Safe Haskell	Safe-Inferred
Language	Haskell98

Morte.Core

Contents

Syntax
Core functions
Utilities
Errors
Builders

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!":

http://homepages.inf.ed.ac.uk/wadler/papers/free-rectypes/free-rectypes.txt

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

Eq Var
Show Var
IsString Var
Binary Var
NFData Var

data Const Source

Constants for the calculus of constructions

The only axiom is:

⊦ * : □

... and all four rule pairs are valid:

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

Constructors

Star
Box

Instances

Bounded Const
Enum Const
Eq Const
Show Const
Binary Const
NFData Const

data Expr 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 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

Instances

Eq Expr
Show Expr
IsString Expr
Binary Expr
NFData Expr

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.

Utilities

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: