State Machine (continued)
Yes, how about mX : (A × (A × X)A)A → (A × X)A?

We have another collection of state machines, mX : (A × (A × X)A)A, which has A as a state set, and which outputs to another collection of state machines (this one too has A as a state set, and X as output set).How do we find for such a compound state machine a match in a state machine on A?
Write it in Lisp:

(define (mx f)
  (let (tr1 out1)
       ((car f)(cadr f)));two components of f:A → (A × (A × X)A
  (lambda (a)
   (let a1 (tr1 a));state after first transition
   (let f2 (out1 a)) ;mapped a state to another state machine
   (let (tr2 out2) ((car f2) (cadr f2))) ;second machine components
      (list (tr2 a1) (out2 a1))))
See what happens here: we have a function from A to A×(A×X)A, which consists of transition A → A and output A → (A×X)A, that is, for each a we have another state a1 and an output function; the resulting function from A to A×X should just apply that output function to the new state.

Boring Exercise: Prove that this is actually a monad.