Commit d64e67b0 authored by Ralf Jung's avatar Ralf Jung

change notation for view shifts to ={E}=>

parent f6909092
......@@ -36,7 +36,7 @@ Proof.
by rewrite -wp_value; apply const_intro.
Qed.
Lemma ht_vs E P P' Q Q' e :
((P >{E}=> P') {{ P' }} e @ E {{ Q' }} v, Q' v >{E}=> Q v)
((P ={E}=> P') {{ P' }} e @ E {{ Q' }} v, Q' v ={E}=> Q v)
{{ P }} e @ E {{ Q }}.
Proof.
apply (always_intro' _ _), impl_intro_l.
......@@ -47,7 +47,7 @@ Proof.
Qed.
Lemma ht_atomic E1 E2 P P' Q Q' e :
E2 E1 atomic e
((P >{E1,E2}=> P') {{ P' }} e @ E2 {{ Q' }} v, Q' v >{E2,E1}=> Q v)
((P ={E1,E2}=> P') {{ P' }} e @ E2 {{ Q' }} v, Q' v ={E2,E1}=> Q v)
{{ P }} e @ E1 {{ Q }}.
Proof.
intros ??; apply (always_intro' _ _), impl_intro_l.
......
......@@ -20,8 +20,8 @@ Lemma ht_lift_step E1 E2
E1 E2 to_val e1 = None
reducible e1 σ1
( e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef φ e2 σ2 ef)
((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)
((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 }})
{{ P }} e1 @ E2 {{ R }}.
......
......@@ -6,22 +6,22 @@ Definition vs {Λ Σ} (E1 E2 : coPset) (P Q : iProp Λ Σ) : iProp Λ Σ :=
( (P pvs E1 E2 Q))%I.
Arguments vs {_ _} _ _ _%I _%I.
Instance: Params (@vs) 4.
Notation "P >{ E1 , E2 }=> Q" := (vs E1 E2 P%I Q%I)
Notation "P ={ E1 , E2 }=> Q" := (vs E1 E2 P%I Q%I)
(at level 199, E1 at level 1, E2 at level 1,
format "P >{ E1 , E2 }=> Q") : uPred_scope.
Notation "P >{ E1 , E2 }=> Q" := (True vs E1 E2 P%I Q%I)
format "P ={ E1 , E2 }=> Q") : uPred_scope.
Notation "P ={ E1 , E2 }=> Q" := (True vs E1 E2 P%I Q%I)
(at level 199, E1 at level 1, E2 at level 1,
format "P >{ E1 , E2 }=> Q") : C_scope.
Notation "P >{ E }=> Q" := (vs E E P%I Q%I)
(at level 199, E at level 1, format "P >{ E }=> Q") : uPred_scope.
Notation "P >{ E }=> Q" := (True vs E E P%I Q%I)
(at level 199, E at level 1, format "P >{ E }=> Q") : C_scope.
format "P ={ E1 , E2 }=> Q") : C_scope.
Notation "P ={ E }=> Q" := (vs E E P%I Q%I)
(at level 199, E at level 1, format "P ={ E }=> Q") : uPred_scope.
Notation "P ={ E }=> Q" := (True vs E E P%I Q%I)
(at level 199, E at level 1, format "P ={ E }=> Q") : C_scope.
Section vs.
Context {Λ : language} {Σ : iFunctor}.
Implicit Types P Q R : iProp Λ Σ.
Lemma vs_alt E1 E2 P Q : (P pvs E1 E2 Q) P >{E1,E2}=> Q.
Lemma vs_alt E1 E2 P Q : (P pvs E1 E2 Q) P ={E1,E2}=> Q.
Proof.
intros; rewrite -{1}always_const; apply always_intro, impl_intro_l.
by rewrite always_const (right_id _ _).
......@@ -35,60 +35,60 @@ Global Instance vs_proper E1 E2 : Proper ((≡) ==> (≡) ==> (≡)) (@vs Λ Σ
Proof. apply ne_proper_2, _. Qed.
Lemma vs_mono E1 E2 P P' Q Q' :
P P' Q' Q (P' >{E1,E2}=> Q') (P >{E1,E2}=> Q).
P P' Q' Q (P' ={E1,E2}=> Q') (P ={E1,E2}=> Q).
Proof. by intros HP HQ; rewrite /vs -HP HQ. Qed.
Global Instance vs_mono' E1 E2 :
Proper (flip () ==> () ==> ()) (@vs Λ Σ E1 E2).
Proof. by intros until 2; apply vs_mono. Qed.
Lemma vs_false_elim E1 E2 P : False >{E1,E2}=> P.
Lemma vs_false_elim E1 E2 P : False ={E1,E2}=> P.
Proof. apply vs_alt, False_elim. Qed.
Lemma vs_timeless E P : TimelessP P P >{E}=> P.
Lemma vs_timeless E P : TimelessP P P ={E}=> P.
Proof. by intros ?; apply vs_alt, pvs_timeless. Qed.
Lemma vs_transitive E1 E2 E3 P Q R :
E2 E1 E3 ((P >{E1,E2}=> Q) (Q >{E2,E3}=> R)) (P >{E1,E3}=> R).
E2 E1 E3 ((P ={E1,E2}=> Q) (Q ={E2,E3}=> R)) (P ={E1,E3}=> R).
Proof.
intros; rewrite -always_and; apply always_intro, impl_intro_l.
rewrite always_and (associative _) (always_elim (P _)) impl_elim_r.
by rewrite pvs_impl_r; apply pvs_trans.
Qed.
Lemma vs_transitive' E P Q R : ((P >{E}=> Q) (Q >{E}=> R)) (P >{E}=> R).
Lemma vs_transitive' E P Q R : ((P ={E}=> Q) (Q ={E}=> R)) (P ={E}=> R).
Proof. apply vs_transitive; solve_elem_of. Qed.
Lemma vs_reflexive E P : P >{E}=> P.
Lemma vs_reflexive E P : P ={E}=> P.
Proof. apply vs_alt, pvs_intro. Qed.
Lemma vs_impl E P Q : (P Q) (P >{E}=> Q).
Lemma vs_impl E P Q : (P Q) (P ={E}=> Q).
Proof.
apply always_intro, impl_intro_l.
by rewrite always_elim impl_elim_r -pvs_intro.
Qed.
Lemma vs_frame_l E1 E2 P Q R : (P >{E1,E2}=> Q) (R P >{E1,E2}=> R Q).
Lemma vs_frame_l E1 E2 P Q R : (P ={E1,E2}=> Q) (R P ={E1,E2}=> R Q).
Proof.
apply always_intro, impl_intro_l.
rewrite -pvs_frame_l always_and_sep_r -always_wand_impl -(associative _).
by rewrite always_elim wand_elim_r.
Qed.
Lemma vs_frame_r E1 E2 P Q R : (P >{E1,E2}=> Q) (P R >{E1,E2}=> Q R).
Lemma vs_frame_r E1 E2 P Q R : (P ={E1,E2}=> Q) (P R ={E1,E2}=> Q R).
Proof. rewrite !(commutative _ _ R); apply vs_frame_l. Qed.
Lemma vs_mask_frame E1 E2 Ef P Q :
Ef (E1 E2) = (P >{E1,E2}=> Q) (P >{E1 Ef,E2 Ef}=> Q).
Ef (E1 E2) = (P ={E1,E2}=> Q) (P ={E1 Ef,E2 Ef}=> Q).
Proof.
intros ?; apply always_intro, impl_intro_l; rewrite (pvs_mask_frame _ _ Ef)//.
by rewrite always_elim impl_elim_r.
Qed.
Lemma vs_mask_frame' E Ef P Q : Ef E = (P >{E}=> Q) (P >{E Ef}=> Q).
Lemma vs_mask_frame' E Ef P Q : Ef E = (P ={E}=> Q) (P ={E Ef}=> Q).
Proof. intros; apply vs_mask_frame; solve_elem_of. Qed.
Lemma vs_open_close N E P Q R :
nclose N E
(inv N R ( R P >{E nclose N}=> R Q)) (P >{E}=> Q).
(inv N R ( R P ={E nclose N}=> R Q)) (P ={E}=> Q).
Proof.
intros; apply (always_intro' _ _), impl_intro_l.
rewrite associative (commutative _ P) -associative.
......@@ -99,7 +99,7 @@ Proof.
by rewrite /vs always_elim impl_elim_r.
Qed.
Lemma vs_alloc (N : namespace) P : P >{N}=> inv N P.
Lemma vs_alloc (N : namespace) P : P ={N}=> inv N P.
Proof. by intros; apply vs_alt, pvs_alloc. Qed.
End vs.
......@@ -110,14 +110,14 @@ Implicit Types a : A.
Implicit Types P Q R : iProp Λ (globalC Σ).
Lemma vs_own_updateP E γ a φ :
a ~~>: φ own i γ a >{E}=> a', φ a' own i γ a'.
a ~~>: φ own i γ a ={E}=> a', φ a' own i γ a'.
Proof. by intros; apply vs_alt, own_updateP. Qed.
Lemma vs_own_updateP_empty `{Empty A, !CMRAIdentity A} E γ φ :
~~>: φ True >{E}=> a', φ a' own i γ a'.
~~>: φ True ={E}=> a', φ a' own i γ a'.
Proof. by intros; eapply vs_alt, own_updateP_empty. Qed.
Lemma vs_update E γ a a' : a ~~> a' own i γ a >{E}=> own i γ a'.
Lemma vs_update E γ a a' : a ~~> a' own i γ a ={E}=> own i γ a'.
Proof. by intros; apply vs_alt, own_update. Qed.
End vs_ghost.
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