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#lang scribble/manual

@begin[(require "../utils.rkt"
		scribble/core scribble/example
		(for-label (only-meta-in 0 typed/racket)))]

@(define the-eval (make-base-eval))
@(the-eval '(require typed/racket))

@title[#:tag "occurrence-typing"]{Occurrence Typing}

@section{Basic Occurrence Typing}

One of Typed Racket's distinguishing type system features is
@deftech{occurrence typing}, which allows the type system to ascribe
more precise types based on whether a predicate check succeeds or
fails.

To illustrate, consider the following code:

@examples[#:no-result #:eval the-eval
  (: flexible-length (-> (U String (Listof Any)) Integer))
  (define (flexible-length str-or-lst)
    (if (string? str-or-lst)
        (string-length str-or-lst)
        (length str-or-lst)))
]

The @racket[_flexible-length] function above computes the length of
either a string or a list. The function body uses the typical Racket
idiom of dispatching using a predicate (e.g., @racket[string?]).

Typed Racket successfully type-checks this function because the type
system understands that in the "then" branch of the @racket[if]
expression, the predicate @racket[string?] must have returned a true
value.  The type system further knows that if @racket[string?]
returns true, then the @racket[_str-or-lst] variable must have type
@racket[String] and can narrow the type from its original union of
@racket[String] and @racket[(Listof Any)]. This allows the call to
@racket[string-length] in the "then" branch to type-check
successfully.

Furthermore, the type system also knows that in the "else" branch of
the @racket[if] expression, the predicate must have returned
@racket[#f].  This implies that the variable @racket[_str-or-lst] must
have type @racket[(Listof Any)] by process of elimination, and thus
the call @racket[(length str-or-lst)] type-checks.

To summarize, if Typed Racket can determine the type a variable must
have based on a predicate check in a conditional expression, it can
narrow the type of the variable within the appropriate branch of the
conditional.

@section[#:tag "propositions-and-predicates"]{Propositions and Predicates}

In the previous section, we demonstrated that a Typed Racket programmer
can take advantage of occurrence typing to type-check functions
with union types and conditionals. This may raise the question: how
does Typed Racket know how to narrow the type based on the predicate?

The answer is that predicate types in Typed Racket are annotated
with logical @deftech{propositions} that tell the typechecker what additional
information is gained when a predicate check succeeds or fails.

For example, consider the REPL's type printout for @racket[string?]:

@examples[#:label #f #:eval the-eval string?]

The type @racket[(-> Any Boolean : String)] has three parts. The first
two are the same as any other function type and indicate that the
predicate takes any value and returns a boolean. The third part, after
the @racket[_:], represents the logical @tech{propositions}
the typechecker learns from the result of applying the function:

@itemlist[#:style 'ordered

  @item{If the predicate check succeeds (i.e. produces a
  non-@racket[#f] value), the argument variable has type
  @racket[String]}

@item{If the predicate check fails (i.e. produces @racket[#f]), the
  argument variable @emph{does not} have type @racket[String]} ]

Predicates for all built-in types are annotated with similar propositions
that allow the type system to reason logically about predicate checks.

@subsection{One-sided Propositions}

Sometimes, a predicate may provide information when it
succeeds, but not when it fails. For instance, consider this
function:

@examples[#:no-result #:eval the-eval
(define (legal-id? s)
  (and (symbol? s)
       (not (member s '(cond else if)))))
]

This function only returns @racket[#t] when given a symbol, so the type of something
that satisfies this predicate can be refined to Symbol.

However, values that fail this predicate can’t be refined to non-symbols;
symbols such as @racket['else] also fail to satisfy this predicate.

In cases such as these, it’s possible to provide a proposition that's
applied only to the ``positive'' assertion. Specifically, this type

@racketblock[
(: legal-id? (Any -> Boolean : #:+ Symbol))
 ]

... captures the idea that if this predicate returns @racket[#t],
the argument is known to be a Symbol, without making any claim at
all about values for which this predicate returns @racket[#f].

There is a negative form as well, which allows types that specify
propositions only about values that cause a predicate to return @racket[#f].

@section{Other conditionals and assertions}

@margin-note{After all, these control flow constructs macro-expand
  to @racket[if] in the end.}

So far, we have seen that occurrence typing allows precise reasoning
about @racket[if] expressions. Occurrence typing works for most
control flow constructs that are present in Racket such as
@racket[cond], @racket[when], and others.

For example, the @racket[_flexible-length] function from earlier can
be re-written to use @racket[cond] with no additional effort:

@examples[#:no-result #:eval the-eval
  (: flexible-length/cond (-> (U String (Listof Any)) Integer))
  (define (flexible-length/cond str-or-lst)
    (cond [(string? str-or-lst) (string-length str-or-lst)]
          [else (length str-or-lst)]))
]

In some cases, the type system does not have enough information or is
too conservative to typecheck an expression. For example, consider
the following interaction:

@examples[#:label #f #:eval the-eval
(: a Positive-Integer)
(define a 15)
(: b Positive-Integer)
(define b 20)
(: c Positive-Integer)
(eval:error (define c (- b a)))
]

In this case, the type system only knows that @racket[_a] and
@racket[_b] are positive integers and cannot conclude that their
difference will always be positive in defining @racket[_c].  In cases
like this, occurrence typing can be used to make the code type-check
using an @emph{assertion}. For example,

@examples[#:no-result #:eval the-eval
(: d Positive-Integer)
(define d (assert (- b a) positive?))
]

Using the logical propositions on @racket[positive?], Typed Racket can
assign the type @racket[Positive-Integer] to the whole @racket[assert]
expression.  This type-checks, but note that the assertion may raise
an exception at run-time if the predicate returns @racket[#f].

Note that @racket[assert] is a derived concept in Typed Racket and is
a natural consequence of occurrence typing. The assertion above is
essentially equivalent to the following:

@examples[#:no-result #:eval the-eval
(: e Positive-Integer)
(define e (let ([diff (- b a)])
            (if (positive? diff)
                diff
                (error "Assertion failed"))))
]

@section{A caveat about @racket[set!]}

If a variable is ever mutated with @racket[set!]  in the scope in
which it is defined, Typed Racket cannot use occurrence typing with
that variable. This precaution is needed to ensure that concurrent
modification of a variable does not invalidate Typed Racket's
knowledge of the type of that variable.  Also see
@Secref["using-set!"  #:doc '(lib "scribblings/guide/guide.scrbl")].

Furthermore, this means that the types of @rtech{top-level variable}s
in the @gtech{REPL} cannot be refined by Typed Racket either. This is
because the scope of a top-level variable includes future top-level
interactions, which may include mutations. It is possible to work
around this by moving the variable inside of a module or into a local
binding form like @racket[let].

@section{@racket[let]-aliasing}

Typed Racket is able to reason about some cases when variables introduced 
by let-expressions alias other values (e.g. when they alias non-mutated identifiers, 
@racket[car]/@racket[cdr]/@racket[struct] accesses into immutable values, etc...). 
This allows programs which explicitly rely on occurrence typing and aliasing to 
typecheck:

@examples[#:no-result #:eval the-eval
(: f (Any -> Number))
(define (f x) 
  (let ([y x])
    (cond
      [(number? y) x]
      [(and (pair? y) 
            (number? (car y)))
       (car x)]
      [else 42])))
]

It also allows the typechecker to check programs which use macros 
that heavily rely on let-bindings internally (such as @racket[match]):

@examples[#:no-result #:eval the-eval
(: g (Any -> Number))
(define (g x) 
  (match x
    [(? number?) x]
    [`(_ . (_ . ,(? number?))) (cddr x)]
    [`(_ . (_ . ,(? pair? p))) 
     (if (number? (caddr x))
         (car p)
         41)]
    [_ 42]))    
]

