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Commit ffa92c50 authored by Robbert Krebbers's avatar Robbert Krebbers
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Introduce notation || e @ E {{ Φ }} for weakest pre.

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barrier/barrier.v
+ 4
− 4
View file @ ffa92c50
......@@ -146,7 +146,7 @@ Section proof.
Lemma newchan_spec (P : iProp) (Φ : val → iProp) :
(heap_ctx heapN ★ ∀ l, recv l P ★ send l P -★ Φ (LocV l))
⊑ wp ⊤ (newchan '()) Φ.
⊑ || newchan '() {{ Φ }}.
Proof.
rewrite /newchan. wp_rec. (* TODO: wp_seq. *)
rewrite -wp_pvs. wp> eapply wp_alloc; eauto with I ndisj.
......@@ -196,7 +196,7 @@ Section proof.
Qed.
Lemma signal_spec l P (Φ : val → iProp) :
heapN ⊥ N → (send l P ★ P ★ Φ '()) ⊑ wp ⊤ (signal (LocV l)) Φ.
heapN ⊥ N → (send l P ★ P ★ Φ '()) ⊑ || signal (LocV l) {{ Φ }}.
Proof.
intros Hdisj. rewrite /signal /send /barrier_ctx. rewrite sep_exist_r.
apply exist_elim=>γ. wp_rec. (* FIXME wp_let *)
......@@ -226,12 +226,12 @@ Section proof.
Qed.
Lemma wait_spec l P (Φ : val → iProp) :
heapN ⊥ N → (recv l P ★ (P -★ Φ '())) ⊑ wp ⊤ (wait (LocV l)) Φ.
heapN ⊥ N → (recv l P ★ (P -★ Φ '())) ⊑ || wait (LocV l) {{ Φ }}.
Proof.
Abort.
Lemma split_spec l P1 P2 Φ :
(recv l (P1 ★ P2) ★ (recv l P1 ★ recv l P2 -★ Φ '())) ⊑ wp ⊤ Skip Φ.
(recv l (P1 ★ P2) ★ (recv l P1 ★ recv l P2 -★ Φ '())) ⊑ || Skip {{ Φ }}.
Proof.
Abort.
......
heap_lang/derived.v
+ 16
− 13
View file @ ffa92c50
......@@ -17,44 +17,47 @@ Implicit Types Φ : val → iProp heap_lang Σ.
(** Proof rules for the sugar *)
Lemma wp_lam' E x ef e v Φ :
to_val e = Some v → ▷ wp E (subst ef x v) Φ ⊑ wp E (App (Lam x ef) e) Φ.
to_val e = Some v →
▷ || subst ef x v @ E {{ Φ }} ⊑ || App (Lam x ef) e @ E {{ Φ }}.
Proof. intros. by rewrite -wp_rec' ?subst_empty. Qed.
Lemma wp_let' E x e1 e2 v Φ :
to_val e1 = Some v → ▷ wp E (subst e2 x v) Φ ⊑ wp E (Let x e1 e2) Φ.
to_val e1 = Some v →
▷ || subst e2 x v @ E {{ Φ }} ⊑ || Let x e1 e2 @ E {{ Φ }}.
Proof. apply wp_lam'. Qed.
Lemma wp_seq E e1 e2 Φ : wp E e1 (λ _, ▷ wp E e2 Φ) ⊑ wp E (Seq e1 e2) Φ.
Lemma wp_seq E e1 e2 Φ :
|| e1 @ E {{ λ _, ▷ || e2 @ E {{ Φ }} }} ⊑ || Seq e1 e2 @ E {{ Φ }}.
Proof.
rewrite -(wp_bind [LetCtx "" e2]). apply wp_mono=>v.
by rewrite -wp_let' //= ?to_of_val ?subst_empty.
Qed.
Lemma wp_skip E Φ : ▷ (Φ (LitV LitUnit)) ⊑ wp E Skip Φ.
Lemma wp_skip E Φ : ▷ Φ (LitV LitUnit) ⊑ || Skip @ E {{ Φ }}.
Proof. rewrite -wp_seq -wp_value // -wp_value //. Qed.
Lemma wp_le E (n1 n2 : Z) P Φ :
(n1 ≤ n2 → P ⊑ ▷ Φ (LitV $ LitBool true)) →
(n2 < n1 → P ⊑ ▷ Φ (LitV $ LitBool false)) →
P ⊑ wp E (BinOp LeOp (Lit $ LitInt n1) (Lit $ LitInt n2)) Φ.
(n1 ≤ n2 → P ⊑ ▷ Φ (LitV (LitBool true))) →
(n2 < n1 → P ⊑ ▷ Φ (LitV (LitBool false))) →
P ⊑ || BinOp LeOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
Proof.
intros. rewrite -wp_bin_op //; [].
destruct (bool_decide_reflect (n1 ≤ n2)); by eauto with omega.
Qed.
Lemma wp_lt E (n1 n2 : Z) P Φ :
(n1 < n2 → P ⊑ ▷ Φ (LitV $ LitBool true)) →
(n2 ≤ n1 → P ⊑ ▷ Φ (LitV $ LitBool false)) →
P ⊑ wp E (BinOp LtOp (Lit $ LitInt n1) (Lit $ LitInt n2)) Φ.
(n1 < n2 → P ⊑ ▷ Φ (LitV (LitBool true))) →
(n2 ≤ n1 → P ⊑ ▷ Φ (LitV (LitBool false))) →
P ⊑ || BinOp LtOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
Proof.
intros. rewrite -wp_bin_op //; [].
destruct (bool_decide_reflect (n1 < n2)); by eauto with omega.
Qed.
Lemma wp_eq E (n1 n2 : Z) P Φ :
(n1 = n2 → P ⊑ ▷ Φ (LitV $ LitBool true)) →
(n1 ≠ n2 → P ⊑ ▷ Φ (LitV $ LitBool false)) →
P ⊑ wp E (BinOp EqOp (Lit $ LitInt n1) (Lit $ LitInt n2)) Φ.
(n1 = n2 → P ⊑ ▷ Φ (LitV (LitBool true))) →
(n1 ≠ n2 → P ⊑ ▷ Φ (LitV (LitBool false))) →
P ⊑ || BinOp EqOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
Proof.
intros. rewrite -wp_bin_op //; [].
destruct (bool_decide_reflect (n1 = n2)); by eauto with omega.
......
heap_lang/heap.v
+ 6
− 6
View file @ ffa92c50
......@@ -65,7 +65,7 @@ Section heap.
(** Allocation *)
Lemma heap_alloc E N σ :
authG heap_lang Σ heapRA → nclose N ⊆ E →
ownP σ ⊑ (|={E}=> ∃ (_ : heapG Σ), heap_ctx N ∧ Π★{map σ} heap_mapsto).
ownP σ ⊑ (|={E}=> ∃ _ : heapG Σ, heap_ctx N ∧ Π★{map σ} heap_mapsto).
Proof.
intros. rewrite -{1}(from_to_heap σ). etransitivity.
{ rewrite [ownP _]later_intro.
......@@ -100,7 +100,7 @@ Section heap.
to_val e = Some v → nclose N ⊆ E →
P ⊑ heap_ctx N →
P ⊑ (▷ ∀ l, l ↦ v -★ Φ (LocV l)) →
P ⊑ wp E (Alloc e) Φ.
P ⊑ || Alloc e @ E {{ Φ }}.
Proof.
rewrite /heap_ctx /heap_inv /heap_mapsto=> ?? Hctx HP.
transitivity (|={E}=> auth_own heap_name ∅ ★ P)%I.
......@@ -127,7 +127,7 @@ Section heap.
nclose N ⊆ E →
P ⊑ heap_ctx N →
P ⊑ (▷ l ↦ v ★ ▷ (l ↦ v -★ Φ v)) →
P ⊑ wp E (Load (Loc l)) Φ.
P ⊑ || Load (Loc l) @ E {{ Φ }}.
Proof.
rewrite /heap_ctx /heap_inv /heap_mapsto=>HN ? HPΦ.
apply (auth_fsa' heap_inv (wp_fsa _) id)
......@@ -146,7 +146,7 @@ Section heap.
to_val e = Some v → nclose N ⊆ E →
P ⊑ heap_ctx N →
P ⊑ (▷ l ↦ v' ★ ▷ (l ↦ v -★ Φ (LitV LitUnit))) →
P ⊑ wp E (Store (Loc l) e) Φ.
P ⊑ || Store (Loc l) e @ E {{ Φ }}.
Proof.
rewrite /heap_ctx /heap_inv /heap_mapsto=>? HN ? HPΦ.
apply (auth_fsa' heap_inv (wp_fsa _) (alter (λ _, Excl v) l))
......@@ -167,7 +167,7 @@ Section heap.
nclose N ⊆ E →
P ⊑ heap_ctx N →
P ⊑ (▷ l ↦ v' ★ ▷ (l ↦ v' -★ Φ (LitV (LitBool false)))) →
P ⊑ wp E (Cas (Loc l) e1 e2) Φ.
P ⊑ || Cas (Loc l) e1 e2 @ E {{ Φ }}.
Proof.
rewrite /heap_ctx /heap_inv /heap_mapsto=>??? HN ? HPΦ.
apply (auth_fsa' heap_inv (wp_fsa _) id)
......@@ -187,7 +187,7 @@ Section heap.
nclose N ⊆ E →
P ⊑ heap_ctx N →
P ⊑ (▷ l ↦ v1 ★ ▷ (l ↦ v2 -★ Φ (LitV (LitBool true)))) →
P ⊑ wp E (Cas (Loc l) e1 e2) Φ.
P ⊑ || Cas (Loc l) e1 e2 @ E {{ Φ }}.
Proof.
rewrite /heap_ctx /heap_inv /heap_mapsto=> ?? HN ? HPΦ.
apply (auth_fsa' heap_inv (wp_fsa _) (alter (λ _, Excl v2) l))
......
heap_lang/lifting.v
+ 21
− 22
View file @ ffa92c50
......@@ -16,18 +16,14 @@ Implicit Types ef : option expr.
(** Bind. *)
Lemma wp_bind {E e} K Φ :
wp E e (λ v, wp E (fill K (of_val v)) Φ) ⊑ wp E (fill K e) Φ.
Proof. apply weakestpre.wp_bind. Qed.
Lemma wp_bindi {E e} Ki Φ :
wp E e (λ v, wp E (fill_item Ki (of_val v)) Φ) ⊑ wp E (fill_item Ki e) Φ.
|| e @ E {{ λ v, || fill K (of_val v) @ E {{ Φ }}}} ⊑ || fill K e @ E {{ Φ }}.
Proof. apply weakestpre.wp_bind. Qed.
(** Base axioms for core primitives of the language: Stateful reductions. *)
Lemma wp_alloc_pst E σ e v Φ :
to_val e = Some v →
(ownP σ ★ ▷ (∀ l, σ !! l = None ∧ ownP (<[l:=v]>σ) -★ Φ (LocV l)))
⊑ wp E (Alloc e) Φ.
⊑ || Alloc e @ E {{ Φ }}.
Proof.
(* TODO RJ: This works around ssreflect bug #22. *)
intros. set (φ v' σ' ef := ∃ l,
......@@ -44,7 +40,7 @@ Qed.
Lemma wp_load_pst E σ l v Φ :
σ !! l = Some v →
(ownP σ ★ ▷ (ownP σ -★ Φ v)) ⊑ wp E (Load (Loc l)) Φ.
(ownP σ ★ ▷ (ownP σ -★ Φ v)) ⊑ || Load (Loc l) @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_atomic_det_step σ v σ None) ?right_id //;
last by intros; inv_step; eauto using to_of_val.
......@@ -52,7 +48,8 @@ Qed.
Lemma wp_store_pst E σ l e v v' Φ :
to_val e = Some v → σ !! l = Some v' →
(ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Φ (LitV LitUnit))) ⊑ wp E (Store (Loc l) e) Φ.
(ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Φ (LitV LitUnit)))
⊑ || Store (Loc l) e @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_atomic_det_step σ (LitV LitUnit) (<[l:=v]>σ) None)
?right_id //; last by intros; inv_step; eauto.
......@@ -60,7 +57,8 @@ Qed.
Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Φ :
to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v' → v' ≠ v1 →
(ownP σ ★ ▷ (ownP σ -★ Φ (LitV $ LitBool false))) ⊑ wp E (Cas (Loc l) e1 e2) Φ.
(ownP σ ★ ▷ (ownP σ -★ Φ (LitV $ LitBool false)))
⊑ || Cas (Loc l) e1 e2 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool false) σ None)
?right_id //; last by intros; inv_step; eauto.
......@@ -69,15 +67,15 @@ Qed.
Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Φ :
to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v1 →
(ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Φ (LitV $ LitBool true)))
⊑ wp E (Cas (Loc l) e1 e2) Φ.
⊑ || Cas (Loc l) e1 e2 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool true) (<[l:=v2]>σ) None)
?right_id //; last by intros; inv_step; eauto.
intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool true)
(<[l:=v2]>σ) None) ?right_id //; last by intros; inv_step; eauto.
Qed.
(** Base axioms for core primitives of the language: Stateless reductions *)
Lemma wp_fork E e Φ :
(▷ Φ (LitV LitUnit) ★ ▷ wp (Σ:=Σ) ⊤ e (λ _, True)) ⊑ wp E (Fork e) Φ.
(▷ Φ (LitV LitUnit) ★ ▷ || e {{ λ _, True }}) ⊑ || Fork e @ E {{ Φ }}.
Proof.
rewrite -(wp_lift_pure_det_step (Fork e) (Lit LitUnit) (Some e)) //=;
last by intros; inv_step; eauto.
......@@ -88,7 +86,8 @@ Qed.
The final version is defined in substitution.v. *)
Lemma wp_rec' E f x e1 e2 v Φ :
to_val e2 = Some v →
▷ wp E (subst (subst e1 f (RecV f x e1)) x v) Φ ⊑ wp E (App (Rec f x e1) e2) Φ.
▷ || subst (subst e1 f (RecV f x e1)) x v @ E {{ Φ }}
⊑ || App (Rec f x e1) e2 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_pure_det_step (App _ _)
(subst (subst e1 f (RecV f x e1)) x v) None) ?right_id //=;
......@@ -97,7 +96,7 @@ Qed.
Lemma wp_un_op E op l l' Φ :
un_op_eval op l = Some l' →
▷ Φ (LitV l') ⊑ wp E (UnOp op (Lit l)) Φ.
▷ Φ (LitV l') ⊑ || UnOp op (Lit l) @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_pure_det_step (UnOp op _) (Lit l') None)
?right_id -?wp_value //; intros; inv_step; eauto.
......@@ -105,21 +104,21 @@ Qed.
Lemma wp_bin_op E op l1 l2 l' Φ :
bin_op_eval op l1 l2 = Some l' →
▷ Φ (LitV l') ⊑ wp E (BinOp op (Lit l1) (Lit l2)) Φ.
▷ Φ (LitV l') ⊑ || BinOp op (Lit l1) (Lit l2) @ E {{ Φ }}.
Proof.
intros Heval. rewrite -(wp_lift_pure_det_step (BinOp op _ _) (Lit l') None)
?right_id -?wp_value //; intros; inv_step; eauto.
Qed.
Lemma wp_if_true E e1 e2 Φ :
▷ wp E e1 Φ ⊑ wp E (If (Lit $ LitBool true) e1 e2) Φ.
▷ || e1 @ E {{ Φ }} ⊑ || If (Lit (LitBool true)) e1 e2 @ E {{ Φ }}.
Proof.
rewrite -(wp_lift_pure_det_step (If _ _ _) e1 None)
?right_id //; intros; inv_step; eauto.
Qed.
Lemma wp_if_false E e1 e2 Φ :
▷ wp E e2 Φ ⊑ wp E (If (Lit $ LitBool false) e1 e2) Φ.
▷ || e2 @ E {{ Φ }} ⊑ || If (Lit (LitBool false)) e1 e2 @ E {{ Φ }}.
Proof.
rewrite -(wp_lift_pure_det_step (If _ _ _) e2 None)
?right_id //; intros; inv_step; eauto.
......@@ -127,7 +126,7 @@ Qed.
Lemma wp_fst E e1 v1 e2 v2 Φ :
to_val e1 = Some v1 → to_val e2 = Some v2 →
▷ Φ v1 ⊑ wp E (Fst $ Pair e1 e2) Φ.
▷ Φ v1 ⊑ || Fst (Pair e1 e2) @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_pure_det_step (Fst _) e1 None)
?right_id -?wp_value //; intros; inv_step; eauto.
......@@ -135,7 +134,7 @@ Qed.
Lemma wp_snd E e1 v1 e2 v2 Φ :
to_val e1 = Some v1 → to_val e2 = Some v2 →
▷ Φ v2 ⊑ wp E (Snd $ Pair e1 e2) Φ.
▷ Φ v2 ⊑ || Snd (Pair e1 e2) @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_pure_det_step (Snd _) e2 None)
?right_id -?wp_value //; intros; inv_step; eauto.
......@@ -143,7 +142,7 @@ Qed.
Lemma wp_case_inl' E e0 v0 x1 e1 x2 e2 Φ :
to_val e0 = Some v0 →
▷ wp E (subst e1 x1 v0) Φ ⊑ wp E (Case (InjL e0) x1 e1 x2 e2) Φ.
▷ || subst e1 x1 v0 @ E {{ Φ }} ⊑ || Case (InjL e0) x1 e1 x2 e2 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_pure_det_step (Case _ _ _ _ _)
(subst e1 x1 v0) None) ?right_id //; intros; inv_step; eauto.
......@@ -151,7 +150,7 @@ Qed.
Lemma wp_case_inr' E e0 v0 x1 e1 x2 e2 Φ :
to_val e0 = Some v0 →
▷ wp E (subst e2 x2 v0) Φ ⊑ wp E (Case (InjR e0) x1 e1 x2 e2) Φ.
▷ || subst e2 x2 v0 @ E {{ Φ }} ⊑ || Case (InjR e0) x1 e1 x2 e2 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_pure_det_step (Case _ _ _ _ _)
(subst e2 x2 v0) None) ?right_id //; intros; inv_step; eauto.
......
heap_lang/notation.v
+ 6
− 1
View file @ ffa92c50
From heap_lang Require Export derived.
(* What about Arguments for hoare triples?. *)
Arguments wp {_ _} _ _%L _.
Notation "|| e @ E {{ Φ } }" := (wp E e%L Φ)
(at level 20, e, Φ at level 200,
format "|| e @ E {{ Φ } }") : uPred_scope.
Notation "|| e {{ Φ } }" := (wp ⊤ e%L Φ)
(at level 20, e, Φ at level 200,
format "|| e {{ Φ } }") : uPred_scope.
Coercion LitInt : Z >-> base_lit.
Coercion LitBool : bool >-> base_lit.
......
heap_lang/substitution.v
+ 9
− 8
View file @ ffa92c50
......@@ -26,10 +26,10 @@ to be unfolded. For example, consider the rule [wp_rec'] from below:
<<
Definition foo : val := rec: "f" "x" := ... .
Lemma wp_rec' E e1 f x erec e2 v Q :
Lemma wp_rec E e1 f x erec e2 v Φ :
e1 = Rec f x erec →
to_val e2 = Some v →
▷ wp E (gsubst (gsubst erec f e1) x e2) Q ⊑ wp E (App e1 e2) Q.
▷ || gsubst (gsubst erec f e1) x e2 @ E {{ Φ }} ⊑ || App e1 e2 @ E {{ Φ }}.
>>
We ensure that [e1] is substituted instead of [RecV f x erec]. So, for example
......@@ -123,7 +123,7 @@ Hint Resolve to_of_val.
Lemma wp_rec E e1 f x erec e2 v Φ :
e1 = Rec f x erec →
to_val e2 = Some v →
▷ wp E (gsubst (gsubst erec f e1) x e2) Φ ⊑ wp E (App e1 e2) Φ.
▷ || gsubst (gsubst erec f e1) x e2 @ E {{ Φ }} ⊑ || App e1 e2 @ E {{ Φ }}.
Proof.
intros -> <-%of_to_val.
rewrite (gsubst_correct _ _ (RecV _ _ _)) gsubst_correct.
......@@ -131,21 +131,22 @@ Proof.
Qed.
Lemma wp_lam E x ef e v Φ :
to_val e = Some v → ▷ wp E (gsubst ef x e) Φ ⊑ wp E (App (Lam x ef) e) Φ.
to_val e = Some v →
▷ || gsubst ef x e @ E {{ Φ }} ⊑ || App (Lam x ef) e @ E {{ Φ }}.
Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_lam'. Qed.
Lemma wp_let E x e1 e2 v Φ :
to_val e1 = Some v → ▷ wp E (gsubst e2 x e1) Φ ⊑ wp E (Let x e1 e2) Φ.
to_val e1 = Some v →
▷ || gsubst e2 x e1 @ E {{ Φ }} ⊑ || Let x e1 e2 @ E {{ Φ }}.
Proof. apply wp_lam. Qed.
Lemma wp_case_inl E e0 v0 x1 e1 x2 e2 Φ :
to_val e0 = Some v0 →
▷ wp E (gsubst e1 x1 e0) Φ ⊑ wp E (Case (InjL e0) x1 e1 x2 e2) Φ.
▷ || gsubst e1 x1 e0 @ E {{ Φ }} ⊑ || Case (InjL e0) x1 e1 x2 e2 @ E {{ Φ }}.
Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_case_inl'. Qed.
Lemma wp_case_inr E e0 v0 x1 e1 x2 e2 Φ :
to_val e0 = Some v0 →
▷ wp E (gsubst e2 x2 e0) Φ ⊑ wp E (Case (InjR e0) x1 e1 x2 e2) Φ.
▷ || gsubst e2 x2 e0 @ E {{ Φ }} ⊑ || Case (InjR e0) x1 e1 x2 e2 @ E {{ Φ }}.
Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_case_inr'. Qed.
End wp.
heap_lang/tests.v
+ 5
− 4
View file @ ffa92c50
......@@ -29,7 +29,8 @@ Section LiftingTests.
Definition heap_e : expr :=
let: "x" := ref '1 in "x" <- !"x" + '1;; !"x".
Lemma heap_e_spec E N : nclose N ⊆ E → heap_ctx N ⊑ wp E heap_e (λ v, v = '2).
Lemma heap_e_spec E N :
nclose N ⊆ E → heap_ctx N ⊑ || heap_e @ E {{ λ v, v = '2 }}.
Proof.
rewrite /heap_e=>HN. rewrite -(wp_mask_weaken N E) //.
wp> eapply wp_alloc; eauto. apply forall_intro=>l; apply wand_intro_l.
......@@ -48,7 +49,7 @@ Section LiftingTests.
if: "x" ≤ '0 then -FindPred (-"x" + '2) '0 else FindPred "x" '0.
Lemma FindPred_spec n1 n2 E Φ :
(■ (n1 < n2) ∧ Φ '(n2 - 1)) ⊑ wp E (FindPred 'n2 'n1) Φ.
(■ (n1 < n2) ∧ Φ '(n2 - 1)) ⊑ || FindPred 'n2 'n1 @ E {{ Φ }}.
Proof.
revert n1; apply löb_all_1=>n1.
rewrite (comm uPred_and (■ _)%I) assoc; apply const_elim_r=>?.
......@@ -62,7 +63,7 @@ Section LiftingTests.
- wp_value. assert (n1 = n2 - 1) as -> by omega; auto with I.
Qed.
Lemma Pred_spec n E Φ : ▷ Φ (LitV (n - 1)) ⊑ wp E (Pred 'n)%L Φ.
Lemma Pred_spec n E Φ : ▷ Φ (LitV (n - 1)) ⊑ || Pred 'n @ E {{ Φ }}.
Proof.
wp_rec>; apply later_mono; wp_bin_op=> ?; wp_if.
- wp_un_op. wp_bin_op.
......@@ -73,7 +74,7 @@ Section LiftingTests.
Qed.
Lemma Pred_user E :
True ⊑ wp (Σ:=globalF Σ) E (let: "x" := Pred '42 in Pred "x") (λ v, v = '40).
(True : iProp) ⊑ || let: "x" := Pred '42 in Pred "x" @ E {{ λ v, v = '40 }}.
Proof.
intros. ewp> apply Pred_spec. wp_rec. ewp> apply Pred_spec. auto with I.
Qed.
......
program_logic/hoare.v
+ 3
− 2
View file @ ffa92c50
From program_logic Require Export weakestpre viewshifts.
Definition ht {Λ Σ} (E : coPset) (P : iProp Λ Σ)
(e : expr Λ) (Φ : val Λ → iProp Λ Σ) : iProp Λ Σ := (□ (P → wp E e Φ))%I.
(e : expr Λ) (Φ : val Λ → iProp Λ Σ) : iProp Λ Σ :=
(□ (P → || e @ E {{ Φ }}))%I.
Instance: Params (@ht) 3.
Notation "{{ P } } e @ E {{ Φ } }" := (ht E P e Φ)
......@@ -37,7 +38,7 @@ Global Instance ht_mono' E :
Proper (flip (⊑) ==> eq ==> pointwise_relation _ (⊑) ==> (⊑)) (@ht Λ Σ E).
Proof. by intros P P' HP e ? <- Φ Φ' HΦ; apply ht_mono. Qed.
Lemma ht_alt E P Φ e : (P ⊑ wp E e Φ) → {{ P }} e @ E {{ Φ }}.
Lemma ht_alt E P Φ e : (P ⊑ || e @ E {{ Φ }}) → {{ P }} e @ E {{ Φ }}.
Proof.
intros; rewrite -{1}always_const. apply: always_intro. apply impl_intro_l.
by rewrite always_const right_id.
......
program_logic/invariants.v
+ 2
− 2
View file @ ffa92c50
......@@ -64,8 +64,8 @@ Proof. intros. by apply: (inv_fsa pvs_fsa). Qed.
Lemma wp_open_close E e N P Φ R :
atomic e → nclose N ⊆ E →
R ⊑ inv N P →
R ⊑ (▷ P -★ wp (E ∖ nclose N) e (λ v, ▷ P ★ Φ v)) →
R ⊑ wp E e Φ.
R ⊑ (▷ P -★ || e @ E ∖ nclose N {{ λ v, ▷ P ★ Φ v }}) →
R ⊑ || e @ E {{ Φ }}.
Proof. intros. by apply: (inv_fsa (wp_fsa e)). Qed.
Lemma inv_alloc N P : ▷ P ⊑ pvs N N (inv N P).
......
program_logic/lifting.v
+ 6
− 6
View file @ ffa92c50
......@@ -24,8 +24,8 @@ Lemma wp_lift_step E1 E2
reducible e1 σ1 →
(∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
(|={E2,E1}=> ownP σ1 ★ ▷ ∀ e2 σ2 ef,
(■ φ e2 σ2 ef ∧ ownP σ2) -★ |={E1,E2}=> wp E2 e2 Φ ★ wp_fork ef)
⊑ wp E2 e1 Φ.
(■ φ e2 σ2 ef ∧ ownP σ2) -★ |={E1,E2}=> || e2 @ E2 {{ Φ }} ★ wp_fork ef)
⊑ || e1 @ E2 {{ Φ }}.
Proof.
intros ? He Hsafe Hstep n r ? Hvs; constructor; auto.
intros rf k Ef σ1' ???; destruct (Hvs rf (S k) Ef σ1')
......@@ -45,7 +45,7 @@ Lemma wp_lift_pure_step E (φ : expr Λ → option (expr Λ) → Prop) Φ 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 Φ ★ wp_fork ef) ⊑ wp E e1 Φ.
(▷ ∀ e2 ef, ■ φ e2 ef → || e2 @ E {{ Φ }} ★ wp_fork ef) ⊑ || e1 @ E {{ Φ }}.
Proof.
intros He Hsafe Hstep n r ? Hwp; constructor; auto.
intros rf k Ef σ1 ???; split; [done|]. destruct n as [|n]; first lia.
......@@ -65,7 +65,7 @@ Lemma wp_lift_atomic_step {E Φ} e1
(∀ 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 -★ Φ v2 ★ wp_fork ef)
⊑ wp E e1 Φ.
⊑ || e1 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_step E E (λ e2 σ2 ef, ∃ v2,
to_val e2 = Some v2 ∧ φ v2 σ2 ef) _ e1 σ1) //; [].
......@@ -84,7 +84,7 @@ Lemma wp_lift_atomic_det_step {E Φ e1} σ1 v2 σ2 ef :
reducible e1 σ1 →
(∀ e2' σ2' ef', prim_step e1 σ1 e2' σ2' ef' →
σ2 = σ2' ∧ to_val e2' = Some v2 ∧ ef = ef') →
(ownP σ1 ★ ▷ (ownP σ2 -★ Φ v2 ★ wp_fork ef)) ⊑ wp E e1 Φ.
(ownP σ1 ★ ▷ (ownP σ2 -★ Φ v2 ★ wp_fork ef)) ⊑ || e1 @ E {{ Φ }}.
Proof.
intros. rewrite -(wp_lift_atomic_step _ (λ v2' σ2' ef',
σ2 = σ2' ∧ v2 = v2' ∧ ef = ef') σ1) //; last naive_solver.
......@@ -99,7 +99,7 @@ Lemma wp_lift_pure_det_step {E Φ} e1 e2 ef :
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')→
▷ (wp E e2 Φ ★ wp_fork ef) ⊑ wp E e1 Φ.
▷ (|| e2 @ E {{ Φ }} ★ wp_fork ef) ⊑ || e1 @ E {{ Φ }}.
Proof.
intros.
rewrite -(wp_lift_pure_step E (λ e2' ef', e2 = e2' ∧ ef = ef') _ e1) //=.
......
program_logic/weakestpre.v
+ 35
− 21
View file @ ffa92c50
......@@ -51,6 +51,13 @@ Next Obligation.
Qed.
Instance: Params (@wp) 4.
Notation "|| e @ E {{ Φ } }" := (wp E e Φ)
(at level 20, e, Φ at level 200,
format "|| e @ E {{ Φ } }") : uPred_scope.
Notation "|| e {{ Φ } }" := (wp ⊤ e Φ)
(at level 20, e, Φ at level 200,
format "|| e {{ Φ } }") : uPred_scope.
Section wp.
Context {Λ : language} {Σ : iFunctor}.
Implicit Types P : iProp Λ Σ.
......@@ -81,7 +88,7 @@ Proof.
by intros Φ Φ' ?; apply equiv_dist=>n; apply wp_ne=>v; apply equiv_dist.
Qed.
Lemma wp_mask_frame_mono E1 E2 e Φ Ψ :
E1 ⊆ E2 → (∀ v, Φ v ⊑ Ψ v) → wp E1 e Φ ⊑ wp E2 e Ψ.
E1 ⊆ E2 → (∀ v, Φ v ⊑ Ψ v) → || e @ E1 {{ Φ }} ⊑ || e @ E2 {{ Ψ }}.
Proof.
intros HE HΦ n r; revert e r; induction n as [n IH] using lt_wf_ind=> e r.
destruct 2 as [n' r v HpvsQ|n' r e1 ? Hgo].
......@@ -95,19 +102,20 @@ Proof.
exists r2, r2'; split_and?; [rewrite HE'|eapply IH|]; eauto.
Qed.
Lemma wp_value_inv E Φ v n r : wp E (of_val v) Φ n r → (|={E}=> Φ v)%I n r.
Lemma wp_value_inv E Φ v n r :
|| of_val v @ E {{ Φ }}%I n r → (|={E}=> Φ v)%I n r.
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 = ∅ →
wp E e Φ n r → wsat (S k) (E ∪ Ef) σ (r ⋅ rf) →
|| e @ E {{ Φ }}%I n r → wsat (S k) (E ∪ Ef) σ (r ⋅ rf) →
wp_go (E ∪ Ef) (λ e, wp E e Φ) (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 Φ v : Φ v ⊑ wp E (of_val v) Φ.
Lemma wp_value' E Φ v : Φ v ⊑ || of_val v @ E {{ Φ }}.
Proof. by constructor; apply pvs_intro. Qed.
Lemma pvs_wp E e Φ : (|={E}=> wp E e Φ) ⊑ wp E e Φ.
Lemma pvs_wp E e Φ : (|={E}=> || e @ E {{ Φ }}) ⊑ || e @ E {{ Φ }}.
Proof.
intros n r ? Hvs.
destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
......@@ -117,7 +125,7 @@ Proof.
destruct (Hvs rf (S k) Ef σ1) as (r'&Hwp&?); auto.
eapply wp_step_inv with (S k) r'; eauto.
Qed.
Lemma wp_pvs E e Φ : wp E e (λ v, |={E}=> Φ v) ⊑ wp E e Φ.
Lemma wp_pvs E e Φ : || e @ E {{ λ v, |={E}=> Φ v }} ⊑ || e @ E {{ Φ }}.
Proof.
intros n r; revert e r; induction n as [n IH] using lt_wf_ind=> e r Hr HΦ.
destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
......@@ -129,7 +137,8 @@ Proof.
exists r2, r2'; split_and?; [|apply (IH k)|]; auto.
Qed.
Lemma wp_atomic E1 E2 e Φ :
E2 ⊆ E1 → atomic e → (|={E1,E2}=> wp E2 e (λ v, |={E2,E1}=> Φ v)) ⊑ wp E1 e Φ.
E2 ⊆ E1 → atomic e →
(|={E1,E2}=> || e @ E2 {{ λ v, |={E2,E1}=> Φ v }}) ⊑ || e @ E1 {{ Φ }}.
Proof.
intros ? He n r ? Hvs; constructor; eauto using atomic_not_val.
intros rf k Ef σ1 ???.
......@@ -146,7 +155,7 @@ Proof.
- by rewrite -assoc.
- constructor; apply pvs_intro; auto.
Qed.
Lemma wp_frame_r E e Φ R : (wp E e Φ ★ R) ⊑ wp E e (λ v, Φ v ★ R).
Lemma wp_frame_r E e Φ R : (|| e @ E {{ Φ }} ★ R) ⊑ || e @ E {{ λ v, Φ v ★ R }}.
Proof.
intros n r' Hvalid (r&rR&Hr&Hwp&?); revert Hvalid.
rewrite Hr; clear Hr; revert e r Hwp.
......@@ -164,7 +173,7 @@ Proof.
- apply IH; eauto using uPred_weaken.
Qed.
Lemma wp_frame_later_r E e Φ R :
to_val e = None → (wp E e Φ ★ ▷ R) ⊑ wp E e (λ v, Φ v ★ R).
to_val e = None → (|| e @ E {{ Φ }} ★ ▷ R) ⊑ || e @ E {{ λ v, Φ v ★ R }}.
Proof.
intros He n r' Hvalid (r&rR&Hr&Hwp&?); revert Hvalid; rewrite Hr; clear Hr.
destruct Hwp as [|n r e ? Hgo]; [by rewrite to_of_val in He|].
......@@ -178,7 +187,7 @@ Proof.
eapply uPred_weaken with n rR; eauto.
Qed.
Lemma wp_bind `{LanguageCtx Λ K} E e Φ :
wp E e (λ v, wp E (K (of_val v)) Φ) ⊑ wp E (K e) Φ.
|| e @ E {{ λ v, || K (of_val v) @ E {{ Φ }} }} ⊑ || K e @ E {{ Φ }}.
Proof.
intros n r; revert e r; induction n as [n IH] using lt_wf_ind=> e r ?.
destruct 1 as [|n r e ? Hgo]; [by apply pvs_wp|].
......@@ -196,39 +205,44 @@ Qed.
(** * Derived rules *)
Opaque uPred_holds.
Import uPred.
Lemma wp_mono E e Φ Ψ : (∀ v, Φ v ⊑ Ψ v) → wp E e Φ ⊑ wp E e Ψ.
Lemma wp_mono E e Φ Ψ : (∀ v, Φ v ⊑ Ψ v) → || e @ E {{ Φ }} ⊑ || e @ E {{ Ψ }}.
Proof. by apply wp_mask_frame_mono. Qed.
Global Instance wp_mono' E e :
Proper (pointwise_relation _ (⊑) ==> (⊑)) (@wp Λ Σ E e).
Proof. by intros Φ Φ' ?; apply wp_mono. Qed.
Lemma wp_strip_pvs E e P Φ : P ⊑ wp E e Φ → (|={E}=> P) ⊑ wp E e Φ.
Lemma wp_strip_pvs E e P Φ :
P ⊑ || e @ E {{ Φ }} → (|={E}=> P) ⊑ || e @ E {{ Φ }}.
Proof. move=>->. by rewrite pvs_wp. Qed.
Lemma wp_value E Φ e v : to_val e = Some v → Φ v ⊑ wp E e Φ.
Lemma wp_value E Φ e v : to_val e = Some v → Φ v ⊑ || e @ E {{ Φ }}.
Proof. intros; rewrite -(of_to_val e v) //; by apply wp_value'. Qed.
Lemma wp_value_pvs E Φ e v : to_val e = Some v → (|={E}=> Φ v) ⊑ wp E e Φ.
Lemma wp_value_pvs E Φ e v :
to_val e = Some v → (|={E}=> Φ v) ⊑ || e @ E {{ Φ }}.
Proof. intros. rewrite -wp_pvs. rewrite -wp_value //. Qed.
Lemma wp_frame_l E e Φ R : (R ★ wp E e Φ) ⊑ wp E e (λ v, R ★ Φ v).
Lemma wp_frame_l E e Φ R : (R ★ || e @ E {{ Φ }}) ⊑ || e @ E {{ λ v, R ★ Φ v }}.
Proof. setoid_rewrite (comm _ R); apply wp_frame_r. Qed.
Lemma wp_frame_later_l E e Φ R :
to_val e = None → (▷ R ★ wp E e Φ) ⊑ wp E e (λ v, R ★ Φ v).
to_val e = None → (▷ R ★ || e @ E {{ Φ }}) ⊑ || e @ E {{ λ v, R ★ Φ v }}.
Proof.
rewrite (comm _ (▷ R)%I); setoid_rewrite (comm _ R).
apply wp_frame_later_r.
Qed.
Lemma wp_always_l E e Φ R `{!AlwaysStable R} :
(R ∧ wp E e Φ) ⊑ wp E e (λ v, R ∧ Φ v).
(R ∧ || e @ E {{ Φ }}) ⊑ || e @ E {{ λ v, R ∧ Φ v }}.
Proof. by setoid_rewrite (always_and_sep_l _ _); rewrite wp_frame_l. Qed.
Lemma wp_always_r E e Φ R `{!AlwaysStable R} :
(wp E e Φ ∧ R) ⊑ wp E e (λ v, Φ v ∧ R).
(|| e @ E {{ Φ }} ∧ R) ⊑ || e @ E {{ λ v, Φ v ∧ R }}.
Proof. by setoid_rewrite (always_and_sep_r _ _); rewrite wp_frame_r. Qed.
Lemma wp_impl_l E e Φ Ψ : ((□ ∀ v, Φ v → Ψ v) ∧ wp E e Φ) ⊑ wp E e Ψ.
Lemma wp_impl_l E e Φ Ψ :
((□ ∀ v, Φ v → Ψ v) ∧ || e @ E {{ Φ }}) ⊑ || e @ E {{ Ψ }}.
Proof.
rewrite wp_always_l; apply wp_mono=> // v.
by rewrite always_elim (forall_elim v) impl_elim_l.
Qed.
Lemma wp_impl_r E e Φ Ψ : (wp E e Φ ∧ □ ∀ v, Φ v → Ψ v) ⊑ wp E e Ψ.
Lemma wp_impl_r E e Φ Ψ :
(|| e @ E {{ Φ }} ∧ □ (∀ v, Φ v → Ψ v)) ⊑ || e @ E {{ Ψ }}.
Proof. by rewrite comm wp_impl_l. Qed.
Lemma wp_mask_weaken E1 E2 e Φ : E1 ⊆ E2 → wp E1 e Φ ⊑ wp E2 e Φ.
Lemma wp_mask_weaken E1 E2 e Φ :
E1 ⊆ E2 → || e @ E1 {{ Φ }} ⊑ || e @ E2 {{ Φ }}.
Proof. auto using wp_mask_frame_mono. Qed.
(** * Weakest-pre is a FSA. *)
......
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