morte-1.1.2: A bare-bones calculus of constructions

Safe HaskellSafe-Inferred




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".

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.



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.


V Text Int 

data Const Source

Constants for the calculus of constructions

The only axiom is:

⊦ * : □

... and all four rule pairs are valid:

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



data Expr Source

Syntax tree for expressions


Const Const
Const c        ~  c
Var Var
Var (V x 0)    ~  x
Var (V x n)    ~  x@n
Lam Text Expr Expr
Lam x     A b  ~  λ(x : A) → b
Pi Text Expr Expr
Pi x      A B  ~  ∀(x : A) → B
Pi unused A B  ~        A  → B
App Expr Expr
App f a        ~  f a

type Context = [(Text, Expr)] 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).

Core functions

typeWith :: Context -> Expr -> Either TypeError Expr Source

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 :: Expr -> Either TypeError Expr Source

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 -> Expr 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.


used :: Text -> Expr -> Bool Source

Determine whether a Pi-bound variable should be displayed

Notice that if any variable within the body of a Pi shares the same name and an equal or greater DeBruijn index we display the Pi-bound variable. To illustrate why we don't just check for equality, consider this type:

forall (a : *) -> forall (a : *) -> a@1

The a@1 refers to the outer a (i.e. the left one), but if we hid the inner a (the right one), the type would make no sense:

forall (a : *) -> * -> a@1

... because the a@1 would misleadingly appear to be an unbound variable.

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

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

prettyExpr :: Expr -> Text Source

Pretty-print an expression

The result is a syntactically valid Morte program

prettyTypeError :: TypeError -> Text Source

Pretty-print a type error


data TypeError Source

A structured type error that includes context




buildConst :: Const -> Builder Source

Render a pretty-printed Const as a Builder

buildVar :: Var -> Builder Source

Render a pretty-printed Var as a Builder

buildExpr :: Expr -> Builder Source

Render a pretty-printed Expr as a Builder

buildTypeMessage :: TypeMessage -> Builder Source

Render a pretty-printed TypeMessage as a Builder

buildTypeError :: TypeError -> Builder Source

Render a pretty-printed TypeError as a Builder