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This document is the primary reference for the Carth programming language. It is updated on a best-effort basis. It should be valid and complete, but it may not be.


How to use this document


Here is this file in the github repository.



Lexical structure

For details about the syntax that are not covered explicitly in this document or cannot be inferred from the provided examples, please consult the source of the parser directly in Parse.hs.


Packages, modules, and source files


Global variable definitions

Global variables are defined at the top-level by using either one of the two implicitly typed variable ~define~ and function ~define~ special forms, or their explicitly typed define: counterparts.

The function definition special forms allows us to name parameters and deconstruct them with irrefutable patterns. This is essentially syntactic sugar for binding a variable to a fun-match expression.

Global variables are practically equivalent to local variables, in all respects but scope.


;; First form. Implicitly typed variable definition.
;; Bind the global variable ~fst-iv~ to the function that extracts the
;; first value of a pair. No explicit type signature is given - it is
;; statically inferred by the typechecker to be the equivalent of
;; ~(forall (a b) (Fun (Pair a b) a))~
(define fst-iv
  (fun ((Pair x _)) x))

;; Second form. Implicitly typed function definition.
;; Again, the fst function, equivalent to the defition above, but
;; using the more convenient function definition syntax.
(define (fst-if p)
  (match p
    (case (Pair x _) x)))
;; ... and again, together with the even more conventient
;; destructuring syntax
(define (fst-ifd (Pair x _))

;; Third form. Explicitly typed variable definition.
;; Explicit type signature is given. The type must properly be a
;; polytype (aka /type scheme/) to be valid and instantiable for any
;; monotype.
(define: fst-ev
    (forall (a b) (Fun (Pair a b) a))
  (fun-match (case (Pair x _) x)))

;; Fourth form. Explicitly typed function definition.
(define: (fst-ef (Pair x _))
    (forall (a b) (Fun (Pair a b) a))

Type definitions

Algebraic datatypes (aka tagged/discriminated unions) are defined at the top-level with the type special form. Roughly equivalent to data in Haskell and enum in Rust.

Recursive datatypes must contain a Box indirection to be representable in memory with a finite size.

Datatypes may be uninhabited, i.e. defined without any variants, like (type Void). As these types have no variants, they have no values, and we can’t construct them – thus the name “uninhabited”.


;; First form. Monomorphic datatype definition.
;; ~Age~ only has one variant, and as such can be seen as a "wrapper"
;; around ~Int~, restricting its usage.
(type Age
  (Age Int))

;; Second form. Polymorphic datatype definition.
;; ~List~ has two variants, representing that a list can either be
;; empty, or a pair of a head and a tail. Note that we must have a
;; ~Box~ indirection so that it doesn't have infinite size.
(type (List' a)
  (Cons' a (Box (List' a)))

;; An uninhabited type that can't be constructed. Useful when you want
;; to employ the type system to make invalid states unrepresentable,
;; or model propositions as types.
(type Void')



unit is the only value inhibiting the type Unit, equivalent to () in Haskell and Rust.
64-bit signed integer literal. Example: 42.
64-bit double precision floating point literal. Example: -13.37.
4-byte UTF-32 Character literal. Example: ~’a’~, ~’維’~, ~’🔥’~.
UTF-8 string literals. At the moment, generates to static arrays. Will likely be changed. Example: ~”Hello, World!”~, ~”😄😦🐱”~.
True or False.


Function application


Anonymous-function / Lambda expression / Closure


Type ascription

Type ascriptions are primarily used to:

  • increase readability when the type of an expression is not obvious;
  • assert at compile-time that an expression is of or can specialize to the given type;
  • or specialize the type of a generic expression, restricting its usage.


(define (id-int x)
  (: x Int))
;; Inferred type of ~id-int~: (Fun Int Int)


Pattern matching. Can match against literals to test for equality, against constructions to deconstruct datatypes, against names to bind a variable to (a substructure of) the matchee.

The literal-types that can be matched against are integers, bools, and strings.

The cases of a match-expression must be exhaustive and non-redundant.

When pattern matching on an uninhabited type, no cases can be given as the type has no constructors, and the match-expression as a whole is absurd. Absurdity, like panic or undefined, implies any type, as it’s unreachable.


(type Foo
(type (Pair' a b)
  (Pair' a b))

;; Ok
(define (fst pair)
  (match pair
    (case (Pair' a _) a)))

;; Matching on an uninhabited type corresponds to the elimination rule
;; for ⊥ (it implies anything).
(define: (absurd void)
    (forall (a) (Fun Void' a))
  (match void))

(define read-binop
    (case "plus" +)
    (case "times" *)
    (case s (panic (str-append "Undefined binop " s)))))
;; Error. Redundant pattern. ~Pair _ _~ already covered by previous
;; pattern ~_~
(define (redundant pair)
  (match pair
    (case _ 1)
    (case (Pair' x y) 2)))

;; Error. Inexhaustive pattern. All cases not covered, specifically
;; ~Bar~
(define (inexhaustive foo)
  (match foo
    (case Baz 123)))


Syntax sugar for a match in a lambda. Equivalent to \case (LambdaCase) in Haskell. (fun-match cases...) translates to (fun VAR (match VAR cases...)) where VAR is a uniquely internally generated variable that cannot be expressed by the user (which means it won’t shadow any other binding).


;; Two versions of `fst`, which returns the first value of a pair
;; using normal `match`
(define (fst-nofun p)
  (match p
    (case (Pair a _) a)))
;; and using `fun-match`
(define fst-fun
    (case (Pair a _) a)))


By applying a constructor to some arguments, or just presenting it literally in the case of a nullary constructor, a value of the associated algebraic datatype is produced. Constructors of arity > 0 behave like n-ary functions: curried and the whole shebang.


;; The following datatype definition will make available the
;; constructors ~UPUnit~ and ~UPPair~ in the environment.
(type UnitOrPair
  (UPPair Int Int))

;; The ~UPUnit~ constructor is nullary, and will construct a
;; ~UnitOrPair~ just presented literally.
(define: upunit

;; The ~UPPair~ constructor is binary, and takes two arguments to
;; construct a ~UnitOrPair~. It behaves like a function of two ~Int~
;; arguments, returning a ~UnitOrPair~.
(define: uppair''
    (Fun Int Int UnitOrPair)
(define: uppair'
    (Fun Int UnitOrPair)
  (UPPair 3))
(define: uppair
  (uppair' 5))


Patterns are used to conditionally deconstruct values of algebraic datatypes in pattern-matching contexts.

There are 3 kinds of patterns: nullary constructors, n-ary constructions, and variable bindings.


See Match/Example.

Type system

Memory model



Compile time evaluation


Literate Carth

Carth has native support for literate programming with Org mode. Either use Emacs with Babel in Org-mode for an interactive session, or interpret/compile the file with carth just like a normal .carth file!


Consider a file with the following content:

#+TITLE: Literate Programming Rules!

Literate programming is just really cool!

~carth~ will assume ~tangle~ = ~yes~ by default, but setting it
explicitly won't hurt.

#+BEGIN_SRC carth :tangle yes
(define (main _)
  (printInt (id 1337)))

* The ~id~ function
  ~id~ is the identity function. It returns its argument unchanged.

  #+BEGIN_SRC carth
  (define (id x) x)

* How not to use ~id~
  Here is an example of how not to use ~id~. Note that this won't
  compile. We show this in a SRC block to get syntax highlighting etc,
  but as ~tangle~ is ~no~, this source block will be ignored by carth.

  #+BEGIN_SRC carth :tangle no
  (printInt id)

When compiling this file with carth c, the Carth source will be untangled from the rest of the document. Line numbers are preserved. The result of the untangling stage will be the following:

(define (main _)
  (printInt (id 1337)))

(define (id x) x)

And for completeness, the result of interpreting that will be 1337.

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