Split monotonicity and closedness fields of uPred.

program_logic/adequacy.v

@@ -18,7 +18,7 @@ Implicit Types m : iGst Λ Σ.
 Notation wptp n := (Forall3 (λ e Φ r, uPred_holds (wp ⊤ e Φ) n r)).
 Lemma wptp_le Φs es rs n n' :
   ✓{n'} (big_op rs) → wptp n es Φs rs → n' ≤ n → wptp n' es Φs rs.
-Proof. induction 2; constructor; eauto using uPred_weaken. Qed.
+Proof. induction 2; constructor; eauto using uPred_closed. Qed.
 Lemma nsteps_wptp Φs k n tσ1 tσ2 rs1 :
   nsteps step k tσ1 tσ2 →
   1 < n → wptp (k + n) (tσ1.1) Φs rs1 →
@@ -51,7 +51,8 @@ Proof.
     { rewrite /option_list right_id_L.
       apply Forall3_app, Forall3_cons; eauto using wptp_le.
       rewrite wp_eq.
-      apply uPred_weaken with (k + n) r2; eauto using cmra_included_l. }
+      apply uPred_closed with (k + n);
+        first apply uPred_mono with r2; eauto using cmra_included_l. }
     by rewrite -Permutation_middle /= big_op_app.
 Qed.
 Lemma wp_adequacy_steps P Φ k n e1 t2 σ1 σ2 r1 :

program_logic/pviewshifts.v

@@ -19,11 +19,12 @@ Next Obligation.
   apply HP; auto. by rewrite (dist_le _ _ _ _ Hr); last lia.
 Qed.
 Next Obligation.
-  intros Λ Σ E1 E2 P r1 r2 n1 n2 HP [r3 ?] Hn ? rf k Ef σ ?? Hws; setoid_subst.
-  destruct (HP (r3⋅rf) k Ef σ) as (r'&?&Hws'); rewrite ?(assoc op); auto.
+  intros Λ Σ E1 E2 P n r1 r2 HP [r3 ?] rf k Ef σ ?? Hws; setoid_subst.
+  destruct (HP (r3 ⋅ rf) k Ef σ) as (r'&?&Hws'); rewrite ?(assoc op); auto.
   exists (r' ⋅ r3); rewrite -assoc; split; last done.
-  apply uPred_weaken with k r'; eauto using cmra_included_l.
+  apply uPred_mono with r'; eauto using cmra_included_l.
 Qed.
+Next Obligation. naive_solver. Qed.
 
 Definition pvs_aux : { x | x = @pvs_def }. by eexists. Qed.
 Definition pvs := proj1_sig pvs_aux.
@@ -62,7 +63,7 @@ Proof. apply ne_proper, _. Qed.
 Lemma pvs_intro E P : P ⊢ |={E}=> P.
 Proof.
   rewrite pvs_eq. split=> n r ? HP rf k Ef σ ???; exists r; split; last done.
-  apply uPred_weaken with n r; eauto.
+  apply uPred_closed with n; eauto.
 Qed.
 Lemma pvs_mono E1 E2 P Q : P ⊢ Q → (|={E1,E2}=> P) ⊢ (|={E1,E2}=> Q).
 Proof.
@@ -75,7 +76,7 @@ Proof.
   rewrite pvs_eq uPred.timelessP_spec=> HP.
   uPred.unseal; split=>-[|n] r ? HP' rf k Ef σ ???; first lia.
   exists r; split; last done.
-  apply HP, uPred_weaken with n r; eauto using cmra_validN_le.
+  apply HP, uPred_closed with n; eauto using cmra_validN_le.
 Qed.
 Lemma pvs_trans E1 E2 E3 P :
   E2 ⊆ E1 ∪ E3 → (|={E1,E2}=> |={E2,E3}=> P) ⊢ (|={E1,E3}=> P).
@@ -96,7 +97,7 @@ Proof.
   destruct (HP (r2 ⋅ rf) k Ef σ) as (r'&?&?); eauto.
   { by rewrite assoc -(dist_le _ _ _ _ Hr); last lia. }
   exists (r' ⋅ r2); split; last by rewrite -assoc.
-  exists r', r2; split_and?; auto; apply uPred_weaken with n r2; auto.
+  exists r', r2; split_and?; auto. apply uPred_closed with n; auto.
 Qed.
 Lemma pvs_openI i P : ownI i P ⊢ (|={{[i]},∅}=> ▷ P).
 Proof.
@@ -105,17 +106,17 @@ Proof.
   destruct (wsat_open k Ef σ (r ⋅ rf) i P) as (rP&?&?); auto.
   { rewrite lookup_wld_op_l ?Hinv; eauto; apply dist_le with (S n); eauto. }
   exists (rP ⋅ r); split; last by rewrite (left_id_L _ _) -assoc.
-  eapply uPred_weaken with (S k) rP; eauto using cmra_included_l.
+  eapply uPred_mono with rP; eauto using cmra_included_l.
 Qed.
 Lemma pvs_closeI i P : (ownI i P ∧ ▷ P) ⊢ (|={∅,{[i]}}=> True).
 Proof.
   rewrite pvs_eq. uPred.unseal; split=> -[|n] r ? [? HP] rf [|k] Ef σ ? HE ?; try lia.
   exists ∅; split; [done|].
   rewrite left_id; apply wsat_close with P r.
-  - apply ownI_spec, uPred_weaken with (S n) r; auto.
+  - apply ownI_spec, uPred_closed with (S n); auto.
   - set_solver +HE.
   - by rewrite -(left_id_L ∅ (∪) Ef).
-  - apply uPred_weaken with n r; auto.
+  - apply uPred_closed with n; auto.
 Qed.
 Lemma pvs_ownG_updateP E m (P : iGst Λ Σ → Prop) :
   m ~~>: P → ownG m ⊢ (|={E}=> ∃ m', ■ P m' ∧ ownG m').
@@ -131,7 +132,7 @@ Proof.
   rewrite pvs_eq. intros ?; rewrite /ownI; uPred.unseal.
   split=> -[|n] r ? HP rf [|k] Ef σ ???; try lia.
   destruct (wsat_alloc k E Ef σ rf P r) as (i&?&?&?); auto.
-  { apply uPred_weaken with n r; eauto. }
+  { apply uPred_closed with n; eauto. }
   exists (Res {[ i := to_agree (Next (iProp_unfold P)) ]} ∅ ∅).
   split; [|done]. by exists i; split; rewrite /uPred_holds /=.
 Qed.

program_logic/weakestpre.v

@@ -38,17 +38,19 @@ Next Obligation.
   intros rf k Ef σ1 ?; rewrite -(dist_le _ _ _ _ Hr); naive_solver.
 Qed.
 Next Obligation.
-  intros Λ Σ E e Φ n1 n2 r1 r2; revert Φ E e n2 r1 r2.
-  induction n1 as [n1 IH] using lt_wf_ind; intros Φ E e n2 r1 r1'.
-  destruct 1 as [|n1 r1 e1 ? Hgo].
-  - constructor; eauto using uPred_weaken.
-  - intros [rf' Hr] ??; constructor; [done|intros rf k Ef σ1 ???].
+  intros Λ Σ E e Φ n r1 r2; revert Φ E e r1 r2.
+  induction n as [n IH] using lt_wf_ind; intros Φ E e r1 r1'.
+  destruct 1 as [|n r1 e1 ? Hgo].
+  - constructor; eauto using uPred_mono.
+  - intros [rf' Hr]; constructor; [done|intros rf k Ef σ1 ???].
     destruct (Hgo (rf' ⋅ rf) k Ef σ1) as [Hsafe Hstep];
       rewrite ?assoc -?Hr; auto; constructor; [done|].
     intros e2 σ2 ef ?; destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
     exists r2, (r2' ⋅ rf'); split_and?; eauto 10 using (IH k), cmra_included_l.
     by rewrite -!assoc (assoc _ r2).
 Qed.
+Next Obligation. destruct 1; constructor; eauto using uPred_closed. Qed.
+
 (* Perform sealing. *)
 Definition wp_aux : { x | x = @wp_def }. by eexists. Qed.
 Definition wp := proj1_sig wp_aux.
@@ -194,7 +196,7 @@ Proof.
   destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
   exists (r2 ⋅ rR), r2'; split_and?; auto.
   - by rewrite -(assoc _ r2) (comm _ rR) !assoc -(assoc _ _ rR).
-  - apply IH; eauto using uPred_weaken.
+  - apply IH; eauto using uPred_closed.
 Qed.
 Lemma wp_frame_step_r E E1 E2 e Φ R :
   to_val e = None → E ⊥ E1 → E2 ⊆ E1 →

program_logic/weakestpre_fix.v

@@ -36,18 +36,19 @@ Next Obligation.
   by rewrite (dist_le _ _ _ _ Hr1); last omega.
 Qed.
 Next Obligation.
-  intros wp E e1 Φ n1 n2 r1 ? Hwp [r2 ?] ?? rf k Ef σ1 ???; setoid_subst.
+  intros wp E e1 Φ n r1 ? Hwp [r2 ?] rf k Ef σ1 ???; setoid_subst.
   destruct (Hwp (r2 ⋅ rf) k Ef σ1) as [Hval Hstep]; rewrite ?assoc; auto.
   split.
   - intros v Hv. destruct (Hval v Hv) as [r3 [??]].
-    exists (r3 ⋅ r2). rewrite -assoc. eauto using uPred_weaken, cmra_included_l.
+    exists (r3 ⋅ r2). rewrite -assoc. eauto using uPred_mono, cmra_included_l.
   - intros ??. destruct Hstep as [Hred Hpstep]; auto.
     split; [done|]=> e2 σ2 ef ?.
     edestruct Hpstep as (r3&r3'&?&?&?); eauto.
     exists r3, (r3' ⋅ r2); split_and?; auto.
     + by rewrite assoc -assoc.
-    + destruct ef; simpl in *; eauto using uPred_weaken, cmra_included_l.
+    + destruct ef; simpl in *; eauto using uPred_mono, cmra_included_l.
 Qed.
+Next Obligation. repeat intro; eauto. Qed.
 
 Lemma wp_pre_contractive' n E e Φ1 Φ2 r
     (wp1 wp2 : coPsetC -n> exprC Λ -n> (valC Λ -n> iProp) -n> iProp) :

program_logic/wsat.v

@@ -63,7 +63,7 @@ Proof.
   destruct (Hwld i (iProp_fold (later_car (P' (S n))))) as (r'&?&?); auto.
   { by rewrite HP' -HPiso. }
   assert (✓{S n} r') by (apply (big_opM_lookup_valid _ rs i); auto).
-  exists r'; split; [done|apply HPP', uPred_weaken with n r'; auto].
+  exists r'; split; [done|]. apply HPP', uPred_closed with n; auto.
 Qed.
 Lemma wsat_valid n E σ r : n ≠ 0 → wsat n E σ r → ✓{n} r.
 Proof.