: Monoid m => Expr m -> m #foldMap :: Monoid m => (a -> m) -> Expr a -> m #foldr :: (a -> b -> b) -> b -> Expr a -> b #foldr' :: (a -> b -> b) -> b -> Expr a -> b #foldl :: (b -> a -> b) -> b -> Expr a -> b #foldl' :: (b -> a -> b) -> b -> Expr a -> b #foldr1 :: (a -> a -> a) -> Expr a -> a #foldl1 :: (a -> a -> a) -> Expr a -> a #toList :: Expr a -> [a] #null :: Expr a -> Bool #length :: Expr a -> Int #elem :: Eq a => a -> Expr a -> Bool #maximum :: Ord a => Expr a -> a #minimum :: Ord a => Expr a -> a #sum :: Num a => Expr a -> a #product :: Num a => Expr a -> a # Source # Instance detailsDefined in Morte.Core Methodstraverse :: Applicative f => (a -> f b) -> Expr a -> f (Expr b) #sequenceA :: Applicative f => Expr (f a) -> f (Expr a) #mapM :: Monad m => (a -> m b) -> Expr a -> m (Expr b) #sequence :: Monad m => Expr (m a) -> m (Expr a) # Eq a => Eq (Expr a) Source # Instance detailsDefined in Morte.Core Methods(==) :: Expr a -> Expr a -> Bool #(/=) :: Expr a -> Expr a -> Bool # Show a => Show (Expr a) Source # Instance detailsDefined in Morte.Core MethodsshowsPrec :: Int -> Expr a -> ShowS #show :: Expr a -> String #showList :: [Expr a] -> ShowS # IsString (Expr a) Source # Instance detailsDefined in Morte.Core Methods Binary a => Binary (Expr a) Source # Instance detailsDefined in Morte.Core Methodsput :: Expr a -> Put #get :: Get (Expr a) #putList :: [Expr a] -> Put # NFData a => NFData (Expr a) Source # Instance detailsDefined in Morte.Core Methodsrnf :: Expr a -> () # Buildable a => Buildable (Expr a) Source # Generates a syntactically valid Morte program Instance detailsDefined in Morte.Core Methodsbuild :: Expr a -> Builder # data Context a Source # Bound variable names and their types Variable names may appear more than once in the Context. The Var x@n refers to the nth occurrence of x in the Context (using 0-based numbering). Instances  Source # Instance detailsDefined in Morte.Context Methodsfmap :: (a -> b) -> Context a -> Context b #(<$) :: a -> Context b -> Context a # NFData a => NFData (Context a) Source # Instance detailsDefined in Morte.Context Methodsrnf :: Context a -> () #

# Core functions

Type-check an expression and return the expression's type if type-checking suceeds or an error if type-checking fails

typeWith does not necessarily normalize the type since full normalization is not necessary for just type-checking. If you actually care about the returned type then you may want to normalize it afterwards.

typeOf is the same as typeWith with an empty context, meaning that the expression must be closed (i.e. no free variables), otherwise type-checking will fail.

normalize :: Expr a -> Expr a Source #

Reduce an expression to its normal form, performing both beta reduction and eta reduction

normalize does not type-check the expression. You may want to type-check expressions before normalizing them since normalization can convert an ill-typed expression into a well-typed expression.

# Utilities

shift :: Int -> Text -> Expr a -> Expr a Source #

shift n x adds n to the index of all free variables named x within an Expr

subst :: Text -> Int -> Expr a -> Expr a -> Expr a Source #

Substitute all occurrences of a variable with an expression

subst x n C B  ~  B[x@n := C]

pretty :: Buildable a => a -> Text Source #

Pretty-print a value

buildExpr :: Buildable a => Expr a -> Builder Source #

Pretty-print an expression as a Builder, using Unicode symbols

Pretty-print an expression as a Builder, using ASCII symbols

# Errors

data TypeError Source #

A structured type error that includes context

Constructors

 TypeError Fieldscontext :: Context (Expr Void) current :: Expr Void typeMessage :: TypeMessage
Instances
 Source # Instance detailsDefined in Morte.Core MethodsshowList :: [TypeError] -> ShowS # Source # Instance detailsDefined in Morte.Core Methods Source # Instance detailsDefined in Morte.Core Methodsrnf :: TypeError -> () # Source # Instance detailsDefined in Morte.Core Methods

The specific type error

Constructors

 UnboundVariable InvalidInputType (Expr Void) InvalidOutputType (Expr Void) NotAFunction TypeMismatch (Expr Void) (Expr Void) Untyped Const
Instances
 Source # Instance detailsDefined in Morte.Core MethodsshowList :: [TypeMessage] -> ShowS # Source # Instance detailsDefined in Morte.Core Methodsrnf :: TypeMessage -> () # Source # Instance detailsDefined in Morte.Core Methods