Lifting lemmas : get rid of \phi when it has became useless

program_logic/ectx_lifting.v

@@ -18,19 +18,17 @@ Lemma wp_ectx_bind {E e} K Φ :
   WP e @ E {{ v, WP fill K (of_val v) @ E {{ Φ }} }} ⊢ WP fill K e @ E {{ Φ }}.
 Proof. apply: weakestpre.wp_bind. Qed.
 
-Lemma wp_lift_head_step E1 E2
-    (φ : expr → state → option expr → Prop) Φ e1 :
+Lemma wp_lift_head_step E1 E2 Φ e1 :
   E2 ⊆ E1 → to_val e1 = None →
-  (|={E1,E2}=> ∃ σ1,
-      ■ head_reducible e1 σ1 ∧
-      ■ (∀ e2 σ2 ef, head_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) ∧
-      ▷ ownP σ1 ★ ▷ ∀ e2 σ2 ef,
-       (■ φ e2 σ2 ef ∧ ownP σ2) ={E2,E1}=★ WP e2 @ E1 {{ Φ }} ★ wp_fork ef)
+  (|={E1,E2}=> ∃ σ1, ■ head_reducible e1 σ1 ∧
+       ▷ ownP σ1 ★ ▷ ∀ e2 σ2 ef, (■ head_step e1 σ1 e2 σ2 ef ∧ ownP σ2)
+                                 ={E2,E1}=★ WP e2 @ E1 {{ Φ }} ★ wp_fork ef)
   ⊢ WP e1 @ E1 {{ Φ }}.
 Proof.
-  iIntros {??} "H". iApply (wp_lift_step E1 E2 φ); try done.
-  iPvs "H" as {σ1} "(%&%&Hσ1&?)". set_solver. iPvsIntro. iExists σ1.
-  repeat iSplit; eauto. by iFrame.
+  iIntros {??} "H". iApply (wp_lift_step E1 E2); try done.
+  iPvs "H" as {σ1} "(%&Hσ1&Hwp)". set_solver. iPvsIntro. iExists σ1.
+  iSplit; first by eauto. iFrame. iNext. iIntros {e2 σ2 ef} "[% ?]".
+  iApply "Hwp". by eauto.
 Qed.
 
 Lemma wp_lift_pure_head_step E (φ : expr → option expr → Prop) Φ e1 :

program_logic/hoare_lifting.v

@@ -18,24 +18,22 @@ Implicit Types e : expr Λ.
 Implicit Types P Q R : iProp Λ Σ.
 Implicit Types Ψ : val Λ → iProp Λ Σ.
 
-Lemma ht_lift_step E1 E2
-    (φ : expr Λ → state Λ → option (expr Λ) → Prop) P P' Φ1 Φ2 Ψ e1 :
+Lemma ht_lift_step E1 E2 P Pσ1 Φ1 Φ2 Ψ e1 :
   E2 ⊆ E1 → to_val e1 = None →
-  (P ={E1,E2}=> ∃ σ1,
-      ■ reducible e1 σ1 ∧
-      ■ (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) ∧
-      ▷ ownP σ1 ★ ▷ P') ∧
-   (∀ e2 σ2 ef, ■ φ e2 σ2 ef ★ ownP σ2 ★ P' ={E2,E1}=> Φ1 e2 σ2 ef ★ Φ2 e2 σ2 ef) ∧
+  (P ={E1,E2}=> ∃ σ1, ■ reducible e1 σ1 ∧
+      ▷ ownP σ1 ★ ▷ Pσ1 σ1) ∧
+   (∀ σ1 e2 σ2 ef, ■ prim_step e1 σ1 e2 σ2 ef ★ ownP σ2 ★ Pσ1 σ1
+                 ={E2,E1}=> Φ1 e2 σ2 ef ★ Φ2 e2 σ2 ef) ∧
    (∀ e2 σ2 ef, {{ Φ1 e2 σ2 ef }} e2 @ E1 {{ Ψ }}) ∧
    (∀ e2 σ2 ef, {{ Φ2 e2 σ2 ef }} ef ?@ ⊤ {{ _, True }})
   ⊢ {{ P }} e1 @ E1 {{ Ψ }}.
 Proof.
   iIntros {??} "#(#Hvs&HΦ&He2&Hef) ! HP".
-  iApply (wp_lift_step E1 E2 φ _ e1); auto.
-  iPvs ("Hvs" with "HP") as {σ1} "(%&%&Hσ&HP)"; first set_solver.
-  iPvsIntro. iExists σ1. repeat iSplit; eauto. iFrame.
+  iApply (wp_lift_step E1 E2 _ e1); auto.
+  iPvs ("Hvs" with "HP") as {σ1} "(%&Hσ&HP)"; first set_solver.
+  iPvsIntro. iExists σ1. repeat iSplit. by eauto. iFrame.
   iNext. iIntros {e2 σ2 ef} "[#Hφ Hown]".
-  iSpecialize ("HΦ" $! e2 σ2 ef with "[-]"). by iFrame "Hφ HP Hown".
+  iSpecialize ("HΦ" $! σ1 e2 σ2 ef with "[-]"). by iFrame "Hφ HP Hown".
   iPvs "HΦ" as "[H1 H2]"; first by set_solver. iPvsIntro. iSplitL "H1".
   - by iApply "He2".
   - destruct ef as [e|]; last done. by iApply ("Hef" $! _ _ (Some e)).
@@ -50,15 +48,15 @@ Lemma ht_lift_atomic_step
   {{ ▷ ownP σ1 ★ ▷ P }} e1 @ E {{ v, ∃ σ2 ef, ownP σ2 ★ ■ φ (of_val v) σ2 ef }}.
 Proof.
   iIntros {? Hsafe Hstep} "#Hef".
-  set (φ' e σ ef := is_Some (to_val e) ∧ φ e σ ef).
-  iApply (ht_lift_step E E φ'  _ P
-    (λ e2 σ2 ef, ownP σ2 ★ ■ (φ' e2 σ2 ef))%I (λ e2 σ2 ef, ■ φ e2 σ2 ef ★ P)%I);
+  set (φ' (_:state Λ) e σ ef := is_Some (to_val e) ∧ φ e σ ef).
+  iApply (ht_lift_step E E _ (λ σ1', P ∧ σ1 = σ1')%I
+    (λ e2 σ2 ef, ownP σ2 ★ ■ (φ' σ1 e2 σ2 ef))%I (λ e2 σ2 ef, ■ φ e2 σ2 ef ★ P)%I);
     try by (eauto using atomic_not_val).
   repeat iSplit.
-  - iIntros "![Hσ1 HP]". iExists σ1. iPvsIntro. unfold φ'.
-    repeat iSplit; eauto using atomic_step. by iFrame.
-  - iIntros {e2 σ2 ef} "! (#Hφ&Hown&HP)"; iPvsIntro.
-    iSplitL "Hown". by iSplit. iSplit. by iDestruct "Hφ" as %[_ ?]. done.
+  - iIntros "![Hσ1 HP]". iExists σ1. iPvsIntro.
+    iSplit. by eauto using atomic_step. iFrame. by auto.
+  - iIntros {? e2 σ2 ef} "! (%&Hown&HP&%)". iPvsIntro. subst.
+    iFrame. iSplit; iPureIntro; auto. split; eauto using atomic_step.
   - iIntros {e2 σ2 ef} "! [Hown #Hφ]"; iDestruct "Hφ" as %[[v2 <-%of_to_val] ?].
     iApply wp_value'. iExists σ2, ef. by iSplit.
   - done.

program_logic/lifting.v

@@ -18,19 +18,16 @@ Implicit Types Φ : val Λ → iProp Λ Σ.
 
 Notation wp_fork ef := (default True ef (flip (wp ⊤) (λ _, True)))%I.
 
-Lemma wp_lift_step E1 E2
-    (φ : expr Λ → state Λ → option (expr Λ) → Prop) Φ e1 :
+Lemma wp_lift_step E1 E2 Φ e1 :
   E2 ⊆ E1 → to_val e1 = None →
-  (|={E1,E2}=> ∃ σ1,
-      ■ reducible e1 σ1 ∧
-      ■ (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) ∧
-      ▷ ownP σ1 ★ ▷ ∀ e2 σ2 ef,
-        (■ φ e2 σ2 ef ∧ ownP σ2) ={E2,E1}=★ WP e2 @ E1 {{ Φ }} ★ wp_fork ef)
+  (|={E1,E2}=> ∃ σ1, ■ reducible e1 σ1 ∧ ▷ ownP σ1 ★
+       ▷ ∀ e2 σ2 ef, (■ prim_step e1 σ1 e2 σ2 ef ∧ ownP σ2)
+                     ={E2,E1}=★ WP e2 @ E1 {{ Φ }} ★ wp_fork ef)
   ⊢ WP e1 @ E1 {{ Φ }}.
 Proof.
   intros ? He. rewrite pvs_eq wp_eq.
   uPred.unseal; split=> n r ? Hvs; constructor; auto. intros k Ef σ1' rf ???.
-  destruct (Hvs (S k) Ef σ1' rf) as (r'&(σ1&Hsafe&Hstep&r1&r2&?&?&Hwp)&Hws);
+  destruct (Hvs (S k) Ef σ1' rf) as (r'&(σ1&Hsafe&r1&r2&?&?&Hwp)&Hws);
     auto; clear Hvs; cofe_subst r'.
   destruct (wsat_update_pst k (E2 ∪ Ef) σ1 σ1' r1 (r2 ⋅ rf)) as [-> Hws'].
   { apply equiv_dist. rewrite -(ownP_spec k); auto. }
@@ -38,7 +35,7 @@ Proof.
   constructor; [done|intros e2 σ2 ef ?; specialize (Hws' σ2)].
   destruct (λ H1 H2 H3, Hwp e2 σ2 ef k (update_pst σ2 r1) H1 H2 H3 k Ef σ2 rf)
     as (r'&(r1'&r2'&?&?&?)&?); auto; cofe_subst r'.
-  { split. by eapply Hstep. apply ownP_spec; auto. }
+  { split. done. apply ownP_spec; auto. }
   { rewrite (comm _ r2) -assoc; eauto using wsat_le. }
   exists r1', r2'; split_and?; try done. by uPred.unseal; intros ? ->.
 Qed.
@@ -71,11 +68,10 @@ Lemma wp_lift_atomic_step {E Φ} e1
     ■ φ (of_val v2) σ2 ef ∧ ownP σ2 -★ (|={E}=> Φ v2) ★ wp_fork ef)
   ⊢ WP e1 @ E {{ Φ }}.
 Proof.
-  iIntros {???} "[Hσ1 Hwp]". iApply (wp_lift_step E E (λ e2 σ2 ef,
-    is_Some (to_val e2) ∧ φ e2 σ2 ef) _ e1); auto using atomic_not_val.
-  iApply pvs_intro. iExists σ1. repeat iSplit; eauto using atomic_step.
-  iFrame. iNext. iIntros {e2 σ2 ef} "[#He2 Hσ2]".
-  iDestruct "He2" as %[[v2 Hv%of_to_val]?]. subst e2.
+  iIntros {???} "[Hσ1 Hwp]". iApply (wp_lift_step E E _ e1); auto using atomic_not_val.
+  iPvsIntro. iExists σ1. repeat iSplit; eauto 10 using atomic_step.
+  iFrame. iNext. iIntros {e2 σ2 ef} "[% Hσ2]".
+  edestruct @atomic_step as [v2 Hv%of_to_val]; eauto. subst e2.
   iDestruct ("Hwp" $! v2 σ2 ef with "[Hσ2]") as "[HΦ ?]". by eauto.
   iFrame. iPvs "HΦ". iPvsIntro. iApply wp_value; auto using to_of_val.
 Qed.