Module L := LAT.(** The state of the iteration has three components:
- [aval]: A mapping from program points to values of type [L.t] representing
  the candidate solution found so far.
- [worklist]: A worklist of program points that remain to be considered.
- [visited]: A set of program points that were visited already
  (i.e. put on the worklist at some point in the past).

Only the first two components are computationally relevant.  The third
is a ghost variable used only for stating and proving invariants.
For this reason, [visited] is defined at sort [Prop] so that it is
erased during program extraction.
*)

Record state : Type :=
  mkstate { aval: PTree.t L.t; worklist: NS.t; visited: positive -> Prop }.(** The state of the iteration has three components:
- [aval]: A mapping from program points to values of type [L.t] representing
  the candidate solution found so far.
- [worklist]: A worklist of program points that remain to be considered.
- [visited]: A set of program points that were visited already
  (i.e. put on the worklist at some point in the past).

Only the first two components are computationally relevant.  The third
is a ghost variable used only for stating and proving invariants.
For this reason, [visited] is defined at sort [Prop] so that it is
erased during program extraction.
*)

Record state : Type :=
  mkstate { aval: PTree.t L.t; worklist: NS.t; visited: positive -> Prop }.(** The state of the iteration has three components:
- [aval]: A mapping from program points to values of type [L.t] representing
  the candidate solution found so far.
- [worklist]: A worklist of program points that remain to be considered.
- [visited]: A set of program points that were visited already
  (i.e. put on the worklist at some point in the past).

Only the first two components are computationally relevant.  The third
is a ghost variable used only for stating and proving invariants.
For this reason, [visited] is defined at sort [Prop] so that it is
erased during program extraction.
*)

Record state : Type :=
  mkstate { aval: PTree.t L.t; worklist: NS.t; visited: positive -> Prop }.(** The state of the iteration has three components:
- [aval]: A mapping from program points to values of type [L.t] representing
  the candidate solution found so far.
- [worklist]: A worklist of program points that remain to be considered.
- [visited]: A set of program points that were visited already
  (i.e. put on the worklist at some point in the past).

Only the first two components are computationally relevant.  The third
is a ghost variable used only for stating and proving invariants.
For this reason, [visited] is defined at sort [Prop] so that it is
erased during program extraction.
*)

Record state : Type :=
  mkstate { aval: PTree.t L.t; worklist: NS.t; visited: positive -> Prop }.(** The state of the iteration has three components:
- [aval]: A mapping from program points to values of type [L.t] representing
  the candidate solution found so far.
- [worklist]: A worklist of program points that remain to be considered.
- [visited]: A set of program points that were visited already
  (i.e. put on the worklist at some point in the past).

Only the first two components are computationally relevant.  The third
is a ghost variable used only for stating and proving invariants.
For this reason, [visited] is defined at sort [Prop] so that it is
erased during program extraction.
*)

Record state : Type :=
  mkstate { aval: PTree.t L.t; worklist: NS.t; visited: positive -> Prop }.(** [propagate_succ_list] corresponds, in the pseudocode,
  to the [for] loop iterating over all successors. *)

Fixpoint propagate_succ_list (s: state) (out: L.t) (succs: list positive)
                             {struct succs} : state :=
  match succs with
  | nil => s
  | n :: rem => propagate_succ_list (propagate_succ s out n) out rem
  end.(** The whole fixpoint computation is the iteration of [step] from
  an initial state. *)

Definition fixpoint_from (start: state) : option (PMap.t L.t) :=
  PrimIter.iterate _ _ step start.(** There are several ways to build the initial state.  For forward
  dataflow analyses, the initial worklist is the function entry point,
  and the initial mapping sets the function entry point to the given
  abstract value, and leaves unset all other program points, which
  corresponds to [L.bot] initial abstract values. *)

Definition start_state (enode: positive) (eval: L.t) :=
  {| aval := PTree.set enode eval (PTree.empty L.t);
     worklist := NS.add enode NS.empty;
     visited := fun n => n = enode |}.Definition fixpoint (enode: positive) (eval: L.t) :=
  fixpoint_from (start_state enode eval).(** For backward analyses (viewed as forward analyses on the reversed CFG),
  the following two variants are more useful.  Both start with an
  empty initial mapping, where all program points start at [L.bot].
  The first initializes the worklist with a given set of entry points
  in the reversed CFG.  (See the backward dataflow solver below for
  how this list is computed.)  The second start state construction
  initializes the worklist with all program points of the given CFG. *)

Definition start_state_nodeset (enodes: NS.t) :=
  {| aval := PTree.empty L.t;
     worklist := enodes;
     visited := fun n => NS.In n enodes |}.Definition fixpoint_nodeset (enodes: NS.t) :=
  fixpoint_from (start_state_nodeset enodes).Definition fixpoint_allnodes :=
  fixpoint_from start_state_allnodes.(** ** Characterization of the propagation functions *)

Inductive optge: option L.t -> option L.t -> Prop :=
  | optge_some: forall l l',
      L.ge l l' -> optge (Some l) (Some l')
  | optge_none: forall ol,
      optge ol None.(** ** Characterization of the propagation functions *)

Inductive optge: option L.t -> option L.t -> Prop :=
  | optge_some: forall l l',
      L.ge l l' -> optge (Some l) (Some l')
  | optge_none: forall ol,
      optge ol None.(** ** Characterization of the propagation functions *)

Inductive optge: option L.t -> option L.t -> Prop :=
  | optge_some: forall l l',
      L.ge l l' -> optge (Some l) (Some l')
  | optge_none: forall ol,
      optge ol None.Remark optge_refl: forall ol, optge ol ol.Remark optge_trans: forall ol1 ol2 ol3, optge ol1 ol2 -> optge ol2 ol3 -> optge ol1 ol3.(** Characterization of [fixpoint_from]. *)

Inductive steps: state -> state -> Prop :=
  | steps_base: forall s, steps s s
  | steps_right: forall s1 s2 s3, steps s1 s2 -> step s2 = inr s3 -> steps s1 s3.(** Characterization of [fixpoint_from]. *)

Inductive steps: state -> state -> Prop :=
  | steps_base: forall s, steps s s
  | steps_right: forall s1 s2 s3, steps s1 s2 -> step s2 = inr s3 -> steps s1 s3.(** Characterization of [fixpoint_from]. *)

Inductive steps: state -> state -> Prop :=
  | steps_base: forall s, steps s s
  | steps_right: forall s1 s2 s3, steps s1 s2 -> step s2 = inr s3 -> steps s1 s3.Scheme steps_ind := Induction for steps Sort Prop.

Lemma start_state_nodeset_good: forall enodes, good_state (start_state_nodeset enodes).