use 'proph' instead of a notation, and rename type of prophecy variables to proph_id

theories/heap_lang/adequacy.v

@@ -7,10 +7,10 @@ Set Default Proof Using "Type".
 Class heapPreG Σ := HeapPreG {
   heap_preG_iris :> invPreG Σ;
   heap_preG_heap :> gen_heapPreG loc val Σ;
-  heap_preG_proph :> proph_mapPreG proph val Σ
+  heap_preG_proph :> proph_mapPreG proph_id val Σ
 }.
 
-Definition heapΣ : gFunctors := #[invΣ; gen_heapΣ loc val; proph_mapΣ proph val].
+Definition heapΣ : gFunctors := #[invΣ; gen_heapΣ loc val; proph_mapΣ proph_id val].
 Instance subG_heapPreG {Σ} : subG heapΣ Σ → heapPreG Σ.
 Proof. solve_inG. Qed.
 
@@ -20,8 +20,8 @@ Definition heap_adequacy Σ `{heapPreG Σ} s e σ φ :
 Proof.
   intros Hwp; eapply (wp_adequacy _ _); iIntros (??) "".
   iMod (gen_heap_init σ.(heap)) as (?) "Hh".
-  iMod (proph_map_init κs σ.(used_proph)) as (?) "Hp".
+  iMod (proph_map_init κs σ.(used_proph_id)) as (?) "Hp".
   iModIntro.
-  iExists (λ σ κs, (gen_heap_ctx σ.(heap) ∗ proph_map_ctx κs σ.(used_proph))%I). iFrame.
+  iExists (λ σ κs, (gen_heap_ctx σ.(heap) ∗ proph_map_ctx κs σ.(used_proph_id))%I). iFrame.
   iApply (Hwp (HeapG _ _ _ _)).
 Qed.

theories/heap_lang/lang.v

@@ -29,10 +29,11 @@ Open Scope Z_scope.
 
 (** Expressions and vals. *)
 Definition loc := positive. (* Really, any countable type. *)
-Definition proph := positive.
+Definition proph_id := positive.
 
 Inductive base_lit : Set :=
-  | LitInt (n : Z) | LitBool (b : bool) | LitUnit | LitLoc (l : loc) | LitProphecy (p: proph).
+  | LitInt (n : Z) | LitBool (b : bool) | LitUnit
+  | LitLoc (l : loc) | LitProphecy (p: proph_id).
 Inductive un_op : Set :=
   | NegOp | MinusUnOp.
 Inductive bin_op : Set :=
@@ -117,7 +118,7 @@ Inductive val :=
 
 Bind Scope val_scope with val.
 
-Definition observation : Set := proph * val.
+Definition observation : Set := proph_id * val.
 
 Fixpoint of_val (v : val) : expr :=
   match v with
@@ -174,7 +175,7 @@ Definition val_is_unboxed (v : val) : Prop :=
 (** The state: heaps of vals. *)
 Record state : Type := {
   heap: gmap loc val;
-  used_proph: gset proph;
+  used_proph_id: gset proph_id;
 }.
 
 (** Equality and other typeclass stuff *)
@@ -295,7 +296,7 @@ Instance val_countable : Countable val.
 Proof. refine (inj_countable of_val to_val _); auto using to_of_val. Qed.
 
 Instance state_inhabited : Inhabited state :=
-  populate {| heap := inhabitant; used_proph := inhabitant |}.
+  populate {| heap := inhabitant; used_proph_id := inhabitant |}.
 Instance expr_inhabited : Inhabited expr := populate (Lit LitUnit).
 Instance val_inhabited : Inhabited val := populate (LitV LitUnit).
 
@@ -442,11 +443,11 @@ Definition vals_cas_compare_safe (vl v1 : val) : Prop :=
 Arguments vals_cas_compare_safe !_ !_ /.
 
 Definition state_upd_heap (f: gmap loc val → gmap loc val) (σ: state) :=
-  {| heap := f σ.(heap); used_proph := σ.(used_proph) |}.
+  {| heap := f σ.(heap); used_proph_id := σ.(used_proph_id) |}.
 Arguments state_upd_heap _ !_ /.
-Definition state_upd_used_proph (f: gset proph → gset proph) (σ: state) :=
-  {| heap := σ.(heap); used_proph := f σ.(used_proph) |}.
-Arguments state_upd_used_proph _ !_ /.
+Definition state_upd_used_proph_id (f: gset proph_id → gset proph_id) (σ: state) :=
+  {| heap := σ.(heap); used_proph_id := f σ.(used_proph_id) |}.
+Arguments state_upd_used_proph_id _ !_ /.
 
 Inductive head_step : expr → state → list observation → expr → state → list (expr) → Prop :=
   | BetaS f x e1 e2 v2 e' σ :
@@ -516,10 +517,10 @@ Inductive head_step : expr → state → list observation → expr → state →
                (Lit $ LitInt i1) (state_upd_heap <[l:=LitV (LitInt (i1 + i2))]> σ)
                []
   | NewProphS σ p :
-     p ∉ σ.(used_proph) →
+     p ∉ σ.(used_proph_id) →
      head_step NewProph σ
                []
-               (Lit $ LitProphecy p) (state_upd_used_proph ({[ p ]} ∪) σ)
+               (Lit $ LitProphecy p) (state_upd_used_proph_id ({[ p ]} ∪) σ)
                []
   | ResolveProphS e1 p e2 v σ :
      to_val e1 = Some (LitV $ LitProphecy p) →
@@ -557,9 +558,9 @@ Lemma alloc_fresh e v σ :
   head_step (Alloc e) σ [] (Lit (LitLoc l)) (state_upd_heap <[l:=v]> σ) [].
 Proof. by intros; apply AllocS, (not_elem_of_dom (D:=gset loc)), is_fresh. Qed.
 
-Lemma new_proph_fresh σ :
-  let p := fresh σ.(used_proph) in
-  head_step NewProph σ [] (Lit $ LitProphecy p) (state_upd_used_proph ({[ p ]} ∪) σ) [].
+Lemma new_proph_id_fresh σ :
+  let p := fresh σ.(used_proph_id) in
+  head_step NewProph σ [] (Lit $ LitProphecy p) (state_upd_used_proph_id ({[ p ]} ∪) σ) [].
 Proof. constructor. apply is_fresh. Qed.
 
 (* Misc *)

theories/heap_lang/lifting.v

@@ -11,13 +11,13 @@ Set Default Proof Using "Type".
 Class heapG Σ := HeapG {
   heapG_invG : invG Σ;
   heapG_gen_heapG :> gen_heapG loc val Σ;
-  heapG_proph_mapG :> proph_mapG proph val Σ
+  heapG_proph_mapG :> proph_mapG proph_id val Σ
 }.
 
 Instance heapG_irisG `{heapG Σ} : irisG heap_lang Σ := {
   iris_invG := heapG_invG;
   state_interp σ κs :=
-    (gen_heap_ctx σ.(heap) ∗ proph_map_ctx κs σ.(used_proph))%I
+    (gen_heap_ctx σ.(heap) ∗ proph_map_ctx κs σ.(used_proph_id))%I
 }.
 
 (** Override the notations so that scopes and coercions work out *)
@@ -29,9 +29,6 @@ Notation "l ↦{ q } -" := (∃ v, l ↦{q} v)%I
   (at level 20, q at level 50, format "l  ↦{ q }  -") : bi_scope.
 Notation "l ↦ -" := (l ↦{1} -)%I (at level 20) : bi_scope.
 
-Notation "p ⥱ v" := (proph_mapsto p v) (at level 20, format "p ⥱ v") : bi_scope.
-Notation "p ⥱ -" := (∃ v, p ⥱ v)%I (at level 20) : bi_scope.
-
 (** The tactic [inv_head_step] performs inversion on hypotheses of the shape
 [head_step]. The tactic will discharge head-reductions starting from values, and
 simplifies hypothesis related to conversions from and to values, and finite map
@@ -56,7 +53,7 @@ Local Hint Extern 1 (head_step _ _ _ _ _ _) => econstructor.
 Local Hint Extern 0 (head_step (CAS _ _ _) _ _ _ _ _) => eapply CasSucS.
 Local Hint Extern 0 (head_step (CAS _ _ _) _ _ _ _ _) => eapply CasFailS.
 Local Hint Extern 0 (head_step (Alloc _) _ _ _ _ _) => apply alloc_fresh.
-Local Hint Extern 0 (head_step NewProph _ _ _ _ _) => apply new_proph_fresh.
+Local Hint Extern 0 (head_step NewProph _ _ _ _ _) => apply new_proph_id_fresh.
 Local Hint Resolve to_of_val.
 
 Local Ltac solve_exec_safe := intros; subst; do 3 eexists; econstructor; eauto.
@@ -270,7 +267,7 @@ Qed.
 
 (** Lifting lemmas for creating and resolving prophecy variables *)
 Lemma wp_new_proph :
-  {{{ True }}} NewProph {{{ v (p : proph), RET (LitV (LitProphecy p)); p ⥱ v }}}.
+  {{{ True }}} NewProph {{{ v (p : proph_id), RET (LitV (LitProphecy p)); proph p v }}}.
 Proof.
   iIntros (Φ) "_ HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
   iIntros (σ1 κ κs) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
@@ -289,7 +286,7 @@ Qed.
 Lemma wp_resolve_proph e1 e2 p v w:
   IntoVal e1 (LitV (LitProphecy p)) →
   IntoVal e2 w →
-  {{{ p ⥱ v }}} ResolveProph e1 e2 {{{ RET (LitV LitUnit); ⌜v = Some w⌝ }}}.
+  {{{ proph p v }}} ResolveProph e1 e2 {{{ RET (LitV LitUnit); ⌜v = Some w⌝ }}}.
 Proof.
   iIntros (<- <- Φ) "Hp HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
   iIntros (σ1 κ κs) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".

theories/heap_lang/notation.v

@@ -8,7 +8,7 @@ Delimit Scope val_scope with V.
 Coercion LitInt : Z >-> base_lit.
 Coercion LitBool : bool >-> base_lit.
 Coercion LitLoc : loc >-> base_lit.
-Coercion LitProphecy : proph >-> base_lit.
+Coercion LitProphecy : proph_id >-> base_lit.
 
 Coercion App : expr >-> Funclass.
 Coercion of_val : val >-> expr.

theories/heap_lang/proph_map.v

@@ -89,18 +89,15 @@ Section definitions.
           dom (gset _) R ⊆ ps⌝ ∗
           proph_map_auth R)%I.
 
-  Definition proph_mapsto_def (p : P) (v: option V) : iProp Σ :=
+  Definition proph_def (p : P) (v: option V) : iProp Σ :=
     own (proph_map_name pG) (◯ {[ p := Excl (v : option $ leibnizC V) ]}).
-  Definition proph_mapsto_aux : seal (@proph_mapsto_def). by eexists. Qed.
-  Definition proph_mapsto := proph_mapsto_aux.(unseal).
-  Definition proph_mapsto_eq :
-    @proph_mapsto = @proph_mapsto_def := proph_mapsto_aux.(seal_eq).
+  Definition proph_aux : seal (@proph_def). by eexists. Qed.
+  Definition proph := proph_aux.(unseal).
+  Definition proph_eq :
+    @proph = @proph_def := proph_aux.(seal_eq).
 
 End definitions.
 
-Local Notation "p ⥱ v" := (proph_mapsto p v) (at level 20, format "p ⥱ v") : bi_scope.
-Local Notation "p ⥱ -" := (∃ v, p ⥱ v)%I (at level 20) : bi_scope.
-
 Section to_proph_map.
   Context (P V : Type) `{Countable P}.
   Implicit Types p : P.
@@ -148,14 +145,14 @@ Section proph_map.
   Implicit Types R : proph_map P V.
 
   (** General properties of mapsto *)
-  Global Instance p_mapsto_timeless p v : Timeless (p ⥱ v).
-  Proof. rewrite proph_mapsto_eq /proph_mapsto_def. apply _. Qed.
+  Global Instance proph_timeless p v : Timeless (proph p v).
+  Proof. rewrite proph_eq /proph_def. apply _. Qed.
 
   Lemma proph_map_alloc R p v :
     p ∉ dom (gset _) R →
-    proph_map_auth R ==∗ proph_map_auth (<[p := v]> R) ∗ p ⥱ v.
+    proph_map_auth R ==∗ proph_map_auth (<[p := v]> R) ∗ proph p v.
   Proof.
-    iIntros (Hp) "HR". rewrite /proph_map_ctx proph_mapsto_eq /proph_mapsto_def.
+    iIntros (Hp) "HR". rewrite /proph_map_ctx proph_eq /proph_def.
     iMod (own_update with "HR") as "[HR Hl]".
     { eapply auth_update_alloc,
         (alloc_singleton_local_update _ _ (Excl $ (v : option (leibnizC _))))=> //.
@@ -164,16 +161,16 @@ Section proph_map.
   Qed.
 
   Lemma proph_map_remove R p v :
-    proph_map_auth R -∗ p ⥱ v ==∗ proph_map_auth (delete p R).
+    proph_map_auth R -∗ proph p v ==∗ proph_map_auth (delete p R).
   Proof.
-    iIntros "HR Hp". rewrite /proph_map_ctx proph_mapsto_eq /proph_mapsto_def.
+    iIntros "HR Hp". rewrite /proph_map_ctx proph_eq /proph_def.
     rewrite /proph_map_auth to_proph_map_delete. iApply (own_update_2 with "HR Hp").
     apply auth_update_dealloc, (delete_singleton_local_update _ _ _).
   Qed.
 
-  Lemma proph_map_valid R p v : proph_map_auth R -∗ p ⥱ v -∗ ⌜R !! p = Some v⌝.
+  Lemma proph_map_valid R p v : proph_map_auth R -∗ proph p v -∗ ⌜R !! p = Some v⌝.
   Proof.
-    iIntros "HR Hp". rewrite /proph_map_ctx proph_mapsto_eq /proph_mapsto_def.
+    iIntros "HR Hp". rewrite /proph_map_ctx proph_eq /proph_def.
     iDestruct (own_valid_2 with "HR Hp")
       as %[HH%proph_map_singleton_included _]%auth_valid_discrete_2; auto.
   Qed.

theories/heap_lang/total_adequacy.v

@@ -10,8 +10,8 @@ Definition heap_total Σ `{heapPreG Σ} s e σ φ :
 Proof.
   intros Hwp; eapply (twp_total _ _); iIntros (?) "".
   iMod (gen_heap_init σ.(heap)) as (?) "Hh".
-  iMod (proph_map_init [] σ.(used_proph)) as (?) "Hp".
+  iMod (proph_map_init [] σ.(used_proph_id)) as (?) "Hp".
   iModIntro.
-  iExists (λ σ κs, (gen_heap_ctx σ.(heap) ∗ proph_map_ctx κs σ.(used_proph))%I). iFrame.
+  iExists (λ σ κs, (gen_heap_ctx σ.(heap) ∗ proph_map_ctx κs σ.(used_proph_id))%I). iFrame.
   iApply (Hwp (HeapG _ _ _ _)).
 Qed.