### Change the level of view shifts so it is more like the implication.

```It is now slightly below implication. In order to do this, I had to change
the notation from P ={E1,E2}=> Q to P >{E1,E2}=> Q because the prefer ={n
is already used at level 70 for the distance of the metric.```
parent 28c4a0bf
 ... ... @@ -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 }}. ... ...
 ... ... @@ -6,20 +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) (at level 69, E1 at level 1, format "P ={ E1 , E2 }=> Q") : uPred_scope. Notation "P ={ E1 , E2 }=> Q" := (True ⊑ vs E1 E2 P%I Q%I) (at level 69, E1 at level 1, format "P ={ E1 , E2 }=> Q") : C_scope. Notation "P ={ E }=> Q" := (vs E E P%I Q%I) (at level 69, E at level 1, format "P ={ E }=> Q") : uPred_scope. Notation "P ={ E }=> Q" := (True ⊑ vs E E P%I Q%I) (at level 69, E at level 1, format "P ={ E }=> Q") : C_scope. 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) (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. 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 _ _). ... ... @@ -33,73 +35,71 @@ 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. (* FIXME I really should not need parenthesis around the pre- and postcondition of a view shift. *) 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 HN. apply always_intro'; first by apply _. apply impl_intro_l. intros; apply (always_intro' _ _), impl_intro_l. rewrite associative (commutative _ P) -associative. rewrite -pvs_open_close; first (apply and_mono; first done); last done. rewrite -(pvs_open_close E N) //; apply and_mono; first done. apply wand_intro_l. (* Oh wow, this is annyoing... *) rewrite always_and_sep_r' associative -always_and_sep_r'. 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. ... ... @@ -109,16 +109,15 @@ Context {Λ : language} {Σ : gid → iFunctor} (i : gid) `{!InG Λ Σ i A}. Implicit Types a : A. Implicit Types P Q R : iProp Λ (globalC Σ). Lemma vs_own_updateP E γ a (P : A → Prop) : a ~~>: P → own i γ a ={E}=> (∃ a', ■ P a' ∧ own i γ a'). Lemma vs_own_updateP E γ a φ : 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 (P : A → Prop) : ∅ ~~>: P → True ={E}=> (∃ a', ■ P a' ∧ own i γ a'). Lemma vs_own_updateP_empty `{Empty A, !CMRAIdentity A} E γ φ : ∅ ~~>: 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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