Definition pTree_from_elements {A} (el: list (positive * A)) : PTree.t A :=
 fold_right (fun ia t => PTree.set (fst ia) (snd ia) t) (PTree.empty _) el.Definition local_trees :=
   (PTree.t val * PTree.t (type * val) * list Prop * option globals)%type.Fixpoint local2ptree_aux (Q: list localdef)
   (T1: PTree.t val) (T2: PTree.t (type * val)) (P': list Prop) (Q': option globals):
   local_trees :=
match Q with
| Q1 :: Qr => local2ptree1 Q1 T1 T2 P' Q' (local2ptree_aux Qr)
| nil => (T1,T2,P',Q')
end.Definition local2ptree (Q: list localdef)
     : (PTree.t val * PTree.t (type * val) * list Prop * option globals) :=
local2ptree_aux Q PTree.Leaf PTree.Leaf nil None.Definition CLEAR_ME {T} (x:T) := x.Ltac hide_it z := let x := fresh "x" in set (x:=z); change z with (CLEAR_ME z) in x.(* TODO: compare this with unfold_name method in symbolic_go_lower. *)

Ltac hnf_localdef_list A :=
  match A with
 | temp _ ?v :: ?Q' => hide_it v; hnf_localdef_list Q'
 | lvar _ ?t ?v :: ?Q' => hide_it t; hide_it v; hnf_localdef_list Q'
 | gvars ?v :: ?Q' => hide_it v; hnf_localdef_list Q'
  | ?B :: ?C => let x := eval hnf in B in change B with x; hnf_localdef_list (x::C)
  | nil => idtac
  | _ => try (is_evar A; fail 1);
            let x := eval hnf in A in (change A with x); hnf_localdef_list x
  end.Ltac grab_gvars L := 
 match L with gvars ?A :: ?B => let x := grab_gvars B in let z := constr:(A::x) in z
                | nil => let x := constr:(@nil globals) in x
  end.Ltac prove_local2ptree :=
 clear;
 match goal with |- local2ptree ?A = _ => hnf_localdef_list A end;
 etransitivity;
 [unfold local2ptree, local2ptree_aux; simpl;
  repeat match goal with x := CLEAR_ME _ |- _ => unfold CLEAR_ME in x; subst x end;
  reflexivity |
  repeat match goal with x := CLEAR_ME _ |- _ => unfold CLEAR_ME in x; subst x end;
  apply f_equal;
  try reflexivity;
  match goal with |- ?L = _ => let x := grab_gvars L in instantiate(1:=x); reflexivity end
 ].Goal exists x,  local2ptree (
      temp 1%positive Vundef
   :: temp 3%positive (Vint (Int.repr (3+4)))
   :: lvar 1%positive tint (Vint (Int.repr (1 + 2)))
   :: nil) = x.Ltac construct_local2ptree Q H :=
  let t := fresh "t" in
  evar (t: (PTree.t val * PTree.t (type * val) * list Prop * option globals)%type);
  assert (H: local2ptree Q = t); subst t;
   [ prove_local2ptree | ].Definition LocalD (T1: PTree.t val) (T2: PTree.t (type * val)) (Q: option globals) :=
  PTree.fold (fun Q i v => temp i v :: Q) T1
  (PTree.fold (fun Q i tv => match tv with (t, v) => lvar i t v end :: Q) T2
   match Q with Some gv => (gvars gv) :: nil | None => nil end).

Lemma LocalD_complete : forall q T1 T2 Q, In q (LocalD T1 T2 Q) -> (exists i v, PTree.get i T1 = Some v /\ q = temp i v) \/ (exists i t v, PTree.get i T2 = Some (t, v) /\ q = lvar i t v) \/ (exists gv, Q = Some gv /\ q = gvars gv).