module React:sig..end
React is a module for functional reactive programming (frp). It provides support to program with time varying values : declarative events and signals. React doesn't define any primitive event or signal, this lets the client choose the concrete timeline.
Consult the semantics, the basics and examples. Open the module to use it, this defines only two types and modules in your scope.
    Release 1.2.0 - Daniel Bünzli <daniel.buenzl i@erratique.ch> 
type 'a event 
'a.type 'a signal 
'a.type 
module E:sig..end
module S:sig..end
module Step:sig..end
The following notations are used to give precise meaning to the combinators. It is important to note that in these semantic descriptions the origin of time t = 0 is always fixed at the time at which the combinator creates the event or the signal and the semantics of the dependents is evaluated relative to this timeline.
We use dt to denote an infinitesimal amount of time.
An event is a value with discrete occurrences over time.
    The semantic function [] : 'a event -> time -> 'a option gives
    meaning to an event e by mapping it to a function of time
    [e] returning Some v whenever the event occurs with value
    v and None otherwise. We write [e]t the evaluation of
    this semantic function at time t.
    As a shortcut notation we also define []<t : 'a event -> 'a option
    (resp. []<=t) to denote the last occurrence, if any, of an
    event before (resp. before or at) t. More precisely :
    
e]<t = [e]t' with t' the greatest t' < t
      (resp. <=) such that
       [e]t' <> None.e]<t = None if there is no such t'.
A signal is a value that varies continuously over time. In contrast to events which occur at specific point in time, a signal has a value at every point in time.
    The semantic function [] : 'a signal -> time -> 'a gives
    meaning to a signal s by mapping it to a function of time
    [s] that returns its value at a given time. We write [s]t
    the evaluation of this semantic function at time t.
    
    Most signal combinators have an optional eq parameter that
    defaults to structural equality. eq specifies the equality
    function used to detect changes in the value of the resulting
    signal. This function is needed for the efficient update of
    signals and to deal correctly with signals that perform
    side effects.
Given an equality function on a type the combinators can be automatically specialized via a functor.
Ultimately signal updates depend on primitives updates. Thus a signal can only approximate a real continuous signal. The accuracy of the approximation depends on the variation rate of the real signal and the primitive's update frequency.
React doesn't define primitive events and signals, they must be created and updated by the client.
    Primitive events are created with React.E.create. This function
    returns a new event and an update function that generates an
    occurrence for the event at the time it is called. The following
    code creates a primitive integer event x and generates three
    occurrences with value 1, 2, 3. Those occurrences are printed
    on stdout by the effectful event pr_x.  
open React;;
let x, send_x = E.create ()
let pr_x = E.map print_int x
let () = List.iter send_x [1; 2; 3]React.S.create. This function
    returns a new signal and an update function that sets the signal's value
    at the time it is called. The following code creates an
    integer signal x initially set to 1 and updates it three time with
    values 2, 2, 3. The signal's values are printed on stdout by the
    effectful signal pr_x. Note that only updates that change
    the signal's value are printed, hence the program prints 123, not 1223.
    See the discussion on
    side effects for more details.
open React;;
let x, set_x = S.create 1
let pr_x = S.map print_int x
let () = List.iter set_x [2; 2; 3]
   The React.E.create and React.S.create functions return update functions
   used to generate primitive event occurences and set the value of
   primitive signals. Upon invocation as in the preceding section
   these functions immediatly create and invoke an update step.
   The update step automatically updates events and signals that
   transitively depend on the updated primitive. The dependents of a
   signal are updated iff the signal's value changed according to its
   equality function.
   The update functions have an optional step argument. If they are
   given a concrete step value created with React.Step.create, then it
   updates the event or signal but doesn't update its dependencies. It
   will only do so whenever step is executed with
   React.Step.execute. This allows to make primitive event occurences and
   signal changes simultaneous. See next section for an example.
Update steps are made under a synchrony hypothesis : the update step takes no time, it is instantenous. Two event occurrences are simultaneous if they occur in the same update step.
    In the code below w, x and y will always have simultaneous
    occurrences. They may have simulatenous occurences with z
    if send_w and send_z are used with the same update step.
let w, send_w = E.create ()
let x = E.map succ w
let y = E.map succ x
let z, send_z = E.create ()
let () =
  let () = send_w 3 (* w x y occur simultaneously, z doesn't occur *) in
  let step = Step.create () in
  send_w ~step 3;
  send_z ~step 4;
  Step.execute step (* w x z y occur simultaneously *)
    Primitives are the only mean to drive the reactive
    system and they are entirely under the control of the client. When
    the client invokes a primitive's update function without the
    step argument or when it invokes React.Step.execute on a step
    value, React performs an update step.
    To ensure correctness in the presence of threads, update steps
    must be executed in a critical section. Let uset(p) be the set
    of events and signals that need to be updated whenever the
    primitive p is updated.  Updating two primitives p and p'
    concurrently is only allowed if uset(p) and uset(p') are
    disjoint. Otherwise the updates must be properly serialized.
    Below, concurrent, updates to x and y must be serialized (or
    performed on the same step if it makes sense semantically), but z
    can be updated concurently to both x and y.
open React;;
let x, set_x = S.create 0
let y, send_y = E.create ()
let z, set_z = S.create 0
let max_xy = S.l2 (fun x y -> if x > y then x else y) x (S.hold 0 y)
let succ_z = S.map succ z
Effectful events and signals perform their side effect exactly once in each update step in which there is an update of at least one of the event or signal it depends on.
Remember that a signal updates in a step iff its equality function determined that the signal value changed. Signal initialization is unconditionally considered as an update.
    It is important to keep references on effectful events and
    signals. Otherwise they may be reclaimed by the garbage collector.
    The following program prints only a 1.
let x, set_x = S.create 1
let () = ignore (S.map print_int x)
let () = Gc.full_major (); List.iter set_x [2; 2; 3]
    Lifting transforms a regular function to make it act on signals.
    The combinators
    React.S.const and React.S.app allow to lift functions of arbitrary arity n,
    but this involves the inefficient creation of n-1 intermediary
    closure signals. The fixed arity lifting
    functions are more efficient. For example :
let f x y = x mod y
let fl x y = S.app (S.app ~eq:(==) (S.const f) x) y (* inefficient *)
let fl' x y = S.l2 f x y                            (* efficient *)
Pervasives's functions and operators are
    already lifted and availables in submodules of React.S. They can be
    be opened in specific scopes. For example if you are dealing with
    float signals you can open React.S.Float.
open React
open React.S.Float
let f t = sqrt t *. sin t (* f is defined on float signals *)
...
open Pervasives (* back to pervasives floats *)
let open
   construct
let open React.S.Float in
let f t = sqrt t *. sin t in (* f is defined on float signals *)
...
Mutual and self reference among time varying values occurs naturally in programs. However a mutually recursive definition of two signals in which both need the value of the other at time t to define their value at time t has no least fixed point. To break this tight loop one signal must depend on the value the other had at time t-dt where dt is an infinitesimal delay.
  The fixed point combinators React.E.fix and React.S.fix allow to refer to
  the value an event or signal had an infinitesimal amount of time
  before. These fixed point combinators act on a function f that takes
  as argument the infinitesimally delayed event or signal that f
  itself returns.
  In the example below history s returns a signal whose value
  is the history of s as a list.
let history ?(eq = ( = )) s =
  let push v = function
    | [] -> [ v ]
    | v' :: _ as l when eq v v' -> l
    | l -> v :: l
  in
  let define h =
    let h' = S.l2 push s h in
    h', h'
  in
  S.fix [] defines is infinitesimally delayed, then
  its update in a step c will trigger a new step c' at the end
  of the step in which the delayed signal of s will have the value
  s had in c. This means that the recursion occuring between a
  signal (or event) and its infinitesimally delayed counterpart must
  be well-founded otherwise this may trigger an infinite number
  of update steps, like in the following examples.
let start, send_start = E.create ()
let diverge =
  let define e =
    let e' = E.select [e; start] in
    e', e'
  in
  E.fix define
let () = send_start ()        (* diverges *)
let diverge =                 (* diverges *)
  let define s =
    let s' = S.Int.succ s in
    s', s'
  in
  S.fix 0 defineInvalid_argument if
  such dependencies are created. This limitation can be
  circumvented by mapping these values with the identity.
  Strong stops should only be used on platforms where weak arrays have
  a strong semantics (i.e. JavaScript). You can safely ignore that
  section and the strong argument of React.E.stop and React.S.stop
  if that's not the case.
  Whenever React.E.stop and React.S.stop is called with ~strong:true on a
  reactive value v, it is first stopped and then it walks over the
  list prods of events and signals that it depends on and
  unregisters itself from these ones as a dependent (something that is
  normally automatically done when v is garbage collected since
  dependents are stored in a weak array). Then for each element of
  prod that has no dependents anymore and is not a primitive it
  stops them aswell and recursively.
  A stop call with ~strong:true is more involved. But it allows to
  prevent memory leaks when used judiciously on the leaves of the
  reactive system that are no longer used.
  Warning. It should be noted that if direct references are kept
  on an intermediate event or signal of the reactive system it may
  suddenly stop updating if all its dependents were strongly stopped. In
  the example below, e1 will never occur:
let e, e_send = E.create ()
let e1 = E.map (fun x -> x + 1) e (* never occurs *)
let () =
  let e2 = E.map (fun x -> x + 1) e1 in
  E.stop ~strong:true e2
let e, e_send = E.create ()
let e1 = E.map (fun x -> x + 1) e (* may still occur *)
let e1_ref = E.map (fun x -> x) e1
let () =
  let e2 = E.map (fun x -> x + 1) e1 in
  E.stop ~strong:true e2
  The following program defines a primitive event seconds holding
  the UNIX time and occuring on every second. An effectful event
  converts these occurences to local time and prints them on stdout
  along with an
  ANSI
  escape sequence to control the cursor position.
let pr_time t =
  let tm = Unix.localtime t in
  Printf.printf "\x1B[8D%02d:%02d:%02d%!"
    tm.Unix.tm_hour tm.Unix.tm_min tm.Unix.tm_sec
open React;;
let seconds, run =
  let e, send = E.create () in
  let run () =
    while true do send (Unix.gettimeofday ()); Unix.sleep 1 done
  in
  e, run
let printer = E.map pr_time seconds
let () = run ()