(* /ADMITTED *)



(*************************************************************)
(** * An abstract machine for cbn evaluation                 *)
(*************************************************************)

(** The advantage of named terms is an efficient operational
    semantics! Compared to LN or de Bruijn representation, we
    don't need always need to modify terms (such as via open or
    shifting) as we traverse binders. Instead, as long as the
    binder is "sufficiently fresh" we can use the name as is,
    and only rename (via swapping) when absolutely
    necessary. *)

(** Below, we define an operational semantics based on an
    abstract machine. As before, it will model execution as a
    sequence of small-step transitions. However, transition
    will be between _machine configurations_, not just
    expressions.

    Our abstract machine configurations have three components:

    - the current expression being evaluated the heap (aka
    - environment): a mapping between variables and expressions
    - the stack: the evaluation context of the current
    - expression

    Because we have a heap, we don't need to use substitution
    to define our semantics! To implement [x ~> e] we add a
    definition that [x] maps to [e] in the heap and then look
    up the definition when we get to [x] during evaluation.  *)


Definition heap := list (atom * n_exp).Inductive frame : Set := | n_app2 : n_exp -> frame.Inductive frame : Set := | n_app2 : n_exp -> frame.Notation  stack := (list frame).Definition configuration := (heap * n_exp * stack)%type.(** The (small-step) semantics is a _function_ from
    configurations to configurations, until completion or
    error. *)

Inductive Step a := Error    : Step a
                  | Done     : Step a
                  | TakeStep : a -> Step a.(** The (small-step) semantics is a _function_ from
    configurations to configurations, until completion or
    error. *)

Inductive Step a := Error    : Step a
                  | Done     : Step a
                  | TakeStep : a -> Step a.(** The (small-step) semantics is a _function_ from
    configurations to configurations, until completion or
    error. *)

Inductive Step a := Error    : Step a
                  | Done     : Step a
                  | TakeStep : a -> Step a.(** The (small-step) semantics is a _function_ from
    configurations to configurations, until completion or
    error. *)

Inductive Step a := Error    : Step a
                  | Done     : Step a
                  | TakeStep : a -> Step a.Definition isVal (t : n_exp) :=
  match t with
  | n_abs _ _ => true
  | _         => false
  end.Definition initconf (t : n_exp) : configuration := (nil,t,nil).

(* Specifications: *)
Lemma values_are_done : forall D t, isVal t = true -> machine_step D (initconf t) = Done _.