Stateful[F, S] is the capability to access and modify a state value from inside the F[_] context, using set(s: S): F[Unit] and get: F[S].

Stateful has four external laws:

def getThenSetDoesNothing = {
  get >>= set <-> pure(())
}
def setThenGetReturnsSetted(s: S) = {
  set(s) *> get <-> set(s) *> pure(s)
}
def setThenSetSetsLast(s1: S, s2: S) = {
  set(s1) *> set(s2) <-> set(s2)
}
def getThenGetGetsOnce = {
  get *> get <-> get
}

Stateful has two internal law:

def modifyIsGetThenSet(f: S => S) = {
  modify(f) <-> (inspect(f) flatMap set)
}

def inspectLaw[A](f: S => A) = {
  inspect(f) <-> (get map f)
}