Misc tweaking of consistency of coding style.

heap_lang/lifting.v

 Require Import prelude.gmap program_logic.lifting.
 Require Export program_logic.weakestpre heap_lang.heap_lang_tactics.
-Import uPred.
-Import heap_lang.
+Import uPred heap_lang.
 Local Hint Extern 0 (language.reducible _ _) => do_step ltac:(eauto 2).
 
 Section lifting.
@@ -9,6 +8,7 @@ Context {Σ : iFunctor}.
 Implicit Types P : iProp heap_lang Σ.
 Implicit Types Q : val → iProp heap_lang Σ.
 Implicit Types K : ectx.
+Implicit Types ef : option expr.
 
 (** Bind. *)
 Lemma wp_bind {E e} K Q :
@@ -26,7 +26,8 @@ Lemma wp_alloc_pst E σ e v Q :
        ⊑ wp E (Alloc e) Q.
 Proof.
   (* TODO RJ: This works around ssreflect bug #22. *)
-  intros. set (φ v' σ' ef := ∃ l, ef = @None expr ∧ v' = LocV l ∧ σ' = <[l:=v]>σ ∧ σ !! l = None).
+  intros. set (φ v' σ' ef := ∃ l,
+    ef = None ∧ v' = LocV l ∧ σ' = <[l:=v]>σ ∧ σ !! l = None).
   rewrite -(wp_lift_atomic_step (Alloc e) φ σ) // /φ;
     last by intros; inv_step; eauto 8.
   apply sep_mono, later_mono; first done.
@@ -41,24 +42,23 @@ Lemma wp_load_pst E σ l v Q :
   σ !! l = Some v →
   (ownP σ ★ ▷ (ownP σ -★ Q v)) ⊑ wp E (Load (Loc l)) Q.
 Proof.
-  intros; rewrite -(wp_lift_atomic_det_step σ v σ None) ?right_id //;
-    last (by intros; inv_step; eauto).
+  intros. rewrite -(wp_lift_atomic_det_step σ v σ None) ?right_id //;
+    last by intros; inv_step; eauto using to_of_val.
 Qed.
 
 Lemma wp_store_pst E σ l e v v' Q :
   to_val e = Some v → σ !! l = Some v' →
   (ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Q LitUnitV)) ⊑ wp E (Store (Loc l) e) Q.
 Proof.
-  intros.
-  rewrite -(wp_lift_atomic_det_step σ LitUnitV (<[l:=v]>σ) None) ?right_id //;
-    last by intros; inv_step; eauto.
+  intros. rewrite -(wp_lift_atomic_det_step σ LitUnitV (<[l:=v]>σ) None)
+    ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v' → v' ≠ v1 →
   (ownP σ ★ ▷ (ownP σ -★ Q LitFalseV)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
-  intros; rewrite -(wp_lift_atomic_det_step σ LitFalseV σ None) ?right_id //;
+  intros. rewrite -(wp_lift_atomic_det_step σ LitFalseV σ None) ?right_id //;
     last by intros; inv_step; eauto.
 Qed.
 
@@ -66,9 +66,8 @@ Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v1 →
   (ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Q LitTrueV)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
-  intros.
-  rewrite -(wp_lift_atomic_det_step σ LitTrueV (<[l:=v2]>σ) None) ?right_id //;
-    last by intros; inv_step; eauto.
+  intros. rewrite -(wp_lift_atomic_det_step σ LitTrueV (<[l:=v2]>σ) None)
+    ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 (** Base axioms for core primitives of the language: Stateless reductions *)
@@ -81,20 +80,19 @@ Proof.
   by rewrite -(wp_value' _ _ LitUnit) //; apply const_intro.
 Qed.
 
-Lemma wp_rec E ef e v Q :
+Lemma wp_rec E erec e v Q :
   to_val e = Some v →
-  ▷ wp E ef.[Rec ef, e /] Q ⊑ wp E (App (Rec ef) e) Q.
+  ▷ wp E erec.[Rec erec, e /] Q ⊑ wp E (App (Rec erec) e) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (App _ _) ef.[Rec ef, e /] None)
-                     ?right_id //=;
-    last by intros; inv_step; eauto.
+  intros. rewrite -(wp_lift_pure_det_step (App _ _) erec.[Rec erec, e /] None)
+    ?right_id //=; last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_plus E n1 n2 Q :
   ▷ Q (LitNatV (n1 + n2)) ⊑ wp E (Plus (LitNat n1) (LitNat n2)) Q.
 Proof.
-  rewrite -(wp_lift_pure_det_step (Plus _ _) (LitNat (n1 + n2)) None) ?right_id //;
-    last by intros; inv_step; eauto.
+  rewrite -(wp_lift_pure_det_step (Plus _ _) (LitNat (n1 + n2)) None)
+    ?right_id //; last by intros; inv_step; eauto.
   by rewrite -wp_value'.
 Qed.
 
@@ -102,7 +100,7 @@ Lemma wp_le_true E n1 n2 Q :
   n1 ≤ n2 →
   ▷ Q LitTrueV ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (Le _ _) LitTrue None) ?right_id //;
+  intros. rewrite -(wp_lift_pure_det_step (Le _ _) LitTrue None) ?right_id //;
     last by intros; inv_step; eauto with omega.
   by rewrite -wp_value'.
 Qed.
@@ -111,7 +109,7 @@ Lemma wp_le_false E n1 n2 Q :
   n1 > n2 →
   ▷ Q LitFalseV ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (Le _ _) LitFalse None) ?right_id //;
+  intros. rewrite -(wp_lift_pure_det_step (Le _ _) LitFalse None) ?right_id //;
     last by intros; inv_step; eauto with omega.
   by rewrite -wp_value'.
 Qed.
@@ -120,7 +118,7 @@ Lemma wp_fst E e1 v1 e2 v2 Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 →
   ▷Q v1 ⊑ wp E (Fst (Pair e1 e2)) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (Fst _) e1 None) ?right_id //;
+  intros. rewrite -(wp_lift_pure_det_step (Fst _) e1 None) ?right_id //;
     last by intros; inv_step; eauto.
   by rewrite -wp_value'.
 Qed.
@@ -129,7 +127,7 @@ Lemma wp_snd E e1 v1 e2 v2 Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 →
   ▷ Q v2 ⊑ wp E (Snd (Pair e1 e2)) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (Snd _) e2 None) ?right_id //;
+  intros. rewrite -(wp_lift_pure_det_step (Snd _) e2 None) ?right_id //;
     last by intros; inv_step; eauto.
   by rewrite -wp_value'.
 Qed.
@@ -138,16 +136,16 @@ Lemma wp_case_inl E e0 v0 e1 e2 Q :
   to_val e0 = Some v0 →
   ▷ wp E e1.[e0/] Q ⊑ wp E (Case (InjL e0) e1 e2) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (Case _ _ _) e1.[e0/] None) ?right_id //;
-    last by intros; inv_step; eauto.
+  intros. rewrite -(wp_lift_pure_det_step (Case _ _ _) e1.[e0/] None)
+    ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_case_inr E e0 v0 e1 e2 Q :
   to_val e0 = Some v0 →
   ▷ wp E e2.[e0/] Q ⊑ wp E (Case (InjR e0) e1 e2) Q.
 Proof.
-  intros; rewrite -(wp_lift_pure_det_step (Case _ _ _) e2.[e0/] None) ?right_id //;
-    last by intros; inv_step; eauto.
+  intros. rewrite -(wp_lift_pure_det_step (Case _ _ _) e2.[e0/] None)
+    ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 (** Some derived stateless axioms *)

program_logic/hoare_lifting.v

 Require Export program_logic.hoare program_logic.lifting.
+Import uPred.
 
 Local Notation "{{ P } } ef ?@ E {{ Q } }" :=
   (default True%I ef (λ e, ht E P e Q))
@@ -12,7 +13,6 @@ Context {Λ : language} {Σ : iFunctor}.
 Implicit Types e : expr Λ.
 Implicit Types P : iProp Λ Σ.
 Implicit Types R : val Λ → iProp Λ Σ.
-Import uPred.
 
 Lemma ht_lift_step E1 E2
     (φ : expr Λ → state Λ → option (expr Λ) → Prop) P P' Q1 Q2 R e1 σ1 :

program_logic/lifting.v

@@ -14,13 +14,15 @@ 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.
+
 Lemma wp_lift_step E1 E2
     (φ : expr Λ → state Λ → option (expr Λ) → Prop) Q e1 σ1 :
   E1 ⊆ E2 → to_val e1 = None →
   reducible e1 σ1 →
   (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
-  pvs E2 E1 (ownP σ1 ★ ▷ ∀ e2 σ2 ef, (■ φ e2 σ2 ef ∧ ownP σ2) -★
-    pvs E1 E2 (wp E2 e2 Q ★ default True ef (flip (wp coPset_all) (λ _, True))))
+  pvs E2 E1 (ownP σ1 ★ ▷ ∀ e2 σ2 ef,
+    (■ φ e2 σ2 ef ∧ ownP σ2) -★ pvs E1 E2 (wp E2 e2 Q ★ wp_fork ef))
   ⊑ wp E2 e1 Q.
 Proof.
   intros ? He Hsafe Hstep r n ? Hvs; constructor; auto.
@@ -41,9 +43,7 @@ Lemma wp_lift_pure_step E (φ : expr Λ → option (expr Λ) → Prop) Q e1 :
   to_val e1 = None →
   (∀ σ1, reducible e1 σ1) →
   (∀ σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → σ1 = σ2 ∧ φ e2 ef) →
-  (▷ ∀ e2 ef, ■ φ e2 ef →
-    wp E e2 Q ★ default True ef (flip (wp coPset_all) (λ _, True)))
-  ⊑ wp E e1 Q.
+  (▷ ∀ e2 ef, ■ φ e2 ef → wp E e2 Q ★ wp_fork ef) ⊑ wp E e1 Q.
 Proof.
   intros He Hsafe Hstep r [|n] ?; [done|]; intros Hwp; constructor; auto.
   intros rf k Ef σ1 ???; split; [done|].
@@ -56,17 +56,17 @@ Qed.
 Opaque uPred_holds.
 Import uPred.
 
-Lemma wp_lift_atomic_step {E Q} e1 (φ : val Λ → state Λ → option (expr Λ) → Prop) σ1 :
+Lemma wp_lift_atomic_step {E Q} e1
+    (φ : val Λ → state Λ → option (expr Λ) → Prop) σ1 :
   to_val e1 = None →
   reducible e1 σ1 →
-  (∀ e' σ' ef, prim_step e1 σ1 e' σ' ef → ∃ v', to_val e' = Some v' ∧ φ v' σ' ef) →
-  (ownP σ1 ★ ▷ ∀ v2 σ2 ef, ■ φ v2 σ2 ef ∧ ownP σ2 -★
-     Q v2 ★ default True ef (flip (wp coPset_all) (λ _, True)))
-    ⊑ wp E e1 Q.
+  (∀ e2 σ2 ef,
+    prim_step e1 σ1 e2 σ2 ef → ∃ v2, to_val e2 = Some v2 ∧ φ v2 σ2 ef) →
+  (ownP σ1 ★ ▷ ∀ v2 σ2 ef, ■ φ v2 σ2 ef ∧ ownP σ2 -★ Q v2 ★ wp_fork ef)
+  ⊑ wp E e1 Q.
 Proof.
-  intros He Hsafe Hstep.
-  rewrite -(wp_lift_step E E
-    (λ e' σ' ef, ∃ v', to_val e' = Some v' ∧ φ v' σ' ef) _ e1 σ1) //; [].
+  intros. rewrite -(wp_lift_step E E (λ e2 σ2 ef, ∃ v2,
+    to_val e2 = Some v2 ∧ φ v2 σ2 ef) _ e1 σ1) //; [].
   rewrite -pvs_intro. apply sep_mono, later_mono; first done.
   apply forall_intro=>e2'; apply forall_intro=>σ2'.
   apply forall_intro=>ef; apply wand_intro_l.
@@ -80,33 +80,29 @@ Qed.
 Lemma wp_lift_atomic_det_step {E Q e1} σ1 v2 σ2 ef :
   to_val e1 = None →
   reducible e1 σ1 →
-  (∀ e' σ' ef', prim_step e1 σ1 e' σ' ef' → ef' = ef ∧ e' = of_val v2 ∧ σ' = σ2) →
-  (ownP σ1 ★ ▷ (ownP σ2 -★ Q v2 ★
-                       default True ef (flip (wp coPset_all) (λ _, True))))
-    ⊑ wp E e1 Q.
+  (∀ e2' σ2' ef', prim_step e1 σ1 e2' σ2' ef' →
+    σ2 = σ2' ∧ to_val e2' = Some v2 ∧ ef = ef') →
+  (ownP σ1 ★ ▷ (ownP σ2 -★ Q v2 ★ wp_fork ef)) ⊑ wp E e1 Q.
 Proof.
-  intros He Hsafe Hstep.
-  rewrite -(wp_lift_atomic_step _ (λ v' σ' ef', v' = v2 ∧ σ' = σ2 ∧ ef' = ef) σ1) //;
-    last first.
-  { intros. exists v2. apply Hstep in H. destruct_conjs; subst.
-    eauto using to_of_val. }
+  intros. rewrite -(wp_lift_atomic_step _ (λ v2' σ2' ef',
+    σ2 = σ2' ∧ v2 = v2' ∧ ef = ef') σ1) //; last naive_solver.
   apply sep_mono, later_mono; first done.
   apply forall_intro=>e2'; apply forall_intro=>σ2'; apply forall_intro=>ef'.
   apply wand_intro_l.
   rewrite always_and_sep_l' -associative -always_and_sep_l'.
-  apply const_elim_l=>-[-> [-> ->]] /=.
-  by rewrite wand_elim_r.
+  apply const_elim_l=>-[-> [-> ->]] /=. by rewrite wand_elim_r.
 Qed.
 
 Lemma wp_lift_pure_det_step {E Q} e1 e2 ef :
   to_val e1 = None →
   (∀ σ1, reducible e1 σ1) →
-  (∀ σ1 e' σ' ef', prim_step e1 σ1 e' σ' ef' → σ1 = σ' ∧ ef' = ef ∧ e' = e2) →
-  ▷ (wp E e2 Q ★ default True ef (flip (wp coPset_all) (λ _, True))) ⊑ wp E e1 Q.
+  (∀ σ1 e2' σ2 ef', prim_step e1 σ1 e2' σ2 ef' → σ1 = σ2 ∧ e2 = e2' ∧ ef = ef')→
+  ▷ (wp E e2 Q ★ wp_fork ef) ⊑ wp E e1 Q.
 Proof.
-  intros. rewrite -(wp_lift_pure_step E (λ e' ef', ef' = ef ∧ e' = e2) _ e1) //=.
+  intros.
+  rewrite -(wp_lift_pure_step E (λ e2' ef', e2 = e2' ∧ ef = ef') _ e1) //=.
   apply later_mono, forall_intro=>e'; apply forall_intro=>ef'.
-  apply impl_intro_l, const_elim_l=>-[-> ->] /=; done.
+  by apply impl_intro_l, const_elim_l=>-[-> ->].
 Qed.
 
 End lifting.