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

Whether or not a path is relative to the user's home directory

The type of path to import (i.e. local vs. remote vs. environment)

How to interpret the path's contents (i.e. as Dhall code or raw text)

Path to an external resource

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             │
λ(x : Type) → λ(y : Type) → λ(x : Type) → x@0

┌─────────────────refers to─────────────────┐
│                                           │
v                                           │
λ(x : Type) → λ(y : Type) → λ(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           │
λ(x : Type) → λ(y : Type) → λ(x : Type) → x

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

Syntax tree for expressions

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.

Reduce an expression to its normal form, performing beta reduction and applying any custom definitions.

normalizeWith is designed to be used with function typeWith. The typeWith function allows typing of Dhall functions in a custom typing context whereas normalizeWith allows evaluating Dhall expressions in a custom context.

To be more precise normalizeWith applies the given normalizer when it finds an application term that it cannot reduce by other means.

Note that the context used in normalization will determine the properties of normalization. That is, if the functions in custom context are not total then the Dhall language, evaluated with those functions is not total either.

Use this to wrap you embedded functions (see normalizeWith) to make them polymorphic enough to be used.

Substitute all occurrences of a variable with an expression

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

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.

Quickly check if an expression is in normal form

Check if an expression is in a normal form given a context of evaluation. Unlike isNormalized, this will fully normalize and traverse through the expression.

It is much more efficient to use isNormalized.

Remove all Note constructors from an Expr (i.e. de-Note)

Pretty-print a value

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

The set of reserved identifiers for the Dhall language

Escape a Builder literal using Dhall's escaping rules

Note that the result does not include surrounding quotes