Dhall.Core

This module contains the core calculus for the Dhall language.

Dhall is essentially a fork of the morte compiler but with more built-in functionality, better error messages, and Haskell integration

Syntax

data Const

Constants for a pure type system

The only axiom is:

⊦ Type : Kind

... and the valid rule pairs are:

⊦ Type ↝ Type : Type  -- Functions from terms to terms (ordinary functions)
⊦ Kind ↝ Type : Type  -- Functions from types to terms (polymorphic functions)
⊦ Kind ↝ Kind : Kind  -- Functions from types to types (type constructors)

These are the same rule pairs as System Fω

Note that Dhall does not support functions from terms to types and therefore Dhall is not a dependently typed language

Constructors

Type

Kind

Instances

Bounded Const

Enum Const

Show Const

Buildable Const

data Path

Path to an external resource

Constructors

File FilePath

URL Text

Instances

Eq Path

Ord Path

Show Path

Buildable Path

data Var

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 : Type) -> \(y : Type) -> \(x : Type) -> x@0

  +------------------refers to-----------------+
  |                                            |
  v                                            |
\(x : Type) -> \(y : Type) -> \(x : Type) -> 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 !Integer

Instances

Eq Var

Show Var

IsString Var

Buildable Var

data Expr s a

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 s a) (Expr s a)

Lam x     A b                            ~  λ(x : A) -> b

Pi Text (Expr s a) (Expr s a)

Pi "_" A B                               ~        A  -> B
Pi x   A B                               ~  ∀(x : A) -> B

App (Expr s a) (Expr s a)

App f a                                  ~  f a

Let Text (Maybe (Expr s a)) (Expr s a) (Expr s a)

Let x Nothing  r e  ~  let x     = r in e
Let x (Just t) r e  ~  let x : t = r in e

Annot (Expr s a) (Expr s a)

Annot x t                                ~  x : t

Bool

Bool                                     ~  Bool

BoolLit Bool

BoolLit b                                ~  b

BoolAnd (Expr s a) (Expr s a)

BoolAnd x y                              ~  x && y

BoolOr (Expr s a) (Expr s a)

BoolOr  x y                              ~  x || y

BoolEQ (Expr s a) (Expr s a)

BoolEQ  x y                              ~  x == y

BoolNE (Expr s a) (Expr s a)

BoolNE  x y                              ~  x != y

BoolIf (Expr s a) (Expr s a) (Expr s a)

BoolIf x y z                             ~  if x then y else z

Natural

Natural                                  ~  Natural

NaturalLit Natural

NaturalLit n                             ~  +n

NaturalFold

NaturalFold                              ~  Natural/fold

NaturalBuild

NaturalBuild                             ~  Natural/build

NaturalIsZero

NaturalIsZero                            ~  Natural/isZero

NaturalEven

NaturalEven                              ~  Natural/even

NaturalOdd

NaturalOdd                               ~  Natural/odd

NaturalPlus (Expr s a) (Expr s a)

NaturalPlus x y                          ~  x + y

NaturalTimes (Expr s a) (Expr s a)

NaturalTimes x y                         ~  x * y

Integer

Integer                                  ~  Integer

IntegerLit Integer

IntegerLit n                             ~  n

Double

Double                                   ~  Double

DoubleLit Double

DoubleLit n                              ~  n

Text

Text                                     ~  Text

TextLit Builder

TextLit t                                ~  t

TextAppend (Expr s a) (Expr s a)

TextAppend x y                           ~  x ++ y

List

List                                     ~  List

ListLit (Expr s a) (Vector (Expr s a))

ListLit t [x, y, z]                      ~  [x, y, z] : List t

ListBuild

ListBuild                                ~  List/build

ListFold

ListFold                                 ~  List/fold

ListLength

ListLength                               ~  List/length

ListHead

ListHead                                 ~  List/head

ListLast

ListLast                                 ~  List/last

ListIndexed

ListIndexed                              ~  List/indexed

ListReverse

ListReverse                              ~  List/reverse

Optional

Optional                                 ~  Optional

OptionalLit (Expr s a) (Vector (Expr s a))

OptionalLit t [e]                        ~  [e] : Optional t
OptionalLit t []                         ~  []  : Optional t

OptionalFold

OptionalFold                             ~  Optional/fold

Record (Map Text (Expr s a))

Record            [(k1, t1), (k2, t2)]   ~  { k1 : t1, k2 : t1 }

RecordLit (Map Text (Expr s a))

RecordLit         [(k1, v1), (k2, v2)]   ~  { k1 = v1, k2 = v2 }

Union (Map Text (Expr s a))

Union             [(k1, t1), (k2, t2)]   ~  < k1 : t1, k2 : t2 >

UnionLit Text (Expr s a) (Map Text (Expr s a))

UnionLit (k1, v1) [(k2, t2), (k3, t3)]   ~  < k1 = t1, k2 : t2, k3 : t3 > 

Combine (Expr s a) (Expr s a)

Combine x y                              ~  x ∧ y

Merge (Expr s a) (Expr s a) (Expr s a)

Merge x y t                              ~  merge x y : t

Field (Expr s a) Text

Field e x                                ~  e.x

Note s (Expr s a)

Note s x                                 ~  e

Embed a

Embed path                               ~  path

Instances

Bifunctor Expr

Monad (Expr s)

Functor (Expr s)

Applicative (Expr s)

Foldable (Expr s)

Traversable (Expr s)

(Show s, Show a) => Show (Expr s a)

IsString (Expr s a)

Buildable a => Buildable (Expr s a)

Generates a syntactically valid Dhall program

Normalization

normalize :: Expr s a -> Expr t a

Reduce an expression to its normal form, performing beta 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.

However, normalize will not fail if the expression is ill-typed and will leave ill-typed sub-expressions unevaluated.

subst :: Var -> Expr s a -> Expr t a -> Expr s a

Substitute all occurrences of a variable with an expression

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

shift :: Integer -> Var -> Expr s a -> Expr t a

shift is used by both normalization and type-checking to avoid variable capture by shifting variable indices

For example, suppose that you were to normalize the following expression:

λ(a : Type) → λ(x : a) → (λ(y : a) → λ(x : a) → y) x

If you were to substitute y with x without shifting any variable indices, then you would get the following incorrect result:

λ(a : Type) → λ(x : a) → λ(x : a) → x  -- Incorrect normalized form

In order to substitute x in place of y we need to shift x by 1 in order to avoid being misinterpreted as the x bound by the innermost lambda. If we perform that shift then we get the correct result:

λ(a : Type) → λ(x : a) → λ(x : a) → x@1

As a more worked example, suppose that you were to normalize the following expression:

    λ(a : Type)
→   λ(f : a → a → a)
→   λ(x : a)
→   λ(x : a)
→   (λ(x : a) → f x x@1) x@1

The correct normalized result would be:

    λ(a : Type)
→   λ(f : a → a → a)
→   λ(x : a)
→   λ(x : a)
→   f x@1 x

The above example illustrates how we need to both increase and decrease variable indices as part of substitution:

• We need to increase the index of the outer x@1 to x@2 before we substitute it into the body of the innermost lambda expression in order to avoid variable capture. This substitution changes the body of the lambda expression to (f x@2 x@1)
• We then remove the innermost lambda and therefore decrease the indices of both xs in (f x@2 x@1) to (f x@1 x) in order to reflect that one less x variable is now bound within that scope

Formally, (shift d (V x n) e) modifies the expression e by adding d to the indices of all variables named x whose indices are greater than (n + m), where m is the number of bound variables of the same name within that scope

In practice, d is always 1 or -1 because we either:

• increment variables by 1 to avoid variable capture during substitution
• decrement variables by 1 when deleting lambdas after substitution

n starts off at 0 when substitution begins and increments every time we descend into a lambda or let expression that binds a variable of the same name in order to avoid shifting the bound variables by mistake.

Pretty-printing

pretty :: Buildable a => a -> Text

Pretty-print a value

Miscellaneous

internalError :: Text -> forall b. b

Utility function used to throw internal errors that should never happen (in theory) but that are not enforced by the type system