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(* main methods *)
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Definition def h := if h is Def _ _ then true else false.
Definition empty := @Def (finmap.nil _ _) is_true_true.
Definition upd h x d := nosimpl
(if h is Def hs ns then
if decP (@idP (x != null)) is left pf then
Def (@upd_nullP _ _ d pf ns)
else Undef
else Undef).
Definition dom h : pred ptr := nosimpl
(if h is Def f _ then mem (supp f) else pred0).
Definition free x h : heap :=
(if h is Def hs ns then Def (free_nullP x ns) else Undef).
Definition look (x : ptr) h :=
(if h is Def hs _ then
if fnd x hs is Some d then d else dyn tt
else dyn tt).
Definition union2 h1 h2 := nosimpl
(if (h1, h2) is (Def hs1 ns1, Def hs2 ns2) then
if disj hs1 hs2 then
Def (@un_nullP _ _ ns1 ns2)
else Undef
else Undef).
Definition empb h :=
if h is Def hs _ then supp hs == [::] else false.
Definition fresh h :=
(if h is Def hs _ then last null (supp hs) else null) .+ 1.
Definition subdom h1 h2 :=
if (h1, h2) is (Def hs1 _, Def hs2 _) then
all (fun x => x \in supp hs2) (supp hs1)
else false.
Definition subheap h1 h2 :=
subdom h1 h2 /\ forall x, x \in dom h1 -> look x h1 = look x h2.
(* Definition subheap h1 h2 p :=  *)
(*   if (h1, h2) is (Def hs1 _, Def hs2 _) then  *)
(*     all (fun x => p (look x h1) (look x h2)) (supp hs1) *)
(*   else false. *)
Definition subtract h1 h2 :=
if h1 is (Def hs1 _) then
foldl (fun h x => free x h) h2 (supp hs1)
else Undef.
Definition pick h :=
if h is Def hs _ then head null (supp hs) else null.
Definition pts A (x : ptr) (v : A) := upd empty x (dyn v).
Notation "h1 :+ h2" := (union2 h1 h2) (at level 43, left associativity).
Notation "h2 :- h1" := (subtract h1 h2) (at level 43, left associativity).
Notation "x :-> v" := (pts x v) (at level 30).
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(* monoidal properties *)
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Lemma unA : forall h1 h2 h3, h1 :+ (h2 :+ h3) = h1 :+ h2 :+ h3.