Benjamin Pierce's F-omega checker, from
http://www.cis.upenn.edu/~bcpierce/tapl/checkers/fullomega/
Available online at
http://ocamllabs.github.io/fomega/
Syntax
There are two types of variables. Term variables begin with a lowercase letter
a, b, c
x, y, z
Type variables begin with an uppercase or Greek letter
A, B, C
α, β, γ
A program is a sequence of definitions, terminated by semicolons:
program ::= definition ; definition ; ... definition ;
Definitions bind types or terms.
definition ::= α :: kind = type
x : type = term
You can leave out either the :: kind or the = type component (but not both) from a type binding, and you can leave out either the : type or the = term component (but not both) from a term binding.
Kinds are either *, the type of kinds, or arrow kinds.
kind ::= *
kind ⇒ kind
( kind )
ASCII alternative: you can use => in place of ⇒.
Types are defined as follows:
type ::= type → type (function types)
type × type (product types)
type + type (sum types)
∀ α :: kind . type (polymorphic types)
∃ α :: kind . type (existential types)
λ α :: kind . type (type abstractions)
type type (type applications)
( type )
ASCII alternatives: -> for →, * for ×, All for ∀, EXISTS for ∃, lambda for λ. Kind annotations (:: kind) can be omitted, and default to *.
Terms are defined as follows:
term ::= λ x : type . term
term term
⟨ term , term , ... term ⟩
@n term (nth projection: @1 is fst, @2 is snd)
inl [ type ] term
inr [ type ] term
case term of x.term | y . term
Λ α :: kind . term
term [ type ]
pack type , term, as ∃ α :: kind . type
open term as α , x in term
( term )
ASCII alternatives: lambda for λ, LAMBDA for Λ. Kind annotations (:: kind) can be omitted, and default to *.