Type class for ⊤ to get overloaded notation for entire set.

barrier/barrier.v

@@ -131,12 +131,12 @@ Section proof.
         saved_prop_own i Q ★ ▷(Q -★ R))%I.
 
   Lemma newchan_spec (P : iProp) (Q : val → iProp) :
-    (∀ l, recv l P ★ send l P -★ Q (LocV l)) ⊑ wp coPset_all (newchan '()) Q.
+    (∀ l, recv l P ★ send l P -★ Q (LocV l)) ⊑ wp ⊤ (newchan '()) Q.
   Proof.
   Abort.
 
   Lemma signal_spec l P (Q : val → iProp) :
-    (send l P ★ P ★ Q '()) ⊑ wp coPset_all (signal (LocV l)) Q.
+    (send l P ★ P ★ Q '()) ⊑ wp ⊤ (signal (LocV l)) Q.
   Proof.
     rewrite /signal /send /barrier_ctx. rewrite sep_exist_r.
     apply exist_elim=>γ. wp_rec. (* FIXME wp_let *)
@@ -167,12 +167,12 @@ Section proof.
   Abort.
 
   Lemma wait_spec l P (Q : val → iProp) :
-    (recv l P ★ (P -★ Q '())) ⊑ wp coPset_all (wait (LocV l)) Q.
+    (recv l P ★ (P -★ Q '())) ⊑ wp ⊤ (wait (LocV l)) Q.
   Proof.
   Abort.
 
   Lemma split_spec l P1 P2 Q :
-    (recv l (P1 ★ P2) ★ (recv l P1 ★ recv l P2 -★ Q '())) ⊑ wp coPset_all Skip Q.
+    (recv l (P1 ★ P2) ★ (recv l P1 ★ recv l P2 -★ Q '())) ⊑ wp ⊤ Skip Q.
   Proof.
   Abort.
 

heap_lang/lifting.v

@@ -77,7 +77,7 @@ Qed.
 
 (** Base axioms for core primitives of the language: Stateless reductions *)
 Lemma wp_fork E e Q :
-  (▷ Q (LitV LitUnit) ★ ▷ wp (Σ:=Σ) coPset_all e (λ _, True)) ⊑ wp E (Fork e) Q.
+  (▷ Q (LitV LitUnit) ★ ▷ wp (Σ:=Σ) ⊤ e (λ _, True)) ⊑ wp E (Fork e) Q.
 Proof.
   rewrite -(wp_lift_pure_det_step (Fork e) (Lit LitUnit) (Some e)) //=;
     last by intros; inv_step; eauto.

prelude/base.v

@@ -209,6 +209,9 @@ intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
 Class Empty A := empty: A.
 Notation "∅" := empty : C_scope.
 
+Class Top A := top : A.
+Notation "⊤" := top : C_scope.
+
 Class Union A := union: A → A → A.
 Instance: Params (@union) 2.
 Infix "∪" := union (at level 50, left associativity) : C_scope.
@@ -311,7 +314,7 @@ Instance: Params (@disjoint) 2.
 Infix "⊥" := disjoint (at level 70) : C_scope.
 Notation "(⊥)" := disjoint (only parsing) : C_scope.
 Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
-Notation "(.⊥ X )" := (λ Y, Y ⊥  X) (only parsing) : C_scope.
+Notation "(.⊥ X )" := (λ Y, Y ⊥ X) (only parsing) : C_scope.
 Infix "⊥*" := (Forall2 (⊥)) (at level 70) : C_scope.
 Notation "(⊥*)" := (Forall2 (⊥)) (only parsing) : C_scope.
 Infix "⊥**" := (Forall2 (⊥*)) (at level 70) : C_scope.

prelude/bsets.v

@@ -6,7 +6,7 @@ From prelude Require Export prelude.
 Record bset (A : Type) : Type := mkBSet { bset_car : A → bool }.
 Arguments mkBSet {_} _.
 Arguments bset_car {_} _ _.
-Definition bset_all {A} : bset A := mkBSet (λ _, true).
+Instance bset_top {A} : Top (bset A) := mkBSet (λ _, true).
 Instance bset_empty {A} : Empty (bset A) := mkBSet (λ _, false).
 Instance bset_singleton {A} `{∀ x y : A, Decision (x = y)} :
   Singleton A (bset A) := λ x, mkBSet (λ y, bool_decide (y = x)).

prelude/co_pset.v

@@ -148,7 +148,7 @@ Instance coPset_singleton : Singleton positive coPset := λ p,
   coPset_singleton_raw p ↾ coPset_singleton_wf _.
 Instance coPset_elem_of : ElemOf positive coPset := λ p X, e_of p (`X).
 Instance coPset_empty : Empty coPset := coPLeaf false ↾ I.
-Definition coPset_all : coPset := coPLeaf true ↾ I.
+Instance coPset_top : Top coPset := coPLeaf true ↾ I.
 Instance coPset_union : Union coPset := λ X Y,
   let (t1,Ht1) := X in let (t2,Ht2) := Y in
   (t1 ∪ t2) ↾ coPset_union_wf _ _ Ht1 Ht2.

prelude/sets.v

@@ -6,7 +6,7 @@ From prelude Require Export prelude.
 Record set (A : Type) : Type := mkSet { set_car : A → Prop }.
 Arguments mkSet {_} _.
 Arguments set_car {_} _ _.
-Definition set_all {A} : set A := mkSet (λ _, True).
+Instance set_all {A} : Top (set A) := mkSet (λ _, True).
 Instance set_empty {A} : Empty (set A) := mkSet (λ _, False).
 Instance set_singleton {A} : Singleton A (set A) := λ x, mkSet (x =).
 Instance set_elem_of {A} : ElemOf A (set A) := λ x X, set_car X x.

program_logic/adequacy.v

@@ -14,16 +14,15 @@ Implicit Types Q : val Λ → iProp Λ Σ.
 Implicit Types m : iGst Λ Σ.
 Transparent uPred_holds.
 
-Notation wptp n := (Forall3 (λ e Q r, uPred_holds (wp coPset_all e Q) n r)).
+Notation wptp n := (Forall3 (λ e Q r, uPred_holds (wp ⊤ e Q) n r)).
 Lemma wptp_le Qs es rs n n' :
   ✓{n'} (big_op rs) → wptp n es Qs rs → n' ≤ n → wptp n' es Qs rs.
 Proof. induction 2; constructor; eauto using uPred_weaken. Qed.
 Lemma nsteps_wptp Qs k n tσ1 tσ2 rs1 :
   nsteps step k tσ1 tσ2 →
   1 < n → wptp (k + n) (tσ1.1) Qs rs1 →
-  wsat (k + n) coPset_all (tσ1.2) (big_op rs1) →
-  ∃ rs2 Qs', wptp n (tσ2.1) (Qs ++ Qs') rs2 ∧
-             wsat n coPset_all (tσ2.2) (big_op rs2).
+  wsat (k + n) ⊤ (tσ1.2) (big_op rs1) →
+  ∃ rs2 Qs', wptp n (tσ2.1) (Qs ++ Qs') rs2 ∧ wsat n ⊤ (tσ2.2) (big_op rs2).
 Proof.
   intros Hsteps Hn; revert Qs rs1.
   induction Hsteps as [|k ?? tσ3 [e1 σ1 e2 σ2 ef t1 t2 ?? Hstep] Hsteps IH];
@@ -31,7 +30,7 @@ Proof.
   { by intros; exists rs, []; rewrite right_id_L. }
   intros (Qs1&?&rs1&?&->&->&?&
     (Q&Qs2&r&rs2&->&->&Hwp&?)%Forall3_cons_inv_l)%Forall3_app_inv_l ?.
-  destruct (wp_step_inv coPset_all ∅ Q e1 (k + n) (S (k + n)) σ1 r
+  destruct (wp_step_inv ⊤ ∅ Q e1 (k + n) (S (k + n)) σ1 r
     (big_op (rs1 ++ rs2))) as [_ Hwpstep]; eauto using values_stuck.
   { by rewrite right_id_L -big_op_cons Permutation_middle. }
   destruct (Hwpstep e2 σ2 ef) as (r2&r2'&Hwsat&?&?); auto; clear Hwpstep.
@@ -52,11 +51,11 @@ Proof.
     by rewrite -Permutation_middle /= big_op_app.
 Qed.
 Lemma ht_adequacy_steps P Q k n e1 t2 σ1 σ2 r1 :
-  {{ P }} e1 @ coPset_all {{ Q }} →
+  {{ P }} e1 @ ⊤ {{ Q }} →
   nsteps step k ([e1],σ1) (t2,σ2) →
-  1 < n → wsat (k + n) coPset_all σ1 r1 →
+  1 < n → wsat (k + n) ⊤ σ1 r1 →
   P (k + n) r1 →
-  ∃ rs2 Qs', wptp n t2 (Q :: Qs') rs2 ∧ wsat n coPset_all σ2 (big_op rs2).
+  ∃ rs2 Qs', wptp n t2 (Q :: Qs') rs2 ∧ wsat n ⊤ σ2 (big_op rs2).
 Proof.
   intros Hht ????; apply (nsteps_wptp [Q] k n ([e1],σ1) (t2,σ2) [r1]);
     rewrite /big_op ?right_id; auto.
@@ -66,9 +65,9 @@ Proof.
 Qed.
 Lemma ht_adequacy_own Q e1 t2 σ1 m σ2 :
   ✓m →
-  {{ ownP σ1 ★ ownG m }} e1 @ coPset_all {{ Q }} →
+  {{ ownP σ1 ★ ownG m }} e1 @ ⊤ {{ Q }} →
   rtc step ([e1],σ1) (t2,σ2) →
-  ∃ rs2 Qs', wptp 2 t2 (Q :: Qs') rs2 ∧ wsat 2 coPset_all σ2 (big_op rs2).
+  ∃ rs2 Qs', wptp 2 t2 (Q :: Qs') rs2 ∧ wsat 2 ⊤ σ2 (big_op rs2).
 Proof.
   intros Hv ? [k ?]%rtc_nsteps.
   eapply ht_adequacy_steps with (r1 := (Res ∅ (Excl σ1) (Some m))); eauto; [|].
@@ -87,7 +86,7 @@ Proof.
   intros Hv ? Hs.
   destruct (ht_adequacy_own (λ v', ■ φ v')%I e (of_val v :: t2) σ1 m σ2)
              as (rs2&Qs&Hwptp&?); auto.
-  { by rewrite -(ht_mask_weaken E coPset_all). }
+  { by rewrite -(ht_mask_weaken E ⊤). }
   inversion Hwptp as [|?? r ?? rs Hwp _]; clear Hwptp; subst.
   apply wp_value_inv in Hwp; destruct (Hwp (big_op rs) 2 ∅ σ2) as [r' []]; auto.
   by rewrite right_id_L.
@@ -100,10 +99,10 @@ Lemma ht_adequacy_reducible E Q e1 e2 t2 σ1 m σ2 :
 Proof.
   intros Hv ? Hs [i ?]%elem_of_list_lookup He.
   destruct (ht_adequacy_own Q e1 t2 σ1 m σ2) as (rs2&Qs&?&?); auto.
-  { by rewrite -(ht_mask_weaken E coPset_all). }
-  destruct (Forall3_lookup_l (λ e Q r, wp coPset_all e Q 2 r) t2
+  { by rewrite -(ht_mask_weaken E ⊤). }
+  destruct (Forall3_lookup_l (λ e Q r, wp ⊤ e Q 2 r) t2
     (Q :: Qs) rs2 i e2) as (Q'&r2&?&?&Hwp); auto.
-  destruct (wp_step_inv coPset_all ∅ Q' e2 1 2 σ2 r2 (big_op (delete i rs2)));
+  destruct (wp_step_inv ⊤ ∅ Q' e2 1 2 σ2 r2 (big_op (delete i rs2)));
     rewrite ?right_id_L ?big_op_delete; auto.
 Qed.
 Theorem ht_adequacy_safe E Q e1 t2 σ1 m σ2 :

program_logic/hoare_lifting.v

@@ -23,7 +23,7 @@ Lemma ht_lift_step E1 E2
   ((P ={E2,E1}=> ownP σ1 ★ ▷ P') ∧ ∀ e2 σ2 ef,
     (■ φ e2 σ2 ef ★ ownP σ2 ★ P' ={E1,E2}=> Q1 e2 σ2 ef ★ Q2 e2 σ2 ef) ∧
     {{ Q1 e2 σ2 ef }} e2 @ E2 {{ R }} ∧
-    {{ Q2 e2 σ2 ef }} ef ?@ coPset_all {{ λ _, True }})
+    {{ Q2 e2 σ2 ef }} ef ?@ ⊤ {{ λ _, True }})
   ⊑ {{ P }} e1 @ E2 {{ R }}.
 Proof.
   intros ?? Hsafe Hstep; apply (always_intro _ _), impl_intro_l.
@@ -51,7 +51,7 @@ Lemma ht_lift_atomic_step
   atomic e1 →
   reducible e1 σ1 →
   (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
-  (∀ e2 σ2 ef, {{ ■ φ e2 σ2 ef ★ P }} ef ?@ coPset_all {{ λ _, True }}) ⊑
+  (∀ e2 σ2 ef, {{ ■ φ e2 σ2 ef ★ P }} ef ?@ ⊤ {{ λ _, True }}) ⊑
   {{ ownP σ1 ★ ▷ P }} e1 @ E {{ λ v, ∃ σ2 ef, ownP σ2 ★ ■ φ (of_val v) σ2 ef }}.
 Proof.
   intros ? Hsafe Hstep; set (φ' e σ ef := is_Some (to_val e) ∧ φ e σ ef).
@@ -79,7 +79,7 @@ Lemma ht_lift_pure_step E (φ : expr Λ → option (expr Λ) → Prop) P P' Q e1
   (∀ σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → σ1 = σ2 ∧ φ e2 ef) →
   (∀ e2 ef,
     {{ ■ φ e2 ef ★ P }} e2 @ E {{ Q }} ∧
-    {{ ■ φ e2 ef ★ P' }} ef ?@ coPset_all {{ λ _, True }})
+    {{ ■ φ e2 ef ★ P' }} ef ?@ ⊤ {{ λ _, True }})
   ⊑ {{ ▷(P ★ P') }} e1 @ E {{ Q }}.
 Proof.
   intros ? Hsafe Hstep; apply (always_intro _ _), impl_intro_l.
@@ -104,7 +104,7 @@ Lemma ht_lift_pure_det_step
   to_val e1 = None →
   (∀ σ1, reducible e1 σ1) →
   (∀ σ1 e2' σ2 ef', prim_step e1 σ1 e2' σ2 ef' → σ1 = σ2 ∧ e2 = e2' ∧ ef = ef')→
-  ({{ P }} e2 @ E {{ Q }} ∧ {{ P' }} ef ?@ coPset_all {{ λ _, True }})
+  ({{ P }} e2 @ E {{ Q }} ∧ {{ P' }} ef ?@ ⊤ {{ λ _, True }})
   ⊑ {{ ▷(P ★ P') }} e1 @ E {{ Q }}.
 Proof.
   intros ? Hsafe Hdet.

program_logic/lifting.v

@@ -15,7 +15,7 @@ Implicit Types σ : state Λ.
 Implicit Types Q : val Λ → iProp Λ Σ.
 Transparent uPred_holds.
 
-Notation wp_fork ef := (default True ef (flip (wp coPset_all) (λ _, True)))%I.
+Notation wp_fork ef := (default True ef (flip (wp ⊤) (λ _, True)))%I.
 
 Lemma wp_lift_step E1 E2
     (φ : expr Λ → state Λ → option (expr Λ) → Prop) Q e1 σ1 :

program_logic/namespaces.v

@@ -9,7 +9,7 @@ Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N).
 
 Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _).
 Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed.
-Lemma nclose_nnil : nclose nnil = coPset_all.
+Lemma nclose_nnil : nclose nnil = ⊤.
 Proof. by apply (sig_eq_pi _). Qed.
 Lemma encode_nclose N : encode N ∈ nclose N.
 Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed.

program_logic/sts.v

@@ -64,16 +64,14 @@ Section sts.
   Proof. intros. by apply own_update, sts_update_frag_up. Qed.
 
   Lemma sts_alloc N s :
-    φ s ⊑ pvs N N (∃ γ, sts_ctx γ N φ ∧
-                        sts_own γ s (set_all ∖ sts.tok s)).
+    φ s ⊑ pvs N N (∃ γ, sts_ctx γ N φ ∧ sts_own γ s (⊤ ∖ sts.tok s)).
   Proof.
     eapply sep_elim_True_r.
-    { apply (own_alloc (sts_auth s (set_all ∖ sts.tok s)) N).
+    { apply (own_alloc (sts_auth s (⊤ ∖ sts.tok s)) N).
       apply sts_auth_valid; solve_elem_of. }
     rewrite pvs_frame_l. apply pvs_strip_pvs.
     rewrite sep_exist_l. apply exist_elim=>γ. rewrite -(exist_intro γ).
-    transitivity (▷ sts_inv γ φ ★
-                    sts_own γ s (set_all ∖ sts.tok s))%I.
+    transitivity (▷ sts_inv γ φ ★ sts_own γ s (⊤ ∖ sts.tok s))%I.
     { rewrite /sts_inv -later_intro -(exist_intro s).
       rewrite [(_ ★ φ _)%I]comm -assoc. apply sep_mono_r.
       by rewrite -own_op sts_op_auth_frag_up; last solve_elem_of. }

program_logic/weakestpre.v

@@ -28,7 +28,7 @@ CoInductive wp_pre {Λ Σ} (E : coPset)
        0 < k < n → E ∩ Ef = ∅ →
        wsat (S k) (E ∪ Ef) σ1 (r1 ⋅ rf) →
        wp_go (E ∪ Ef) (wp_pre E Q)
-                      (wp_pre coPset_all (λ _, True%I)) k rf e1 σ1) →
+                      (wp_pre ⊤ (λ _, True%I)) k rf e1 σ1) →
      wp_pre E Q e1 n r1.
 Program Definition wp {Λ Σ} (E : coPset) (e : expr Λ)
   (Q : val Λ → iProp Λ Σ) : iProp Λ Σ := {| uPred_holds := wp_pre E Q e |}.
@@ -102,7 +102,7 @@ Qed.
 Lemma wp_step_inv E Ef Q e k n σ r rf :
   to_val e = None → 0 < k < n → E ∩ Ef = ∅ →
   wp E e Q n r → wsat (S k) (E ∪ Ef) σ (r ⋅ rf) →
-  wp_go (E ∪ Ef) (λ e, wp E e Q) (λ e, wp coPset_all e (λ _, True%I)) k rf e σ.
+  wp_go (E ∪ Ef) (λ e, wp E e Q) (λ e, wp ⊤ e (λ _, True%I)) k rf e σ.
 Proof. intros He; destruct 3; [by rewrite ?to_of_val in He|eauto]. Qed.
 
 Lemma wp_value' E Q v : Q v ⊑ wp E (of_val v) Q.