Make IParam a canonical structure.

This enables us to remove a whole bunch of type annotations.

barrier/lifting.v

@@ -2,14 +2,14 @@ Require Import prelude.gmap iris.lifting.
 Require Export iris.weakestpre barrier.parameter.
 Import uPred.
 
-(* TODO RJ: Figure out a way to to always use our Σ. *)
+Section lifting.
+Implicit Types P : iProp Σ.
+Implicit Types Q : ival Σ → iProp Σ.
 
 (** Bind. *)
 Lemma wp_bind {E e} K Q :
-  wp (Σ:=Σ) E e (λ v, wp (Σ:=Σ) E (fill K (v2e v)) Q) ⊑ wp (Σ:=Σ) E (fill K e) Q.
-Proof.
-  by apply (wp_bind (Σ:=Σ) (K := fill K)), fill_is_ctx.
-Qed.
+  wp E e (λ v, wp E (fill K (v2e v)) Q) ⊑ wp E (fill K e) Q.
+Proof. apply (wp_bind (K:=fill K)), fill_is_ctx. Qed.
 
 (** Base axioms for core primitives of the language: Stateful reductions. *)
 
@@ -17,9 +17,9 @@ Lemma wp_lift_step E1 E2 (φ : expr → state → Prop) Q e1 σ1 :
   E1 ⊆ E2 → to_val e1 = None →
   reducible e1 σ1 →
   (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → ef = None ∧ φ e2 σ2) →
-  pvs E2 E1 (ownP (Σ:=Σ) σ1 ★ ▷ ∀ e2 σ2, (■ φ e2 σ2 ∧ ownP (Σ:=Σ) σ2) -★
-    pvs E1 E2 (wp (Σ:=Σ) E2 e2 Q))
-  ⊑ wp (Σ:=Σ) E2 e1 Q.
+  pvs E2 E1 (ownP σ1 ★ ▷ ∀ e2 σ2, (■ φ e2 σ2 ∧ ownP σ2) -★
+    pvs E1 E2 (wp E2 e2 Q))
+  ⊑ wp E2 e1 Q.
 Proof.
   intros ? He Hsafe Hstep.
   (* RJ: working around https://coq.inria.fr/bugs/show_bug.cgi?id=4536 *)
@@ -45,8 +45,8 @@ Qed.
    postcondition a predicate over a *location* *)
 Lemma wp_alloc_pst E σ e v Q :
   e2v e = Some v →
-  (ownP (Σ:=Σ) σ ★ ▷(∀ l, ■(σ !! l = None) ∧ ownP (Σ:=Σ) (<[l:=v]>σ) -★ Q (LocV l)))
-       ⊑ wp (Σ:=Σ) E (Alloc e) Q.
+  (ownP σ ★ ▷(∀ l, ■(σ !! l = None) ∧ ownP (<[l:=v]>σ) -★ Q (LocV l)))
+       ⊑ wp E (Alloc e) Q.
 Proof.
   (* RJ FIXME (also for most other lemmas in this file): rewrite would be nicer... *)
   intros Hvl. etransitivity; last eapply wp_lift_step with (σ1 := σ)
@@ -72,7 +72,7 @@ Qed.
 
 Lemma wp_load_pst E σ l v Q :
   σ !! l = Some v →
-  (ownP (Σ:=Σ) σ ★ ▷(ownP σ -★ Q v)) ⊑ wp (Σ:=Σ) E (Load (Loc l)) Q.
+  (ownP σ ★ ▷(ownP σ -★ Q v)) ⊑ wp E (Load (Loc l)) Q.
 Proof.
   intros Hl. etransitivity; last eapply wp_lift_step with (σ1 := σ)
     (φ := λ e' σ', e' = v2e v ∧ σ' = σ); last first.
@@ -93,7 +93,7 @@ Qed.
 Lemma wp_store_pst E σ l e v v' Q :
   e2v e = Some v →
   σ !! l = Some v' →
-  (ownP (Σ:=Σ) σ ★ ▷(ownP (<[l:=v]>σ) -★ Q LitUnitV)) ⊑ wp (Σ:=Σ) E (Store (Loc l) e) Q.
+  (ownP σ ★ ▷(ownP (<[l:=v]>σ) -★ Q LitUnitV)) ⊑ wp E (Store (Loc l) e) Q.
 Proof.
   intros Hvl Hl. etransitivity; last eapply wp_lift_step with (σ1 := σ)
     (φ := λ e' σ', e' = LitUnit ∧ σ' = <[l:=v]>σ); last first.
@@ -114,7 +114,7 @@ Qed.
 Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Q :
   e2v e1 = Some v1 → e2v e2 = Some v2 →
   σ !! l = Some v' → v' <> v1 →
-  (ownP (Σ:=Σ) σ ★ ▷(ownP σ -★ Q LitFalseV)) ⊑ wp (Σ:=Σ) E (Cas (Loc l) e1 e2) Q.
+  (ownP σ ★ ▷(ownP σ -★ Q LitFalseV)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
   intros Hvl Hl. etransitivity; last eapply wp_lift_step with (σ1 := σ)
     (φ := λ e' σ', e' = LitFalse ∧ σ' = σ); last first.
@@ -137,7 +137,7 @@ Qed.
 Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Q :
   e2v e1 = Some v1 → e2v e2 = Some v2 →
   σ !! l = Some v1 →
-  (ownP (Σ:=Σ) σ ★ ▷(ownP (<[l:=v2]>σ) -★ Q LitTrueV)) ⊑ wp (Σ:=Σ) E (Cas (Loc l) e1 e2) Q.
+  (ownP σ ★ ▷(ownP (<[l:=v2]>σ) -★ Q LitTrueV)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
   intros Hvl Hl. etransitivity; last eapply wp_lift_step with (σ1 := σ)
     (φ := λ e' σ', e' = LitTrue ∧ σ' = <[l:=v2]>σ); last first.
@@ -162,7 +162,7 @@ Qed.
 (** Base axioms for core primitives of the language: Stateless reductions *)
 
 Lemma wp_fork E e :
-  ▷ wp (Σ:=Σ) coPset_all e (λ _, True) ⊑ wp (Σ:=Σ) E (Fork e) (λ v, ■(v = LitUnitV)).
+  ▷ wp coPset_all e (λ _, True) ⊑ wp E (Fork e) (λ v, ■(v = LitUnitV)).
 Proof.
   etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e' ef, e' = LitUnit ∧ ef = Some e);
@@ -189,7 +189,7 @@ Lemma wp_lift_pure_step E (φ : expr → Prop) Q e1 :
   to_val e1 = None →
   (∀ σ1, reducible e1 σ1) →
   (∀ σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → σ1 = σ2 ∧ ef = None ∧ φ e2) →
-  (▷ ∀ e2, ■ φ e2 → wp (Σ:=Σ) E e2 Q) ⊑ wp (Σ:=Σ) E e1 Q.
+  (▷ ∀ e2, ■ φ e2 → wp E e2 Q) ⊑ wp E e1 Q.
 Proof.
   intros He Hsafe Hstep.
   (* RJ: working around https://coq.inria.fr/bugs/show_bug.cgi?id=4536 *)
@@ -209,7 +209,7 @@ Qed.
 
 Lemma wp_rec E ef e v Q :
   e2v e = Some v →
-  ▷wp (Σ:=Σ) E ef.[Rec ef, e /] Q ⊑ wp (Σ:=Σ) E (App (Rec ef) e) Q.
+  ▷wp E ef.[Rec ef, e /] Q ⊑ wp E (App (Rec ef) e) Q.
 Proof.
   etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = ef.[Rec ef, e /]); last first.
@@ -221,7 +221,7 @@ Proof.
 Qed.
 
 Lemma wp_plus n1 n2 E Q :
-  ▷Q (LitNatV (n1 + n2)) ⊑ wp (Σ:=Σ) E (Plus (LitNat n1) (LitNat n2)) Q.
+  ▷Q (LitNatV (n1 + n2)) ⊑ wp E (Plus (LitNat n1) (LitNat n2)) Q.
 Proof.
   etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = LitNat (n1 + n2)); last first.
@@ -235,7 +235,7 @@ Qed.
 
 Lemma wp_le_true n1 n2 E Q :
   n1 ≤ n2 →
-  ▷Q LitTrueV ⊑ wp (Σ:=Σ) E (Le (LitNat n1) (LitNat n2)) Q.
+  ▷Q LitTrueV ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
 Proof.
   intros Hle. etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = LitTrue); last first.
@@ -250,7 +250,7 @@ Qed.
 
 Lemma wp_le_false n1 n2 E Q :
   n1 > n2 →
-  ▷Q LitFalseV ⊑ wp (Σ:=Σ) E (Le (LitNat n1) (LitNat n2)) Q.
+  ▷Q LitFalseV ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
 Proof.
   intros Hle. etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = LitFalse); last first.
@@ -265,7 +265,7 @@ Qed.
 
 Lemma wp_fst e1 v1 e2 v2 E Q :
   e2v e1 = Some v1 → e2v e2 = Some v2 →
-  ▷Q v1 ⊑ wp (Σ:=Σ) E (Fst (Pair e1 e2)) Q.
+  ▷Q v1 ⊑ wp E (Fst (Pair e1 e2)) Q.
 Proof.
   intros Hv1 Hv2. etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = e1); last first.
@@ -279,7 +279,7 @@ Qed.
 
 Lemma wp_snd e1 v1 e2 v2 E Q :
   e2v e1 = Some v1 → e2v e2 = Some v2 →
-  ▷Q v2 ⊑ wp (Σ:=Σ) E (Snd (Pair e1 e2)) Q.
+  ▷Q v2 ⊑ wp E (Snd (Pair e1 e2)) Q.
 Proof.
   intros Hv1 Hv2. etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = e2); last first.
@@ -293,7 +293,7 @@ Qed.
 
 Lemma wp_case_inl e0 v0 e1 e2 E Q :
   e2v e0 = Some v0 →
-  ▷wp (Σ:=Σ) E e1.[e0/] Q ⊑ wp (Σ:=Σ) E (Case (InjL e0) e1 e2) Q.
+  ▷wp E e1.[e0/] Q ⊑ wp E (Case (InjL e0) e1 e2) Q.
 Proof.
   intros Hv0. etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = e1.[e0/]); last first.
@@ -306,7 +306,7 @@ Qed.
 
 Lemma wp_case_inr e0 v0 e1 e2 E Q :
   e2v e0 = Some v0 →
-  ▷wp (Σ:=Σ) E e2.[e0/] Q ⊑ wp (Σ:=Σ) E (Case (InjR e0) e1 e2) Q.
+  ▷wp E e2.[e0/] Q ⊑ wp E (Case (InjR e0) e1 e2) Q.
 Proof.
   intros Hv0. etransitivity; last eapply wp_lift_pure_step with
     (φ := λ e', e' = e2.[e0/]); last first.
@@ -322,7 +322,7 @@ Qed.
 Lemma wp_le n1 n2 E P Q :
   (n1 ≤ n2 → P ⊑ ▷Q LitTrueV) →
   (n1 > n2 → P ⊑ ▷Q LitFalseV) →
-  P ⊑ wp (Σ:=Σ) E (Le (LitNat n1) (LitNat n2)) Q.
+  P ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
 Proof.
   intros HPle HPgt.
   assert (Decision (n1 ≤ n2)) as Hn12 by apply _.
@@ -330,3 +330,4 @@ Proof.
   - rewrite -wp_le_true; auto.
   - assert (n1 > n2) by omega. rewrite -wp_le_false; auto.
 Qed.
+End lifting.

barrier/parameter.v

 Require Export barrier.heap_lang.
 Require Import iris.parameter.
 
-Definition Σ := iParam_const heap_lang unitRA.
+Canonical Structure Σ := iParam_const heap_lang unitRA.

barrier/sugar.v

@@ -16,10 +16,13 @@ Definition LamV (e : {bind expr}) := RecV e.[ren(+1)].
 Definition LetCtx (K1 : ectx) (e2 : {bind expr}) := AppRCtx (LamV e2) K1.
 Definition SeqCtx (K1 : ectx) (e2 : expr) := LetCtx K1 (e2.[ren(+1)]).
 
+Section suger.
+Implicit Types P : iProp Σ.
+Implicit Types Q : ival Σ → iProp Σ.
+
 (** Proof rules for the sugar *)
 Lemma wp_lam E ef e v Q :
-  e2v e = Some v →
-  ▷wp (Σ:=Σ) E ef.[e/] Q ⊑ wp (Σ:=Σ) E (App (Lam ef) e) Q.
+  e2v e = Some v → ▷wp E ef.[e/] Q ⊑ wp E (App (Lam ef) e) Q.
 Proof.
   intros Hv. rewrite -wp_rec; last eassumption.
   (* RJ: This pulls in functional extensionality. If that bothers us, we have
@@ -28,20 +31,20 @@ Proof.
 Qed.
 
 Lemma wp_let e1 e2 E Q :
-  wp (Σ:=Σ) E e1 (λ v, ▷wp (Σ:=Σ) E (e2.[v2e v/]) Q) ⊑ wp (Σ:=Σ) E (Let e1 e2) Q.
+  wp E e1 (λ v, ▷wp E (e2.[v2e v/]) Q) ⊑ wp E (Let e1 e2) Q.
 Proof.
   rewrite -(wp_bind (LetCtx EmptyCtx e2)). apply wp_mono=>v.
   rewrite -wp_lam //. by rewrite v2v.
 Qed.
 
 Lemma wp_if_true e1 e2 E Q :
-  ▷wp (Σ:=Σ) E e1 Q ⊑ wp (Σ:=Σ) E (If LitTrue e1 e2) Q.
+  ▷wp E e1 Q ⊑ wp E (If LitTrue e1 e2) Q.
 Proof.
   rewrite -wp_case_inl //. by asimpl.
 Qed.
 
 Lemma wp_if_false e1 e2 E Q :
-  ▷wp (Σ:=Σ) E e2 Q ⊑ wp (Σ:=Σ) E (If LitFalse e1 e2) Q.
+  ▷wp E e2 Q ⊑ wp E (If LitFalse e1 e2) Q.
 Proof.
   rewrite -wp_case_inr //. by asimpl.
 Qed.
@@ -49,7 +52,7 @@ Qed.
 Lemma wp_lt n1 n2 E P Q :
   (n1 < n2 → P ⊑ ▷Q LitTrueV) →
   (n1 ≥ n2 → P ⊑ ▷Q LitFalseV) →
-  P ⊑ wp (Σ:=Σ) E (Lt (LitNat n1) (LitNat n2)) Q.
+  P ⊑ wp E (Lt (LitNat n1) (LitNat n2)) Q.
 Proof.
   intros HPlt HPge.
   rewrite -(wp_bind (LeLCtx EmptyCtx _)) -wp_plus -later_intro. simpl.
@@ -59,7 +62,7 @@ Qed.
 Lemma wp_eq n1 n2 E P Q :
   (n1 = n2 → P ⊑ ▷Q LitTrueV) →
   (n1 ≠ n2 → P ⊑ ▷Q LitFalseV) →
-  P ⊑ wp (Σ:=Σ) E (Eq (LitNat n1) (LitNat n2)) Q.
+  P ⊑ wp E (Eq (LitNat n1) (LitNat n2)) Q.
 Proof.
   intros HPeq HPne.
   rewrite -wp_let -wp_value' // -later_intro. asimpl.
@@ -72,3 +75,4 @@ Proof.
   - asimpl. rewrite -wp_case_inr // -later_intro -wp_value' //.
     apply HPne; omega.
 Qed.
+End suger.

barrier/tests.v

@@ -30,11 +30,14 @@ Module ParamTests.
 End ParamTests.
 
 Module LiftingTests.
+  Implicit Types P : iProp Σ.
+  Implicit Types Q : ival Σ → iProp Σ.
+
   (* TODO RJ: Some syntactic sugar for language expressions would be nice. *)
   Definition e3 := Load (Var 0).
   Definition e2 := Seq (Store (Var 0) (Plus (Load $ Var 0) (LitNat 1))) e3.
   Definition e := Let (Alloc (LitNat 1)) e2.
-  Goal ∀ σ E, (ownP (Σ:=Σ) σ) ⊑ (wp (Σ:=Σ) E e (λ v, ■(v = LitNatV 2))).
+  Goal ∀ σ E, (ownP σ) ⊑ (wp E e (λ v, ■(v = LitNatV 2))).
   Proof.
     move=> σ E. rewrite /e.
     rewrite -wp_let. rewrite -wp_alloc_pst; last done.
@@ -74,7 +77,7 @@ Module LiftingTests.
 
   Lemma FindPred_spec n1 n2 E Q :
     (■(n1 < n2) ∧ Q (LitNatV $ pred n2)) ⊑
-       wp (Σ:=Σ) E (App (FindPred (LitNat n2)) (LitNat n1)) Q.
+       wp E (App (FindPred (LitNat n2)) (LitNat n1)) Q.
   Proof.
     revert n1. apply löb_all_1=>n1.
     rewrite -wp_rec //. asimpl.
@@ -104,7 +107,7 @@ Module LiftingTests.
   Qed.
 
   Lemma Pred_spec n E Q :
-    ▷Q (LitNatV $ pred n) ⊑ wp (Σ:=Σ) E (App Pred (LitNat n)) Q.
+    ▷Q (LitNatV $ pred n) ⊑ wp E (App Pred (LitNat n)) Q.
   Proof.
     rewrite -wp_lam //. asimpl.
     rewrite -(wp_bind (CaseCtx EmptyCtx _ _)).
@@ -118,7 +121,7 @@ Module LiftingTests.
       + done.
   Qed.
 
-  Goal ∀ E, True ⊑ wp (Σ:=Σ) E (Let (App Pred (LitNat 42)) (App Pred (Var 0))) (λ v, ■(v = LitNatV 40)).
+  Goal ∀ E, True ⊑ wp E (Let (App Pred (LitNat 42)) (App Pred (Var 0))) (λ v, ■(v = LitNatV 40)).
   Proof.
     intros E. rewrite -wp_let. rewrite -Pred_spec -!later_intro.
     asimpl. (* TODO RJ: Can we somehow make it so that Pred gets folded again? *)

iris/parameter.v

 Require Export modures.cmra iris.language.
 
-Record iParam := IParam {
+Structure iParam := IParam {
   iexpr : Type;
   ival : Type;
   istate : Type;