simply-typed-with-subtyping

simply-typed-lambda-calculus(with extensions)

Terms and values:

t ::=
  x        // variable
  \x:T.t   // abstraction
  t t      // application
  true
  false
  if t then t else t
  unit
  t as T
  let x=t in t
  {t_i}    // with i=1..n
  t.i
  <l=t> as T
  case t of <l_i=x_i> => t_i // with i=1..n

v ::=
  \x:T.t   // abstraction value
  true
  false
  unit
  {v_i}    // with i=1..n

Types:

T ::=
  Bool
  Unit
  T -> T
  {T_i}     // with i=1..n
  <l_i:T_i> // with i=1..n

Derived forms:

t1;t2 === (λx:Unit.t2) t1, where x is not a free variable on t2

Examples

Trying with booleans:

• and = λx:Bool.λy:Bool. if x then y else false

and true true evaluates to true:

enter term:
(\x:Bool.\y:Bool. if x then y else false) true false
Term before evaluation: (λx:Bool.(λy:Bool.if x then y else false)) true true
Succesfully typechecked: Bool
Term after evaluation: true

and true false evaluates to false:

enter term:
(\x:Bool.\y:Bool. if x then y else false) true false
Term before evaluation: (λx:Bool.(λy:Bool.if x then y else false)) true false
Succesfully typechecked: Bool
Term after evaluation: false

and true (λx:Bool.x) doesn't type check:

(\x:Bool.\y:Bool. if x then y else false) true (\x:Bool.x)
Term before evaluation: (λx:Bool.(λy:Bool.if x then y else false)) true (λx:Bool.x)
Expression does not typecheck: TypeAppArgumentMustMatch