# Library ddifc-coq

Require Import Omega.
Require Import Arith.
Require Import ZArith.
Require Import List.
Require Import Classical.
Require Import ProofIrrelevance.
Require Import FunctionalExtensionality.
Require Import Coq.Bool.Bool.

Ltac inv H := inversion H; try subst; try clear H.
Ltac dup H := generalize H; intro.
Ltac intuit := try solve [intuition].
Ltac decomp H := decompose [and or] H; try clear H.

Notation "[ ]" := nil (at level 1).
Notation "[ a ; .. ; b ]" := (a :: .. (b :: []) ..) (at level 1).

Proposition app_assoc {A} : forall l1 l2 l3 : list A, (l1 ++ l2) ++ l3 = l1 ++ l2 ++ l3.
Proof.
induction l1; simpl; intros; auto.
rewrite IHl1; auto.
Qed.

Proposition in_app_iff {A} : forall (l1 l2 : list A) x, In x (l1++l2) <-> In x l1 \/ In x l2.
Proof.
intros; split; intros.
apply in_app_or; auto.
apply in_or_app; auto.
Qed.

Proposition app_nil_r {A} : forall l : list A, l ++ [] = l.
Proof.
induction l; auto.
simpl; rewrite IHl; auto.
Qed.

Proposition list_finite {A} : forall (l : list A) x, l <> x :: l.
Proof.
induction l; intros; intro.
inv H.
inv H.
Qed.

Proposition list_finite' {A} : forall l l' : list A, l' <> [] -> l <> l' ++ l.
Proof.
induction l; intros; intro.
rewrite app_nil_r in H0; subst.
destruct l'.
inv H0.
subst a.
intro.
destruct l'; inv H0.
rewrite app_assoc; auto.
Qed.

Proposition app_cancel_l {A} : forall l l1 l2 : list A, l ++ l1 = l ++ l2 -> l1 = l2.
Proof.
induction l; intros; auto.
inv H; intuit.
Qed.

Proposition app_cancel_r_help {A} : forall (l1 l2 : list A) x, l1 ++ [x] = l2 ++ [x] -> l1 = l2.
Proof.
induction l1; intros.
destruct l2; auto; inv H.
destruct l2; inv H2.
destruct l2; inv H.
destruct l1; inv H2.
apply IHl1 in H2; subst; auto.
Qed.

Proposition app_cancel_r {A} : forall l l1 l2 : list A, l1 ++ l = l2 ++ l -> l1 = l2.
Proof.
induction l; intros.
repeat rewrite app_nil_r in H; auto.
change (l1++([a]++l) = l2++([a]++l)) in H.
repeat rewrite <- app_assoc in H; apply IHl in H.
apply app_cancel_r_help in H; auto.
Qed.

Definition var := nat.
Definition lvar1 := nat.
Definition lvar2 := nat.
Definition fname := nat.

Open Scope Z_scope.

Definition nat_of_Z (v : Z) (pf : v >= 0) : nat.
intros.
destruct v.
apply O.
apply (nat_of_P p).
assert (~ Zneg p >= 0).
clear pf; induction p.
intro H; contradiction H; simpl; auto.
Defined.

Proposition Zneg_dec : forall v : Z, {v >= 0} + {v < 0}.
Proof.
intros.
destruct v.
left; omega.
left.
induction p; auto.
omega.
right.
induction p; auto.
omega.
Qed.

Record poset {A : Set} : Type :=
{leq : A -> A -> bool;
leq_refl : forall x : A, leq x x = true;
leq_antisym : forall x y : A, leq x y = true -> leq y x = true -> x = y;
leq_trans : forall x y z : A, leq x y = true -> leq y z = true -> leq x z = true}.

Record join_semi {A : Set} : Type :=
{po : poset (A:=A);
lub : A -> A -> A;
lub_l : forall x y : A, leq po x (lub x y) = true;
lub_r : forall x y : A, leq po y (lub x y) = true;
lub_least : forall x y z : A, leq po x z = true -> leq po y z = true -> leq po (lub x y) z = true}.

Record join_semi' {A : Set} (js : join_semi (A:=A)) : Type :=
{lub_idem : forall x : A, lub js x x = x;
lub_comm : forall x y : A, lub js x y = lub js y x;
lub_assoc : forall x y z : A, lub js (lub js x y) z = lub js x (lub js y z);
lub_leq : forall x y z : A, leq (po js) (lub js x y) z = true <-> leq (po js) x z = true /\ leq (po js) y z = true}.

Definition join_semi_extend {A : Set} (js : join_semi (A:=A)) : join_semi' (A:=A) js.
intros; split; intros.
apply (leq_antisym (po js)).
apply lub_least; apply leq_refl.
apply lub_l.
apply (leq_antisym (po js)); solve [apply lub_least; [apply lub_r | apply lub_l]].
apply (leq_antisym (po js)).
apply lub_least.
apply lub_least.
apply lub_l.
apply (leq_trans _ _ (lub js y z) _); [apply lub_l | apply lub_r].
apply (leq_trans _ _ (lub js y z) _); [apply lub_r | apply lub_r].
apply lub_least.
apply (leq_trans _ _ (lub js x y) _); [apply lub_l | apply lub_l].
apply lub_least.
apply (leq_trans _ _ (lub js x y) _); [apply lub_r | apply lub_l].
apply lub_r.
split; intros; try split.
apply (leq_trans _ _ (lub js x y) _); [apply lub_l | auto].
apply (leq_trans _ _ (lub js x y) _); [apply lub_r | auto].
apply lub_least; intuit.
Qed.

Record bounded_join_semi {A : Set} : Type :=
{js : join_semi (A:=A);
bot : A;
leq_bot : forall x : A, leq (po js) bot x = true}.

Record bounded_join_semi' {A : Set} (bjs : bounded_join_semi (A:=A)) : Type :=
{bot_unit : forall x : A, lub (js bjs) x (bot bjs) = x}.

Definition bounded_join_semi_extend {A : Set} (bjs : bounded_join_semi (A:=A)) : bounded_join_semi' (A:=A) bjs.
intros; split; intros.
apply (leq_antisym (po (js bjs))).
apply lub_least; [apply leq_refl | apply leq_bot].
apply lub_l.
Qed.

Coercion po : join_semi >-> poset.
Coercion js : bounded_join_semi >-> join_semi.

Parameter lbl : Set.
Parameter lbl_lattice : bounded_join_semi (A:=lbl).
Definition lbl_lattice' := join_semi_extend lbl_lattice.
Definition lbl_lattice'' := bounded_join_semi_extend lbl_lattice.
Definition bottom := bot lbl_lattice.
Definition llub := lub lbl_lattice.
Definition lleq := leq lbl_lattice.

Ltac llub_simpl H := apply (lub_leq lbl_lattice lbl_lattice') in H; destruct H.

Inductive glbl := Lo | Hi.
Definition grp (L l : lbl) := if lleq l L then Lo else Hi.

Definition glbl_poset : poset (A:=glbl).
apply Build_poset with (leq := fun l1 l2 : glbl => if l1 then true else (if l2 then false else true)); intros.
destruct x; auto.
destruct x; destruct y; simpl in *; auto; inv H; inv H0.
destruct x; destruct y; destruct z; simpl in *; auto.
Defined.

Definition glbl_join_semi : join_semi (A:=glbl).
apply Build_join_semi with (po := glbl_poset) (lub := fun l1 l2 : glbl => if l1 then l2 else Hi); intros.
destruct x; destruct y; auto.
destruct x; destruct y; auto.
destruct x; destruct y; auto.
Defined.

Definition glbl_lattice : bounded_join_semi (A:=glbl).
apply Build_bounded_join_semi with (js := glbl_join_semi) (bot := Lo); auto.
Defined.

Definition glbl_lattice' := join_semi_extend glbl_lattice.
Definition glbl_lattice'' := bounded_join_semi_extend glbl_lattice.
Definition gleq := leq glbl_lattice.
Definition glub := lub glbl_lattice.

Delimit Scope glbl_scope with glbl.
Bind Scope glbl_scope with glbl.
Delimit Scope lbl_scope with lbl.
Bind Scope lbl_scope with lbl.
Notation "x <<= y" := (gleq x y = true) (at level 70) : glbl_scope.
Notation "x \_/ y" := (glub x y) (at level 50) : glbl_scope.
Notation "x <<= y" := (lleq x y = true) (at level 70) : lbl_scope.
Notation "x \_/ y" := (llub x y) (at level 50) : lbl_scope.

Open Scope lbl_scope.

Proposition glub_homo : forall l l1 l2, grp l (llub l1 l2) = glub (grp l l1) (grp l l2).
Proof.
intros; case_eq (lleq l1 l); intros.
case_eq (lleq l2 l); intros; unfold grp; rewrite H; rewrite H0; simpl.
assert (l1 \_/ l2 <<= l).
rewrite (lub_leq lbl_lattice lbl_lattice'); split; auto.
rewrite H1; auto.
assert (~ l1 \_/ l2 <<= l).
rewrite (lub_leq lbl_lattice lbl_lattice'); intro.
destruct H1.
unfold lleq in H0; rewrite H2 in H0; inv H0.
destruct (lleq (l1 \_/ l2)%lbl l); auto.
unfold grp; rewrite H; simpl.
assert (~ l1 \_/ l2 <<= l).
rewrite (lub_leq lbl_lattice lbl_lattice'); intro.
destruct H0.
unfold lleq in H; rewrite H0 in H; inv H.
destruct (lleq (l1 \_/ l2) l); auto.
Qed.

Close Scope lbl_scope.

Proposition glub_leq : forall l l1 l2, glub (grp l l1) (grp l l2) = Lo <-> grp l l1 = Lo /\ grp l l2 = Lo.
Proof.
intros; unfold grp; destruct (lleq l1 l); destruct (lleq l2 l); simpl; intuit.
Qed.

Proposition glub_lo : forall l1 l2, glub l1 l2 = Lo <-> l1 = Lo /\ l2 = Lo.
Proof.
destruct l1; destruct l2; intuit.
Qed.

Ltac glub_simpl H := apply glub_lo in H; destruct H.
Ltac glub_simpl_grp H := try (rewrite glub_homo in H); apply glub_leq in H; destruct H.

Inductive binop := Plus | Minus | Mult | Div | Mod.
Inductive bbinop := And | Or | Impl.

Inductive exp :=
| Var : var -> exp
| LVar : lvar1 -> exp
| Num : Z -> exp
| BinOp : binop -> exp -> exp -> exp.

Fixpoint expvars (e : exp) (x : var) : bool :=
match e with
| Var y => if eq_nat_dec y x then true else false
| BinOp _ e1 e2 => if expvars e1 x then true else expvars e2 x
| _ => false
end.

Fixpoint no_lvars_exp (e : exp) :=
match e with
| LVar _ => False
| BinOp _ e1 e2 => no_lvars_exp e1 /\ no_lvars_exp e2
| _ => True
end.

Proposition exp_eq_dec : forall e1 e2 : exp, {e1 = e2} + {e1 <> e2}.
Proof.
induction e1; destruct e2; try solve [right; discriminate].
destruct (eq_nat_dec v v0); subst.
left; auto.
right; intro H; inv H; contradiction n; auto.
destruct (eq_nat_dec l l0); subst.
left; auto.
right; intro H; inv H; contradiction n; auto.
destruct (Z_eq_dec z z0); subst.
left; auto.
right; intro H; inv H; contradiction n; auto.
assert ({b = b0} + {BinOp b e1_1 e1_2 <> BinOp b0 e2_1 e2_2}).
destruct b; destruct b0; auto; try solve [right; intro H; inv H].
destruct H; auto; subst.
destruct (IHe1_1 e2_1); subst.
destruct (IHe1_2 e2_2); subst; auto.
right; intro H; inv H; contradiction n; auto.
right; intro H; inv H; contradiction n; auto.
Qed.

Inductive bexp :=
| FF : bexp
| TT : bexp
| Eq : exp -> exp -> bexp
| Not : bexp -> bexp
| BBinOp : bbinop -> bexp -> bexp -> bexp.

Fixpoint bexpvars (b : bexp) (x : var) : bool :=
match b with
| Eq e1 e2 => if expvars e1 x then true else expvars e2 x
| Not b => bexpvars b x
| BBinOp _ b1 b2 => if bexpvars b1 x then true else bexpvars b2 x
| _ => false
end.

Fixpoint no_lvars_bexp (b : bexp) :=
match b with
| Eq e1 e2 => no_lvars_exp e1 /\ no_lvars_exp e2
| Not b => no_lvars_bexp b
| BBinOp _ b1 b2 => no_lvars_bexp b1 /\ no_lvars_bexp b2
| _ => True
end.

Inductive cmd :=
| Skip : cmd
| Output : exp -> cmd
| Assign : var -> exp -> cmd
| Read : var -> exp -> cmd
| Write : exp -> exp -> cmd
| Seq : cmd -> cmd -> cmd
| If : bexp -> cmd -> cmd -> cmd
| While : bexp -> cmd -> cmd.

Fixpoint mods (C : cmd) : list var :=
match C with
| Assign x _ => [x]
| Read x _ => [x]
| Seq C1 C2 => mods C1 ++ mods C2
| If _ C1 C2 => mods C1 ++ mods C2
| While _ C => mods C
| _ => []
end.

Fixpoint modifies (K : list cmd) : list var :=
match K with
| [] => []
| C::K => mods C ++ modifies K
end.

Fixpoint no_lvars_cmd (C : cmd) :=
match C with
| Skip => True
| Output e => no_lvars_exp e
| Assign _ e => no_lvars_exp e
| Read _ e => no_lvars_exp e
| Write e1 e2 => no_lvars_exp e1 /\ no_lvars_exp e2
| Seq C1 C2 => no_lvars_cmd C1 /\ no_lvars_cmd C2
| If b C1 C2 => no_lvars_bexp b /\ no_lvars_cmd C1 /\ no_lvars_cmd C2
| While b C => no_lvars_bexp b /\ no_lvars_cmd C
end.

Fixpoint no_lvars (K : list cmd) :=
match K with
| [] => True
| C::K => no_lvars_cmd C /\ no_lvars K
end.

Definition val := prod Z glbl.
Definition lmap := prod (lvar1 -> Z) (lvar2 -> glbl).
Definition store := var -> option val.
Definition heap := addr -> option val.
Inductive state := St : lmap -> store -> heap -> state.
Definition getLmap (st : state) := let (i,_,_) := st in i.
Coercion getLmap : state >-> lmap.
Definition getStore (st : state) := let (_,s,_) := st in s.
Coercion getStore : state >-> store.
Definition getHeap (st : state) := let (_,_,h) := st in h.
Coercion getHeap : state >-> heap.

Proposition val_eq_dec : forall v1 v2 : val, {v1 = v2} + {v1 <> v2}.
Proof.
destruct v1; destruct v2.
destruct g; destruct g0; try solve [right; intro H; inv H].
destruct (Z_eq_dec z z0); subst.
left; auto.
right; intro H; inv H; contradiction n; auto.
destruct (Z_eq_dec z z0); subst.
left; auto.
right; intro H; inv H; contradiction n; auto.
Qed.

Proposition opt_eq_dec {A} : (forall a1 a2 : A, {a1 = a2} + {a1 <> a2}) -> forall o1 o2 : option A, {o1 = o2} + {o1 <> o2}.
Proof.
intros.
destruct o1; destruct o2.
destruct (X a a0); subst; auto.
right; intro H; inv H; contradiction n; auto.
right; discriminate.
right; discriminate.
left; auto.
Qed.

Definition upd {A} (x : nat -> option A) y z : nat -> option A := fun w => if eq_nat_dec w y then Some z else x w.

Record SepAlg : Type := mkSepAlg {
sepstate : Set;
unit : sepstate -> Prop;
dot : sepstate -> sepstate -> sepstate -> Prop;
dot_func : forall x y z1 z2, dot x y z1 -> dot x y z2 -> z1 = z2;
dot_comm : forall x y z, dot x y z -> dot y x z;
dot_assoc : forall x y z a b, dot x y a -> dot a z b -> exists c, dot y z c /\ dot x c b;
dot_unit : forall x, exists u, unit u /\ dot u x x;
dot_unit_min : forall u x y, unit u -> dot u x y -> x = y}.

Definition mycombine {A} (s1 s2 : nat -> option A) (n : nat) : option A :=
match s1 n, s2 n with
| Some a, _ => Some a
| None, Some a => Some a
| None, None => None
end.

Definition mydot {A} (s1 s2 s : nat -> option A) : Prop := forall n,
match s n with
| None => s1 n = None /\ s2 n = None
| Some a => (s1 n = Some a /\ s2 n = None) \/ (s1 n = None /\ s2 n = Some a)
end.

Definition mysep : SepAlg.
apply (mkSepAlg state (fun st => match st with St _ _ h => h = (fun _ => None) end)
(fun st1 st2 st3 =>
match st1, st2, st3 with St i1 s1 h1, St i2 s2 h2, St i3 s3 h3 =>
i1 = i2 /\ i1 = i3 /\ s1 = s2 /\ s1 = s3 /\ mydot h1 h2 h3
end)); intros.
destruct x as [i1 s1 h1]; destruct y as [i2 s2 h2]; destruct z1 as [i3 s3 h3]; destruct z2 as [i4 s4 h4].
decomp H; decomp H0; repeat subst.
apply f_equal; apply functional_extensionality; intro n.
specialize (H6 n); specialize (H10 n).
destruct (h3 n); destruct (h4 n); auto.
decomp H6; decomp H10.
rewrite H1 in H3; auto.
rewrite H1 in H3; inv H3.
rewrite H1 in H3; inv H3.
rewrite H2 in H4; auto.
decomp H6; decomp H10.
rewrite H1 in H0; inv H0.
rewrite H2 in H3; inv H3.
decomp H6; decomp H10.
rewrite H0 in H3; inv H3.
rewrite H1 in H4; inv H4.
destruct x as [i1 s1 h1]; destruct y as [i2 s2 h2]; destruct z as [i3 s3 h3].
decomp H; repeat split; repeat subst; auto.
intro n; specialize (H5 n).
destruct (h3 n); intuit.
destruct x as [i1 s1 h1]; destruct y as [i2 s2 h2]; destruct z as [i3 s3 h3]; destruct a as [i4 s4 h4]; destruct b as [i5 s5 h5].
decomp H; decomp H0; repeat subst.
exists (St i5 s5 (mycombine h2 h3)).
repeat split; auto.
intro n; unfold mycombine; specialize (H6 n); specialize (H10 n).
destruct (h2 n); destruct (h3 n); auto.
destruct (h4 n); destruct (h5 n); intuit.
decomp H6.
inv H1.
decomp H10.
inv H3.
inv H2.
destruct H6.
inv H0.
intro n; unfold mycombine; specialize (H6 n); specialize (H10 n).
destruct (h4 n); destruct (h5 n).
decomp H6.
decomp H10.
inv H2; rewrite H0; left; split; auto.
rewrite H1; rewrite H3; auto.
inv H2.
right; split; auto.
rewrite H1.
decomp H10; auto.
inv H2.
destruct H10.
inv H.
decomp H10.
inv H0.
destruct H6; right; split; auto.
rewrite H2; rewrite H1; auto.
destruct H6; destruct H10.
rewrite H0; rewrite H2; auto.
destruct x as [i s h].
exists (St i s (fun _ => None)); repeat split.
intro n.
destruct (h n); auto.
destruct u as [i s h]; subst.
destruct x as [i1 s1 h1]; destruct y as [i2 s2 h2].
decomp H0; repeat subst.
apply f_equal; apply functional_extensionality; intro n; specialize (H5 n).
destruct (h1 n); destruct (h2 n); intuit.
decomp H5; auto.
inv H1.
Defined.

Proposition mydot_upd {A} : forall (x y z : nat -> option A) n v,
mydot x y z -> y n = None -> mydot (upd x n v) y (upd z n v).
Proof.
unfold mydot; unfold upd; intros.
destruct (eq_nat_dec n0 n); subst; intuit.
apply (H n0).
Qed.

Definition option_map2 {A B C} (op : A -> B -> C) x y : option C :=
match x, y with
| Some x, Some y => Some (op x y)
| _, _ => None
end.

Open Scope Z_scope.
Open Scope glbl_scope.

Definition opden (bop : binop) : Z -> Z -> Z :=
match bop with
| Plus => Zplus
| Minus => Zminus
| Mult => Zmult
| Div => Zdiv
| Mod => Zmod
end.

Fixpoint eden (e : exp) (i : lmap) (s : store) : option val :=
match e with
| Var x => s x
| LVar X => Some (fst i X, Lo)
| Num c => Some (c,Lo)
| BinOp bop e1 e2 => option_map2 (fun v1 v2 => (opden bop (fst v1) (fst v2), snd v1 \_/ snd v2)) (eden e1 i s) (eden e2 i s)
end.

Fixpoint edenZ (e : exp) (i : lmap) (s : store) : option Z :=
match e with
| Var x => option_map (fun v => fst v) (s x)
| LVar X => Some (fst i X)
| Num c => Some c
| BinOp bop e1 e2 => option_map2 (fun v1 v2 => opden bop v1 v2) (edenZ e1 i s) (edenZ e2 i s)
end.

Proposition edenZ_some : forall e i s v, edenZ e i s = Some v <-> exists l, eden e i s = Some (v,l).
Proof.
induction e; simpl; intros; split; intros.
destruct (s v) as [[v1 l1]|]; inv H.
exists l1; auto.
destruct H as [l]; rewrite H; auto.
inv H; exists Lo; auto.
destruct H as [l0]; inv H; auto.
inv H; exists Lo; auto.
destruct H as [l]; inv H; auto.
case_eq (edenZ e1 i s); intros.
case_eq (edenZ e2 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite IHe1 in H0; rewrite IHe2 in H1.
destruct H0 as [l1]; destruct H1 as [l2].
rewrite H; rewrite H0; exists (l1 \_/ l2); auto.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite H0 in H; inv H.
destruct H as [l].
case_eq (eden e1 i s); intros.
case_eq (eden e2 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
destruct v0 as [v0 l0]; destruct v1 as [v1 l1].
assert (exists l, eden e1 i s = Some (v0,l)).
exists l0; auto.
assert (exists l, eden e2 i s = Some (v1,l)).
exists l1; auto.
rewrite <- IHe1 in H; rewrite <- IHe2 in H2.
rewrite H; rewrite H2; auto.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite H0 in H; inv H.
Qed.

Proposition edenZ_none : forall e i s, edenZ e i s = None <-> eden e i s = None.
Proof.
induction e; simpl; intros; split; intros.
destruct (s v); inv H; auto.
rewrite H; auto.
inv H.
inv H.
inv H.
inv H.
case_eq (edenZ e1 i s); intros.
case_eq (edenZ e2 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite IHe2 in H1; rewrite H1.
destruct (eden e1 i s); auto.
rewrite IHe1 in H0; rewrite H0; auto.
case_eq (eden e1 i s); intros.
case_eq (eden e2 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite <- IHe2 in H1; rewrite H1.
destruct (edenZ e1 i s); auto.
rewrite <- IHe1 in H0; rewrite H0; auto.
Qed.

Definition bopden (bop : bbinop) : bool -> bool -> bool :=
match bop with
| And => andb
| Or => orb
| Impl => fun v1 v2 => if v1 then v2 else true
end.

Fixpoint bden (b : bexp) (i : lmap) (s : store) : option (bool * glbl) :=
match b with
| FF => Some (false,Lo)
| TT => Some (true,Lo)
| Eq e1 e2 => option_map2 (fun v1 v2 => (if Z_eq_dec (fst v1) (fst v2) then true else false, snd v1 \_/ snd v2)) (eden e1 i s) (eden e2 i s)
| Not b => option_map (fun v => (negb (fst v), snd v)) (bden b i s)
| BBinOp bop b1 b2 => option_map2 (fun v1 v2 => (bopden bop (fst v1) (fst v2), snd v1 \_/ snd v2)) (bden b1 i s) (bden b2 i s)
end.

Fixpoint bdenZ (b : bexp) (i : lmap) (s : store) : option bool :=
match b with
| FF => Some false
| TT => Some true
| Eq e1 e2 => option_map2 (fun v1 v2 => if Z_eq_dec v1 v2 then true else false) (edenZ e1 i s) (edenZ e2 i s)
| Not b => option_map (fun v => negb v) (bdenZ b i s)
| BBinOp bop b1 b2 => option_map2 (fun v1 v2 => bopden bop v1 v2) (bdenZ b1 i s) (bdenZ b2 i s)
end.

Proposition bdenZ_some : forall b i s v, bdenZ b i s = Some v <-> exists l, bden b i s = Some (v,l).
Proof.
induction b; simpl; intros; split; intros.
inv H; exists Lo; auto.
destruct H as [l]; inv H; auto.
inv H; exists Lo; auto.
destruct H as [l]; inv H; auto.
case_eq (edenZ e i s); intros.
case_eq (edenZ e0 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite edenZ_some in H0; rewrite edenZ_some in H1.
destruct H0 as [l]; destruct H1 as [l0]; rewrite H; rewrite H0.
exists (l \_/ l0); auto.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite H0 in H; inv H.
destruct H as [l].
case_eq (eden e i s); intros.
case_eq (eden e0 i s); intros.
destruct v0 as [v0 l0]; destruct v1 as [v1 l1].
rewrite H0 in H; rewrite H1 in H; inv H.
assert (exists l, eden e i s = Some (v0,l)).
exists l0; auto.
assert (exists l, eden e0 i s = Some (v1,l)).
exists l1; auto.
rewrite <- edenZ_some in H; rewrite <- edenZ_some in H2.
rewrite H; rewrite H2; auto.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite H0 in H; inv H.
case_eq (bdenZ b i s); intros.
rewrite H0 in H; inv H.
rewrite IHb in H0; destruct H0 as [l]; exists l.
rewrite H; auto.
rewrite H0 in H; inv H.
destruct H as [l].
case_eq (bden b i s); intros.
destruct p as [v1 l1].
assert (exists l, bden b i s = Some (v1,l)).
exists l1; auto.
rewrite H0 in H; inv H.
rewrite <- IHb in H1; rewrite H1; auto.
rewrite H0 in H; inv H.
case_eq (bdenZ b2 i s); intros.
case_eq (bdenZ b3 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite IHb1 in H0; rewrite IHb2 in H1.
destruct H0 as [l1]; destruct H1 as [l2].
rewrite H; rewrite H0; exists (l1 \_/ l2); auto.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite H0 in H; inv H.
destruct H as [l].
case_eq (bden b2 i s); intros.
case_eq (bden b3 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
destruct p as [v0 l0]; destruct p0 as [v1 l1].
assert (exists l, bden b2 i s = Some (v0,l)).
exists l0; auto.
assert (exists l, bden b3 i s = Some (v1,l)).
exists l1; auto.
rewrite <- IHb1 in H; rewrite <- IHb2 in H2.
rewrite H; rewrite H2; auto.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite H0 in H; inv H.
Qed.

Proposition bdenZ_none : forall b i s, bdenZ b i s = None <-> bden b i s = None.
Proof.
induction b; simpl; intros; split; intros.
inv H.
inv H.
inv H.
inv H.
case_eq (edenZ e i s); intros.
case_eq (edenZ e0 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite edenZ_none in H1; rewrite H1.
destruct (eden e i s); auto.
rewrite edenZ_none in H0; rewrite H0; auto.
case_eq (eden e i s); intros.
case_eq (eden e0 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite <- edenZ_none in H1; rewrite H1.
destruct (edenZ e i s); auto.
rewrite <- edenZ_none in H0; rewrite H0; auto.
case_eq (bdenZ b i s); intros.
rewrite H0 in H; inv H.
rewrite IHb in H0; rewrite H0; auto.
case_eq (bden b i s); intros.
rewrite H0 in H; inv H.
rewrite <- IHb in H0; rewrite H0; auto.
case_eq (bdenZ b2 i s); intros.
case_eq (bdenZ b3 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite IHb2 in H1; rewrite H1.
destruct (bden b2 i s); auto.
rewrite IHb1 in H0; rewrite H0; auto.
case_eq (bden b2 i s); intros.
case_eq (bden b3 i s); intros.
rewrite H0 in H; rewrite H1 in H; inv H.
rewrite <- IHb2 in H1; rewrite H1.
destruct (bdenZ b2 i s); auto.
rewrite <- IHb1 in H0; rewrite H0; auto.
Qed.

Proposition eden_local : forall e i1 s1 h1 i2 s2 h2 i3 s3 h3 v,
dot mysep (St i1 s1 h1) (St i2 s2 h2) (St i3 s3 h3) -> eden e i1 s1 = Some v -> eden e i3 s3 = Some v.
Proof.
intros.
simpl in H; decomp H; repeat subst; auto.
Qed.

Proposition bden_local : forall b i1 s1 h1 i2 s2 h2 i3 s3 h3 v,
dot mysep (St i1 s1 h1) (St i2 s2 h2) (St i3 s3 h3) -> bden b i1 s1 = Some v -> bden b i3 s3 = Some v.
Proof.
intros.
simpl in H; decomp H; repeat subst; auto.
Qed.

Proposition eden_no_lvars : forall e i i' s, no_lvars_exp e -> eden e i s = eden e i' s.
Proof.
induction e; simpl; intros; intuit.
rewrite (IHe1 _ i'); intuit; rewrite (IHe2 _ i'); intuit.
Qed.

Proposition bden_no_lvars : forall b i i' s, no_lvars_bexp b -> bden b i s = bden b i' s.
Proof.
induction b; simpl; intros; intuit.
rewrite (eden_no_lvars e _ i'); intuit; rewrite (eden_no_lvars e0 _ i'); intuit.
rewrite (IHb _ i'); intuit.
rewrite (IHb1 _ i'); intuit; rewrite (IHb2 _ i'); intuit.
Qed.

Definition context := glbl.
Inductive config := Cf : state -> cmd -> list cmd -> config.
Definition getStoreFromConfig (cf : config) := match cf with Cf (St _ s _) _ _ => s end.
Coercion getStoreFromConfig : config >-> store.

Definition taint_vars (K : list cmd) (s : store) : store :=
fun x => if In_dec eq_nat_dec x (modifies K) then
match s x with Some (v,_) => Some (v,Hi) | None => Some (0,Hi) end
else s x.

Definition taint_vars_cf (cf : config) : config :=
match cf with Cf (St i s h) C K => Cf (St i (taint_vars (C::K) s) h) C K end.

Inductive hstep : config -> config -> Prop :=
| HStep_skip : forall st C K, hstep (Cf st Skip (C::K)) (Cf st C K)
| HStep_assign : forall i s h K x e v l,
eden e i s = Some (v,l) ->
hstep (Cf (St i s h) (Assign x e) K) (Cf (St i (upd s x (v, Hi)) h) Skip K)
| HStep_read : forall i s h K x e v1 l1 v2 l2 (pf : v1 >= 0),
eden e i s = Some (v1,l1) -> h (nat_of_Z v1 pf) = Some (v2,l2) ->
hstep (Cf (St i s h) (Read x e) K) (Cf (St i (upd s x (v2, Hi)) h) Skip K)
| HStep_write : forall i s h K e1 e2 v1 l1 v2 l2 (pf : v1 >= 0),
eden e1 i s = Some (v1,l1) -> eden e2 i s = Some (v2,l2) -> h (nat_of_Z v1 pf) <> None ->
hstep (Cf (St i s h) (Write e1 e2) K) (Cf (St i s (upd h (nat_of_Z v1 pf) (v2, Hi))) Skip K)
| HStep_seq : forall st C1 C2 K, hstep (Cf st (Seq C1 C2) K) (Cf st C1 (C2::K))
| HStep_if_true : forall i s h C1 C2 K b l,
bden b i s = Some (true,l) -> hstep (Cf (St i s h) (If b C1 C2) K) (Cf (St i s h) C1 K)
| HStep_if_false : forall i s h C1 C2 K b l,
bden b i s = Some (false,l) -> hstep (Cf (St i s h) (If b C1 C2) K) (Cf (St i s h) C2 K)
| HStep_while_true : forall i s h C K b l,
bden b i s = Some (true,l) -> hstep (Cf (St i s h) (While b C) K) (Cf (St i s h) C (While b C :: K))
| HStep_while_false : forall i s h C K b l,
bden b i s = Some (false,l) -> hstep (Cf (St i s h) (While b C) K) (Cf (St i s h) Skip K).

Inductive hstepn : nat -> config -> config -> Prop :=
| HStep_zero : forall cf, hstepn 0 cf cf
| HStep_succ : forall n cf cf' cf'', hstep cf cf' -> hstepn n cf' cf'' -> hstepn (S n) cf cf''.

Definition halt_config cf := match cf with Cf _ Skip [] => true | _ => false end.
Inductive can_hstep : config -> Prop := Can_hstep : forall cf cf', hstep cf cf' -> can_hstep cf.
Definition hsafe cf := forall n cf', hstepn n cf cf' -> halt_config cf' = false -> can_hstep cf'.

Inductive lstep : config -> config -> list Z -> Prop :=
| LStep_skip : forall st C K, lstep (Cf st Skip (C::K)) (Cf st C K) []
| LStep_output : forall i s h K e v,
eden e i s = Some (v,Lo) ->
lstep (Cf (St i s h) (Output e) K) (Cf (St i s h) Skip K) [v]
| LStep_assign : forall i s h K x e v l,
eden e i s = Some (v,l) ->
lstep (Cf (St i s h) (Assign x e) K) (Cf (St i (upd s x (v, l)) h) Skip K) []
| LStep_read : forall i s h K x e v1 l1 v2 l2 (pf : v1 >= 0),
eden e i s = Some (v1,l1) -> h (nat_of_Z v1 pf) = Some (v2,l2) ->
lstep (Cf (St i s h) (Read x e) K) (Cf (St i (upd s x (v2, l1 \_/ l2)) h) Skip K) []
| LStep_write : forall i s h K e1 e2 v1 l1 v2 l2 (pf : v1 >= 0),
eden e1 i s = Some (v1,l1) -> eden e2 i s = Some (v2,l2) -> h (nat_of_Z v1 pf) <> None ->
lstep (Cf (St i s h) (Write e1 e2) K) (Cf (St i s (upd h (nat_of_Z v1 pf) (v2, l1 \_/ l2))) Skip K) []
| LStep_seq : forall st C1 C2 K, lstep (Cf st (Seq C1 C2) K) (Cf st C1 (C2::K)) []
| LStep_if_true : forall i s h C1 C2 K b,
bden b i s = Some (true,Lo) -> lstep (Cf (St i s h) (If b C1 C2) K) (Cf (St i s h) C1 K) []
| LStep_if_false : forall i s h C1 C2 K b,
bden b i s = Some (false,Lo) -> lstep (Cf (St i s h) (If b C1 C2) K) (Cf (St i s h) C2 K) []
| LStep_while_true : forall i s h C K b,
bden b i s = Some (true,Lo) -> lstep (Cf (St i s h) (While b C) K) (Cf (St i s h) C (While b C :: K)) []
| LStep_while_false : forall i s h C K b,
bden b i s = Some (false,Lo) -> lstep (Cf (St i s h) (While b C) K) (Cf (St i s h) Skip K) []
| LStep_if_hi : forall i s h st' C1 C2 K b v n,
bden b i s = Some (v,Hi) -> hsafe (taint_vars_cf (Cf (St i s h) (If b C1 C2) [])) ->
hstepn n (taint_vars_cf (Cf (St i s h) (If b C1 C2) [])) (Cf st' Skip []) ->
lstep (Cf (St i s h) (If b C1 C2) K) (Cf st' Skip K) []
| LStep_if_hi_dvg : forall i s h C1 C2 K b v,
bden b i s = Some (v,Hi) -> hsafe (taint_vars_cf (Cf (St i s h) (If b C1 C2) [])) ->
(forall n st', ~ hstepn n (taint_vars_cf (Cf (St i s h) (If b C1 C2) [])) (Cf st' Skip [])) ->
lstep (Cf (St i s h) (If b C1 C2) K) (Cf (St i s h) (If b C1 C2) K) []
| LStep_while_hi : forall i s h st' C K b v n,
bden b i s = Some (v,Hi) -> hsafe (taint_vars_cf (Cf (St i s h) (While b C) [])) ->
hstepn n (taint_vars_cf (Cf (St i s h) (While b C) [])) (Cf st' Skip []) ->
lstep (Cf (St i s h) (While b C) K) (Cf st' Skip K) []
| LStep_while_hi_dvg : forall i s h C K b v,
bden b i s = Some (v,Hi) -> hsafe (taint_vars_cf (Cf (St i s h) (While b C) [])) ->
(forall n st', ~ hstepn n (taint_vars_cf (Cf (St i s h) (While b C) [])) (Cf st' Skip [])) ->
lstep (Cf (St i s h) (While b C) K) (Cf (St i s h) (While b C) K) [].

Inductive lstepn : nat -> config -> config -> list Z -> Prop :=
| LStep_zero : forall cf, lstepn 0 cf cf []
| LStep_succ : forall n cf cf' cf'' o o', lstep cf cf' o -> lstepn n cf' cf'' o' -> lstepn (S n) cf cf'' (o++o').

Inductive can_lstep : config -> Prop := Can_lstep : forall cf cf' o, lstep cf cf' o -> can_lstep cf.
Definition lsafe cf := forall n cf' o, lstepn n cf cf' o -> halt_config cf' = false -> can_lstep cf'.

Definition side_condition C (st1 st2 : state) :=
match C, st1, st2 with
| Read _ e, St i1 s1 h1, St i2 s2 h2 =>
match (eden e i1 s1), (eden e i2 s2) with
| Some (v1,_), Some (v2,_) =>
match Zneg_dec v1, Zneg_dec v2 with
| left pf1, left pf2 =>
match h1 (nat_of_Z v1 pf1), h2 (nat_of_Z v2 pf2) with
| Some (_,l1), Some (_,l2) => l1 = l2
| _, _ => False
end
| _, _ => False
end
| _, _ => False
end
| _, _, _ => True
end.

Close Scope Z_scope.

Proposition dvg_ex_mid : forall cf,
(forall n st, ~ hstepn n cf (Cf st Skip [])) \/ exists n, exists st, hstepn n cf (Cf st Skip []).
Proof.
intros.
dup (classic (exists n, exists st, hstepn n cf (Cf st Skip []))).
destruct H; [right | left]; auto.
exists n; exists st; auto.
Qed.

Lemma hstep_trans : forall n1 n2 cf1 cf2 cf3, hstepn n1 cf1 cf2 -> hstepn n2 cf2 cf3 -> hstepn (n1+n2) cf1 cf3.
Proof.
induction n1 using (well_founded_induction lt_wf); intros.
inv H0; simpl; auto.
apply HStep_succ with (cf' := cf'); auto.
apply H with (cf2 := cf2); auto.
Qed.

Lemma lstep_trans : forall n1 n2 cf1 cf2 cf3 o1 o2, lstepn n1 cf1 cf2 o1 -> lstepn n2 cf2 cf3 o2 -> lstepn (n1+n2) cf1 cf3 (o1++o2).
Proof.
induction n1 using (well_founded_induction lt_wf); intros.
inv H0; simpl; auto.
rewrite app_assoc; apply LStep_succ with (cf' := cf'); auto.
apply H with (cf2 := cf2); auto.
Qed.

Lemma hstep_extend : forall st C K st' C' K' K0,
hstep (Cf st C K) (Cf st' C' K') -> hstep (Cf st C (K++K0)) (Cf st' C' (K'++K0)).
Proof.
intros.
inv H.
apply HStep_skip.
apply HStep_assign with (l := l); auto.
apply HStep_read with (v1 := v1) (pf := pf) (l1 := l1) (l2 := l2); auto.
apply HStep_write with (l1 := l1) (l2 := l2); auto.
apply HStep_seq.
apply HStep_if_true with (l := l); auto.
apply HStep_if_false with (l := l); auto.
apply HStep_while_true with (l := l); auto.
apply HStep_while_false with (l := l); auto.
Qed.

Lemma hstepn_extend : forall n st C K st' C' K' K0,
hstepn n (Cf st C K) (Cf st' C' K') -> hstepn n (Cf st C (K++K0)) (Cf st' C' (K'++K0)).
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0.
apply HStep_zero.
destruct cf' as [st'' C'' K''].
apply HStep_succ with (cf' := Cf st'' C'' (K''++K0)).
apply hstep_extend; auto.
apply H; auto.
Qed.

Lemma lstep_extend : forall st C K st' C' K' K0 o,
lstep (Cf st C K) (Cf st' C' K') o -> lstep (Cf st C (K++K0)) (Cf st' C' (K'++K0)) o.
Proof.
intros.
inv H.
apply LStep_skip.
apply LStep_output; auto.
apply LStep_assign with (l := l); auto.
apply LStep_read with (v1 := v1) (pf := pf) (l1 := l1) (l2 := l2); auto.
apply LStep_write with (l1 := l1) (l2 := l2); auto.
apply LStep_seq.
apply LStep_if_true; auto.
apply LStep_if_false; auto.
apply LStep_while_true; auto.
apply LStep_while_false; auto.
apply LStep_if_hi with (b := b) (v := v) (n := n); auto.
apply LStep_if_hi_dvg with (b := b) (v := v); auto.
apply LStep_while_hi with (b := b) (v := v) (n := n); auto.
apply LStep_while_hi_dvg with (b := b) (v := v); auto.
Qed.

Lemma lstepn_extend : forall n st C K st' C' K' K0 o,
lstepn n (Cf st C K) (Cf st' C' K') o -> lstepn n (Cf st C (K++K0)) (Cf st' C' (K'++K0)) o.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0.
apply LStep_zero.
destruct cf' as [st'' C'' K''].
apply LStep_succ with (cf' := Cf st'' C'' (K''++K0)).
apply lstep_extend; auto.
apply H; auto.
Qed.

Lemma hstep_trans_inv : forall n st st' C C' K0 K K',
hstepn n (Cf st C (K0++K)) (Cf st' C' K') ->
(exists K'', hstepn n (Cf st C K0) (Cf st' C' K'') /\ K' = K''++K) \/
exists st'', exists n1, exists n2,
hstepn n1 (Cf st C K0) (Cf st'' Skip []) /\ hstepn n2 (Cf st'' Skip K) (Cf st' C' K') /\
n = n1 + n2.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0.
left; exists K0.
split; auto; apply HStep_zero.
inv H1.

destruct K0.
simpl in H5; subst.
right; exists st; exists 0; exists (S n0); repeat (split; auto).
apply HStep_zero.
apply HStep_succ with (cf' := Cf st C0 K1); auto.
apply HStep_skip.
inv H5.
apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf st c K0); auto.
apply HStep_skip.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf st c K0); auto.
apply HStep_skip.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i (upd s x (v,Hi)) h) Skip K0); auto.
apply HStep_assign with (l := l); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i (upd s x (v,Hi)) h) Skip K0); auto.
apply HStep_assign with (l := l); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i (upd s x (v2,Hi)) h) Skip K0); auto.
apply HStep_read with (v1 := v1) (l1 := l1) (l2 := l2) (pf := pf); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i (upd s x (v2, Hi)) h) Skip K0); auto.
apply HStep_read with (v1 := v1) (l1 := l1) (l2 := l2) (pf := pf); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i s (upd h (nat_of_Z v1 pf) (v2,Hi))) Skip K0); auto.
apply HStep_write with (l1 := l1) (l2 := l2); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i s (upd h (nat_of_Z v1 pf) (v2,Hi))) Skip K0); auto.
apply HStep_write with (l1 := l1) (l2 := l2); auto.

change (hstepn n0 (Cf st C1 ((C2 :: K0) ++ K)) (Cf st' C' K')) in H2.
apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf st C1 (C2::K0)); auto.
apply HStep_seq.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf st C1 (C2::K0)); auto.
apply HStep_seq.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i s h) C1 K0); auto.
apply HStep_if_true with (l := l); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i s h) C1 K0); auto.
apply HStep_if_true with (l := l); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i s h) C2 K0); auto.
apply HStep_if_false with (l := l); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i s h) C2 K0); auto.
apply HStep_if_false with (l := l); auto.

change (hstepn n0 (Cf (St i s h) C0 ((While b C0 :: K0) ++ K)) (Cf st' C' K')) in H2.
apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i s h) C0 (While b C0 :: K0)); auto.
apply HStep_while_true with (l := l); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i s h) C0 (While b C0 :: K0)); auto.
apply HStep_while_true with (l := l); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply HStep_succ with (cf' := Cf (St i s h) Skip K0); auto.
apply HStep_while_false with (l := l); auto.
destruct H0 as [st'' [n1 [n2 [H0 [H1]]]]]; subst.
right; exists st''; exists (S n1); exists n2; repeat (split; auto).
apply HStep_succ with (cf' := Cf (St i s h) Skip K0); auto.
apply HStep_while_false with (l := l); auto.
Qed.

Lemma lstep_trans_inv : forall n st st' C C' K0 K K' o,
lstepn n (Cf st C (K0++K)) (Cf st' C' K') o ->
(exists K'', lstepn n (Cf st C K0) (Cf st' C' K'') o /\ K' = K''++K) \/
exists st'', exists n1, exists n2, exists o1, exists o2,
lstepn n1 (Cf st C K0) (Cf st'' Skip []) o1 /\ lstepn n2 (Cf st'' Skip K) (Cf st' C' K') o2 /\
n = n1 + n2 /\ o = o1 ++ o2.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0.
left; exists K0.
split; auto; apply LStep_zero.
inv H1.

destruct K0.
simpl in H5; subst.
right; exists st; exists 0; exists (S n0); exists []; exists ([]++o'); repeat (split; auto).
apply LStep_zero.
apply LStep_succ with (cf' := Cf st C0 K1); auto.
apply LStep_skip.
inv H5.
apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf st c K0); auto.
apply LStep_skip.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf st c K0); auto.
apply LStep_skip.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) Skip K0); auto.
apply LStep_output; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([v]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) Skip K0); auto.
apply LStep_output; auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i (upd s x (v,l)) h) Skip K0); auto.
apply LStep_assign; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i (upd s x (v, l)) h) Skip K0); auto.
apply LStep_assign; auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i (upd s x (v2, l1 \_/ l2)) h) Skip K0); auto.
apply LStep_read with (v1 := v1) (pf := pf); auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i (upd s x (v2, l1 \_/ l2)) h) Skip K0); auto.
apply LStep_read with (v1 := v1) (pf := pf); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s (upd h (nat_of_Z v1 pf) (v2, l1 \_/ l2))) Skip K0); auto.
apply LStep_write; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s (upd h (nat_of_Z v1 pf) (v2, l1 \_/ l2))) Skip K0); auto.
apply LStep_write; auto.

change (lstepn n0 (Cf st C1 ((C2 :: K0) ++ K)) (Cf st' C' K') o') in H2.
apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf st C1 (C2::K0)); auto.
apply LStep_seq.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf st C1 (C2::K0)); auto.
apply LStep_seq.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) C1 K0); auto.
apply LStep_if_true; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) C1 K0); auto.
apply LStep_if_true; auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) C2 K0); auto.
apply LStep_if_false; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) C2 K0); auto.
apply LStep_if_false; auto.

change (lstepn n0 (Cf (St i s h) C0 ((While b C0 :: K0) ++ K)) (Cf st' C' K') o') in H2.
apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) C0 (While b C0 :: K0)); auto.
apply LStep_while_true; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) C0 (While b C0 :: K0)); auto.
apply LStep_while_true; auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) Skip K0); auto.
apply LStep_while_false; auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) Skip K0); auto.
apply LStep_while_false; auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf st'0 Skip K0); auto.
apply LStep_if_hi with (b := b) (v := v) (n := n); auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf st'0 Skip K0); auto.
apply LStep_if_hi with (b := b) (v := v) (n := n); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) (If b C1 C2) K0); auto.
apply LStep_if_hi_dvg with (b := b) (v := v); auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) (If b C1 C2) K0); auto.
apply LStep_if_hi_dvg with (b := b) (v := v); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf st'0 Skip K0); auto.
apply LStep_while_hi with (b := b) (v := v) (n := n); auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf st'0 Skip K0); auto.
apply LStep_while_hi with (b := b) (v := v) (n := n); auto.

apply H in H2; auto.
destruct H2.
destruct H0 as [K'' [H0]]; subst.
left; exists K''; split; auto.
apply LStep_succ with (cf' := Cf (St i s h) (While b C0) K0); auto.
apply LStep_while_hi_dvg with (b := b) (v := v); auto.
destruct H0 as [st'' [n1 [n2 [o1 [o2 [H0 [H1 [H2]]]]]]]]; subst.
right; exists st''; exists (S n1); exists n2; exists ([]++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := Cf (St i s h) (While b C0) K0); auto.
apply LStep_while_hi_dvg with (b := b) (v := v); auto.
Qed.

Lemma hstep_trans_inv' : forall a b cf cf',
hstepn (a+b) cf cf' -> exists cf'', hstepn a cf cf'' /\ hstepn b cf'' cf'.
Proof.
induction a using (well_founded_induction lt_wf); intros.
inv H0.
assert (a = 0); try omega.
assert (b = 0); try omega; subst.
exists cf'; split; apply HStep_zero.
destruct a; simpl in H1; subst.
exists cf; split.
apply HStep_zero.
apply HStep_succ with (cf' := cf'0); auto.
inv H1.
apply H in H3; auto.
destruct H3 as [cf'' [H3]]; exists cf''; split; auto.
apply HStep_succ with (cf' := cf'0); auto.
Qed.

Lemma lstep_trans_inv' : forall a b cf cf' o,
lstepn (a+b) cf cf' o -> exists cf'', exists o1, exists o2,
lstepn a cf cf'' o1 /\ lstepn b cf'' cf' o2 /\ o = o1 ++ o2.
Proof.
induction a using (well_founded_induction lt_wf); intros.
inv H0.
assert (a = 0); try omega.
assert (b = 0); try omega; subst.
exists cf'; exists []; exists []; repeat (split; auto); apply LStep_zero.
destruct a; simpl in H1; subst.
exists cf; exists []; exists (o0++o'); repeat (split; auto).
apply LStep_zero.
apply LStep_succ with (cf' := cf'0); auto.
inv H1.
apply H in H3; auto.
destruct H3 as [cf'' [o1 [o2 [H3 [H4]]]]]; exists cf''; exists (o0++o1); exists o2; repeat (split; auto).
apply LStep_succ with (cf' := cf'0); auto.
subst; rewrite app_assoc; auto.
Qed.

Lemma hstep_det : forall cf cf1 cf2, hstep cf cf1 -> hstep cf cf2 -> cf1 = cf2.
Proof.
intros.
inv H; inv H0; auto.
rewrite H8 in H1; inv H1; auto.
rewrite H9 in H1; inv H1.
rewrite (proof_irrelevance _ pf0 pf) in H10; rewrite H10 in H2; inv H2; auto.
rewrite H10 in H1; inv H1; rewrite H11 in H2; inv H2.
rewrite (proof_irrelevance _ pf0 pf); auto.
rewrite H9 in H1; inv H1.
rewrite H9 in H1; inv H1.
rewrite H8 in H1; inv H1.
rewrite H8 in H1; inv H1.
Qed.

Lemma hstepn_det : forall n cf cf1 cf2, hstepn n cf cf1 -> hstepn n cf cf2 -> cf1 = cf2.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; inv H1; auto.
dup (hstep_det _ _ _ H2 H4); subst.
apply H with (y := n0) (cf := cf'0); auto.
Qed.

Lemma hstepn_det_term : forall n1 n2 cf st1 st2, hstepn n1 cf (Cf st1 Skip []) -> hstepn n2 cf (Cf st2 Skip []) -> n1 = n2.
Proof.
intros.
assert (n1 = n2 \/ n1 < n2 \/ n2 < n1); try omega.
decomp H1; auto.
assert (n1 + (n2-n1) = n2); try omega.
rewrite <- H1 in H0; apply hstep_trans_inv' in H0.
destruct H0 as [cf' [H0]].
dup (hstepn_det _ _ _ _ H H0); subst cf'.
inv H2; try omega.
inv H5.
assert (n2 + (n1-n2) = n1); try omega.
rewrite <- H1 in H; apply hstep_trans_inv' in H.
destruct H as [cf' [H]].
dup (hstepn_det _ _ _ _ H H0); subst cf'.
inv H2; try omega.
inv H5.
Qed.

Lemma lstep_det : forall cf cf1 cf2 o1 o2, lstep cf cf1 o1 -> lstep cf cf2 o2 -> cf1 = cf2 /\ o1 = o2.
Proof.
intros.
inv H.
inv H0; auto.
inv H0.
rewrite H8 in H1; inv H1; auto.
inv H0.
rewrite H9 in H1; inv H1; auto.
inv H0.
rewrite H10 in H1; inv H1.
rewrite (proof_irrelevance _ pf0 pf) in H11; rewrite H11 in H2; inv H2; auto.
inv H0.
rewrite H11 in H1; inv H1; rewrite H12 in H2; inv H2.
rewrite (proof_irrelevance _ pf0 pf); auto.
inv H0; auto.
inv H0; auto.
rewrite H10 in H1; inv H1.
rewrite H10 in H1; inv H1.
rewrite H10 in H1; inv H1.
inv H0; auto.
rewrite H10 in H1; inv H1.
rewrite H10 in H1; inv H1.
rewrite H10 in H1; inv H1.
inv H0; auto.
rewrite H9 in H1; inv H1.
rewrite H9 in H1; inv H1.
rewrite H9 in H1; inv H1.
inv H0; auto.
rewrite H9 in H1; inv H1.
rewrite H9 in H1; inv H1.
rewrite H9 in H1; inv H1.
inv H0; auto.
rewrite H12 in H1; inv H1.
rewrite H12 in H1; inv H1.
dup (hstepn_det_term _ _ _ _ _ H3 H14); subst.
dup (hstepn_det _ _ _ _ H3 H14).
inv H; auto.
inv H0; auto.
rewrite H12 in H1; inv H1.
rewrite H12 in H1; inv H1.
inv H0; auto.
rewrite H11 in H1; inv H1.
rewrite H11 in H1; inv H1.
dup (hstepn_det_term _ _ _ _ _ H3 H13); subst.
dup (hstepn_det _ _ _ _ H3 H13).
inv H; auto.
inv H0; auto.
rewrite H11 in H1; inv H1.
rewrite H11 in H1; inv H1.
Qed.

Lemma lstepn_det : forall n cf cf1 cf2 o1 o2, lstepn n cf cf1 o1 -> lstepn n cf cf2 o2 -> cf1 = cf2 /\ o1 = o2.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; inv H1; auto.
destruct (lstep_det _ _ _ _ _ H2 H4); subst.
assert (n0 < S n0); try omega.
destruct (H _ H0 _ _ _ _ _ H3 H5); subst; auto.
Qed.

Lemma lstepn_det_term : forall n1 n2 cf st1 st2 o1 o2, lstepn n1 cf (Cf st1 Skip []) o1 -> lstepn n2 cf (Cf st2 Skip []) o2 -> n1 = n2.
Proof.
intros.
assert (n1 = n2 \/ n1 < n2 \/ n2 < n1); try omega.
decomp H1; auto.
assert (n1 + (n2-n1) = n2); try omega.
rewrite <- H1 in H0; clear H1; apply lstep_trans_inv' in H0.
destruct H0 as [cf' [o3 [o4 [H0 [H2]]]]]; subst.
destruct (lstepn_det _ _ _ _ _ _ H H0); subst.
inv H2; try omega.
inv H4.
assert (n2 + (n1-n2) = n1); try omega.
rewrite <- H1 in H; clear H1; apply lstep_trans_inv' in H.
destruct H as [cf' [o3 [o4 [H [H1]]]]]; subst.
destruct (lstepn_det _ _ _ _ _ _ H H0); subst.
inv H1; try omega.
inv H4.
Qed.

Definition diverge cf := forall n, exists cf', exists o, lstepn n cf cf' o.

Corollary diverge_halt : forall n cf st o, diverge cf -> lstepn n cf (Cf st Skip []) o -> False.
Proof.
intros.
destruct (H (n+1)) as [cf' [o']].
apply lstep_trans_inv' in H1.
destruct H1 as [cf'' [o1 [o2]]]; decomp H1; subst.
destruct (lstepn_det _ _ _ _ _ _ H0 H2); subst; inv H4.
inv H3.
Qed.

Proposition diverge_same_cf : forall cf o, lstep cf cf o -> diverge cf.
Proof.
intros.
assert (forall n, exists o, lstepn n cf cf o).
induction n; intros.
exists []; apply LStep_zero.
destruct IHn as [o']; exists (o++o'); apply LStep_succ with (cf' := cf); auto.
intro n; destruct (H0 n) as [o'].
exists cf; exists o'; auto.
Qed.

Lemma diverge_seq1 : forall C1 C2 K st, diverge (Cf st C1 []) -> diverge (Cf st (Seq C1 C2) K).
Proof.
intros; intro n.
destruct n.
exists (Cf st (Seq C1 C2) K); exists []; apply LStep_zero.
destruct (H n) as [[st' C' K'] [o]].
exists (Cf st' C' (K'++[C2]++K)); exists ([]++o).
apply LStep_succ with (cf' := Cf st C1 ([]++[C2]++K)).
apply LStep_seq.
apply lstepn_extend; auto.
Qed.

Lemma diverge_seq2 : forall C1 C2 K st st' n o,
lstepn n (Cf st C1 []) (Cf st' Skip []) o -> diverge (Cf st' C2 K) -> diverge (Cf st (Seq C1 C2) K).
Proof.
intros; intro n'.
assert (n' <= S n \/ n' > S n); try omega.
destruct H1.
destruct n'.
exists (Cf st (Seq C1 C2) K); exists []; apply LStep_zero.
assert (n = n'+(n-n')); try omega.
rewrite H2 in H; apply lstep_trans_inv' in H.
destruct H as [[st'' C'' K''] [o1'' [o2'']]]; decomp H.
exists (Cf st'' C'' (K''++[C2]++K)); exists ([]++o1'').
apply LStep_succ with (cf' := Cf st C1 ([]++[C2]++K)).
apply LStep_seq.
apply lstepn_extend; auto.
destruct (H0 (n' - S (S n))) as [cf [o']].
exists cf; exists ([]++o++[]++o').
assert (n' = S (n + S (n' - S (S n)))); try omega.
rewrite H3; apply LStep_succ with (cf' := Cf st C1 ([]++[C2]++K)).
apply LStep_seq.
apply lstep_trans with (cf2 := Cf st' Skip ([]++[C2]++K)).
apply lstepn_extend; auto.
apply LStep_succ with (cf' := Cf st' C2 K); auto.
apply LStep_skip.
Qed.

Lemma hstep_ff : forall C K C' K' i s h1 h2 h3 i' s' h1',
mydot h1 h2 h3 -> hstep (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K') ->
exists h3', mydot h1' h2 h3' /\ hstep (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K').
Proof.
intros.
inv H0.

exists h3; split; auto; apply HStep_skip.

exists h3; split; auto; apply HStep_assign with (l := l); auto.

exists h3; split; auto; apply HStep_read with (l1 := l1) (l2 := l2) (pf := pf); auto.
specialize (H (nat_of_Z v1 pf)); destruct (h3 (nat_of_Z v1 pf)); decomp H.
rewrite H1 in H12; inv H12; auto.
rewrite H1 in H12; inv H12.
rewrite H0 in H12; inv H12.

exists (upd h3 (nat_of_Z v1 pf) (v2,Hi)); split.
apply mydot_upd; auto.
specialize (H (nat_of_Z v1 pf)); destruct (h3 (nat_of_Z v1 pf)); decomp H; auto; try contradiction.
apply HStep_write with (l1 := l1) (l2 := l2); auto.
contradict H13; specialize (H (nat_of_Z v1 pf)).
rewrite H13 in H; intuit.

exists h3; split; auto; apply HStep_seq.

exists h3; split; auto; apply HStep_if_true with (l := l); auto.

exists h3; split; auto; apply HStep_if_false with (l := l); auto.

exists h3; split; auto; apply HStep_while_true with (l := l); auto.

exists h3; split; auto; apply HStep_while_false with (l := l); auto.
Qed.

Lemma hstepn_ff : forall n C K C' K' i s h1 h2 h3 i' s' h1',
mydot h1 h2 h3 -> hstepn n (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K') ->
exists h3', mydot h1' h2 h3' /\ hstepn n (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K').
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H1.
exists h3; split; auto; apply HStep_zero.
destruct cf' as [[i'' s'' h''] C'' K'']; apply hstep_ff with (h2 := h2) (h3 := h3) in H2; auto.
destruct H2 as [h3' [H2]].
assert (n0 < S n0); try omega.
destruct (H _ H4 _ _ _ _ _ _ _ _ _ _ _ _ H2 H3) as [h3'' [H5]]; exists h3''; split; auto.
apply HStep_succ with (cf' := Cf (St i'' s'' h3') C'' K''); auto.
Qed.

Lemma hstep_bf : forall C K C' K' i s h1 h2 h3 i' s' h3',
mydot h1 h2 h3 -> hstep (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K') -> hsafe (Cf (St i s h1) C K) ->
exists h1', mydot h1' h2 h3' /\ hstep (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K').
Proof.
intros.
inv H0.

exists h1; split; auto; apply HStep_skip.

exists h1; split; auto; apply HStep_assign with (l := l); auto.

exists h1; split; auto; apply HStep_read with (l1 := l1) (l2 := l2) (pf := pf); auto.
specialize (H (nat_of_Z v1 pf)); rewrite H13 in H; decomp H; auto.
specialize (H1 0 (Cf (St i' s h1) (Read x e) K') (HStep_zero _) (refl_equal _)).
inv H1.
inv H.
rewrite H10 in H4; inv H4.
rewrite (proof_irrelevance _ pf0 pf) in H11; rewrite H11 in H2; inv H2.

specialize (H1 0 (Cf (St i' s' h1) (Write e1 e2) K') (HStep_zero _) (refl_equal _)).
inv H1.
inv H0.
rewrite H9 in H5; inv H5.
rewrite (proof_irrelevance _ pf0 pf) in H11.
exists (upd h1 (nat_of_Z v1 pf) (v2,Hi)); split.
apply mydot_upd; auto.
specialize (H (nat_of_Z v1 pf)); destruct (h3 (nat_of_Z v1 pf)); decomp H; auto; try contradiction.
apply HStep_write with (l1 := l1) (l2 := l2); auto.

exists h1; split; auto; apply HStep_seq.

exists h1; split; auto; apply HStep_if_true with (l := l); auto.

exists h1; split; auto; apply HStep_if_false with (l := l); auto.

exists h1; split; auto; apply HStep_while_true with (l := l); auto.

exists h1; split; auto; apply HStep_while_false with (l := l); auto.
Qed.

Lemma hstepn_bf : forall n C K C' K' i s h1 h2 h3 i' s' h3',
mydot h1 h2 h3 -> hstepn n (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K') -> hsafe (Cf (St i s h1) C K) ->
exists h1', mydot h1' h2 h3' /\ hstepn n (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K').
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H1.
exists h1; split; auto; apply HStep_zero.
destruct cf' as [[i'' s'' h''] C'' K'']; apply hstep_bf with (h1 := h1) (h2 := h2) in H3; auto.
destruct H3 as [h1' [H3]].
assert (n0 < S n0); try omega.
assert (hsafe (Cf (St i'' s'' h1') C'' K'')).
unfold hsafe; intros.
apply (H2 (S n)); auto.
apply HStep_succ with (cf' := Cf (St i'' s'' h1') C'' K''); auto.
destruct (H _ H5 _ _ _ _ _ _ _ _ _ _ _ _ H3 H4 H6) as [h1'' [H7]]; exists h1''; split; auto.
apply HStep_succ with (cf' := Cf (St i'' s'' h1') C'' K''); auto.
Qed.

Lemma lstep_ff : forall C K C' K' i s h1 h2 h3 i' s' h1' o,
mydot h1 h2 h3 -> lstep (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K') o ->
exists h3', mydot h1' h2 h3' /\ lstep (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K') o.
Proof.
intros.
inv H0.

exists h3; split; auto; apply LStep_skip.

exists h3; split; auto; apply LStep_output; auto.

exists h3; split; auto; apply LStep_assign; auto.

exists h3; split; auto; apply LStep_read with (v1 := v1) (pf := pf); auto.
specialize (H (nat_of_Z v1 pf)); rewrite H13 in H.
destruct (h3 (nat_of_Z v1 pf)); decomp H; auto.
inv H1.

exists (upd h3 (nat_of_Z v1 pf) (v2, l1 \_/ l2)); split.
apply mydot_upd; auto.
specialize (H (nat_of_Z v1 pf)).
destruct (h3 (nat_of_Z v1 pf)); decomp H0; auto; try contradiction.
apply LStep_write; auto.
contradict H14; specialize (H (nat_of_Z v1 pf)).
rewrite H14 in H; intuit.

exists h3; split; auto; apply LStep_seq.

exists h3; split; auto; apply LStep_if_true; auto.

exists h3; split; auto; apply LStep_if_false; auto.

exists h3; split; auto; apply LStep_while_true; auto.

exists h3; split; auto; apply LStep_while_false; auto.

apply hstepn_ff with (h2 := h2) (h3 := h3) in H12; auto.
destruct H12 as [h3' [H12]]; exists h3'; split; auto.
apply LStep_if_hi with (v := v) (n := n); auto.
unfold hsafe; intros.
destruct cf' as [[i'' s'' h''] C'' K'']; apply hstepn_bf with (h1 := h1) (h2 := h2) in H1; auto.
destruct H1 as [h1'' [H1]].
apply H11 in H3; apply H3 in H2.
inv H2.
destruct cf' as [[i''' s''' h'''] C''' K''']; apply hstep_ff with (h2 := h2) (h3 := h'') in H4; auto.
destruct H4 as [h3'' [H4]].
apply (Can_hstep _ _ H2).

exists h3; split; auto.
apply LStep_if_hi_dvg with (v := v); auto.
unfold hsafe; intros.
destruct cf' as [[i'' s'' h''] C'' K'']; apply hstepn_bf with (h1 := h1') (h2 := h2) in H0; auto.
destruct H0 as [h1'' [H0]].
apply H13 in H2; apply H2 in H1.
inv H1.
destruct cf' as [[i''' s''' h'''] C''' K''']; apply hstep_ff with (h2 := h2) (h3 := h'') in H3; auto.
destruct H3 as [h3' [H3]].
apply (Can_hstep _ _ H1).
intros; intro.
destruct st' as [i'' s'' h3']; apply hstepn_bf with (h1 := h1') (h2 := h2) in H0; auto.
destruct H0 as [h1'' [H0]].
contradiction (H14 n (St i'' s'' h1'')).

apply hstepn_ff with (h2 := h2) (h3 := h3) in H12; auto.
destruct H12 as [h3' [H12]]; exists h3'; split; auto.
apply LStep_while_hi with (v := v) (n := n); auto.
unfold hsafe; intros.
destruct cf' as [[i'' s'' h''] C'' K'']; apply hstepn_bf with (h1 := h1) (h2 := h2) in H1; auto.
destruct H1 as [h1'' [H1]].
apply H11 in H3; apply H3 in H2.
inv H2.
destruct cf' as [[i''' s''' h'''] C''' K''']; apply hstep_ff with (h2 := h2) (h3 := h'') in H4; auto.
destruct H4 as [h3'' [H4]].
apply (Can_hstep _ _ H2).

exists h3; split; auto.
apply LStep_while_hi_dvg with (v := v); auto.
unfold hsafe; intros.
destruct cf' as [[i'' s'' h''] C'' K'']; apply hstepn_bf with (h1 := h1') (h2 := h2) in H0; auto.
destruct H0 as [h1'' [H0]].
apply H13 in H2; apply H2 in H1.
inv H1.
destruct cf' as [[i''' s''' h'''] C''' K''']; apply hstep_ff with (h2 := h2) (h3 := h'') in H3; auto.
destruct H3 as [h3' [H3]].
apply (Can_hstep _ _ H1).
intros; intro.
destruct st' as [i'' s'' h3']; apply hstepn_bf with (h1 := h1') (h2 := h2) in H0; auto.
destruct H0 as [h1'' [H0]].
contradiction (H14 n (St i'' s'' h1'')).
Qed.

Lemma lstepn_ff : forall n C K C' K' i s h1 h2 h3 i' s' h1' o,
mydot h1 h2 h3 -> lstepn n (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K') o ->
exists h3', mydot h1' h2 h3' /\ lstepn n (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K') o.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H1.
exists h3; split; auto; apply LStep_zero.
destruct cf' as [[i'' s'' h''] C'' K'']; apply lstep_ff with (h2 := h2) (h3 := h3) in H2; auto.
destruct H2 as [h3' [H2]].
assert (n0 < S n0); try omega.
destruct (H _ H4 _ _ _ _ _ _ _ _ _ _ _ _ _ H2 H3) as [h3'' [H5]]; exists h3''; split; auto.
apply LStep_succ with (cf' := Cf (St i'' s'' h3') C'' K''); auto.
Qed.

Corollary lstepn_nonincreasing : forall n i s h i' s' h' C K C' K' o a,
lstepn n (Cf (St i s h) C K) (Cf (St i' s' h') C' K') o -> h a = None -> h' a = None.
Proof.
intros.
apply lstepn_ff with (h2 := fun n => if eq_nat_dec n a then Some (0%Z,Lo) else None) (h3 := upd h a (0%Z,Lo)) in H.
destruct H as [h3' [H]].
specialize (H a).
destruct (h3' a); decomp H; auto.
destruct (eq_nat_dec a a); inv H4.
intro a'.
unfold upd; destruct (eq_nat_dec a' a); subst; auto.
destruct (h a'); auto.
Qed.

Lemma lstep_bf : forall C K C' K' i s h1 h2 h3 i' s' h3' o,
mydot h1 h2 h3 -> lstep (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K') o -> lsafe (Cf (St i s h1) C K) ->
exists h1', mydot h1' h2 h3' /\ lstep (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K') o.
Proof.
intros.
inv H0.

exists h1; split; auto; apply LStep_skip.

exists h1; split; auto; apply LStep_output; auto.

exists h1; split; auto; apply LStep_assign with (l := l); auto.

exists h1; split; auto; apply LStep_read with (l1 := l1) (l2 := l2) (pf := pf); auto.
specialize (H (nat_of_Z v1 pf)); rewrite H14 in H; decomp H; auto.
specialize (H1 0 (Cf (St i' s h1) (Read x e) K') [] (LStep_zero _) (refl_equal _)).
inv H1.
inv H.
rewrite H10 in H13; inv H13.
rewrite (proof_irrelevance _ pf0 pf) in H11; rewrite H11 in H2; inv H2.

specialize (H1 0 (Cf (St i' s' h1) (Write e1 e2) K') [] (LStep_zero _) (refl_equal _)).
inv H1.
inv H0.
rewrite H9 in H13; inv H13.
rewrite (proof_irrelevance _ pf0 pf) in H11.
exists (upd h1 (nat_of_Z v1 pf) (v2, l1 \_/ l2)); split.
apply mydot_upd; auto.
specialize (H (nat_of_Z v1 pf)); destruct (h3 (nat_of_Z v1 pf)); decomp H; auto; try contradiction.
apply LStep_write with (l1 := l1) (l2 := l2); auto.

exists h1; split; auto; apply LStep_seq.

exists h1; split; auto; apply LStep_if_true; auto.

exists h1; split; auto; apply LStep_if_false; auto.

exists h1; split; auto; apply LStep_while_true; auto.

exists h1; split; auto; apply LStep_while_false; auto.

assert (hsafe (taint_vars_cf (Cf (St i s h1) (If b C1 C2) []))).
specialize (H1 0 (Cf (St i s h1) (If b C1 C2) K') [] (LStep_zero _) (refl_equal _)).
inv H1.
inv H0; auto.
rewrite H10 in H11; inv H11.
rewrite H10 in H11; inv H11.
apply hstepn_bf with (h1 := h1) (h2 := h2) in H13; auto.
destruct H13 as [h1' [H13]]; exists h1'; split; auto.
apply LStep_if_hi with (v := v) (n := n); auto.

exists h1; split; auto.
apply LStep_if_hi_dvg with (v := v); auto.
specialize (H1 0 (Cf (St i' s' h1) (If b C1 C2) K') [] (LStep_zero _) (refl_equal _)).
inv H1.
inv H0; auto.
rewrite H10 in H13; inv H13.
rewrite H10 in H13; inv H13.
intros; intro.
destruct st' as [i'' s'' h'']; apply hstepn_ff with (h2 := h2) (h3 := h3') in H0; auto.
destruct H0 as [h3'' [H0]].
contradiction (H15 n (St i'' s'' h3'')).

assert (hsafe (taint_vars_cf (Cf (St i s h1) (While b C0) []))).
specialize (H1 0 (Cf (St i s h1) (While b C0) K') [] (LStep_zero _) (refl_equal _)).
inv H1.
inv H0; auto.
rewrite H9 in H11; inv H11.
rewrite H9 in H11; inv H11.
apply hstepn_bf with (h1 := h1) (h2 := h2) in H13; auto.
destruct H13 as [h1' [H13]]; exists h1'; split; auto.
apply LStep_while_hi with (v := v) (n := n); auto.

exists h1; split; auto.
apply LStep_while_hi_dvg with (v := v); auto.
specialize (H1 0 (Cf (St i' s' h1) (While b C0) K') [] (LStep_zero _) (refl_equal _)).
inv H1.
inv H0; auto.
rewrite H9 in H13; inv H13.
rewrite H9 in H13; inv H13.
intros; intro.
destruct st' as [i'' s'' h'']; apply hstepn_ff with (h2 := h2) (h3 := h3') in H0; auto.
destruct H0 as [h3'' [H0]].
contradiction (H15 n (St i'' s'' h3'')).
Qed.

Lemma lstepn_bf : forall n C K C' K' i s h1 h2 h3 i' s' h3' o,
mydot h1 h2 h3 -> lstepn n (Cf (St i s h3) C K) (Cf (St i' s' h3') C' K') o -> lsafe (Cf (St i s h1) C K) ->
exists h1', mydot h1' h2 h3' /\ lstepn n (Cf (St i s h1) C K) (Cf (St i' s' h1') C' K') o.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H1.
exists h1; split; auto; apply LStep_zero.
destruct cf' as [[i'' s'' h''] C'' K'']; apply lstep_bf with (h1 := h1) (h2 := h2) in H3; auto.
destruct H3 as [h1' [H3]].
assert (n0 < S n0); try omega.
assert (lsafe (Cf (St i'' s'' h1') C'' K'')).
unfold lsafe; intros.
apply (H2 (S n) _ (o0++o)); auto.
apply LStep_succ with (cf' := Cf (St i'' s'' h1') C'' K''); auto.
destruct (H _ H5 _ _ _ _ _ _ _ _ _ _ _ _ _ H3 H4 H6) as [h1'' [H7]]; exists h1''; split; auto.
apply LStep_succ with (cf' := Cf (St i'' s'' h1') C'' K''); auto.
Qed.

Lemma hstep_modifies_monotonic : forall st st' C C' K K' x,
hstep (Cf st C K) (Cf st' C' K') -> In x (modifies (C'::K')) -> In x (modifies (C::K)).
Proof.
intros.
inv H; simpl in *; auto.
rewrite app_assoc; auto.
repeat rewrite in_app_iff in H0 |- *; intuit.
repeat rewrite in_app_iff in H0 |- *; intuit.
repeat rewrite in_app_iff in H0 |- *; intuit.
rewrite in_app_iff; intuit.
Qed.

Lemma hstepn_modifies_monotonic : forall n st st' C C' K K' x,
hstepn n (Cf st C K) (Cf st' C' K') -> In x (modifies (C'::K')) -> In x (modifies (C::K)).
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; auto.
destruct cf' as [st'' C'' K'']; apply H with (x := x) in H3; auto.
apply hstep_modifies_monotonic with (x := x) in H2; auto.
Qed.

Lemma lstep_modifies_monotonic : forall st st' C C' K K' x o,
lstep (Cf st C K) (Cf st' C' K') o -> In x (modifies (C'::K')) -> In x (modifies (C::K)).
Proof.
intros.
inv H; simpl in *; auto.
rewrite app_assoc; auto.
repeat rewrite in_app_iff in H0 |- *; intuit.
repeat rewrite in_app_iff in H0 |- *; intuit.
repeat rewrite in_app_iff in H0 |- *; intuit.
rewrite in_app_iff; intuit.
repeat rewrite in_app_iff; intuit.
rewrite in_app_iff; intuit.
Qed.

Lemma lstepn_modifies_monotonic : forall n st st' C C' K K' x o,
lstepn n (Cf st C K) (Cf st' C' K') o -> In x (modifies (C'::K')) -> In x (modifies (C::K)).
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; auto.
destruct cf' as [st'' C'' K'']; apply H with (x := x) in H3; auto.
apply lstep_modifies_monotonic with (x := x) in H2; auto.
Qed.

Lemma hstep_modifies_const : forall st st' C C' K K' x,
hstep (Cf st C K) (Cf st' C' K') -> ~ In x (modifies (C::K)) -> (st:store) x = (st':store) x.
Proof.
intros.
inv H; simpl; auto.
unfold upd; destruct (eq_nat_dec x x0); auto.
unfold upd; destruct (eq_nat_dec x x0); auto.
Qed.

Lemma hstepn_modifies_const : forall n st st' C C' K K' x,
hstepn n (Cf st C K) (Cf st' C' K') -> ~ In x (modifies (C::K)) -> (st:store) x = (st':store) x.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; auto.
destruct cf' as [st'' C'' K''].
apply H with (x := x) in H3; auto.
apply hstep_modifies_const with (x := x) in H2; auto.
rewrite H2; rewrite H3; auto.
apply hstep_modifies_monotonic with (x := x) in H2; auto.
Qed.

Lemma lstep_modifies_const : forall st st' C C' K K' x o,
lstep (Cf st C K) (Cf st' C' K') o -> ~ In x (modifies (C::K)) -> (st:store) x = (st':store) x.
Proof.
intros.
inv H; simpl; auto.
unfold upd; destruct (eq_nat_dec x x0); auto.
unfold upd; destruct (eq_nat_dec x x0); auto.
apply hstepn_modifies_const with (x := x) in H10; simpl in *.
rewrite <- H10; unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); auto.
contradiction H0; simpl in i0; rewrite in_app_iff in i0 |- *.
destruct i0; auto.
inv H.
rewrite in_app_iff in H |- *.
destruct H; auto.
inv H.
apply hstepn_modifies_const with (x := x) in H10; simpl in *.
rewrite <- H10; unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [While b C0])); auto.
contradiction H0; simpl in i0; rewrite in_app_iff in i0 |- *.
destruct i0; auto.
inv H.
rewrite in_app_iff in H |- *.
destruct H; auto.
inv H.
Qed.

Lemma lstepn_modifies_const : forall n st st' C C' K K' x o,
lstepn n (Cf st C K) (Cf st' C' K') o -> ~ In x (modifies (C::K)) -> (st:store) x = (st':store) x.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; auto.
destruct cf' as [st'' C'' K''].
apply H with (x := x) in H3; auto.
apply lstep_modifies_const with (x := x) in H2; auto.
rewrite H2; rewrite H3; auto.
apply lstep_modifies_monotonic with (x := x) in H2; auto.
Qed.

Lemma hstep_taints_s : forall i s h i' s' h' C K C' K' x,
hstep (Cf (St i s h) C K) (Cf (St i' s' h') C' K') ->
s x <> s' x -> exists v, s' x = Some (v,Hi).
Proof.
intros.
inv H; try solve [contradiction H0; auto].
destruct (eq_nat_dec x x0); subst.
exists v; unfold upd; destruct (eq_nat_dec x0 x0); auto.
destruct (eq_nat_dec x x0); auto; contradiction.
destruct (eq_nat_dec x x0); subst.
exists v2; unfold upd; destruct (eq_nat_dec x0 x0); auto.
destruct (eq_nat_dec x x0); auto; contradiction.
Qed.

Lemma hstepn_taints_s : forall n i s h i' s' h' C K C' K' x,
hstepn n (Cf (St i s h) C K) (Cf (St i' s' h') C' K') ->
s x <> s' x -> exists v, s' x = Some (v,Hi).
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0.
destruct cf' as [[i'' s'' h''] C'' K''].
destruct (opt_eq_dec val_eq_dec (s'' x) (s' x)).
rewrite <- e in H1 |- *.
apply hstep_taints_s with (x := x) in H2; auto.
assert (n0 < S n0); try omega.
apply (H _ H0 _ _ _ _ _ _ _ _ _ _ _ H3 n).
Qed.

Lemma hstep_taints_h : forall i s h i' s' h' C K C' K' a,
hstep (Cf (St i s h) C K) (Cf (St i' s' h') C' K') ->
h a <> h' a -> exists v, h' a = Some (v,Hi).
Proof.
intros.
inv H; try solve [contradiction H0; auto].
destruct (eq_nat_dec (nat_of_Z v1 pf) a); subst.
exists v2; unfold upd; destruct (eq_nat_dec (nat_of_Z v1 pf) (nat_of_Z v1 pf)); auto.
destruct (eq_nat_dec a (nat_of_Z v1 pf)); auto; subst.
Qed.

Lemma hstepn_taints_h : forall n i s h i' s' h' C K C' K' a,
hstepn n (Cf (St i s h) C K) (Cf (St i' s' h') C' K') ->
h a <> h' a -> exists v, h' a = Some (v,Hi).
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0.
destruct cf' as [[i'' s'' h''] C'' K''].
destruct (opt_eq_dec val_eq_dec (h'' a) (h' a)).
rewrite <- e in H1 |- *.
apply hstep_taints_h with (a := a) in H2; auto.
assert (n0 < S n0); try omega.
apply (H _ H0 _ _ _ _ _ _ _ _ _ _ _ H3 n).
Qed.

Proposition hstep_i_const : forall i s h i' s' h' C C' K K',
hstep (Cf (St i s h) C K) (Cf (St i' s' h') C' K') -> i' = i.
Proof.
intros.
inv H; auto.
Qed.

Proposition hstepn_i_const : forall n i s h i' s' h' C C' K K',
hstepn n (Cf (St i s h) C K) (Cf (St i' s' h') C' K') -> i' = i.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; auto.
destruct cf' as [[i'' s'' h''] C'' K'']; apply H in H2; subst; auto.
apply hstep_i_const in H1; auto.
Qed.

Proposition lstep_i_const : forall i s h i' s' h' C C' K K' o,
lstep (Cf (St i s h) C K) (Cf (St i' s' h') C' K') o -> i' = i.
Proof.
intros.
inv H; auto.
apply hstepn_i_const in H11; auto.
apply hstepn_i_const in H11; auto.
Qed.

Proposition lstepn_i_const : forall n i s h i' s' h' C C' K K' o,
lstepn n (Cf (St i s h) C K) (Cf (St i' s' h') C' K') o -> i' = i.
Proof.
induction n using (well_founded_induction lt_wf); intros.
inv H0; auto.
destruct cf' as [[i'' s'' h''] C'' K'']; apply H in H2; subst; auto.
apply lstep_i_const in H1; auto.
Qed.

Close Scope Z_scope.

Definition obs_eq_s (s1 s2 : store) : Prop := forall x,
match s1 x, s2 x with
| None, None => True
| Some (v1,l1), Some (v2,l2) => l1 = l2 /\ (l1 = Lo -> v1 = v2)
| _, _ => False
end.

Definition obs_eq_h (h1 h2 : heap) : Prop := forall n,
match h1 n, h2 n with
| Some (v1,l1), Some (v2,l2) => l1 = Lo -> l2 = Lo -> v1 = v2
| _, _ => True
end.

Definition obs_eq (st1 st2 : state) : Prop := (st1:lmap) = (st2:lmap) /\ obs_eq_s st1 st2 /\ obs_eq_h st1 st2.

Proposition obs_eq_s_refl : forall s, obs_eq_s s s.
Proof.
unfold obs_eq_s; intros.
destruct (s x) as [[v l]|]; auto.
Qed.

Proposition obs_eq_h_refl : forall h, obs_eq_h h h.
Proof.
unfold obs_eq_h; intros.
destruct (h n) as [[v l]|]; auto.
Qed.

Proposition obs_eq_refl : forall st, obs_eq st st.
Proof.
unfold obs_eq; intuition.
apply obs_eq_s_refl.
apply obs_eq_h_refl.
Qed.

Proposition obs_eq_s_sym : forall s1 s2, obs_eq_s s1 s2 -> obs_eq_s s2 s1.
Proof.
unfold obs_eq_s; intros.
specialize (H x); destruct (s1 x) as [[v1 l1]|]; destruct (s2 x) as [[v2 l2]|]; auto.
destruct H; split; auto; intros.
subst; intuit.
Qed.

Proposition obs_eq_h_sym : forall h1 h2, obs_eq_h h1 h2 -> obs_eq_h h2 h1.
Proof.
unfold obs_eq_h; intros.
specialize (H n); destruct (h1 n) as [[v1 l1]|]; destruct (h2 n) as [[v2 l2]|]; intuit.
Qed.

Proposition obs_eq_sym : forall st1 st2, obs_eq st1 st2 -> obs_eq st2 st1.
Proof.
unfold obs_eq; intuition.
apply obs_eq_s_sym; auto.
apply obs_eq_h_sym; auto.
Qed.

Lemma obs_eq_exp : forall e i1 s1 h1 i2 s2 h2, obs_eq (St i1 s1 h1) (St i2 s2 h2) ->
match eden e i1 s1, eden e i2 s2 with
| None, None => True
| Some (v1,l1), Some (v2,l2) => l1 = l2 /\ (l1 = Lo -> v1 = v2)
| _, _ => False
end.
Proof.
induction e; simpl; intros; auto.
unfold obs_eq in H; decomp H.
apply H2.
unfold obs_eq in H; decomp H; simpl in *; subst; auto.
specialize (IHe1 _ _ _ _ _ _ H); specialize (IHe2 _ _ _ _ _ _ H).
destruct (eden e1 i1 s1) as [[v1 l1]|]; destruct (eden e2 i2 s2) as [[v2 l2]|];
destruct (eden e1 i2 s2) as [[v1' l1']|]; destruct (eden e2 i1 s1) as [[v2' l2']|]; simpl in *; intuit.
destruct IHe1; destruct IHe2; destruct H; subst; split; auto; intros.
glub_simpl H0; rewrite H1; auto; rewrite H3; auto.
Qed.

Lemma obs_eq_bexp : forall b i1 s1 h1 i2 s2 h2, obs_eq (St i1 s1 h1) (St i2 s2 h2) ->
match bden b i1 s1, bden b i2 s2 with
| None, None => True
| Some (v1,l1), Some (v2,l2) => l1 = l2 /\ (l1 = Lo -> v1 = v2)
| _, _ => False
end.
Proof.
induction b; simpl; intros; auto.
dup H; apply (obs_eq_exp e) in H.
apply (obs_eq_exp e0) in H0.
destruct (eden e i1 s1) as [[v1 l1]|]; destruct (eden e0 i2 s2) as [[v2 l2]|];
destruct (eden e i2 s2) as [[v1' l1']|]; destruct (eden e0 i1 s1) as [[v2' l2']|]; simpl in *; intuit.
destruct H; destruct H0; subst; split; auto; intros.
glub_simpl H; rewrite H1; auto; rewrite H2; auto.
apply IHb in H.
destruct (bden b i1 s1) as [[v1 l1]|].
destruct (bden b i2 s2) as [[v2 l2]|]; auto; simpl.
destruct H; subst; split; auto; intros.
rewrite H0; auto.
destruct (bden b i2 s2); intuit.
dup H.
apply IHb1 in H; apply IHb2 in H0.
destruct (bden b2 i1 s1) as [[v1 l1]|]; destruct (bden b3 i2 s2) as [[v2 l2]|];
destruct (bden b2 i2 s2) as [[v1' l1']|]; destruct (bden b3 i1 s1) as [[v2' l2']|]; simpl in *; intuit.
destruct H; destruct H0; subst; split; auto; intros.
glub_simpl H; rewrite H1; auto; rewrite H2; auto.
Qed.

Inductive lexp :=
| Lbl : glbl -> lexp
| Lblvar : nat -> lexp
| Lub : lexp -> lexp -> lexp.

Definition toLexp (l : glbl) : lexp := Lbl l.
Coercion toLexp : glbl >-> lexp.

Fixpoint lden (L : lexp) (i : lmap) : glbl :=
match L with
| Lbl l => l
| Lblvar X => snd i X
| Lub L1 L2 => glub (lden L1 i) (lden L2 i)
end.

Proposition lden_lblvars : forall L i1 i2 i, lden L (i1,i) = lden L (i2,i).
Proof.
induction L; simpl; auto; intros.
rewrite (IHL1 _ i2); rewrite (IHL2 _ i2); auto.
Qed.

Inductive assert :=
| TrueA : assert
| FalseA : assert
| Emp : assert
| Allocated : exp -> assert
| Mapsto : exp -> exp -> lexp -> assert
| BoolExp : bexp -> assert
| EqLbl : lexp -> lexp -> assert
| LblEq : var -> lexp -> assert
| LblLeq : var -> lexp -> assert
| LblLeq' : lexp -> var -> assert
| LblExp : exp -> lexp -> assert
| LblBexp : bexp -> lexp -> assert
| Conj : assert -> assert -> assert
| Disj : assert -> assert -> assert
| Star : assert -> assert -> assert.

Fixpoint vars (P : assert) (x : var) : bool :=
match P with
| TrueA => false
| FalseA => false
| Emp => false
| Allocated e => expvars e x
| Mapsto e e' L => orb (expvars e x) (expvars e' x)
| BoolExp b => bexpvars b x
| EqLbl L1 L2 => false
| LblEq y L => if eq_nat_dec y x then true else false
| LblLeq y L => if eq_nat_dec y x then true else false
| LblLeq' L y => if eq_nat_dec y x then true else false
| LblExp e L => expvars e x
| LblBexp b L => bexpvars b x
| Conj P Q => orb (vars P x) (vars Q x)
| Disj P Q => orb (vars P x) (vars Q x)
| Star P Q => orb (vars P x) (vars Q x)
end.
Notation " P `AND` Q " := (Conj P Q) (at level 91, left associativity).
Notation " P `OR` Q " := (Disj P Q) (at level 91, left associativity).
Notation " P ** Q " := (Star P Q) (at level 91, left associativity).

Fixpoint ereplace e x ex : exp :=
match e with
| Var y => if eq_nat_dec y x then ex else Var y
| BinOp bop e1 e2 => BinOp bop (ereplace e1 x ex) (ereplace e2 x ex)
| _ => e
end.

Proposition ereplace_deletes : forall e x ex, expvars ex x = false -> expvars (ereplace e x ex) x = false.
Proof.
induction e; simpl; intros; auto.
destruct (eq_nat_dec v x); subst; simpl; auto.
destruct (eq_nat_dec v x); try contradiction; auto.
rewrite (IHe1 _ _ H); rewrite (IHe2 _ _ H); auto.
Qed.

Proposition eden_ereplace : forall e x ex i s, eden (Var x) i s = eden ex i s -> eden (ereplace e x ex) i s = eden e i s.
Proof.
induction e; simpl; intros; auto.
destruct (eq_nat_dec v x); subst; auto.
rewrite (IHe1 _ _ _ _ H); rewrite (IHe2 _ _ _ _ H); auto.
Qed.

Proposition edenZ_ereplace : forall e x ex i s, edenZ (Var x) i s = edenZ ex i s -> edenZ (ereplace e x ex) i s = edenZ e i s.
Proof.
induction e; simpl; intros; auto.
destruct (eq_nat_dec v x); subst; auto.
rewrite (IHe1 _ _ _ _ H); rewrite (IHe2 _ _ _ _ H); auto.
Qed.

Fixpoint aden (P : assert) (st : state) : Prop :=
match st with St i s h =>
match P with
| TrueA => True
| FalseA => False
| Emp => h = fun _ => None
| Allocated e => exists v : Z, exists pf : (v>=0)%Z, edenZ e i s = Some v /\
exists v', exists l', h = fun n => if eq_nat_dec n (nat_of_Z v pf) then Some (v',l') else None
| Mapsto e e' L => exists v : Z, exists pf : (v>=0)%Z, edenZ e i s = Some v /\ exists v', edenZ e' i s = Some v' /\
h = fun n => if eq_nat_dec n (nat_of_Z v pf) then Some (v', lden L i) else None
| BoolExp b => bdenZ b i s = Some true
| EqLbl L1 L2 => lden L1 i = lden L2 i
| LblEq x L => exists v, s x = Some (v, lden L i)
| LblLeq x L => exists v, exists l, s x = Some (v,l) /\ gleq l (lden L i) = true
| LblLeq' L x => exists v, exists l, s x = Some (v,l) /\ gleq (lden L i) l = true
| LblExp e L => exists v, eden e i s = Some (v, lden L i)
| LblBexp b L => exists v, bden b i s = Some (v, lden L i)
| Conj P Q => aden P st /\ aden Q st
| Disj P Q => aden P st \/ aden Q st
| Star P Q => exists h1, exists h2, mydot h1 h2 h /\ aden P (St i s h1) /\ aden Q (St i s h2)
end
end.

Definition aden2 (P : assert) (st1 st2 : state) : Prop := aden P st1 /\ aden P st2 /\ obs_eq st1 st2.

Definition implies (P Q : assert) := forall st, aden P st -> aden Q st.

Fixpoint haslbl (P : assert) (x : var) : bool :=
match P with
| LblEq y L => if eq_nat_dec y x then true else false
| LblLeq y L => if eq_nat_dec y x then true else false
| LblLeq' L y => if eq_nat_dec y x then true else false
| LblExp e L => expvars e x
| LblBexp b L => bexpvars b x
| Conj P Q => orb (haslbl P x) (haslbl Q x)
| Disj P Q => orb (haslbl P x) (haslbl Q x)
| Star P Q => orb (haslbl P x) (haslbl Q x)
| _ => false
end.

Proposition eden_upd : forall e x i s v l, expvars e x = false -> eden e i (upd s x (v,l)) = eden e i s.
Proof.
induction e; simpl; intros; auto.
unfold upd; destruct (eq_nat_dec v x); inv H; auto.
rewrite IHe1.
rewrite IHe2; auto.
destruct (expvars e1 x); destruct (expvars e2 x); inv H; auto.
destruct (expvars e1 x); inv H; auto.
Qed.

Proposition edenZ_upd : forall e x i s v l, expvars e x = false -> edenZ e i (upd s x (v,l)) = edenZ e i s.
Proof.
induction e; simpl; intros; auto.
unfold upd; destruct (eq_nat_dec v x); inv H; auto.
rewrite IHe1.
rewrite IHe2; auto.
destruct (expvars e1 x); destruct (expvars e2 x); inv H; auto.
destruct (expvars e1 x); inv H; auto.
Qed.

Proposition bden_upd : forall b x i s v l, bexpvars b x = false -> bden b i (upd s x (v,l)) = bden b i s.
Proof.
induction b; simpl; intros; auto.
repeat rewrite eden_upd; auto.
destruct (expvars e x); destruct (expvars e0 x); inv H; auto.
destruct (expvars e x); inv H; auto.
rewrite IHb; auto.
rewrite IHb1.
rewrite IHb2; auto.
destruct (bexpvars b2 x); destruct (bexpvars b3 x); inv H; auto.
destruct (bexpvars b2 x); inv H; auto.
Qed.

Proposition bdenZ_upd : forall b x i s v l, bexpvars b x = false -> bdenZ b i (upd s x (v,l)) = bdenZ b i s.
Proof.
induction b; simpl; intros; auto.
repeat rewrite edenZ_upd; auto.
destruct (expvars e x); destruct (expvars e0 x); inv H; auto.
destruct (expvars e x); inv H; auto.
rewrite IHb; auto.
rewrite IHb1.
rewrite IHb2; auto.
destruct (bexpvars b2 x); destruct (bexpvars b3 x); inv H; auto.
destruct (bexpvars b2 x); inv H; auto.
Qed.

Proposition aden_upd : forall P x i s h v l, vars P x = false -> aden P (St i s h) -> aden P (St i (upd s x (v,l)) h).
Proof.
induction P; simpl; intros; auto.
rewrite edenZ_upd; auto.
apply orb_false_elim in H.
repeat rewrite edenZ_upd; intuit.
rewrite bdenZ_upd; auto.
unfold upd; destruct (eq_nat_dec v x); inv H; auto.
unfold upd; destruct (eq_nat_dec v x); inv H; auto.
unfold upd; destruct (eq_nat_dec v x); inv H; auto.
rewrite eden_upd; auto.
rewrite bden_upd; auto.
apply orb_false_elim in H; intuit.
apply orb_false_elim in H; intuit.
apply orb_false_elim in H; destruct H0 as [h1 [h2]]; exists h1; exists h2; intuit.
Qed.

Proposition eden_vars_same : forall e i s s',
(forall x, expvars e x = true -> s x = s' x) -> eden e i s = eden e i s'.
Proof.
induction e; simpl; intros; auto.
apply H; destruct (eq_nat_dec v v); auto.
rewrite IHe1 with (s' := s'); intros.
rewrite IHe2 with (s' := s'); auto; intros.
apply H; rewrite H0; destruct (expvars e1 x); auto.
apply H; rewrite H0; auto.
Qed.

Proposition edenZ_vars_same : forall e i s s',
(forall x, expvars e x = true -> s x = s' x) -> edenZ e i s = edenZ e i s'.
Proof.
induction e; simpl; intros; auto.
rewrite H; destruct (eq_nat_dec v v); auto.
rewrite IHe1 with (s' := s'); intros.
rewrite IHe2 with (s' := s'); auto; intros.
apply H; rewrite H0; destruct (expvars e1 x); auto.
apply H; rewrite H0; auto.
Qed.

Proposition bden_vars_same : forall b i s s',
(forall x, bexpvars b x = true -> s x = s' x) -> bden b i s = bden b i s'.
Proof.
induction b; simpl; intros; auto.
rewrite eden_vars_same with (s' := s'); intros.
rewrite (eden_vars_same e0) with (s' := s'); auto; intros.
apply H; rewrite H0; destruct (expvars e x); auto.
apply H; rewrite H0; auto.
rewrite IHb with (s' := s'); auto.
rewrite IHb1 with (s' := s'); intros.
rewrite IHb2 with (s' := s'); auto; intros.
apply H; rewrite H0; destruct (bexpvars b2 x); auto.
apply H; rewrite H0; auto.
Qed.

Proposition bdenZ_vars_same : forall b i s s',
(forall x, bexpvars b x = true -> s x = s' x) -> bdenZ b i s = bdenZ b i s'.
Proof.
induction b; simpl; intros; auto.
rewrite edenZ_vars_same with (s' := s'); intros.
rewrite (edenZ_vars_same e0) with (s' := s'); auto; intros.
apply H; rewrite H0; destruct (expvars e x); auto.
apply H; rewrite H0; auto.
rewrite IHb with (s' := s'); auto.
rewrite IHb1 with (s' := s'); intros.
rewrite IHb2 with (s' := s'); auto; intros.
apply H; rewrite H0; destruct (bexpvars b2 x); auto.
apply H; rewrite H0; auto.
Qed.

Proposition aden_vars_same : forall P i s s' h,
(forall x, vars P x = true -> s x = s' x) -> aden P (St i s h) -> aden P (St i s' h).
Proof.
induction P; simpl; intros; auto.
rewrite edenZ_vars_same with (s' := s') in H0; auto.
rewrite edenZ_vars_same with (s' := s') in H0; intuit.
rewrite (edenZ_vars_same e0) with (s' := s') in H0; intuit.
rewrite bdenZ_vars_same with (s' := s') in H0; auto.
rewrite <- H; auto.
destruct (eq_nat_dec v v); auto.
rewrite <- H; auto.
destruct (eq_nat_dec v v); auto.
rewrite <- H; auto.
destruct (eq_nat_dec v v); auto.
rewrite eden_vars_same with (s' := s') in H0; auto.
rewrite bden_vars_same with (s' := s') in H0; auto.
split; [apply IHP1 with (s := s) | apply IHP2 with (s := s)]; intuit.
destruct H0; [left; apply IHP1 with (s := s) | right; apply IHP2 with (s := s)]; intuit.
destruct H0 as [h1 [h2]]; exists h1; exists h2; intuition.
apply IHP1 with (s := s); intuit.
apply IHP2 with (s := s); intuit.
Qed.

Proposition expvars_none : forall e i s x v l, eden e i s = Some (v,l) -> s x = None -> expvars e x = false.
Proof.
induction e; simpl; intros; auto.
destruct (eq_nat_dec v x); subst; auto.
rewrite H in H0; inv H0.
case_eq (eden e1 i s); intros.
case_eq (eden e2 i s); intros.
destruct v1 as [v2 l2]; destruct v0 as [v1 l1].
rewrite H1 in H; rewrite H2 in H; inv H.
apply IHe1 with (x := x) in H1; auto.
apply IHe2 with (x := x) in H2; auto.
rewrite H1; rewrite H2; auto.
rewrite H2 in H; destruct (eden e1 i s); inv H.
rewrite H1 in H; inv H.
Qed.

Proposition bexpvars_none : forall b i s x v l, bden b i s = Some (v,l) -> s x = None -> bexpvars b x = false.
Proof.
induction b; simpl; intros; auto.
case_eq (eden e i s); intros.
case_eq (eden e0 i s); intros.
destruct v1 as [v2 l2]; destruct v0 as [v1 l1].
rewrite H1 in H; rewrite H2 in H; inv H.
apply expvars_none with (x := x) in H1; auto.
apply expvars_none with (x := x) in H2; auto.
rewrite H1; rewrite H2; auto.
rewrite H2 in H; destruct (eden e i s); inv H.
rewrite H1 in H; inv H.
case_eq (bden b i s); intros.
destruct p; apply IHb with (x := x) in H1; auto.
rewrite H1 in H; inv H.
case_eq (bden b2 i s); intros.
case_eq (bden b3 i s); intros.
destruct p0 as [v2 l2]; destruct p as [v1 l1].
rewrite H1 in H; rewrite H2 in H; inv H.
apply IHb1 with (x := x) in H1; auto.
apply IHb2 with (x := x) in H2; auto.
rewrite H1; rewrite H2; auto.
rewrite H2 in H; destruct (bden b2 i s); inv H.
rewrite H1 in H; inv H.
Qed.

Proposition aden_upd_none : forall P x i s h v l, s x = None -> aden P (St i s h) -> aden P (St i (upd s x (v,l)) h).
Proof.
induction P; simpl; intros; intuit.
rewrite edenZ_upd; auto.
destruct H0 as [v1 [pf [H0]]].
rewrite edenZ_some in H0; destruct H0 as [l1].
apply expvars_none with (x := x) in H0; auto.
repeat rewrite edenZ_upd; auto.
destruct H0 as [v1 [pf [H0 [v2 [H1]]]]].
rewrite edenZ_some in H1; destruct H1 as [l2].
apply expvars_none with (x := x) in H1; auto.
destruct H0 as [v1 [pf [H0]]].
rewrite edenZ_some in H0; destruct H0 as [l1].
apply expvars_none with (x := x) in H0; auto.
rewrite bdenZ_upd; auto.
rewrite bdenZ_some in H0; destruct H0 as [l1].
apply bexpvars_none with (x := x) in H0; auto.
unfold upd; destruct (eq_nat_dec v x); subst; auto.
destruct H0 as [v]; rewrite H in H0; inv H0.
unfold upd; destruct (eq_nat_dec v x); subst; auto.
destruct H0 as [v1 [l1 [H0]]]; rewrite H in H0; inv H0.
unfold upd; destruct (eq_nat_dec v x); subst; auto.
destruct H0 as [v1 [l1 [H0]]]; rewrite H in H0; inv H0.
rewrite eden_upd; auto.
destruct H0 as [v1]; apply expvars_none with (x := x) in H0; auto.
rewrite bden_upd; auto.
destruct H0 as [v1]; apply bexpvars_none with (x := x) in H0; auto.
destruct H0 as [h1 [h2]]; exists h1; exists h2; intuit.
Qed.

Proposition eden_taint_vars : forall e i s K v l, eden e i s = Some (v,l) ->
exists l', eden e i (taint_vars K s) = Some (v,l') /\ l <<= l'.
Proof.
induction e; simpl; intros.
unfold taint_vars.
destruct (In_dec eq_nat_dec v (modifies K)).
exists Hi; rewrite H; split; auto.
destruct l; auto.
exists l; rewrite H; split; auto.
destruct l; auto.
inv H; exists Lo; split; auto.
inv H; exists Lo; split; auto.
case_eq (eden e1 i s); case_eq (eden e2 i s); intros.
rewrite H1 in H; rewrite H0 in H; simpl in H; inv H.
destruct v1 as [v1 l1]; destruct v0 as [v2 l2].
apply IHe1 with (K := K) in H1; apply IHe2 with (K := K) in H0.
destruct H1 as [l1' [H1]]; destruct H0 as [l2' [H0]].
exists (l1' \_/ l2'); simpl; split.
rewrite H1; rewrite H0; simpl; auto.
destruct l1; destruct l1'; destruct l2; destruct l2'; intuit.
rewrite H1 in H; rewrite H0 in H; inv H.
rewrite H1 in H; rewrite H0 in H; inv H.
rewrite H1 in H; rewrite H0 in H; inv H.
Qed.

Proposition bden_taint_vars : forall b i s K v l, bden b i s = Some (v,l) ->
exists l', bden b i (taint_vars K s) = Some (v,l') /\ l <<= l'.
Proof.
induction b; simpl; intros.
inv H; exists Lo; split; auto.
inv H; exists Lo; split; auto.
case_eq (eden e i s); case_eq (eden e0 i s); intros.
destruct v1 as [v1 l1]; destruct v0 as [v2 l2].
rewrite H1 in H; rewrite H0 in H; inv H.
apply eden_taint_vars with (K := K) in H1; apply eden_taint_vars with (K := K) in H0.
destruct H1 as [l1' [H1]]; destruct H0 as [l2' [H0]].
exists (l1' \_/ l2'); split.
rewrite H1; rewrite H0; auto.
destruct l1; destruct l1'; destruct l2; destruct l2'; intuit.
rewrite H1 in H; rewrite H0 in H; inv H.
rewrite H1 in H; rewrite H0 in H; inv H.
rewrite H1 in H; rewrite H0 in H; inv H.
case_eq (bden b i s); intros.
destruct p as [v' l']; rewrite H0 in H; inv H.
apply IHb with (K := K) in H0; destruct H0 as [l' [H0]].
exists l'; split; auto.
rewrite H0; auto.
rewrite H0 in H; inv H.
case_eq (bden b2 i s); case_eq (bden b3 i s); intros.
rewrite H1 in H; rewrite H0 in H; simpl in H; inv H.
destruct p0 as [v1 l1]; destruct p as [v2 l2].
apply IHb1 with (K := K) in H1; apply IHb2 with (K := K) in H0.
destruct H1 as [l1' [H1]]; destruct H0 as [l2' [H0]].
exists (l1' \_/ l2'); simpl; split.
rewrite H1; rewrite H0; simpl; auto.
destruct l1; destruct l1'; destruct l2; destruct l2'; intuit.
rewrite H1 in H; rewrite H0 in H; inv H.
rewrite H1 in H; rewrite H0 in H; inv H.
rewrite H1 in H; rewrite H0 in H; inv H.
Qed.

Proposition edenZ_ignores_lbl : forall e i s x v l l',
s x = Some (v,l) -> edenZ e i (upd s x (v,l')) = edenZ e i s.
Proof.
induction e; simpl; intros; auto.
unfold upd; destruct (eq_nat_dec v x); subst; auto.
rewrite H; auto.
rewrite IHe1 with (l := l); auto.
rewrite IHe2 with (l := l); auto.
Qed.

Proposition bdenZ_ignores_lbl : forall b i s x v l l',
s x = Some (v,l) -> bdenZ b i (upd s x (v,l')) = bdenZ b i s.
Proof.
induction b; simpl; intros; auto.
repeat rewrite edenZ_ignores_lbl with (l := l); auto.
rewrite IHb with (l := l); auto.
rewrite IHb1 with (l := l); auto.
rewrite IHb2 with (l := l); auto.
Qed.

Proposition aden_haslbl : forall P x i s h v l l', haslbl P x = false -> s x = Some (v,l) ->
aden P (St i s h) -> aden P (St i (upd s x (v,l')) h).
Proof.
induction P; simpl; intros; auto.
rewrite edenZ_ignores_lbl with (l := l); auto.
repeat rewrite edenZ_ignores_lbl with (l := l0); auto.
rewrite bdenZ_ignores_lbl with (l := l); auto.
unfold upd; destruct (eq_nat_dec v x); auto; inv H.
unfold upd; destruct (eq_nat_dec v x); auto; inv H.
unfold upd; destruct (eq_nat_dec v x); auto; inv H.
rewrite eden_upd; auto.
rewrite bden_upd; auto.
apply orb_false_elim in H; destruct H; destruct H1; split.
apply IHP1 with (l := l); auto.
apply IHP2 with (l := l); auto.
apply orb_false_elim in H; destruct H; destruct H1; [left | right].
apply IHP1 with (l := l); auto.
apply IHP2 with (l := l); auto.
apply orb_false_elim in H; destruct H; destruct H1 as [h1 [h2]]; decomp H1.
exists h1; exists h2; repeat (split; auto).
apply IHP1 with (l := l); auto.
apply IHP2 with (l := l); auto.
Qed.

Definition taint_vars_assert (P : assert) (xs : list var) (l1 l2 : glbl) : assert :=
if gleq l1 l2 then P else P `AND` fold_right (fun x P => P `AND` LblLeq' (glub l1 l2) x) TrueA xs.

Proposition aden_fold : forall (f : var -> assert) xs st,
(forall x, In x xs -> aden (f x) st) -> aden (fold_right (fun x P => P `AND` f x) TrueA xs) st.
Proof.
induction xs; destruct st as [i s h]; simpl; intros; auto.
Qed.

Proposition aden_fold_inv : forall (f : var -> assert) xs st,
aden (fold_right (fun x P => P `AND` f x) TrueA xs) st -> forall x, In x xs -> aden (f x) st.
Proof.
induction xs; destruct st as [i s h]; simpl; intros; intuit.
destruct H0; subst; intuit.
Qed.

Fixpoint no_lbls (P : assert) (xs : list var) :=
match xs with
| [] => true
| x::xs => andb (negb (haslbl P x)) (no_lbls P xs)
end.

Definition same_values (s1 s2 : store) (xs : list var) := forall x,
if In_dec eq_nat_dec x xs then
match s1 x, s2 x with
| Some (v1,_), Some (v2,_) => v1 = v2
| Some _, None => False
| _, _ => True
end
else s1 x = s2 x.

Proposition no_lbls_same_values : forall P xs i s1 s2 h,
no_lbls P xs = true -> same_values s1 s2 xs -> aden P (St i s1 h) -> aden P (St i s2 h).
Proof.
induction xs; simpl; intros.
assert (s1 = s2).
extensionality x; specialize (H0 x); simpl in H0; auto.
subst; auto.
rewrite andb_true_iff in H; destruct H.
destruct (In_dec eq_nat_dec a xs).
apply IHxs with (s1 := s1); auto; intro x; specialize (H0 x).
simpl in H0.
destruct (eq_nat_dec a x); subst.
destruct (In_dec eq_nat_dec x xs); try contradiction; auto.
destruct (In_dec eq_nat_dec x xs); auto.
dup H0; specialize (H0 a).
simpl in H0; destruct (eq_nat_dec a a).
case_eq (s1 a); case_eq (s2 a); intros.
destruct v as [v2 l2]; destruct v0 as [v1 l1].
rewrite H4 in H0; rewrite H5 in H0; subst.
apply IHxs with (s1 := upd s1 a (v2,l2)); auto.
intro x; specialize (H3 x); simpl in H3; unfold upd.
destruct (In_dec eq_nat_dec x xs).
destruct (eq_nat_dec a x); destruct (eq_nat_dec x a); try subst; try contradiction; auto.
subst x; rewrite H5 in H3; auto.
destruct (eq_nat_dec a x); destruct (eq_nat_dec x a); try subst; try contradiction; auto.
subst x; auto.
apply aden_haslbl with (l := l1); auto.
destruct (haslbl P a); auto; inv H.
destruct v; rewrite H4 in H0; rewrite H5 in H0; inv H0.
destruct v as [v l]; apply IHxs with (s1 := upd s1 a (v,l)); auto.
intro x; specialize (H3 x); simpl in H3; unfold upd.
destruct (In_dec eq_nat_dec x xs).
destruct (eq_nat_dec a x); destruct (eq_nat_dec x a); try subst; try contradiction; auto.
subst x; rewrite H4; auto.
destruct (eq_nat_dec a x); destruct (eq_nat_dec x a); try subst; try contradiction; auto.
subst x; auto.
apply IHxs with (s1 := s1); auto; intro x; specialize (H3 x).
simpl in H3.
destruct (eq_nat_dec a x); subst.
destruct (In_dec eq_nat_dec x xs); try contradiction.
rewrite H4; rewrite H5; auto.
destruct (In_dec eq_nat_dec x xs); auto.
Qed.

Proposition taint_vars_same_values : forall K s, same_values s (taint_vars K s) (modifies K).
Proof.
intros; intro x; unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies K)); destruct (s x) as [[v l]|]; auto.
Qed.

Proposition no_lbls_taint_vars : forall P K i s h,
no_lbls P (modifies K) = true -> aden P (St i s h) -> aden P (St i (taint_vars K s) h).
Proof.
intros; apply no_lbls_same_values with (xs := modifies K) (s1 := s); auto.
apply taint_vars_same_values.
Qed.

Proposition taint_vars_assert_inv : forall P K l l' i s h, gleq l l' = false ->
aden (taint_vars_assert P (modifies K) l l') (St i s h) -> s = taint_vars K s.
Proof.
unfold taint_vars_assert; intros.
rewrite H in H0; simpl in H0; destruct H0.
extensionality x; unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies K)); auto.
apply aden_fold_inv with (x := x) in H1; auto.
simpl in H1.
destruct H1 as [vx [lx [H1]]].
rewrite H1.
destruct l; destruct l'; destruct lx; auto; inv H2; inv H.
Qed.

Proposition taint_vars_idempotent : forall K s, taint_vars K (taint_vars K s) = taint_vars K s.
Proof.
unfold taint_vars; intros.
extensionality x; destruct (In_dec eq_nat_dec x (modifies K)); auto.
destruct (s x) as [[v l]|]; auto.
Qed.

Inductive judge : nat -> context -> assert -> cmd -> assert -> Prop :=
| Judge_skip : forall pc, judge 0 pc Emp Skip Emp
| Judge_output : forall e, judge 0 Lo (LblExp e Lo `AND` Emp) (Output e) (LblExp e Lo `AND` Emp)
| Judge_assign : forall x e e' pc L, expvars e' x = false ->
judge 0 pc (BoolExp (Eq e e') `AND` LblExp e L `AND` Emp) (Assign x e)
(BoolExp (Eq (Var x) e') `AND` LblEq x (Lub L pc) `AND` Emp)
| Judge_read : forall x e e1 e2 pc L1 L2, expvars e1 x = false -> expvars e2 x = false ->
judge 0 pc (BoolExp (Eq (Var x) e1) `AND` LblExp e L1 `AND` Mapsto e e2 L2) (Read x e)
(BoolExp (Eq (Var x) e2) `AND` LblEq x (Lub (Lub L1 L2) pc) `AND` Mapsto (ereplace e x e1) e2 L2)
| Judge_write : forall e1 e2 pc L1 L2,
judge 0 pc (LblExp e1 L1 `AND` LblExp e2 L2 `AND` Allocated e1) (Write e1 e2)
(Mapsto e1 e2 (Lub (Lub L1 L2) pc))
| Judge_seq : forall N1 N2 P Q R C1 C2 pc, judge N1 pc P C1 Q -> judge N2 pc Q C2 R -> judge (S (N1+N2)) pc P (Seq C1 C2) R
| Judge_if : forall N1 N2 P Q b C1 C2 pc (lt lf : glbl),
implies P (BoolExp b `OR` BoolExp (Not b)) ->
implies (BoolExp b `AND` P) (LblBexp b lt) -> implies (BoolExp (Not b) `AND` P) (LblBexp b lf) ->
(gleq (glub lt lf) pc = false -> no_lbls P (modifies [If b C1 C2]) = true) ->
judge N1 (glub lt pc) (BoolExp b `AND` taint_vars_assert P (modifies [If b C1 C2]) lt pc) C1 Q ->
judge N2 (glub lf pc) (BoolExp (Not b) `AND` taint_vars_assert P (modifies [If b C1 C2]) lf pc) C2 Q ->
judge (S (N1+N2)) pc P (If b C1 C2) Q
| Judge_while : forall N P b C pc (l : glbl),
implies P (LblBexp b l) -> (gleq l pc = false -> no_lbls P (modifies [While b C]) = true) ->
judge N (glub l pc) (BoolExp b `AND` taint_vars_assert P (modifies [While b C]) l pc) C
(taint_vars_assert P (modifies [While b C]) l pc) ->
judge (S N) pc P (While b C) (BoolExp (Not b) `AND` taint_vars_assert P (modifies [While b C]) l pc)
| Judge_conseq : forall N P P' Q Q' C pc, implies P' P -> implies Q Q' -> judge N pc P C Q -> judge (S N) pc P' C Q'
| Judge_conj : forall N1 N2 P1 P2 Q1 Q2 C pc, judge N1 pc P1 C Q1 -> judge N2 pc P2 C Q2 ->
judge (S (N1+N2)) pc (P1 `AND` P2) C (Q1 `AND` Q2)
| Judge_frame : forall N P Q R C pc, judge N pc P C Q -> (forall x, In x (modifies [C]) -> vars R x = false) ->
judge (S N) pc (P ** R) C (Q ** R).

Inductive sound : context -> assert -> cmd -> assert -> Prop :=
| Jden_hi : forall P C Q,
(forall st, aden P st -> hsafe (Cf st C [])) ->
(forall n st st', aden P st -> hstepn n (Cf st C []) (Cf st' Skip []) -> aden Q st') ->
sound Hi P C Q
| Jden_lo : forall P C Q,
(forall st, aden P st -> lsafe (Cf st C [])) ->
(forall n st st' o, aden P st -> lstepn n (Cf st C []) (Cf st' Skip []) o -> aden Q st') ->
(forall n st1 st2 st1' st2' C' K' o1 o2, aden2 P st1 st2 ->
lstepn n (Cf st1 C []) (Cf st1' C' K') o1 -> lstepn n (Cf st2 C []) (Cf st2' C' K') o2 ->
diverge (Cf st1 C []) \/ diverge (Cf st2 C []) \/ side_condition C' st1' st2') ->
(forall n1 n2 st1 st2 st1' st2' o1 o2, aden2 P st1 st2 -> side_condition C st1 st2 ->
lstepn n1 (Cf st1 C []) (Cf st1' Skip []) o1 -> lstepn n2 (Cf st2 C []) (Cf st2' Skip []) o2 ->
obs_eq st1' st2' /\ o1 = o2) ->
(forall n st1 st2 st1' C' K' o1, aden2 P st1 st2 ->
lstepn n (Cf st1 C []) (Cf st1' C' K') o1 ->
diverge (Cf st1 C []) \/ diverge (Cf st2 C []) \/
exists st2', exists o2, lstepn n (Cf st2 C []) (Cf st2' C' K') o2) ->
(forall n1 n2 i1 s1 h1 i1' s1' h1' i2 s2 h2 i2' s2' h2' o1 o2 a,
aden2 P (St i1 s1 h1) (St i2 s2 h2) ->
lstepn n1 (Cf (St i1 s1 h1) C []) (Cf (St i1' s1' h1') Skip []) o1 ->
lstepn n2 (Cf (St i2 s2 h2) C []) (Cf (St i2' s2' h2') Skip []) o2 ->
h1 a <> h1' a -> (exists v, h1' a = Some (v,Lo)) -> h2 a <> None) ->
sound Lo P C Q.

Lemma soundness_skip : forall ct, sound ct Emp Skip Emp.
Proof.
destruct ct.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H0.
inv H1.
inv H2.
inv H0; auto.
inv H1.
right; right; inv H0; simpl; auto.
inv H2.
inv H1.
inv H2.
inv H; intuit.
inv H1.
inv H3.
right; right; inv H0.
exists st2; exists []; apply LStep_zero.
inv H1.
inv H0.
inv H4.

apply Jden_hi; intros.
unfold hsafe; intros.
inv H0.
inv H1.
inv H2.
inv H0; auto.
inv H1.
Qed.

Lemma soundness_output : forall e, sound Lo (LblExp e Lo `AND` Emp) (Output e) (LblExp e Lo `AND` Emp).
Proof.
intros.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H0.
destruct st as [i s h].
destruct H as [[v]].
apply (Can_lstep _ (Cf (St i s h) Skip []) [v]); apply LStep_output; auto.
inv H2.
inv H3.
inv H1.
inv H0.
inv H0.
inv H1.
inv H2; auto.
inv H0.
right; right; inv H0; simpl; auto.
inv H2.
inv H3; simpl; auto.
inv H0.
inv H1.
inv H3.
inv H4.
inv H2.
inv H1.
inv H3.
inv H.
destruct H2.
dup (obs_eq_exp e _ _ _ _ _ _ H2).
rewrite H9 in H3; rewrite H8 in H3; destruct H3.
apply H4 in H3; subst; split; auto.
inv H1.
inv H1.
right; right; inv H0.
exists st2; exists []; apply LStep_zero.
inv H1.
inv H2.
destruct H.
destruct H0.
destruct st2 as [i2 s2 h2].
destruct H0 as [[v2]].
exists (St i2 s2 h2); exists ([v2]++[]); apply LStep_succ with (cf' := Cf (St i2 s2 h2) Skip []).
apply LStep_output; auto.
apply LStep_zero.
inv H0.
inv H0.
inv H4.
inv H5.
inv H0.
Qed.

Lemma soundness_assign : forall e e' x L ct, expvars e' x = false ->
sound ct (BoolExp (Eq e e') `AND` LblExp e L `AND` Emp) (Assign x e)
(BoolExp (Eq (Var x) e') `AND` LblEq x (Lub L ct) `AND` Emp).
Proof.
intros; destruct ct.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H1.
destruct st as [i s h]; destruct H0 as [[H0 [v]]].
apply (Can_lstep _ (Cf (St i (upd s x (v, lden L i)) h) Skip []) []).
apply LStep_assign; auto.
inv H3.
inv H4.
inv H2.
inv H1.
inv H1.
inv H2.
inv H3.
simpl in *.
decomp H0; subst.
destruct H4 as [v'].
rewrite H0 in H9; inv H9.
repeat (split; auto).
rewrite edenZ_upd; auto.
unfold upd.
destruct (eq_nat_dec x x); simpl.
destruct (edenZ e' i s).
assert (exists l, eden e i s = Some (v,l)).
exists (lden L i); auto.
rewrite <- edenZ_some in H1; rewrite H1 in H3; simpl in H3.
destruct (Z_eq_dec v z); auto.
destruct (edenZ e i s); inv H3.
exists v; unfold upd.
destruct (eq_nat_dec x x).
destruct (lden L i); auto.
inv H1.
right; right; inv H1; simpl; auto.
inv H3.
inv H4; simpl; auto.
inv H1.
inv H2.
inv H4.
inv H5.
inv H3.
inv H2.
inv H4.
split; auto.
inv H0.
destruct H3.
dup (obs_eq_exp e _ _ _ _ _ _ H3).
rewrite H11 in H4; rewrite H10 in H4.
destruct H4; subst.
dup H3; inv H3.
simpl in *; subst; destruct H7.
repeat (split; auto).
intro y; simpl.
unfold upd; destruct (eq_nat_dec y x); subst; intuit.
apply H4.
inv H2.
inv H2.
right; right; inv H1.
exists st2; exists []; apply LStep_zero.
inv H2.
inv H3.
destruct st2 as [i' s' h'].
inv H0.
destruct H2.
simpl in H0; decomp H0.
destruct H6 as [v']; exists (St i' (upd s' x (v',lden L i')) h'); exists ([]++[]).
apply LStep_succ with (cf' := Cf (St i' (upd s' x (v',lden L i')) h') Skip []).
apply LStep_assign; auto.
apply LStep_zero.
inv H1.
inv H1.
inv H5.
inv H6.
inv H1.

apply Jden_hi; intros.
unfold hsafe; intros.
inv H1.
destruct st as [i s h]; destruct H0 as [[H0 [v]]].
apply (Can_hstep _ (Cf (St i (upd s x (v,Hi)) h) Skip [])).
apply HStep_assign with (l := lden L i); auto.
inv H3.
inv H4.
inv H2.
inv H1.
inv H1.
inv H2.
inv H3.
simpl in *.
decomp H0; subst.
destruct H4 as [v'].
rewrite H0 in H8; inv H8.
repeat (split; auto).
rewrite edenZ_upd; auto.
unfold upd.
destruct (eq_nat_dec x x); simpl.
destruct (edenZ e' i s).
assert (exists l, eden e i s = Some (v,l)).
exists (lden L i); auto.
rewrite <- edenZ_some in H1; rewrite H1 in H3; simpl in H3.
destruct (Z_eq_dec v z); auto.
destruct (edenZ e i s); inv H3.
exists v; unfold upd.
destruct (eq_nat_dec x x).
destruct (lden L i); auto.
inv H1.
Qed.

Lemma soundness_read : forall ct e e1 e2 x L1 L2, expvars e1 x = false -> expvars e2 x = false ->
sound ct (BoolExp (Eq (Var x) e1) `AND` LblExp e L1 `AND` Mapsto e e2 L2)
(Read x e) (BoolExp (Eq (Var x) e2) `AND` LblEq x (Lub (Lub L1 L2) ct)
`AND` Mapsto (ereplace e x e1) e2 L2).
Proof.
destruct ct; intros.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H2.
destruct st as [i s h].
destruct H1 as [[H1 [v]]].
destruct H4 as [v' [pf [H4 [v'' [H5]]]]].
apply (Can_lstep _ (Cf (St i (upd s x (v'', lden L1 i \_/ lden L2 i)) h) Skip []) []).
rewrite edenZ_some in H4; destruct H4 as [l'].
rewrite H4 in H2; inv H2.
apply LStep_read with (v1 := v) (pf := pf); auto.
destruct (eq_nat_dec (nat_of_Z v pf) (nat_of_Z v pf)); auto.
inv H4.
inv H5.
inv H3.
inv H2.
inv H2.
inv H3.
inv H4.
inv H1.
destruct H2 as [H2 [v]].
destruct H3 as [v' [pf' [H3 [v'' [H4]]]]].
rewrite edenZ_some in H3; destruct H3 as [l'].
rewrite H3 in H1; inv H1.
rewrite H3 in H10; inv H10.
rewrite (proof_irrelevance _ pf' pf) in H11.
destruct (eq_nat_dec (nat_of_Z v1 pf) (nat_of_Z v1 pf)); inv H11.
simpl; repeat split.
unfold upd at 1; destruct (eq_nat_dec x x); simpl.
rewrite edenZ_upd; auto; rewrite H4.
destruct (Z_eq_dec v2 v2); auto.
exists v2; unfold upd; destruct (eq_nat_dec x x).
destruct (lden L1 i \_/ lden L2 i); auto.
exists v1; exists pf; split.
rewrite edenZ_upd.
rewrite edenZ_ereplace.
rewrite edenZ_some; exists (lden L1 i); auto.
simpl in H2 |- *.
destruct (s x); destruct (edenZ e1 i s); simpl in H2 |- *; try solve [inv H2].
destruct (Z_eq_dec (fst v) z); inv H2; auto.
apply ereplace_deletes; auto.
exists v2; split.
rewrite edenZ_upd; auto.
rewrite (proof_irrelevance _ pf' pf); auto.
inv H2.
right; right; inv H2.
inv H3; simpl.
inv H1.
destruct H3.
destruct st1' as [i1 s1 h1]; destruct st2' as [i2 s2 h2]; simpl in *.
decomp H2; decomp H1.
destruct H5 as [v1 [pf1 [H5 [v1' [H10]]]]].
destruct H4 as [v2 [pf2 [H4 [v2' [H11]]]]].
apply edenZ_some in H5; destruct H5 as [l1].
apply edenZ_some in H4; destruct H4 as [l2].
destruct H7 as [v3]; destruct H9 as [v4].
rewrite H5 in H7; inv H7; rewrite H4 in H9; inv H9.
rewrite H5; rewrite H4.
rewrite (proof_irrelevance _ g pf1); rewrite (proof_irrelevance _ g0 pf2).
destruct (eq_nat_dec (nat_of_Z v3 pf1) (nat_of_Z v3 pf1)); intuit.
destruct (eq_nat_dec (nat_of_Z v4 pf2) (nat_of_Z v4 pf2)); intuit.
destruct H3.
simpl in H1; subst; auto.
inv H4.
inv H5; simpl; auto.
inv H2.
inv H3.
inv H5.
inv H6.
inv H4.
inv H3.
inv H5.
split; auto.
simpl in H2.
rewrite H12 in H2; rewrite H11 in H2.
rewrite (proof_irrelevance _ g pf) in H2; rewrite H13 in H2.
rewrite (proof_irrelevance _ g0 pf0) in H2; rewrite H14 in H2; subst.
destruct H1.
destruct H2.
dup H3; destruct H3.
destruct H5; repeat (split; auto).
intro y; simpl.
unfold upd; destruct (eq_nat_dec y x); subst.
dup (obs_eq_exp e _ _ _ _ _ _ H4).
rewrite H12 in H7; rewrite H11 in H7.
destruct H7; subst; intuition.
glub_simpl H7; subst.
specialize (H8 (refl_equal _)); subst.
rewrite (proof_irrelevance _ pf0 pf) in H14.
specialize (H6 (nat_of_Z v0 pf)); simpl in H6.
rewrite H13 in H6; rewrite H14 in H6; intuit.
apply H5.
inv H3.
inv H3.
right; right; inv H2.
exists st2; exists []; apply LStep_zero.
inv H3.
inv H4.
inv H1.
destruct H3.
destruct st2 as [i' s' h']; simpl in H1.
decomp H1.
destruct H5 as [v1' [pf1 [H5 [v1'' [H12]]]]].
exists (St i' (upd s' x (v1'', lden L1 i' \_/ lden L2 i')) h'); exists ([]++[]).
apply LStep_succ with (cf' := Cf (St i' (upd s' x (v1'', lden L1 i' \_/ lden L2 i')) h') Skip []).
apply LStep_read with (v1 := v1') (pf := pf1).
apply edenZ_some in H5.
destruct H7 as [v']; destruct H5 as [l'].
rewrite H5 in H4; inv H4; auto.
subst; destruct (eq_nat_dec (nat_of_Z v1' pf1) (nat_of_Z v1' pf1)); auto.
apply LStep_zero.
inv H2.
inv H2.
inv H6.
inv H7.
inv H2.

apply Jden_hi; intros.
unfold hsafe; intros.
inv H2.
destruct st as [i s h].
destruct H1 as [[H1 [v]]].
destruct H4 as [v' [pf [H4 [v'' [H5]]]]].
apply (Can_hstep _ (Cf (St i (upd s x (v'',Hi)) h) Skip [])).
rewrite edenZ_some in H4; destruct H4 as [l'].
rewrite H4 in H2; inv H2.
apply HStep_read with (v1 := v) (pf := pf) (l1 := lden L1 i) (l2 := lden L2 i); auto.
destruct (eq_nat_dec (nat_of_Z v pf) (nat_of_Z v pf)); auto.
inv H4.
inv H5.
inv H3.
inv H2.
inv H2.
inv H3.
inv H4.
inv H1.
destruct H2 as [H2 [v]].
destruct H3 as [v' [pf' [H3 [v'' [H4]]]]].
rewrite edenZ_some in H3; destruct H3 as [l'].
rewrite H3 in H1; inv H1.
rewrite H3 in H9; inv H9.
rewrite (proof_irrelevance _ pf' pf) in H10.
destruct (eq_nat_dec (nat_of_Z v1 pf) (nat_of_Z v1 pf)); inv H10.
simpl; repeat split.
unfold upd at 1; destruct (eq_nat_dec x x); simpl.
rewrite edenZ_upd; auto; rewrite H4.
destruct (Z_eq_dec v2 v2); auto.
exists v2; unfold upd; destruct (eq_nat_dec x x).
destruct (lden L1 i \_/ lden L2 i); auto.
exists v1; exists pf; split.
rewrite edenZ_upd.
rewrite edenZ_ereplace.
rewrite edenZ_some; exists (lden L1 i); auto.
simpl in H2 |- *.
destruct (s x); destruct (edenZ e1 i s); simpl in H2 |- *; try solve [inv H2].
destruct (Z_eq_dec (fst v) z); inv H2; auto.
apply ereplace_deletes; auto.
exists v2; split.
rewrite edenZ_upd; auto.
rewrite (proof_irrelevance _ pf' pf); auto.
inv H2.
Qed.

Lemma soundness_write : forall e1 e2 ct L1 L2,
sound ct (LblExp e1 L1 `AND` LblExp e2 L2 `AND` Allocated e1) (Write e1 e2)
(Mapsto e1 e2 (Lub (Lub L1 L2) ct)).
Proof.
destruct ct; intros.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H0.
destruct st as [i s h].
simpl in H; decomp H.
destruct H2 as [v' [pf [H2 [v'' [l'']]]]].
destruct H3 as [v1]; destruct H4 as [v2].
apply (Can_lstep _ (Cf (St i s (upd h (nat_of_Z v' pf) (v2, lden L1 i \_/ lden L2 i))) Skip []) []).
rewrite edenZ_some in H2; destruct H2 as [l'].
rewrite H0 in H2; inv H2.
apply LStep_write; auto.
destruct (eq_nat_dec (nat_of_Z v' pf) (nat_of_Z v' pf)); auto; try discriminate.
inv H2.
inv H3.
inv H1.
inv H0.
inv H0.
inv H1.
inv H2.
simpl in H; decomp H.
destruct H2 as [v1']; destruct H3 as [v2'].
destruct H1 as [v' [pf' [H1 [v'' [l'']]]]].
rewrite edenZ_some in H1; destruct H1 as [l'].
rewrite H1 in H; inv H.
rewrite H0 in H9; inv H9.
rewrite H1 in H8; inv H8.
simpl.
exists v1; exists pf; split.
rewrite edenZ_some; exists (lden L1 i); auto.
exists v2; split.
rewrite edenZ_some; exists (lden L2 i); auto.
unfold upd; rewrite (proof_irrelevance _ pf' pf); extensionality n.
destruct (eq_nat_dec n (nat_of_Z v1 pf)); auto.
destruct (lden L1 i \_/ lden L2 i); auto.
inv H0.
right; right; inv H0.
inv H1; simpl; auto.
inv H2.
inv H3; simpl; auto.
inv H0.
inv H1.
inv H3.
inv H4.
inv H2.
inv H1.
inv H3.
split; auto.
destruct H.
destruct H1.
dup H2; destruct H2.
destruct H4; repeat (split; auto).
intro n; simpl.
dup (obs_eq_exp e1 _ _ _ _ _ _ H3).
dup (obs_eq_exp e2 _ _ _ _ _ _ H3).
rewrite H10 in H6; rewrite H9 in H6.
rewrite H13 in H7; rewrite H11 in H7.
destruct H6; destruct H7; subst.
simpl in H2; unfold upd; destruct (eq_nat_dec n (nat_of_Z v1 pf)); subst.
destruct l0.
specialize (H8 (refl_equal _)); subst.
rewrite (proof_irrelevance _ pf0 pf).
destruct (eq_nat_dec (nat_of_Z v0 pf) (nat_of_Z v0 pf)); auto.
destruct (eq_nat_dec (nat_of_Z v1 pf) (nat_of_Z v0 pf0)); intros.
inv H2.
destruct (h0 (nat_of_Z v1 pf)) as [[v l]|]; auto; intros.
inv H2.
specialize (H5 n); simpl in H5.
destruct (h n) as [[v l]|]; auto.
destruct l0.
specialize (H8 (refl_equal _)); subst.
destruct (eq_nat_dec n (nat_of_Z v0 pf0)); subst.
contradiction n0; rewrite (proof_irrelevance _ pf0 pf); auto.
destruct (h0 n) as [[v' l']|]; auto.
destruct (eq_nat_dec n (nat_of_Z v0 pf0)); subst; auto.
intros.
inv H6.
inv H1.
inv H1.
right; right; inv H0.
exists st2; exists []; apply LStep_zero.
inv H1.
inv H2.
inv H.
destruct H1.
destruct st2 as [i' s' h']; simpl in H.
decomp H.
destruct H3 as [v1' [pf1 [H3 [v1'' [l1'']]]]].
destruct H4 as [v3]; destruct H5 as [v4].
exists (St i' s' (upd h' (nat_of_Z v1' pf1) (v4, lden L1 i' \_/ lden L2 i'))); exists ([]++[]).
apply LStep_succ with (cf' := Cf (St i' s' (upd h' (nat_of_Z v1' pf1) (v4, lden L1 i' \_/ lden L2 i'))) Skip []).
apply LStep_write; auto.
rewrite edenZ_some in H3; destruct H3 as [l''].
rewrite H2 in H3; inv H3; auto.
subst.
destruct (eq_nat_dec (nat_of_Z v1' pf1) (nat_of_Z v1' pf1)); auto; try discriminate.
apply LStep_zero.
inv H0.
inv H0; inv H1.
inv H4; inv H0.
inv H5.
inv H6.
unfold upd in H2, H3.
destruct H3 as [v]; destruct (eq_nat_dec a (nat_of_Z v1 pf)).
inv H0.
glub_simpl H4; subst.
inv H.
destruct H1.
dup (obs_eq_exp e1 _ _ _ _ _ _ H1).
rewrite H13 in H3; rewrite H14 in H3; destruct H3.
specialize (H4 (refl_equal _)); subst.
rewrite (proof_irrelevance _ pf pf0); auto.
inv H0.
inv H0.

apply Jden_hi; intros.
unfold hsafe; intros.
inv H0.
destruct st as [i s h].
simpl in H; decomp H.
destruct H2 as [v' [pf [H2 [v'' [l'']]]]].
destruct H3 as [v1]; destruct H4 as [v2].
apply (Can_hstep _ (Cf (St i s (upd h (nat_of_Z v' pf) (v2, Hi))) Skip [])).
rewrite edenZ_some in H2; destruct H2 as [l'].
rewrite H0 in H2; inv H2.
apply HStep_write with (l1 := lden L1 i) (l2 := lden L2 i); auto.
destruct (eq_nat_dec (nat_of_Z v' pf) (nat_of_Z v' pf)); auto; try discriminate.
inv H2.
inv H3.
inv H1.
inv H0.
inv H0.
inv H1.
inv H2.
simpl in H; decomp H.
destruct H2 as [v1']; destruct H3 as [v2'].
destruct H1 as [v' [pf' [H1 [v'' [l'']]]]].
rewrite edenZ_some in H1; destruct H1 as [l'].
rewrite H1 in H; inv H.
rewrite H0 in H8; inv H8.
rewrite H1 in H7; inv H7.
simpl.
exists v1; exists pf; split.
rewrite edenZ_some; exists (lden L1 i); auto.
exists v2; split.
rewrite edenZ_some; exists (lden L2 i); auto.
unfold upd; rewrite (proof_irrelevance _ pf' pf); extensionality n.
destruct (eq_nat_dec n (nat_of_Z v1 pf)); auto.
destruct (lden L1 i \_/ lden L2 i); auto.
inv H0.
Qed.

Lemma soundness_seq : forall N1 N2 P Q R C1 C2 ct,
(forall y : nat, y < S (N1 + N2) ->
forall (ct : context) (P : assert) (C : cmd) (Q : assert),
judge y ct P C Q -> sound ct P C Q) ->
judge N1 ct P C1 Q -> judge N2 ct Q C2 R -> sound ct P (Seq C1 C2) R.
Proof.
intros.
rename H1 into H2; rename H0 into H1; destruct ct.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H3.
apply (Can_lstep _ (Cf st C1 [C2]) []); apply LStep_seq.
inv H5.
change (lstepn n0 (Cf st C1 ([]++[C2])) cf' o') in H6.
destruct cf' as [st' C' K']; apply lstep_trans_inv in H6.
destruct H6.
destruct H3 as [K [H3]]; subst.
apply H in H1; try omega; inv H1.
case_eq (halt_config (Cf st' C' K)); intros.
destruct C'; destruct K; inv H1.
apply (Can_lstep _ (Cf st' C2 []) []); apply LStep_skip.
specialize (H5 st H0 _ _ _ H3 H1).
inv H5.
destruct cf' as [st'' C'' K''].
apply lstep_extend with (K0 := [C2]) in H11.
apply (Can_lstep _ (Cf st'' C'' (K''++[C2])) o); auto.
destruct H3 as [st'' [n1 [n2 [o1 [o2]]]]]; decomp H3; subst.
apply H in H1; try omega; apply H in H2; try omega; inv H1; inv H2.
apply H6 in H5; auto.
apply H1 in H5.
inv H7.
apply (Can_lstep _ (Cf st' C2 []) []); apply LStep_skip.
inv H2.
apply H5 in H17; auto.
inv H3.
inv H4.
change (lstepn n0 (Cf st C1 ([]++[C2])) (Cf st' Skip []) o') in H5.
apply lstep_trans_inv in H5; destruct H5.
destruct H3 as [K [H3]].
apply sym_eq in H4; apply app_eq_nil in H4; destruct H4.
inv H5.
destruct H3 as [st'' [n1 [n2 [o1 [o2]]]]]; decomp H3; subst.
apply H in H1; try omega; apply H in H2; try omega; inv H1; inv H2.
inv H6.
inv H2.
apply H5 in H4; auto; apply H11 in H16; auto.
inv H3; simpl; auto.
inv H5.
inv H4.
inv H5.
change (lstepn n0 (Cf st1 C1 ([]++[C2])) (Cf st1' C' K') o') in H6.
change (lstepn n0 (Cf st2 C1 ([]++[C2])) (Cf st2' C' K') o'0) in H7.
apply lstep_trans_inv in H6; apply lstep_trans_inv in H7.
destruct H6.
destruct H7.
destruct H3 as [K1 [H3]]; destruct H4 as [K2 [H4]]; subst.
apply app_cancel_r in H6; subst.
apply H in H1; try omega; inv H1.
dup (H7 _ _ _ _ _ _ _ _ _ H0 H3 H4).
decomp H1; auto.
left; apply diverge_seq1; auto.
right; left; apply diverge_seq1; auto.
destruct H3 as [K1 [H3]].
destruct H4 as [st'' [n1 [n2 [o1 [o2]]]]]; decomp H4; subst.
apply H in H1; try omega; inv H1.
apply H10 with (st2 := st1) in H6.
decomp H6.
right; left; apply diverge_seq1; auto.
left; apply diverge_seq1; auto.
destruct H12 as [st2'' [o2']].
apply lstep_trans_inv' in H3.
destruct H3 as [cf'' [o1'' [o2'']]]; decomp H3.
destruct (lstepn_det _ _ _ _ _ _ H6 H1); subst.
inv H13; simpl; auto.
inv H3.
inv H0.
destruct H12; split; auto; split; auto.
apply obs_eq_sym; auto.
destruct H7.
destruct H4 as [K1 [H5]].
destruct H3 as [st'' [n1 [n2 [o1 [o2]]]]]; decomp H3; subst.
apply H in H1; try omega; inv H1.
apply H10 with (st2 := st2) in H6; auto.
decomp H6.
left; apply diverge_seq1; auto.
right; left; apply diverge_seq1; auto.
destruct H12 as [st2'' [o2']].
apply lstep_trans_inv' in H5.
destruct H5 as [cf'' [o1'' [o2'']]]; decomp H5.
destruct (lstepn_det _ _ _ _ _ _ H6 H1); subst.
inv H13; simpl; auto.
inv H5.
destruct H3 as [st1'' [n1 [n2 [o1 [o2]]]]]; decomp H3; subst.
destruct H4 as [st2'' [n3 [n4 [o3 [o4]]]]].
decomp H3; subst.
apply H in H1; try omega; inv H1.
apply H in H2; try omega; inv H2.
assert (n1 = n3).
dup H5; apply H12 with (st2 := st2) in H5; auto.
decomp H5.
apply (False_ind _ (diverge_halt _ _ _ _ H19 H2)).
apply (False_ind _ (diverge_halt _ _ _ _ H20 H4)).
destruct H20 as [st2''' [o2']].
apply (lstepn_det_term _ _ _ _ _ _ _ H5 H4).
assert (n2 = n4); subst; try omega.
destruct n4.
inv H8; simpl; auto.
inv H8.
inv H19.
inv H7.
inv H8.
dup H0; inv H0.
destruct H8; split; try split.
apply (H9 _ _ _ _ H7 H5).
apply (H9 _ _ _ _ H0 H4).
apply (H11 n3 n3 st1 st2 st1'' st2'' o1 o3); auto.
repeat (split; auto).
decomp (H10 0 st1 st2 st1 st2 C1 [] [] [] H2 (LStep_zero _) (LStep_zero _)); auto.
apply (False_ind _ (diverge_halt _ _ _ _ H21 H5)).
apply (False_ind _ (diverge_halt _ _ _ _ H22 H4)).
decomp (H15 n4 st1'' st2'' st1' st2' C' K' o'0 o' H2 H19 H20); auto.
left; apply diverge_seq2 with (st' := st1'') (n := n3) (o := o1); auto.
right; left; apply diverge_seq2 with (st' := st2'') (n := n3) (o := o3); auto.
inv H4.
inv H6.
inv H5.
inv H4.
change (lstepn n (Cf st1 C1 ([]++[C2])) (Cf st1' Skip []) o') in H7.
change (lstepn n0 (Cf st2 C1 ([]++[C2])) (Cf st2' Skip []) o'0) in H6.
apply lstep_trans_inv in H7; apply lstep_trans_inv in H6.
destruct H7.
destruct H4 as [K1 [H4]].
apply f_equal with (f := fun l => length l) in H5; simpl in H5.
destruct K1; inv H5.
destruct H6.
destruct H5 as [K2 [H5]].
apply f_equal with (f := fun l => length l) in H6; simpl in H6.
destruct K2; inv H6.
destruct H4 as [st1'' [n1 [n2 [o1 [o2]]]]]; decomp H4; subst.
destruct H5 as [st2'' [n3 [n4 [o3 [o4]]]]].
decomp H4; subst.
apply H in H1; try omega; inv H1.
apply H in H2; try omega; inv H2.
assert (n1 = n3).
dup H6; apply H12 with (st2 := st2) in H6; auto.
decomp H6.
apply (False_ind _ (diverge_halt _ _ _ _ H19 H2)).
apply (False_ind _ (diverge_halt _ _ _ _ H20 H5)).
destruct H20 as [st [o]].
apply (lstepn_det_term _ _ _ _ _ _ _ H6 H5).
subst.
inv H8.
inv H2.
inv H9.
inv H2.
assert (obs_eq st1' st2' /\ o' = o'0).
apply (H16 n n0 st1'' st2'' st1' st2' o' o'0); auto.
dup H0; inv H0.
destruct H20; split; try split.
apply H7 in H6; auto.
apply H7 in H5; auto.
apply (H11 n3 n3 st1 st2 st1'' st2'' o1 o3); auto.
repeat (split; auto).
decomp (H10 0 st1 st2 st1 st2 C1 [] [] [] H2 (LStep_zero _) (LStep_zero _)); auto.
apply (False_ind _ (diverge_halt _ _ _ _ H21 H6)).
apply (False_ind _ (diverge_halt _ _ _ _ H22 H5)).
dup H0; inv H0.
destruct H20; split; try split.
apply (H7 _ _ _ _ H9 H6).
apply (H7 _ _ _ _ H0 H5).
apply (H11 n3 n3 st1 st2 st1'' st2'' o1 o3); auto.
decomp (H10 0 st1 st2 st1 st2 C1 [] [] [] H2 (LStep_zero _) (LStep_zero _)); auto.
apply (False_ind _ (diverge_halt _ _ _ _ H21 H6)).
apply (False_ind _ (diverge_halt _ _ _ _ H22 H5)).
decomp (H15 0 st1'' st2'' st1'' st2'' C2 [] [] [] H2 (LStep_zero _) (LStep_zero _)); auto.
apply (False_ind _ (diverge_halt _ _ _ _ H9 H19)).
apply (False_ind _ (diverge_halt _ _ _ _ H20 H8)).
destruct H2; subst; split; auto.
assert (o1 = o3).
apply (H11 n3 n3 st1 st2 st1'' st2'' o1 o3); auto.
decomp (H10 0 st1 st2 st1 st2 C1 [] [] [] H0 (LStep_zero _) (LStep_zero _)); auto.
apply (False_ind _ (diverge_halt _ _ _ _ H9 H6)).
apply (False_ind _ (diverge_halt _ _ _ _ H20 H5)).
subst; auto.
inv H3.
right; right; exists st2; exists []; apply LStep_zero.
inv H4.
change (lstepn n0 (Cf st1 C1 ([]++[C2])) (Cf st1' C' K') o') in H5.
apply lstep_trans_inv in H5; destruct H5.
destruct H3 as [K'' [H3]]; subst.
apply H in H1; try omega; inv H1.
apply H8 with (st2 := st2) in H3; auto.
decomp H3.
left; apply diverge_seq1; auto.
right; left; apply diverge_seq1; auto.
right; right; destruct H10 as [st2' [o2]]; exists st2'; exists ([]++o2).
apply LStep_succ with (cf' := Cf st2 C1 [C2]); auto.
apply LStep_seq.
apply lstepn_extend with (K0 := [C2]) in H1; auto.
destruct H3 as [st1'' [n1 [n2 [o1 [o2]]]]]; decomp H3; subst.
apply H in H1; apply H in H2; try omega; inv H1; inv H2.
dup H4; apply H9 with (st2 := st2) in H4; auto.
decomp H4.
left; apply diverge_seq1; auto.
right; left; apply diverge_seq1; auto.
destruct H17 as [st2'' [o2']].
inv H6.
right; right; exists st2''; exists ([]++o2').
apply LStep_succ with (cf' := Cf st2 C1 [C2]).
apply LStep_seq.
assert (n1 + 0 = n1); try omega.
rewrite H6; apply lstepn_extend with (K0 := [C2]) in H4; auto.
inv H16.
apply H14 with (st2 := st2'') in H17.
decomp H17.
left; apply diverge_seq2 with (st' := st1'') (n := n1) (o := o1); auto.
right; left; apply diverge_seq2 with (st' := st2'') (n := n1) (o := o2'); auto.
right; right; destruct H16 as [st2' [o2'']].
exists st2'; exists ([]++o2'++[]++o2'').
apply LStep_succ with (cf' := Cf st2 C1 [C2]).
apply LStep_seq.
apply lstep_trans with (cf2 := Cf st2'' Skip [C2]).
apply lstepn_extend with (K0 := [C2]) in H4; auto.
apply LStep_succ with (cf' := Cf st2'' C2 []); auto.
apply LStep_skip.
dup H0; inv H0.
destruct H18; split; try split.
apply H5 in H2; auto.
apply H5 in H4; auto.
apply (H8 n1 n1 st1 st2 st1'' st2'' o1 o2'); auto.
decomp (H7 0 st1 st2 st1 st2 C1 [] [] [] H6 (LStep_zero _) (LStep_zero _)); auto.
apply (False_ind _ (diverge_halt _ _ _ _ H19 H2)).
apply (False_ind _ (diverge_halt _ _ _ _ H20 H4)).
apply H in H1; try omega; inv H1.
apply H in H2; try omega; inv H2.
inv H3; inv H4.
inv H2; inv H3.
change (lstepn n (Cf (St i1 s1 h1) C1 ([]++[C2])) (Cf (St i1' s1' h1') Skip []) o') in H18.
change (lstepn n0 (Cf (St i2 s2 h2) C1 ([]++[C2])) (Cf (St i2' s2' h2') Skip []) o'0) in H19.
apply lstep_trans_inv in H18; apply lstep_trans_inv in H19.
destruct H18.
destruct H2 as [K [H2]].
apply f_equal with (f := fun l => length l) in H3; simpl in H3.
destruct K; inv H3.
destruct H19.
destruct H3 as [K [H3]].
apply f_equal with (f := fun l => length l) in H4; simpl in H4.
destruct K; inv H4.
destruct H2 as [[i1'' s1'' h1''] [n1 [n2 [o1 [o2]]]]]; decomp H2.
destruct H3 as [[i2'' s2'' h2''] [n1' [n2' [o1' [o2']]]]].
decomp H2; subst.
inv H19; inv H22.
inv H2; inv H19.
destruct (opt_eq_dec val_eq_dec (h1 a) (h1'' a)).
destruct (opt_eq_dec val_eq_dec (h1'' a) (h1' a)).
apply (H17 _ n0 _ _ _ _ _ _ i2'' s2'' h2'' i2' s2' h2' _ o'0 a) in H18; auto.
intro; apply lstepn_nonincreasing with (a := a) in H3; auto.
split; try split.
destruct H0; apply H8 in H4; intuit.
destruct H0; apply H8 in H3; intuit.
decomp (H9 _ _ _ _ _ _ _ _ _ H0 (LStep_zero _) (LStep_zero _)).
apply (False_ind _ (diverge_halt _ _ _ _ H2 H4)).
apply (False_ind _ (diverge_halt _ _ _ _ H19 H3)).
destruct (H10 _ _ _ _ _ _ _ _ H0 H19 H4 H3); auto.
destruct (opt_eq_dec val_eq_dec (h1'' a) (h1' a)).
apply (H12 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ H0 H4 H3 n2).
rewrite e; auto.
apply (H17 _ n0 _ _ _ _ _ _ i2'' s2'' h2'' i2' s2' h2' _ o'0 a) in H18; auto.
intro; apply lstepn_nonincreasing with (a := a) in H3; auto.
split; try split.
destruct H0; apply H8 in H4; intuit.
destruct H0; apply H8 in H3; intuit.
decomp (H9 _ _ _ _ _ _ _ _ _ H0 (LStep_zero _) (LStep_zero _)).
apply (False_ind _ (diverge_halt _ _ _ _ H2 H4)).
apply (False_ind _ (diverge_halt _ _ _ _ H19 H3)).
destruct (H10 _ _ _ _ _ _ _ _ H0 H19 H4 H3); auto.

apply Jden_hi; intros.
unfold hsafe; intros.
inv H3.
apply (Can_hstep _ (Cf st C1 [C2])); apply HStep_seq.
inv H5.
change (hstepn n0 (Cf st C1 ([]++[C2])) cf') in H6.
destruct cf' as [st' C' K']; apply hstep_trans_inv in H6.
destruct H6.
destruct H3 as [K [H3]]; subst.
apply H in H1; try omega; inv H1.
case_eq (halt_config (Cf st' C' K)); intros.
destruct C'; destruct K; inv H1.
apply (Can_hstep _ (Cf st' C2 [])); apply HStep_skip.
specialize (H5 st H0 _ _ H3 H1).
inv H5.
destruct cf' as [st'' C'' K''].
apply hstep_extend with (K0 := [C2]) in H7.
apply (Can_hstep _ (Cf st'' C'' (K''++[C2]))); auto.
destruct H3 as [st'' [n1 [n2]]]; decomp H3; subst.
apply H in H1; try omega; apply H in H2; try omega; inv H1; inv H2.
apply H6 in H5; auto.
apply H1 in H5.
inv H7.
apply (Can_hstep _ (Cf st' C2 [])); apply HStep_skip.
inv H2.
apply H5 in H9; auto.
inv H3.
inv H4.
change (hstepn n0 (Cf st C1 ([]++[C2])) (Cf st' Skip [])) in H5.
apply hstep_trans_inv in H5; destruct H5.
destruct H3 as [K [H3]].
apply sym_eq in H4; apply app_eq_nil in H4; destruct H4.
inv H5.
destruct H3 as [st'' [n1 [n2]]]; decomp H3; subst.
apply H in H1; try omega; apply H in H2; try omega; inv H1; inv H2.
inv H6.
inv H2.
apply H5 in H4; auto; apply H7 in H8; auto.
Qed.

Lemma soundness_if : forall N1 N2 P Q b C1 C2 ct (lt lf : glbl),
(forall y : nat, y < S (N1 + N2) ->
forall (ct : context) (P : assert) (C : cmd) (Q : assert),
judge y ct P C Q -> sound ct P C Q) ->
implies P (BoolExp b `OR` BoolExp (Not b)) ->
implies (BoolExp b `AND` P) (LblBexp b lt) -> implies (BoolExp (Not b) `AND` P) (LblBexp b lf) ->
(gleq (glub lt lf) ct = false -> no_lbls P (modifies [If b C1 C2]) = true) ->
judge N1 (glub lt ct) (BoolExp b `AND` taint_vars_assert P (modifies [If b C1 C2]) lt ct) C1 Q ->
judge N2 (glub lf ct) (BoolExp (Not b) `AND` taint_vars_assert P (modifies [If b C1 C2]) lf ct) C2 Q ->
sound ct P (If b C1 C2) Q.
Proof.
intros.
rename H5 into H6; rename H4 into H5; rename H3 into H4;
rename H2 into H3; rename H1 into H2; rename H0 into H1; destruct ct.
apply Jden_lo; intros.
unfold lsafe; intros.
inv H7.
dup H0; apply H1 in H0.
destruct st as [i s h]; simpl in H0; destruct H0.
assert (aden (LblBexp b lt) (St i s h)).
apply H2; simpl; split; auto.
destruct H9 as [v].
rewrite bdenZ_some in H0; destruct H0 as [l].
rewrite H9 in H0; inv H0.
destruct l.
apply (Can_lstep _ (Cf (St i s h) C1 []) []).
apply LStep_if_true; auto.
destruct (dvg_ex_mid (taint_vars_cf (Cf (St i s h) (If b C1 C2) []))).
apply (Can_lstep _ (Cf (St i s h) (If b C1 C2) []) []).
apply LStep_if_hi_dvg with (v := true); auto.
unfold hsafe; intros.
apply H in H5; try omega; inv H5.
inv H10.
apply (Can_hstep _ (Cf (St i (taint_vars [If b C1 C2] s) h) C1 [])).
apply HStep_if_true with (l := Hi).
apply bden_taint_vars with (K := [If b C1 C2]) in H9; destruct H9 as [l [H9]].
destruct l; inv H5; auto.
inv H5.
apply H12 in H14; intuit.
simpl; split; try split.
rewrite bdenZ_some; exists l; auto.
apply no_lbls_taint_vars; auto.
simpl.
unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); try contradiction.
destruct (s x) as [[v1 l1]|].
exists v1; exists Hi; split; auto.
exists 0%Z; exists Hi; split; auto.
apply bden_taint_vars with (K := [If b C1 C2]) in H9.
destruct H9 as [l' [H9]].
rewrite H9 in H22; inv H22.
destruct H0 as [n [st]].
apply (Can_lstep _ (Cf st Skip []) []).
apply LStep_if_hi with (v := true) (n := n); auto.
unfold hsafe; intros.
apply H in H5; try omega; inv H5.
inv H10.
apply (Can_hstep _ (Cf (St i (taint_vars [If b C1 C2] s) h) C1 [])).
apply HStep_if_true with (l := Hi).
apply bden_taint_vars with (K := [If b C1 C2]) in H9; destruct H9 as [l [H9]].
destruct l; inv H5; auto.
inv H5.
apply H12 in H14; intuit.
simpl; split; try split.
rewrite bdenZ_some; exists l; auto.
apply no_lbls_taint_vars; auto.
simpl.
unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); try contradiction.
destruct (s x) as [[v1 l1]|].
exists v1; exists Hi; split; auto.
exists 0%Z; exists Hi; split; auto.
apply bden_taint_vars with (K := [If b C1 C2]) in H9.
destruct H9 as [l' [H9]].
rewrite H9 in H22; inv H22.
case_eq (bdenZ b i s); intros.
rewrite H9 in H0; inv H0.
destruct b0; inv H11.
apply bdenZ_some in H9; destruct H9 as [l]; destruct l.
apply (Can_lstep _ (Cf (St i s h) C2 []) []).
apply LStep_if_false; auto.
destruct (dvg_ex_mid (taint_vars_cf (Cf (St i s h) (If b C1 C2) []))).
apply (Can_lstep _ (Cf (St i s h) (If b C1 C2) []) []).
apply LStep_if_hi_dvg with (v := false); auto.
unfold hsafe; intros.
apply H in H6; try omega; inv H6.
inv H10.
apply (Can_hstep _ (Cf (St i (taint_vars [If b C1 C2] s) h) C2 [])).
apply HStep_if_false with (l := Hi).
apply bden_taint_vars with (K := [If b C1 C2]) in H0; destruct H0 as [l [H0]].
destruct l; inv H6; auto.
inv H6.
apply bden_taint_vars with (K := [If b C1 C2]) in H0.
destruct H0 as [l' [H0]].
rewrite H0 in H23; inv H23.
apply H13 in H15; intuit.
destruct lf; inv H12; simpl; split; try split.
assert (exists l, bden b i (taint_vars [If b C1 C2] s) = Some (false,l)).
exists l; auto.
rewrite <- bdenZ_some in H6; rewrite H6; auto.
apply no_lbls_taint_vars; auto.
apply H4.
destruct lt; auto.
simpl.
unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); try contradiction.
destruct (s x) as [[v1 l1]|].
exists v1; exists Hi; split; auto.
exists 0%Z; exists Hi; split; auto.
destruct lf; inv H12.
specialize (H3 (St i s h)); simpl in H3.
assert (exists v, bden b i s = Some (v,Lo)).
apply H3; split; auto.
case_eq (bdenZ b i s); intros.
destruct b0; simpl; auto.
apply bdenZ_some in H6; destruct H6 as [l].
rewrite H6 in H0; inv H0.
apply bdenZ_none in H6; rewrite H6 in H0; inv H0.
destruct H6 as [v]; rewrite H6 in H0; inv H0.
destruct H9 as [n [st]].
apply (Can_lstep _ (Cf st Skip []) []).
apply LStep_if_hi with (v := false) (n := n); auto.
unfold hsafe; intros.
apply H in H6; try omega; inv H6.
inv H10.
apply (Can_hstep _ (Cf (St i (taint_vars [If b C1 C2] s) h) C2 [])).
apply HStep_if_false with (l := Hi).
apply bden_taint_vars with (K := [If b C1 C2]) in H0; destruct H0 as [l [H0]].
destruct l; inv H6; auto.
inv H6.
apply bden_taint_vars with (K := [If b C1 C2]) in H0; destruct H0 as [l' [H0]].
rewrite H23 in H0; inv H0.
apply H13 in H15; intuit.
destruct lf; inv H12.
simpl; split; try split.
assert (exists l, bden b i (taint_vars [If b C1 C2] s) = Some (false,l)).
exists l; auto.
rewrite <- bdenZ_some in H6; rewrite H6; auto.
apply no_lbls_taint_vars; auto.
apply H4; destruct lt; auto.
simpl.
unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); try contradiction.
destruct (s x) as [[v1 l1]|].
exists v1; exists Hi; split; auto.
exists 0%Z; exists Hi; split; auto.
destruct lf; inv H12.
specialize (H3 (St i s h)); simpl in H3.
assert (exists v, bden b i s = Some (v,Lo)).
apply H3; split; auto.
case_eq (bdenZ b i s); intros.
destruct b0; simpl; auto.
apply bdenZ_some in H6; destruct H6 as [l].
rewrite H6 in H0; inv H0.
apply bdenZ_none in H6; rewrite H6 in H0; inv H0.
destruct H6 as [v]; rewrite H6 in H0; inv H0.
rewrite H9 in H0; inv H0.
inv H9.
apply H in H5; try omega; inv H5.
destruct lt; inv H7.
specialize (H2 (St i s h)); simpl in H2.
assert (exists v, bden b i s = Some (v,Hi)).
apply H2; split; auto.
rewrite bdenZ_some; exists Lo; auto.
destruct H5 as [v]; rewrite H5 in H17; inv H17.
apply H9 in H10; intuit.
destruct lt; inv H7.
unfold taint_vars_assert; simpl; split; auto.
rewrite bdenZ_some; exists Lo; auto.
apply H in H6; try omega; inv H6.
destruct lf; inv H7.
specialize (H3 (St i s h)); simpl in H3.
assert (exists v, bden b i s = Some (v,Hi)).
apply H3; split; auto.
case_eq (bdenZ b i s); intros.
destruct b0; auto.
apply bdenZ_some in H6; destruct H6 as [l].
rewrite H6 in H17; inv H17.
apply bdenZ_none in H6; rewrite H6 in H17; inv H17.
destruct H6 as [v]; rewrite H6 in H17; inv H17.
apply H9 in H10; intuit.
destruct lf; inv H7.
unfold taint_vars_assert; simpl; split; auto.
case_eq (bdenZ b i s); intros.
destruct b0; auto.
apply bdenZ_some in H6; destruct H6 as [l].
rewrite H6 in H17; inv H17.
apply bdenZ_none in H6; rewrite H6 in H17; inv H17.
inv H10.
inv H8.
inv H7.
generalize cf' o' H8 H10; clear cf' o' H8 H10.
induction n0; intros.
inv H10.
apply (Can_lstep _ (Cf (St i s h) (If b C1 C2) []) []).
apply LStep_if_hi_dvg with (v := v); auto.
inv H10.
inv H9.
rewrite H22 in H17; inv H17.
rewrite H22 in H17; inv H17.
inv H11.
inv H8.
inv H7.
apply IHn0 with (o' := o'0); auto.
inv H7.
inv H8.
apply H in H5; try omega; inv H5.
destruct lt; inv H7.
specialize (H2 (St i s h)); simpl in H2.
assert (exists v, bden b i s = Some (v,Hi)).
apply H2; split; auto.
rewrite bdenZ_some; exists Lo; auto.
destruct H5 as [v]; rewrite H5 in H16; inv H16.
apply H10 in H9; auto.
destruct lt; inv H7.
simpl; split; auto.
rewrite bdenZ_some; exists Lo; auto.
apply H in H6; try omega; inv H6.
destruct lf; inv H7.
specialize (H3 (St i s h)); simpl in H3.
assert (exists v, bden b i s = Some (v,Hi)).
apply H3; split; auto.
case_eq (bdenZ b i s); intros.
destruct b0; auto.
rewrite bdenZ_some in H6; destruct H6 as [l].
rewrite H6 in H16; inv H16.
rewrite bdenZ_none in H6; rewrite H6 in H16; inv H16.
destruct H6 as [v]; rewrite H6 in H16; inv H16.
apply H10 in H9; auto.
destruct lf; inv H7.
simpl; split; auto.
assert (exists l, bden b i s = Some (false,l)).
exists Lo; auto.
rewrite <- bdenZ_some in H6; rewrite H6; auto.
inv H9.
inv H18.
inv H7.
apply H in H5; try omega; inv H5.
apply H10 in H8; auto.
destruct lt; inv H7.
simpl; split; try split.
rewrite bdenZ_some; exists l; auto.
apply no_lbls_taint_vars; auto.
unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); try contradiction.
destruct (s x) as [[v1 l1]|].
exists v1; exists Hi; split; auto.
exists 0%Z; exists Hi; split; auto.
destruct lt; inv H7.
dup H16; apply bden_taint_vars with (K := [If b C1 C2]) in H16.
destruct H16 as [l' [H16]].
rewrite H16 in H19; inv H19.
destruct l; inv H7.
specialize (H2 (St i s h)); simpl in H2.
assert (exists v, bden b i s = Some (v,Lo)).
apply H2; split; auto.
rewrite bdenZ_some; exists Hi; auto.
destruct H7 as [v].
rewrite H7 in H5; inv H5.
apply H in H6; try omega; inv H6.
apply H10 in H8; auto.
destruct lf; inv H7.
simpl; split; try split.
assert (exists l, bden b i (taint_vars [If b C1 C2] s) = Some (false,l)).
exists l; auto.
rewrite <- bdenZ_some in H6; rewrite H6; auto.
apply no_lbls_taint_vars; auto.
apply H4; destruct lt; auto.
unfold taint_vars.
destruct (In_dec eq_nat_dec x (modifies [If b C1 C2])); try contradiction.
destruct (s x) as [[v1 l1]|].
exists v1; exists Hi; split; auto.
exists 0%Z; exists Hi; split; auto.
destruct lf; inv H7.
dup H16; apply bden_taint_vars with (K := [If b C1 C2]) in H16.
destruct H16 as [l' [H16]].
rewrite H16 in H19; inv H19.
destruct l; inv H7.
specialize (H3 (St i s h)); simpl in H3.
assert (exists v, bden b i s = Some (v,Lo)).
apply H3; split; auto.
assert (exists l, bden b i s = Some (false,l)).
exists Hi; auto.
rewrite <- bdenZ_some in H7; rewrite H7; auto.
destruct H7 as [v].
rewrite H7 in H6; inv H6.
inv H7.
generalize st' o' H9 H18; clear st' o' H9 H18.
induction n0; intros.
inv H9.
inv H9.
inv H8.
rewrite H21 in H16; inv H16.
rewrite H21 in H16; inv H16.