Events. Semantically: time-ordered list of future values. Instances:

• Monoid: mempty is the event that never occurs, and e mappend e' is the event that combines occurrences from e and e'.
• Functor: fmap f e is the event that occurs whenever e occurs, and whose occurrence values come from applying f to the values from e.
• Applicative: pure a is an event with a single occurrence at time -Infinity. ef <*> ex is an event whose occurrences are made from the product of the occurrences of ef and ex. For every occurrence f at time tf of ef and occurrence x at time tx of ex, ef <*> ex has an occurrence f x at time tf max tx. N.B.: I don't expect this instance to be very useful. If ef has nf instances and ex has nx instances, then ef <*> ex has nf*nx instances. However, there are only nf+nx possibilities for tf max tx, so many of the occurrences are simultaneous. If you think you want to use this instance, consider using Reactive instead.
• Monad: return a is the same as pure a (as usual). In e >>= f, each occurrence of e leads, through f, to a new event. Similarly for join ee, which is somehow simpler for me to think about. The occurrences of e >>= f (or join ee) correspond to the union of the occurrences (temporal interleaving) of all such events. For example, suppose we're playing Asteroids and tracking collisions. Each collision can break an asteroid into more of them, each of which has to be tracked for more collisions. Another example: A chat room has an enter event, whose occurrences contain new events like speak. An especially useful monad-based function is joinMaybes, which filters a Maybe-valued event.

Reactive value: a discretely changing value. Reactive values can be understood in terms of (a) a simple denotational semantics of reactive values as functions of time, and (b) the corresponding instances for functions. The semantics is given by the function at :: ReactiveG t a -> (t -> a). A reactive value may also be thought of (and in this module is implemented as) a current value and an event (stream of future values).

The semantics of ReactiveG instances are given by corresponding instances for the semantic model (functions):

• Functor: at (fmap f r) == fmap f (at r), i.e., fmap f r at t == f (r at t).
• Applicative: at (pure a) == pure a, and at (s <*> r) == at s <*> at t. That is, pure a at t == a, and (s <*> r) at t == (s at t) (r at t).
• Monad: at (return a) == return a, and at (join rr) == join (at . at rr). That is, return a at t == a, and join rr at t == (rr at t) at t. As always, (r >>= f) == join (fmap f r). at (r >>= f) == at r >>= at . f.
• Monoid: a typical lifted monoid. If o is a monoid, then Reactive o is a monoid, with mempty == pure mempty, and mappend == liftA2 mappend. That is, mempty at t == mempty, and (r mappend s) at t == (r at t) mappend (s at t).