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low_equivalence.v
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Require Import id_and_loc augmented mmemory mimperative language mlattice bridge types bijection Coq.Program.Tactics Coq.Program.Equality Arith Omega tactics FunctionalExtensionality decision List.
Require Import LibTactics Setoid.
Set Implicit Arguments.
Module LowEquivalence(L: Lattice) (M: Memory L).
Module B := Bridge L M.
Import B Aug Imp TDefs M T MemProp LatProp Lang L.
Lemma onvals_sym:
forall u v φ,
onvals (left φ) v = Some u <->
onvals (right φ) u = Some v.
Proof.
intros.
destruct u eqn:H_u; subst.
- destruct v eqn:H_v; subst.
+ unfold onvals in *.
rewrite_inj.
splits*.
+ splits; intros; unfold onvals in *.
* destruct (left φ l); discriminate.
* discriminate.
- splits; intros.
+ destruct v eqn:H_v; subst; unfold onvals in *.
* discriminate.
* destruct (left φ l0) eqn:H_l1; try discriminate.
rewrite_inj.
assert (right φ l = Some l0) by (destruct φ; eauto).
decide_exist.
reflexivity.
+ unfold onvals in *.
* destruct v eqn:H_v; subst.
{ destruct (right φ l); discriminate. }
{ destruct (right φ l) eqn:H_l; try discriminate.
rewrite_inj.
assert (left φ l0 = Some l) by (destruct φ; eauto).
decide_exist.
reflexivity.
}
Qed.
Hint Resolve onvals_sym.
Lemma onvals_sym1:
forall u v φ,
onvals (left φ) v = Some u ->
onvals (right φ) u = Some v.
Proof.
intros.
eapply onvals_sym; eauto.
Qed.
Hint Resolve onvals_sym1.
Lemma onvals_sym2:
forall u v φ,
onvals (right φ) v = Some u ->
onvals (left φ) u = Some v.
Proof.
intros.
eapply onvals_sym; eauto.
Qed.
Hint Resolve onvals_sym2.
Lemma low_reach_implies_reach:
forall ℓ Γ Σ m h loc,
low_reach ℓ Γ Σ m h loc ->
reach m h loc.
Proof.
intros.
dependent induction H; eauto.
Unshelve.
- constructor; eauto.
Qed.
Hint Resolve low_reach_implies_reach.
Ltac invert_low_reach :=
match goal with
[H: low_reach _ _ _ _ _ _ |- _] =>
inverts H
end.
Inductive reach_from_loc: Memory -> Heap -> loc -> loc -> Prop :=
ReachFromLocRefl:
forall m h loc,
reach_from_loc m h loc loc
| ReachFromLocTrans:
forall m h loc1 loc2 loc3 μ ℓ n,
reach_from_loc m h loc1 loc2 ->
heap_lookup loc2 h = Some (ℓ, μ) ->
lookup μ n = Some (ValLoc loc3) ->
reach_from_loc m h loc1 loc3.
Hint Constructors reach_from_loc.
Inductive low : level_proj1 -> tenv -> stenv -> Memory -> Heap -> loc -> Prop :=
| LowReachable:
forall ℓ_adv m h loc Γ Σ,
low_reach ℓ_adv Γ Σ m h loc ->
low ℓ_adv Γ Σ m h loc
| LowHeapLevel:
forall ℓ_adv m h loc Γ Σ ℓ μ,
heap_lookup loc h = Some (ℓ, μ) ->
ℓ ⊑ ℓ_adv ->
low ℓ_adv Γ Σ m h loc.
Hint Constructors low.
Ltac destruct_low :=
match goal with
[H: low _ _ _ _ _ _ |- _] =>
dependent destruction H
end.
Lemma low_dec:
forall ℓ_adv Γ Σ m h loc,
{ low ℓ_adv Γ Σ m h loc} + { ~ low ℓ_adv Γ Σ m h loc }.
Proof.
intros.
destruct (low_reach_dec ℓ_adv Γ Σ m h loc); eauto.
destruct (heap_lookup loc h) eqn:H_loc.
- destruct p.
destruct (flowsto_dec l ℓ_adv); eauto.
right.
intro.
destruct_low.
+ contradiction.
+ congruence.
- right.
intro.
destruct_low.
+ contradiction.
+ congruence.
Qed.
Hint Resolve low_dec.
Lemma low_left:
forall (T : Type)
(p q : T)
ℓ_adv Γ Σ m h loc,
low ℓ_adv Γ Σ m h loc ->
(if low_dec ℓ_adv Γ Σ m h loc then p else q) = p.
Proof.
intros.
destruct (low_dec ℓ_adv Γ Σ m h loc) eqn:H_loc.
- reflexivity.
- contradiction.
Qed.
Lemma low_right:
forall (T : Type)
(p q : T)
ℓ_adv Γ Σ m h loc,
~ low ℓ_adv Γ Σ m h loc ->
(if low_dec ℓ_adv Γ Σ m h loc then p else q) = q.
Proof.
intros.
destruct (low_dec ℓ_adv Γ Σ m h loc) eqn:H_loc.
- contradiction.
- reflexivity.
Qed.
Definition low_eq_stenv (ℓ_adv : level_proj1) (φ: bijection loc loc) (m1 m2 : Memory) (h1 h2 : Heap) (Γ : tenv) (Σ1 Σ2 : stenv) :=
forall loc1 loc2 τ,
left φ loc1 = Some loc2 ->
(Σ1 loc1 = Some τ /\ low ℓ_adv Γ Σ1 m1 h1 loc1) <->
(Σ2 loc2 = Some τ /\ low ℓ_adv Γ Σ2 m2 h2 loc2).
Hint Unfold low_eq_stenv.
Definition wf_bijection ℓ_adv (φ : bijection loc loc) Γ Σ m h :=
forall loc,
(exists loc', left φ loc = Some loc') <-> low ℓ_adv Γ Σ m h loc.
Hint Unfold wf_bijection.
Lemma low_eq_stenv_refl:
forall ℓ_adv Γ Σ m h,
low_eq_stenv ℓ_adv (identity_bijection loc) m m h h Γ Σ Σ.
Proof.
intros.
unfolds.
intros.
unfold left, identity_bijection in *.
rewrite_inj.
splits*.
Qed.
Hint Resolve low_eq_stenv_refl.
Lemma low_eq_stenv_symmetry:
forall φ ℓ_adv Γ Σ1 Σ2 m1 m2 h1 h2,
low_eq_stenv ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
low_eq_stenv ℓ_adv (inverse φ) m2 m1 h2 h1 Γ Σ2 Σ1.
Proof.
intros.
unfold low_eq_stenv in *.
intros.
assert (left φ loc2 = Some loc1) by (destruct φ; eauto).
eauto using iff_sym.
Qed.
Hint Resolve low_eq_stenv_symmetry.
Lemma low_eq_stenv_trans:
forall ℓ_adv φ ψ m1 m2 m3 h1 h2 h3 Γ Σ1 Σ2 Σ3,
low_eq_stenv ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
low_eq_stenv ℓ_adv ψ m2 m3 h2 h3 Γ Σ2 Σ3 ->
low_eq_stenv ℓ_adv (bijection_compose φ ψ) m1 m3 h1 h3 Γ Σ1 Σ3.
Proof.
intros.
unfold low_eq_stenv in *.
intros.
edestruct (left_compose φ ψ); eauto.
super_destruct'.
rename loc2 into loc3.
destruct x as [loc2 | ]; subst.
- assert (left ψ loc2 = Some loc3) by eauto.
erewrite -> H by solve[eauto].
erewrite -> H0 by solve[eauto].
reflexivity.
- repeat specialize_gen.
discriminate.
Qed.
Definition low_heap_domain_eq (ℓ_adv : level_proj1) (φ: bijection loc loc)
(m1 m2 : Memory) (h1 h2 : Heap) (Γ : tenv) (Σ1 Σ2 : stenv) :=
forall l1 l2 ℓ,
left φ l1 = Some l2 ->
((exists μ, heap_lookup l1 h1 = Some (ℓ, μ)) /\ low ℓ_adv Γ Σ1 m1 h1 l1)
<->
((exists ν, heap_lookup l2 h2 = Some (ℓ, ν)) /\ low ℓ_adv Γ Σ2 m2 h2 l2).
Hint Unfold low_heap_domain_eq.
Lemma low_heap_domain_eq_proj1:
forall ℓ_adv φ h1 h2 l1 l2 ℓ μ m1 m2 Γ Σ1 Σ2,
low_heap_domain_eq ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
left φ l1 = Some l2 ->
heap_lookup l1 h1 = Some (ℓ, μ) ->
low ℓ_adv Γ Σ1 m1 h1 l1 ->
exists ν, heap_lookup l2 h2 = Some (ℓ, ν).
Proof.
intros.
eapply H; eauto.
Qed.
Hint Resolve low_heap_domain_eq_proj1.
Lemma low_heap_domain_eq_proj2:
forall ℓ_adv φ h1 h2 l1 l2 ℓ μ m1 m2 Γ Σ1 Σ2,
low_heap_domain_eq ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
left φ l1 = Some l2 ->
low ℓ_adv Γ Σ2 m2 h2 l2 ->
heap_lookup l2 h2 = Some (ℓ, μ) ->
exists ν, heap_lookup l1 h1 = Some (ℓ, ν).
Proof.
intros.
eapply H; eauto.
Qed.
Hint Resolve low_heap_domain_eq_proj2.
Lemma low_heap_domain_eq_refl:
forall ℓ_adv Γ Σ m h,
low_heap_domain_eq ℓ_adv (identity_bijection loc) m m h h Γ Σ Σ.
Proof.
intros.
unfolds.
intros.
unfold left, identity_bijection in *.
rewrite_inj.
splits*.
Qed.
Hint Resolve low_heap_domain_eq_refl.
Lemma low_heap_domain_eq_sym:
forall ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2,
low_heap_domain_eq ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
low_heap_domain_eq ℓ_adv (inverse φ) m2 m1 h2 h1 Γ Σ2 Σ1.
Proof.
intros.
unfold low_heap_domain_eq in *.
intros.
assert (left φ l2 = Some l1) by (destruct φ; eauto).
eauto using iff_sym.
Qed.
Hint Resolve low_heap_domain_eq_sym.
Lemma low_heap_domain_eq_trans:
forall ℓ_adv φ ψ m1 m2 m3 h1 h2 h3 Γ Σ1 Σ2 Σ3,
low_heap_domain_eq ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
low_heap_domain_eq ℓ_adv ψ m2 m3 h2 h3 Γ Σ2 Σ3 ->
low_heap_domain_eq ℓ_adv (bijection_compose φ ψ) m1 m3 h1 h3 Γ Σ1 Σ3.
Proof.
intros.
unfolds.
intros.
_apply left_compose in *.
super_destruct.
destruct my; try solve[repeat specialize_gen; discriminate].
rename l2 into l3.
rename l into l2.
eapply iff_trans.
- eapply H; eauto.
- eapply H0; eauto.
Qed.
Hint Resolve low_heap_domain_eq_trans.
Lemma low_reach_extend_mem_with_num:
forall ℓ Γ Σ x n m h loc,
low_reach ℓ Γ Σ (extend_memory x (ValNum n) m) h loc ->
low_reach ℓ Γ Σ m h loc.
Proof.
intros.
dependent induction H; eauto 2.
destruct (decide (x = x0)); subst.
- rewrite -> extend_memory_lookup_eq in *.
discriminate.
- rewrite -> extend_memory_lookup_neq in * by solve[eauto].
eauto.
Qed.
Lemma wf_bijection_proj1:
forall loc loc' φ ℓ_adv Γ Σ m h,
wf_bijection ℓ_adv φ Γ Σ m h ->
left φ loc = Some loc' ->
low ℓ_adv Γ Σ m h loc.
Proof.
intros.
eapply H; eauto.
Qed.
Hint Resolve wf_bijection_proj1.
Lemma wf_bijection_proj2:
forall loc φ ℓ_adv Γ Σ m h,
wf_bijection ℓ_adv φ Γ Σ m h ->
low ℓ_adv Γ Σ m h loc ->
exists loc', left φ loc = Some loc'.
Proof.
intros.
eapply H; eauto.
Qed.
Hint Resolve wf_bijection_proj2.
Lemma bijection_implies_in_heap:
forall loc loc' φ ℓ_adv Γ Σ m h,
wf_bijection ℓ_adv φ Γ Σ m h ->
left φ loc = Some loc' ->
dangling_pointer_free m h ->
exists ℓ μ, heap_lookup loc h = Some (ℓ, μ).
Proof.
intros.
assert (low ℓ_adv Γ Σ m h loc) by eauto.
destruct H2; eauto.
Qed.
Hint Resolve bijection_implies_in_heap.
Definition low_domain_eq (ℓ : level_proj1) (Γ : tenv) (m s : Memory) :=
(forall x t l ι,
Γ x = Some (SecType t (l, ι)) ->
l ⊑ ℓ ->
(exists v, memory_lookup m x = Some v) <->
(exists u, memory_lookup s x = Some u)).
Hint Unfold low_domain_eq.
Ltac destruct_low_domain_eq :=
match goal with
[H: low_domain_eq _ _ _ _ _ _ _ |- _] =>
destruct H
end.
Lemma low_domain_eq_refl:
forall ℓ Γ m,
low_domain_eq ℓ Γ m m.
Proof.
intros.
unfolds.
intros.
splits*.
Qed.
Hint Resolve low_domain_eq_refl.
Lemma low_domain_eq_symmetry:
forall ℓ Γ m s,
low_domain_eq ℓ Γ m s ->
low_domain_eq ℓ Γ s m.
Proof.
intros.
unfold low_domain_eq in *.
intros.
splits.
- intros.
erewrite -> H; eauto.
- intros.
erewrite <- H; eauto.
Qed.
Hint Resolve low_domain_eq_symmetry.
Lemma low_domain_eq_trans:
forall ℓ Γ m1 m2 m3,
low_domain_eq ℓ Γ m1 m2 ->
low_domain_eq ℓ Γ m2 m3 ->
low_domain_eq ℓ Γ m1 m3.
Proof.
intros.
unfold low_domain_eq in *.
super_destruct.
intros.
splits.
- intros.
rewrite <- H0 by eauto.
rewrite <- H by eauto.
eauto.
- rewrite -> H by eauto.
rewrite -> H0 by eauto.
eauto.
Qed.
Ltac eapply_low_domain_eq :=
match goal with
[H: low_domain_eq _ _ _ _ |- _] =>
eapply H
end.
Inductive val_low_eq: level_proj1 -> sectype -> value -> value -> bijection loc loc -> Prop :=
| ValNumLowEqH:
forall ℓ_adv ℓ n1 n2 φ ι,
not (ℓ ⊑ ℓ_adv) -> val_low_eq ℓ_adv (SecType Int (ℓ, ι)) (ValNum n1) (ValNum n2) φ
| ValLocLowEqH:
forall ℓ_adv ℓ l1 l2 φ τ ι ℓ_p,
not (ℓ ⊑ ℓ_adv) -> val_low_eq ℓ_adv (SecType (Array τ ℓ_p) (ℓ, ι)) (ValLoc l1) (ValLoc l2) φ
| ValNumLowEqL:
forall ℓ_adv ℓ n φ ι,
ℓ ⊑ ℓ_adv ->
val_low_eq ℓ_adv (SecType Int (ℓ, ι)) (ValNum n) (ValNum n) φ
| ValLocLowEqL:
forall ℓ_adv ℓ φ τ l1 l2 ℓ_p ι,
ℓ ⊑ ℓ_adv ->
left φ l1 = Some l2 ->
val_low_eq ℓ_adv (SecType (Array τ ℓ_p) (ℓ, ι)) (ValLoc l1) (ValLoc l2) φ.
Hint Constructors val_low_eq.
Lemma val_low_eq_symmetry:
forall ℓ τ v1 v2 φ,
val_low_eq ℓ τ v1 v2 φ ->
val_low_eq ℓ τ v2 v1 (inverse φ).
Proof.
intros.
destruct φ.
inverts H; eauto.
Qed.
Hint Resolve val_low_eq_symmetry.
Ltac invert_val_low_eq :=
match goal with
[H: context[val_low_eq] |- _] => inverts H
end.
Lemma val_low_eq_num_refl:
forall ℓ_adv l n,
val_low_eq ℓ_adv (SecType Int l) (ValNum n) (ValNum n)
(identity_bijection loc).
Proof.
intros.
destruct l as [l ι].
destruct (flowsto_dec l ℓ_adv); eauto.
Qed.
Hint Resolve val_low_eq_num_refl.
Lemma val_low_eq_loc_refl:
forall ℓ_adv s loc0 ℓ_p ε,
val_low_eq ℓ_adv (SecType (Array s ℓ_p) ε) (ValLoc loc0) (ValLoc loc0)
(identity_bijection loc).
Proof.
intros.
destruct ε as [ℓ ι].
destruct (flowsto_dec ℓ ℓ_adv); eauto.
Qed.
Hint Resolve val_low_eq_loc_refl.
Lemma val_low_eq_trans:
forall ℓ τ v1 v2 v3 φ ψ,
val_low_eq ℓ τ v1 v2 φ ->
val_low_eq ℓ τ v2 v3 ψ ->
val_low_eq ℓ τ v1 v3 (bijection_compose φ ψ).
Proof.
intros.
destruct τ.
destruct t.
- inverts H; inverts H0; eauto.
- inverts H.
+ invert_val_low_eq; eauto.
+ invert_val_low_eq.
* contradiction.
* apply ValLocLowEqL; eauto.
Qed.
Definition memory_lookup_low_eq ℓ Γ m1 m2 φ :=
(forall x τ v1 v2,
Γ x = Some τ ->
memory_lookup m1 x = Some v1 ->
memory_lookup m2 x = Some v2 ->
val_low_eq ℓ τ v1 v2 φ).
Hint Unfold memory_lookup_low_eq.
Lemma val_low_eq_plus:
forall ℓ l a1 a2 b1 b2 φ,
val_low_eq ℓ (SecType Int l) (ValNum a1) (ValNum a2) φ ->
val_low_eq ℓ (SecType Int l) (ValNum b1) (ValNum b2) φ ->
val_low_eq ℓ (SecType Int l) (ValNum (a1 + b1)) (ValNum (a2 + b2)) φ.
Proof.
intros.
destruct l as [l ι].
destruct (flowsto_dec l ℓ).
- invert_val_low_eq; invert_val_low_eq; contradiction || eauto.
- eauto.
Qed.
Lemma val_low_eq_mult:
forall ℓ l a1 a2 b1 b2 φ,
val_low_eq ℓ (SecType Int l) (ValNum a1) (ValNum a2) φ ->
val_low_eq ℓ (SecType Int l) (ValNum b1) (ValNum b2) φ ->
val_low_eq ℓ (SecType Int l) (ValNum (a1 * b1)) (ValNum (a2 * b2)) φ.
Proof.
intros.
destruct l as [l ι].
destruct (flowsto_dec l ℓ).
- invert_val_low_eq; invert_val_low_eq; contradiction || eauto.
- eauto.
Qed.
Lemma val_low_eq_mon:
forall ℓ τ l1 v1 v2 φ l2,
val_low_eq ℓ (SecType τ l1) v1 v2 φ ->
l1 ≼ l2 ->
val_low_eq ℓ (SecType τ l2) v1 v2 φ.
Proof.
intros.
destruct l1 as [l1 ι1].
destruct l2 as [l2 ι2].
invert_val_low_eq.
- eauto.
- eauto.
- destruct (flowsto_dec l2 ℓ); eauto 2.
- destruct (flowsto_dec l2 ℓ); eauto 2.
Qed.
Hint Resolve val_low_eq_mon.
Lemma eval_low_eq:
forall e φ m1 m2 Γ ℓ τ v u,
wf_tenv Γ m1 ->
wf_tenv Γ m2 ->
memory_lookup_low_eq ℓ Γ m1 m2 φ ->
expr_has_type Γ e τ ->
eval m1 e = Some v ->
eval m2 e = Some u ->
val_low_eq ℓ τ v u φ.
Proof.
intros.
revert v u τ H1 H2 H3 H4.
induction e; intros; subst.
(* e = n *)
{
do 2 invert_eval.
invert_wt_expr.
destruct l0 as [l0 ι].
destruct (flowsto_dec l0 ℓ); intros; eauto.
}
(* e = x *)
{
do 2 invert_eval.
invert_wt_expr.
eauto.
}
(* e = e1 op e2 *)
{
invert_wt_expr.
match goal with
[H1: eval m1 (BinOp ?o ?e1 ?e2) = Some ?v,
H2: eval m2 (BinOp ?o ?e1 ?e2) = Some ?u |- _] =>
destruct (unfold_eval_binop m1 o e1 e2 v H1) as [v1 [v2 [H_v1 H_v2]]];
destruct (unfold_eval_binop m2 o e1 e2 u H2) as [u1 [u2 [H_u1 H_u2]]]
end.
assert (val_low_eq ℓ (SecType Int l1) v1 u1 φ) by eauto.
assert (val_low_eq ℓ (SecType Int l2) v2 u2 φ) by eauto.
edestruct (wt_int_e_is_num Γ m1 e1 v1); eauto; subst.
edestruct (wt_int_e_is_num Γ m1 e2 v2); eauto; subst.
edestruct (wt_int_e_is_num Γ m2 e1 u1); eauto; subst.
edestruct (wt_int_e_is_num Γ m2 e2 u2); eauto; subst.
rewrite -> about_eval in *.
repeat decide_exist in *.
destruct l1 as [l1 ι1].
destruct l2 as [l2 ι2].
rewrite -> ProdL.join_is_pairwise.
destruct o; subst; rewrite_inj.
- destruct (flowsto_dec (l1 ⊔ l2) ℓ); subst.
+ eapply val_low_eq_plus; eapply val_low_eq_mon; eauto.
* rewrite <- ProdL.join_is_pairwise.
eapply ProdLatProp.flowsto_join.
* rewrite <- ProdL.join_is_pairwise.
rewrite -> ProdL.join_symmetry.
eapply ProdLatProp.flowsto_join.
+ eauto.
- destruct (flowsto_dec (l1 ⊔ l2) ℓ); subst.
+ eapply val_low_eq_mult; eapply val_low_eq_mon; eauto.
* rewrite <- ProdL.join_is_pairwise.
eapply ProdLatProp.flowsto_join.
* rewrite <- ProdL.join_is_pairwise.
rewrite -> ProdL.join_symmetry.
eapply ProdLatProp.flowsto_join.
+ eauto.
}
Qed.
Hint Resolve eval_low_eq.
Lemma new_low_reach_implies_flowsto_low:
forall Γ x t ℓ ℓ_x ℓ_adv l n v m h loc Σ ℓ_p H1 H2,
Γ x = Some (SecType (Array (SecType t ℓ) ℓ_p) (ℓ_x, ∘)) ->
dangling_pointer_free m h ->
(forall s ℓ, t = Array s ℓ -> exists loc', v = ValLoc loc' /\ reach m h loc') ->
(t = Int -> exists n, v = ValNum n) ->
low_reach ℓ_adv Γ Σ
(m [x → ValLoc loc])
(h [loc → (n × v, l), H1, H2]) loc ->
ℓ_x ⊑ ℓ_adv.
Proof.
intros.
dependent induction H5.
- destruct (decide (x = x0)); subst.
+ rewrite -> extend_memory_lookup_eq in *.
rewrite_inj.
eauto.
+ rewrite -> extend_memory_lookup_neq in * by solve[eauto].
assert (exists ℓ μ, heap_lookup loc h = Some (ℓ, μ)) by eauto.
super_destruct; subst.
congruence.
- destruct (decide (loc2 = loc1)); subst.
+ rewrite_inj.
eapply IHlow_reach; eauto.
+ rewrite -> heap_lookup_extend_neq in * by solve[eauto].
assert (reach (m [x → ValLoc loc2])
(h [loc2 → (n × v, l), H1, H2 ]) loc1) by eauto 2.
assert (~ reach m h loc2).
{
intro.
assert (exists ℓ μ, heap_lookup loc2 h = Some (ℓ, μ)) by eauto.
super_destruct; subst.
congruence.
}
assert (reach m h loc1).
{
eapply reach_extend_implies_reach_if with (v := v).
- intros; subst.
destruct t.
+ repeat specialize_gen.
super_destruct; discriminate.
+ specialize (H3 s l1 eq_refl).
super_destruct; subst.
injects.
eauto.
- eauto.
- eauto.
}
exfalso; eauto.
Unshelve.
* eauto.
Qed.
Lemma low_reach_extend_implies_low_reach_if:
forall m h loc1 loc2 x v n Γ Σ ℓ_adv ℓ ι ℓ_p σ H1 H2,
(forall s ℓ', σ = Array s ℓ' -> exists loc',
v = ValLoc loc' /\ (ℓ ⊑ ℓ_adv -> low_reach ℓ_adv Γ Σ m h loc')) ->
(σ = Int -> exists n, v = ValNum n) ->
low_reach ℓ_adv Γ (extend_stenv loc1 (SecType σ (ℓ, ι)) Σ)
(m [x → ValLoc loc1])
(h [loc1 → (n × v, ℓ_p), H1, H2]) loc2 ->
loc2 <> loc1 ->
low_reach ℓ_adv Γ Σ m h loc2.
Proof.
intros.
dependent induction H3.
- destruct (decide (x = x0)); subst.
+ rewrite -> extend_memory_lookup_eq in *.
rewrite_inj.
exfalso; eauto.
+ rewrite -> extend_memory_lookup_neq in * by solve[eauto].
eauto 2.
- destruct (decide (loc0 = loc1)); subst.
+ rewrite_inj.
rewrite -> extend_stenv_lookup_eq in *.
rewrite_inj.
apply_heap_lookup_extend_eq.
super_destruct; subst.
rewrite_inj.
assert (v = ValLoc loc2) by congruence; subst.
specialize (H τ ℓ_p0 eq_refl).
super_destruct; subst.
injects.
eauto.
+ rewrite -> heap_lookup_extend_neq in * by solve[eauto].
rewrite -> extend_stenv_lookup_neq in * by solve[eauto].
assert (low_reach ℓ_adv Γ Σ m h loc0) by eauto 2. (* IH *)
eauto 2.
Qed.
Definition low_reach_NI ℓ_adv (φ : bijection loc loc) m1 h1 m2 h2 Γ Σ1 Σ2 :=
forall loc loc',
left φ loc = Some loc' ->
low_reach ℓ_adv Γ Σ1 m1 h1 loc <-> low_reach ℓ_adv Γ Σ2 m2 h2 loc'.
Hint Unfold low_reach_NI.
Lemma low_reach_NI_refl:
forall ℓ_adv m h Γ Σ,
low_reach_NI ℓ_adv (identity_bijection loc) m h m h Γ Σ Σ.
Proof.
intros.
unfolds.
intros.
unfold left, identity_bijection in *.
rewrite_inj.
splits*.
Qed.
Hint Resolve low_reach_NI_refl.
Lemma low_reach_NI_sym:
forall ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2,
low_reach_NI ℓ_adv φ m1 h1 m2 h2 Γ Σ1 Σ2 ->
low_reach_NI ℓ_adv (inverse φ) m2 h2 m1 h1 Γ Σ2 Σ1.
Proof.
intros.
unfolds.
intros.
assert (left φ loc' = Some loc) by (destruct φ; eauto).
eauto using iff_sym.
Qed.
Hint Resolve low_reach_NI_sym.
Lemma low_reach_NI_trans:
forall ℓ_adv φ ψ m1 m2 m3 h1 h2 h3 Γ Σ1 Σ2 Σ3,
low_reach_NI ℓ_adv φ m1 h1 m2 h2 Γ Σ1 Σ2 ->
low_reach_NI ℓ_adv ψ m2 h2 m3 h3 Γ Σ2 Σ3 ->
low_reach_NI ℓ_adv (bijection_compose φ ψ) m1 h1 m3 h3 Γ Σ1 Σ3.
Proof.
intros.
unfolds.
intros.
remember_simple (left_compose φ ψ _ _ H1).
super_destruct; subst.
destruct (left φ loc) eqn:H_loc.
- rename l into loc2.
rewrite -> (H _) by solve[eauto].
eauto.
- repeat specialize_gen.
discriminate.
Qed.
Definition heap_val := prod level_proj1 lookupfunc.
Inductive heapval_low_eq: level_proj1 -> sectype -> loc -> loc ->
Memory -> Memory -> Heap -> Heap ->
bijection loc loc -> Prop :=
| HeapValLowEq:
forall ℓ_adv τ ℓ_τ ℓ μ ν φ m1 m2 h1 h2 loc1 loc2,
heap_lookup loc1 h1 = Some (ℓ, μ) ->
heap_lookup loc2 h2 = Some (ℓ, ν) ->
(forall n, (exists v, lookup μ n = Some v) <-> (exists u, lookup ν n = Some u)) ->
(forall n u v, lookup μ n = Some u -> lookup ν n = Some v ->
reach m1 h1 loc1 ->
reach m2 h2 loc2 ->
val_low_eq ℓ_adv (SecType τ ℓ_τ) u v φ) ->
heapval_low_eq ℓ_adv (SecType τ ℓ_τ) loc1 loc2 m1 m2 h1 h2 φ.
Hint Constructors heapval_low_eq.
Ltac invert_heapval_low_eq :=
match goal with
[H: context[heapval_low_eq] |- _] => inverts H
end.
Lemma heapval_low_eq_num_refl:
forall ℓ_adv m l ℓ h μ loc0,
heap_lookup loc0 h = Some (ℓ, μ) ->
(forall v n, lookup μ n = Some v -> exists n0, lookup μ n = Some (ValNum n0)) ->
heapval_low_eq ℓ_adv (SecType Int l) loc0 loc0 m m h h
(identity_bijection loc).
Proof.
intros.
eapply HeapValLowEq; eauto.
- intros.
splits*.
- intros.
rewrite_inj.
do 3 specialize_gen.
destruct_exists.
rewrite_inj.
eauto.
Qed.
Hint Resolve heapval_low_eq_num_refl.
Lemma heapval_low_eq_loc_refl:
forall ℓ_adv μ s ℓ ℓ_p m loc0 h ε,
heap_lookup loc0 h = Some (ℓ, μ) ->
(forall v n, lookup μ n = Some v -> exists loc0, lookup μ n = Some (ValLoc loc0)) ->
heapval_low_eq ℓ_adv (SecType (Array s ℓ_p) ε) loc0 loc0 m m h h
(identity_bijection loc).
Proof.
intros.
eapply HeapValLowEq; eauto.
- intros.
splits*.
- intros.
rewrite_inj.
do 3 specialize_gen.
super_destruct; subst.
rewrite_inj.
eauto.
Qed.
Hint Resolve heapval_low_eq_loc_refl.
Lemma heapval_low_eq_symmetry:
forall ℓ τ loc1 loc2 φ m1 m2 h1 h2,
heapval_low_eq ℓ τ loc1 loc2 m1 m2 h1 h2 φ ->
heapval_low_eq ℓ τ loc2 loc1 m2 m1 h2 h1 (inverse φ).
Proof.
intros.
invert_heapval_low_eq.
eapply HeapValLowEq; eauto.
intros.
eapply iff_sym.
eauto.
Qed.
Hint Resolve heapval_low_eq_symmetry.
(*
Lemma heapval_low_eq_trans:
forall ℓ_adv τ loc1 loc2 loc3 φ ψ m1 m2 m3 h1 h2 h3,
heapval_low_eq ℓ_adv τ loc1 loc2 m1 m2 h1 h2 φ ->
heapval_low_eq ℓ_adv τ loc2 loc3 m2 m3 h2 h3 ψ ->
heapval_low_eq ℓ_adv τ loc1 loc3 m1 m3 h1 h3 (bijection_compose φ ψ).
Proof.
intros.
do 2 invert_heapval_low_eq.
eapply HeapValLowEq.
- eauto.
-
- intros.
split.
+ intros.
eapply H10.
eapply H12.
eauto.
+ intros.
eapply H12.
eapply H10.
eauto.
- intros.
assert (exists u, lookup ν n = Some u) by (eapply H12; eauto).
destruct_exists.
eapply val_low_eq_trans.
* eapply H13; eauto.
* eapply H11; eauto.
Qed.
*)
Definition heap_lookup_low_eq ℓ φ m1 m2 h1 h2 Γ Σ1 Σ2 :=
forall loc1 loc2 τ,
Σ1 loc1 = Some τ ->
Σ2 loc2 = Some τ ->
left φ loc1 = Some loc2 ->
low ℓ Γ Σ1 m1 h1 loc1 ->
low ℓ Γ Σ2 m2 h2 loc2 ->
heapval_low_eq ℓ τ loc1 loc2 m1 m2 h1 h2 φ.
Hint Unfold heap_lookup_low_eq.
Ltac eapply_lookup_func_domain_eq :=
match goal with
[H: forall _, (exists _, _) <-> (exists _, _) |- _] =>
eapply H
end.
Lemma bijection_implies_lookup:
forall ℓ_adv ℓ ι τ m1 m2 h1 h2 n Γ Σ1 Σ2 loc1 loc2 loc1' loc2' φ l μ ν,
wf_stenv Σ1 h1 ->
wf_stenv Σ2 h2 ->
heap_lookup_low_eq ℓ_adv φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
Σ1 loc1 = Some (SecType τ (ℓ, ι)) ->
Σ2 loc2 = Some (SecType τ (ℓ, ι)) ->
low ℓ_adv Γ Σ1 m1 h1 loc1 ->
low ℓ_adv Γ Σ2 m2 h2 loc2 ->
reach m1 h1 loc1 ->
reach m2 h2 loc2 ->
ℓ ⊑ ℓ_adv ->
heap_lookup loc1 h1 = Some (l, μ) ->
heap_lookup loc2 h2 = Some (l, ν) ->
left φ loc1 = Some loc2 ->
lookup μ n = Some (ValLoc loc1') ->
left φ loc1' = Some loc2' ->
lookup ν n = Some (ValLoc loc2').
Proof.
intros.
super_destruct; subst.
assert (heapval_low_eq ℓ_adv (SecType τ (ℓ, ι)) loc1 loc2 m1 m2 h1 h2 φ).
{
eauto.
}
invert_heapval_low_eq.
rewrite_inj.
assert (exists u, lookup ν0 n = Some u).
{
assert (exists v, lookup μ0 n = Some v) by eauto.
eapply_lookup_func_domain_eq; eauto.
}
super_destruct; subst.
assert (val_low_eq ℓ_adv (SecType τ (ℓ, ι)) (ValLoc loc1') u φ) by eauto.
invert_val_low_eq.
- contradiction.
- congruence.
Qed.
(*
Lemma low_reach_in_extended_memory_and_heap:
forall ℓ Γ Σ1 Σ2 m1 h1 m2 h2 x n1 n2 v1 v2 l1 l2 loc1 loc2 φ ℓ_τ ℓ_p τ ℓ_x ι_τ,
Γ x = Some (SecType (Array (SecType τ (ℓ_τ, ι_τ)) ℓ_p) (ℓ_x, ∘)) ->
wf_bijection ℓ φ Γ Σ1 m1 h1 ->
low_domain_eq ℓ Γ m1 m2 ->
low_eq_stenv ℓ φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
low_reach_NI ℓ φ m1 h1 m2 h2 Γ Σ1 Σ2 ->
low_heap_domain_eq ℓ φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
heap_lookup_low_eq ℓ φ m1 m2 h1 h2 Γ Σ1 Σ2 ->
memory_lookup_low_eq ℓ Γ m1 m2 φ ->
low_reach ℓ Γ (extend_stenv l1 (SecType τ (ℓ_τ, ι_τ)) Σ1)
(m1 [SecType (Array (SecType τ (ℓ_τ, ι_τ)) ℓ_p) (ℓ_x, ∘), x → ValLoc l1])
(h1 [l1 → (n1 × v1, ℓ_p)]) loc1 ->
dangling_pointer_free m1 h1 ->
dangling_pointer_free m2 h2 ->
heap_is_some_implies_in_stenv h1 Σ1 ->
heap_is_some_implies_in_stenv h2 Σ2 ->
wf_tenv Γ m1 ->
wf_tenv Γ m2 ->
wf_stenv Σ1 h1 ->
wf_stenv Σ2 h2 ->
left φ loc1 = Some loc2 ->
heap_lookup l1 h1 = None ->
heap_lookup l2 h2 = None ->
(forall s ℓ', τ = Array s ℓ' -> exists loc', v1 = ValLoc loc' /\ (ℓ_τ ⊑ ℓ -> low_reach ℓ Γ Σ1 m1 h1 loc')) ->
(τ = Int -> exists n, v1 = ValNum n) ->
(ℓ_τ ⊑ ℓ -> onvals (left φ) v1 = Some v2) ->
size ℓ_p h1 + n1 <= maxsize ℓ_p h1 ->
size ℓ_p h2 + n2 <= maxsize ℓ_p h2 ->
low_reach ℓ Γ (extend_stenv l2 (SecType τ (ℓ_τ, ι_τ)) Σ2)
(m2 [SecType (Array (SecType τ (ℓ_τ, ι_τ)) ℓ_p) (ℓ_x, ∘), x → ValLoc l2])
(h2 [l2 → (n2 × v2, ℓ_p)]) loc2.
Proof.
intros.
revert dependent loc2.
match goal with
[H: low_reach _ _ _ _ _ _ |- _] =>
dependent induction H; intros
end.
- destruct (decide (x = x0)); subst.
+ rewrite extend_memory_lookup_eq in *.
rewrite_inj.
assert (exists ℓ μ, heap_lookup loc h1 = Some (ℓ, μ)) by eauto 2.
super_destruct; congruence.
+ rewrite -> extend_memory_lookup_neq in * by solve[eauto].
assert (exists v, memory_lookup m2 x0 = Some v).
{
eapply_low_domain_eq; eauto.
}
super_destruct; subst.
assert (val_low_eq ℓ_adv (SecType (Array τ0 ℓ_p0) (l, ∘))
(ValLoc loc) v φ).
{
eauto 2 using eval_low_eq.
}
invert_val_low_eq; try contradiction.
rewrite_inj.
eapply LowReachMem with (x := x0); eauto.
rewrite -> extend_memory_lookup_neq by solve[eauto].
eauto.
- destruct (decide (l1 = loc1)); subst.
+ rewrite_inj.
rewrite -> extend_stenv_lookup_eq in *.
rewrite_inj.
assert (onvals (left φ) v1 = Some v2) by eauto.
assert ({μ : lookupfunc |
heap_lookup loc1 (extend_heap loc1 ℓ_p n1 v1 h1) = Some (ℓ_p, μ) /\
(forall n : nat, lookup μ n = Some v1)}) by eauto 2.
super_destruct; subst.
rewrite_inj.
assert (v1 = ValLoc loc2) by congruence; subst.
assert (ValLoc loc0 = v2).
{
unfold onvals in *.
decide_exist in *.
rewrite_inj.
reflexivity.
}
subst.
assert ({μ : lookupfunc |
heap_lookup l2 (extend_heap l2 ℓ_p n2 (ValLoc loc0) h2) = Some (ℓ_p, μ)
/\ (forall n : nat, lookup μ n = Some (ValLoc loc0))}) by eauto 2.
super_destruct; subst.
rewrite_inj.