Commit 4064c62e authored by Robbert Krebbers's avatar Robbert Krebbers

Disjointness of sets.

parent 71667413
......@@ -26,17 +26,16 @@ Context {sts : stsT}.
(** ** Step relations *)
Inductive step : relation (state sts * tokens sts) :=
| Step s1 s2 T1 T2 :
(* TODO: This asks for ⊥ on sets: T1 ⊥ T2 := T1 ∩ T2 ⊆ ∅. *)
prim_step s1 s2 tok s1 T1 tok s2 T2
prim_step s1 s2 tok s1 T1 tok s2 T2
tok s1 T1 tok s2 T2 step (s1,T1) (s2,T2).
Notation steps := (rtc step).
Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop :=
| Frame_step T1 T2 :
T1 (tok s1 T) step (s1,T1) (s2,T2) frame_step T s1 s2.
T1 tok s1 T step (s1,T1) (s2,T2) frame_step T s1 s2.
(** ** Closure under frame steps *)
Record closed (S : states sts) (T : tokens sts) : Prop := Closed {
closed_disjoint s : s S tok s T ;
closed_disjoint s : s S tok s T;
closed_step s1 s2 : s1 S frame_step T s1 s2 s2 S
}.
Definition up (s : state sts) (T : tokens sts) : states sts :=
......@@ -50,6 +49,7 @@ Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts.
Hint Extern 50 (¬equiv (A:=set _) _ _) => set_solver : sts.
Hint Extern 50 (_ _) => set_solver : sts.
Hint Extern 50 (_ _) => set_solver : sts.
Hint Extern 50 (_ _) => set_solver : sts.
(** ** Setoids *)
Instance framestep_mono : Proper (flip () ==> (=) ==> (=) ==> impl) frame_step.
......@@ -60,10 +60,7 @@ Qed.
Global Instance framestep_proper : Proper (() ==> (=) ==> (=) ==> iff) frame_step.
Proof. by intros ?? [??] ??????; split; apply framestep_mono. Qed.
Instance closed_proper' : Proper (() ==> () ==> impl) closed.
Proof.
intros ?? HT ?? HS; destruct 1;
constructor; intros until 0; rewrite -?HS -?HT; eauto.
Qed.
Proof. destruct 3; constructor; intros until 0; setoid_subst; eauto. Qed.
Global Instance closed_proper : Proper (() ==> () ==> iff) closed.
Proof. by split; apply closed_proper'. Qed.
Global Instance up_preserving : Proper ((=) ==> flip () ==> ()) up.
......@@ -95,16 +92,16 @@ Proof.
- apply Hstep2 with s3, Frame_step with T3 T4; auto with sts.
Qed.
Lemma step_closed s1 s2 T1 T2 S Tf :
step (s1,T1) (s2,T2) closed S Tf s1 S T1 Tf
s2 S T2 Tf tok s2 T2 .
step (s1,T1) (s2,T2) closed S Tf s1 S T1 Tf
s2 S T2 Tf tok s2 T2.
Proof.
inversion_clear 1 as [???? HR Hs1 Hs2]; intros [? Hstep]??; split_and?; auto.
- eapply Hstep with s1, Frame_step with T1 T2; auto with sts.
- set_solver -Hstep Hs1 Hs2.
Qed.
Lemma steps_closed s1 s2 T1 T2 S Tf :
steps (s1,T1) (s2,T2) closed S Tf s1 S T1 Tf
tok s1 T1 s2 S T2 Tf tok s2 T2 .
steps (s1,T1) (s2,T2) closed S Tf s1 S T1 Tf
tok s1 T1 s2 S T2 Tf tok s2 T2.
Proof.
remember (s1,T1) as sT1 eqn:HsT1; remember (s2,T2) as sT2 eqn:HsT2.
intros Hsteps; revert s1 T1 HsT1 s2 T2 HsT2.
......@@ -120,8 +117,7 @@ Lemma subseteq_up_set S T : S ⊆ up_set S T.
Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed.
Lemma up_up_set s T : up s T up_set {[ s ]} T.
Proof. by rewrite /up_set collection_bind_singleton. Qed.
Lemma closed_up_set S T :
( s, s S tok s T ) closed (up_set S T) T.
Lemma closed_up_set S T : ( s, s S tok s T) closed (up_set S T) T.
Proof.
intros HS; unfold up_set; split.
- intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs').
......@@ -131,7 +127,7 @@ Proof.
- intros s1 s2; rewrite /up; set_unfold; intros (s&?&?) ?; exists s.
split; [eapply rtc_r|]; eauto.
Qed.
Lemma closed_up s T : tok s T closed (up s T) T.
Lemma closed_up s T : tok s T closed (up s T) T.
Proof.
intros; rewrite -(collection_bind_singleton (λ s, up s T) s).
apply closed_up_set; set_solver.
......@@ -188,7 +184,7 @@ Inductive sts_equiv : Equiv (car sts) :=
Global Existing Instance sts_equiv.
Global Instance sts_valid : Valid (car sts) := λ x,
match x with
| auth s T => tok s T
| auth s T => tok s T
| frag S' T => closed S' T S'
end.
Global Instance sts_core : Core (car sts) := λ x,
......@@ -198,11 +194,9 @@ Global Instance sts_core : Core (car sts) := λ x,
end.
Inductive sts_disjoint : Disjoint (car sts) :=
| frag_frag_disjoint S1 S2 T1 T2 :
S1 S2 T1 T2 frag S1 T1 frag S2 T2
| auth_frag_disjoint s S T1 T2 :
s S T1 T2 auth s T1 frag S T2
| frag_auth_disjoint s S T1 T2 :
s S T1 T2 frag S T1 auth s T2.
S1 S2 T1 T2 frag S1 T1 frag S2 T2
| auth_frag_disjoint s S T1 T2 : s S T1 T2 auth s T1 frag S T2
| frag_auth_disjoint s S T1 T2 : s S T1 T2 frag S T1 auth s T2.
Global Existing Instance sts_disjoint.
Global Instance sts_op : Op (car sts) := λ x1 x2,
match x1, x2 with
......@@ -216,6 +210,8 @@ Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts.
Hint Extern 50 (¬equiv (A:=set _) _ _) => set_solver : sts.
Hint Extern 50 (_ _) => set_solver : sts.
Hint Extern 50 (_ _) => set_solver : sts.
Hint Extern 50 (_ _) => set_solver : sts.
Global Instance sts_equivalence: Equivalence (() : relation (car sts)).
Proof.
split.
......@@ -303,11 +299,11 @@ Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s).
Proof. intros T1 T2 HT. by rewrite /sts_frag_up HT. Qed.
(** Validity *)
Lemma sts_auth_valid s T : sts_auth s T tok s T .
Lemma sts_auth_valid s T : sts_auth s T tok s T.
Proof. done. Qed.
Lemma sts_frag_valid S T : sts_frag S T closed S T S .
Proof. done. Qed.
Lemma sts_frag_up_valid s T : tok s T sts_frag_up s T.
Lemma sts_frag_up_valid s T : tok s T sts_frag_up s T.
Proof. intros. by apply sts_frag_valid; auto using closed_up, up_non_empty. Qed.
Lemma sts_auth_frag_valid_inv s S T1 T2 :
......@@ -335,7 +331,7 @@ Proof.
Qed.
Lemma sts_op_frag S1 S2 T1 T2 :
T1 T2 sts.closed S1 T1 sts.closed S2 T2
T1 T2 sts.closed S1 T1 sts.closed S2 T2
sts_frag (S1 S2) (T1 T2) sts_frag S1 T1 sts_frag S2 T2.
Proof.
intros HT HS1 HS2. rewrite /sts_frag.
......@@ -390,7 +386,7 @@ Qed.
Lemma sts_frag_included S1 S2 T1 T2 :
closed S2 T2 → S2 ≢ ∅ →
(sts_frag S1 T1 ≼ sts_frag S2 T2) ↔
(closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ∩ Tf ≡ ∅
(closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ⊥ Tf
S2 ≡ S1 ∩ up_set S2 Tf).
Proof.
intros ??; split.
......
......@@ -5,9 +5,11 @@ importantly, it implements some tactics to automatically solve goals involving
collections. *)
From iris.prelude Require Export base tactics orders.
Instance collection_disjoint `{ElemOf A C} : Disjoint C := λ X Y,
x, x X x Y False.
Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y,
x, x X x Y.
Typeclasses Opaque collection_subseteq.
Typeclasses Opaque collection_disjoint collection_subseteq.
(** * Basic theorems *)
Section simple_collection.
......@@ -36,6 +38,9 @@ Section simple_collection.
Proof. firstorder. Qed.
Lemma elem_of_equiv_empty X : X x, x X.
Proof. firstorder. Qed.
Lemma elem_of_disjoint X Y : X Y x, x X x Y False.
Proof. done. Qed.
Lemma collection_positive_l X Y : X Y X .
Proof.
rewrite !elem_of_equiv_empty. setoid_rewrite elem_of_union. naive_solver.
......@@ -52,11 +57,14 @@ Section simple_collection.
- intros ??. rewrite elem_of_singleton. by intros ->.
- intros Ex. by apply (Ex x), elem_of_singleton.
Qed.
Global Instance singleton_proper : Proper ((=) ==> ()) (singleton (B:=C)).
Proof. by repeat intro; subst. Qed.
Global Instance elem_of_proper :
Proper ((=) ==> () ==> iff) (() : A C Prop) | 5.
Proper ((=) ==> () ==> iff) (@elem_of A C _) | 5.
Proof. intros ???; subst. firstorder. Qed.
Global Instance disjoint_prope : Proper (() ==> () ==> iff) (@disjoint C _).
Proof. intros ??????. by rewrite !elem_of_disjoint; setoid_subst. Qed.
Lemma elem_of_union_list Xs x : x Xs X, X Xs x X.
Proof.
split.
......@@ -196,6 +204,10 @@ Section set_unfold_simple.
constructor. rewrite subset_spec, elem_of_subseteq, elem_of_equiv.
repeat f_equiv; naive_solver.
Qed.
Global Instance set_unfold_disjoint (P Q : A Prop) :
( x, SetUnfold (x X) (P x)) ( x, SetUnfold (x Y) (Q x))
SetUnfold (X Y) ( x, P x Q x False).
Proof. constructor. rewrite elem_of_disjoint. naive_solver. Qed.
Context `{!LeibnizEquiv C}.
Global Instance set_unfold_equiv_same_L X : SetUnfold (X = X) True | 1.
......@@ -387,7 +399,7 @@ Section collection.
Proof. set_solver. Qed.
Lemma difference_intersection_distr_l X Y Z : (X Y) Z X Z Y Z.
Proof. set_solver. Qed.
Lemma disjoint_union_difference X Y : X Y (X Y) X Y.
Lemma disjoint_union_difference X Y : X Y (X Y) X Y.
Proof. set_solver. Qed.
Section leibniz.
......@@ -407,7 +419,7 @@ Section collection.
Lemma difference_intersection_distr_l_L X Y Z :
(X Y) Z = X Z Y Z.
Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
Lemma disjoint_union_difference_L X Y : X Y = (X Y) X = Y.
Lemma disjoint_union_difference_L X Y : X Y (X Y) X = Y.
Proof. unfold_leibniz. apply disjoint_union_difference. Qed.
End leibniz.
......
......@@ -92,9 +92,9 @@ Proof.
- rewrite elem_of_singleton. eauto using size_singleton_inv.
- set_solver.
Qed.
Lemma size_union X Y : X Y size (X Y) = size X + size Y.
Lemma size_union X Y : X Y size (X Y) = size X + size Y.
Proof.
intros [E _]. unfold size, collection_size. simpl. rewrite <-app_length.
intros. unfold size, collection_size. simpl. rewrite <-app_length.
apply Permutation_length, NoDup_Permutation.
- apply NoDup_elements.
- apply NoDup_app; repeat split; try apply NoDup_elements.
......
......@@ -74,15 +74,14 @@ Proof. rewrite not_elem_of_dom. apply delete_partial_alter. Qed.
Lemma delete_insert_dom {A} (m : M A) i x :
i dom D m delete i (<[i:=x]>m) = m.
Proof. rewrite not_elem_of_dom. apply delete_insert. Qed.
Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 m2 dom D m1 dom D m2 .
Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 m2 dom D m1 dom D m2.
Proof.
rewrite map_disjoint_spec, elem_of_equiv_empty.
setoid_rewrite elem_of_intersection.
rewrite map_disjoint_spec, elem_of_disjoint.
setoid_rewrite elem_of_dom. unfold is_Some. naive_solver.
Qed.
Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 m2 dom D m1 dom D m2 .
Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 m2 dom D m1 dom D m2.
Proof. apply map_disjoint_dom. Qed.
Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 dom D m2 m1 m2.
Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 dom D m2 m1 m2.
Proof. apply map_disjoint_dom. Qed.
Lemma dom_union {A} (m1 m2 : M A) : dom D (m1 m2) dom D m1 dom D m2.
Proof.
......@@ -90,8 +89,7 @@ Proof.
unfold is_Some. setoid_rewrite lookup_union_Some_raw.
destruct (m1 !! i); naive_solver.
Qed.
Lemma dom_intersection {A} (m1 m2 : M A) :
dom D (m1 m2) dom D m1 dom D m2.
Lemma dom_intersection {A} (m1 m2: M A) : dom D (m1 m2) dom D m1 dom D m2.
Proof.
apply elem_of_equiv. intros i. rewrite elem_of_intersection, !elem_of_dom.
unfold is_Some. setoid_rewrite lookup_intersection_Some. naive_solver.
......
From iris.program_logic Require Export hoare.
From iris.program_logic Require Import wsat ownership.
Local Hint Extern 10 (_ _) => omega.
Local Hint Extern 100 (@eq coPset _ _) => eassumption || set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 10 ({_} _) =>
repeat match goal with
| H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
......@@ -151,5 +151,4 @@ Proof.
intros ? Hht. eapply wp_adequacy_safe with (E:=E) (Φ:=Φ); first done.
move:Hht. by rewrite /ht uPred.always_elim=>/uPred.impl_entails.
Qed.
End adequacy.
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Import wsat ownership.
Local Hint Extern 10 (_ _) => omega.
Local Hint Extern 100 (@eq coPset _ _) => set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 10 ({_} _) =>
repeat match goal with
| H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
......
......@@ -49,13 +49,11 @@ Section ndisjoint.
Lemma ndot_preserve_disjoint_r N1 N2 x : N1 N2 N1 N2 .@ x .
Proof. rewrite ![N1 _]comm. apply ndot_preserve_disjoint_l. Qed.
Lemma ndisj_disjoint N1 N2 : N1 N2 nclose N1 nclose N2 = .
Lemma ndisj_disjoint N1 N2 : N1 N2 nclose N1 nclose N2.
Proof.
intros (N1' & N2' & [N1'' ->] & [N2'' ->] & Hl & Hne).
apply elem_of_equiv_empty_L=> p; unfold nclose.
rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]].
rewrite !list_encode_app !assoc in Hq.
by eapply Hne, list_encode_suffix_eq.
intros (N1' & N2' & [N1'' ->] & [N2'' ->] & Hl & Hne) p; unfold nclose.
rewrite !elem_coPset_suffixes; intros [q ->] [q' Hq]; destruct Hne.
by rewrite !list_encode_app !assoc in Hq; apply list_encode_suffix_eq in Hq.
Qed.
End ndisjoint.
......@@ -63,20 +61,19 @@ End ndisjoint.
of masks (i.e., coPsets) with set_solver, taking
disjointness of namespaces into account. *)
(* TODO: This tactic is by far now yet as powerful as it should be.
For example, given N1 ⊥ N2, it should be able to solve
nclose (ndot N1 x) ∩ N2 ≡ ∅. It should also solve
(ndot N x) ∩ (ndot N y) ≡ ∅ if x ≠ y is in the context or
For example, given [N1 ⊥ N2], it should be able to solve
[nclose (ndot N1 x) ⊥ N2]. It should also solve
[ndot N x ⊥ ndot N y] if x ≠ y is in the context or
follows from [discriminate]. *)
Ltac set_solver_ndisj :=
repeat match goal with
(* TODO: Restrict these to have type namespace *)
| [ H : (?N1 ?N2) |-_ ] => apply ndisj_disjoint in H
end;
set_solver.
(* TODO: Restrict these to have type namespace *)
| [ H : ?N1 ?N2 |-_ ] => apply ndisj_disjoint in H
end; set_solver.
(* TODO: restrict this to match only if this is ⊆ of coPset *)
Hint Extern 500 (_ _) => set_solver_ndisj : ndisj.
(* The hope is that registering these will suffice to solve most goals
of the form N1 ⊥ N2.
of the form [N1 ⊥ N2].
TODO: Can this prove x ≠ y if discriminate can? *)
Hint Resolve ndot_ne_disjoint : ndisj.
Hint Resolve ndot_preserve_disjoint_l : ndisj.
......
......@@ -2,7 +2,7 @@ From iris.prelude Require Export co_pset.
From iris.program_logic Require Export model.
From iris.program_logic Require Import ownership wsat.
Local Hint Extern 10 (_ _) => omega.
Local Hint Extern 100 (@eq coPset _ _) => set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 10 ({_} _) =>
repeat match goal with
......@@ -11,7 +11,7 @@ Local Hint Extern 10 (✓{_} _) =>
Program Definition pvs_def {Λ Σ} (E1 E2 : coPset) (P : iProp Λ Σ) : iProp Λ Σ :=
{| uPred_holds n r1 := rf k Ef σ,
0 < k n (E1 E2) Ef =
0 < k n E1 E2 Ef
wsat k (E1 Ef) σ (r1 rf)
r2, P k r2 wsat k (E2 Ef) σ (r2 rf) |}.
Next Obligation.
......@@ -84,7 +84,7 @@ Proof.
destruct (HP1 rf k Ef σ) as (r2&HP2&?); auto.
Qed.
Lemma pvs_mask_frame E1 E2 Ef P :
Ef (E1 E2) = (|={E1,E2}=> P) (|={E1 Ef,E2 Ef}=> P).
Ef E1 E2 (|={E1,E2}=> P) (|={E1 Ef,E2 Ef}=> P).
Proof.
rewrite pvs_eq. intros ?; split=> n r ? HP rf k Ef' σ ???.
destruct (HP rf k (EfEf') σ) as (r'&?&?); rewrite ?(assoc_L _); eauto.
......@@ -244,6 +244,5 @@ Proof.
Qed.
Lemma pvs_mk_fsa {Λ Σ} E (P Q : iProp Λ Σ) :
P pvs_fsa E (λ _, Q)
P |={E}=> Q.
P pvs_fsa E (λ _, Q) P |={E}=> Q.
Proof. by intros ?. Qed.
......@@ -72,7 +72,7 @@ Section sts.
Proof. intros ???. by apply own_update, sts_update_frag_up. Qed.
Lemma sts_ownS_op γ S1 S2 T1 T2 :
T1 T2 sts.closed S1 T1 sts.closed S2 T2
T1 T2 sts.closed S1 T1 sts.closed S2 T2
sts_ownS γ (S1 S2) (T1 T2) (sts_ownS γ S1 T1 sts_ownS γ S2 T2).
Proof. intros. by rewrite /sts_ownS -own_op sts_op_frag. Qed.
......
......@@ -78,18 +78,17 @@ Lemma vs_frame_r E1 E2 P Q R : (P ={E1,E2}=> Q) ⊢ (P ★ R ={E1,E2}=> Q ★ R)
Proof. rewrite !(comm _ _ R); apply vs_frame_l. Qed.
Lemma vs_mask_frame E1 E2 Ef P Q :
Ef (E1 E2) = (P ={E1,E2}=> Q) (P ={E1 Ef,E2 Ef}=> Q).
Ef E1 E2 (P ={E1,E2}=> Q) (P ={E1 Ef,E2 Ef}=> Q).
Proof.
intros ?; apply always_intro', impl_intro_l; rewrite (pvs_mask_frame _ _ Ef)//.
by rewrite always_elim impl_elim_r.
Qed.
Lemma vs_mask_frame' E Ef P Q : Ef E = (P ={E}=> Q) (P ={E Ef}=> Q).
Lemma vs_mask_frame' E Ef P Q : Ef E (P ={E}=> Q) (P ={E Ef}=> Q).
Proof. intros; apply vs_mask_frame; set_solver. Qed.
Lemma vs_inv N E P Q R :
nclose N E
(inv N R ( R P ={E nclose N}=> R Q)) (P ={E}=> Q).
nclose N E (inv N R ( R P ={E nclose N}=> R Q)) (P ={E}=> Q).
Proof.
intros; apply: always_intro. apply impl_intro_l.
rewrite always_and_sep_r assoc [(P _)%I]comm -assoc.
......
From iris.program_logic Require Export pviewshifts.
From iris.program_logic Require Import wsat.
Local Hint Extern 10 (_ _) => omega.
Local Hint Extern 100 (@eq coPset _ _) => eassumption || set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 100 (_ _) => set_solver.
Local Hint Extern 100 (@subseteq coPset _ _ _) => set_solver.
Local Hint Extern 10 ({_} _) =>
repeat match goal with
......@@ -25,7 +25,7 @@ CoInductive wp_pre {Λ Σ} (E : coPset)
| wp_pre_step n r1 e1 :
to_val e1 = None
( rf k Ef σ1,
0 < k < n E Ef =
0 < k < n E Ef
wsat (S k) (E Ef) σ1 (r1 rf)
wp_go (E Ef) (wp_pre E Φ)
(wp_pre (λ _, True%I)) k rf e1 σ1)
......@@ -122,7 +122,7 @@ Proof.
by inversion 1 as [|??? He]; [|rewrite ?to_of_val in He]; simplify_eq.
Qed.
Lemma wp_step_inv E Ef Φ e k n σ r rf :
to_val e = None 0 < k < n E Ef =
to_val e = None 0 < k < n E Ef
wp_def E e Φ n r wsat (S k) (E Ef) σ (r rf)
wp_go (E Ef) (λ e, wp_def E e Φ) (λ e, wp_def e (λ _, True%I)) k rf e σ.
Proof.
......
From Coq Require Import Wf_nat.
From iris.program_logic Require Import weakestpre wsat.
Local Hint Extern 10 (_ _) => omega.
Local Hint Extern 10 (@eq coPset _ _) => set_solver.
Local Hint Extern 10 (_ _) => set_solver.
Local Hint Extern 10 ({_} _) =>
repeat match goal with
| H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
......@@ -19,7 +19,7 @@ Program Definition wp_pre
(wp : coPsetC -n> exprC Λ -n> (valC Λ -n> iProp) -n> iProp)
(E : coPset) (e1 : expr Λ) (Φ : valC Λ -n> iProp) : iProp :=
{| uPred_holds n r1 := rf k Ef σ1,
0 k < n E Ef =
0 k < n E Ef
wsat (S k) (E Ef) σ1 (r1 rf)
( v, to_val e1 = Some v
r2, Φ v (S k) r2 wsat (S k) (E Ef) σ1 (r2 rf))
......
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