Lifting lemmas : make the initial state appear under an existential under the...

Lifting lemmas : make the initial state appear under an existential under the view shift. This is still sound, but slightly more expressive.

program_logic/ectx_lifting.v

 (** Some derived lemmas for ectx-based languages *)
 From iris.program_logic Require Export ectx_language weakestpre lifting.
 From iris.program_logic Require Import ownership.
+From iris.proofmode Require Import weakestpre.
 
 Section wp.
 Context {expr val ectx state} {Λ : EctxLanguage expr val ectx state}.
@@ -18,14 +19,19 @@ Lemma wp_ectx_bind {E e} K Φ :
 Proof. apply: weakestpre.wp_bind. Qed.
 
 Lemma wp_lift_head_step E1 E2
-    (φ : expr → state → option expr → Prop) Φ e1 σ1 :
+    (φ : expr → state → option expr → Prop) Φ e1 :
   E2 ⊆ E1 → to_val e1 = None →
-  head_reducible e1 σ1 →
-  (∀ e2 σ2 ef, head_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
-  (|={E1,E2}=> ▷ 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 ∧
+      ■ (∀ 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)
   ⊢ WP e1 @ E1 {{ Φ }}.
-Proof. eauto using wp_lift_step. Qed.
+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.
+Qed.
 
 Lemma wp_lift_pure_head_step E (φ : expr → option expr → Prop) Φ e1 :
   to_val e1 = None →

program_logic/hoare_lifting.v

@@ -19,23 +19,24 @@ 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 σ1 :
+    (φ : expr Λ → state Λ → option (expr Λ) → Prop) P P' Φ1 Φ2 Ψ e1 :
   E2 ⊆ E1 → to_val e1 = None →
-  reducible e1 σ1 →
-  (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
-  (P ={E1,E2}=> ▷ ownP σ1 ★ ▷ P') ∧
+  (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) ∧
    (∀ 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 {?? Hsafe Hstep} "#(#Hvs&HΦ&He2&Hef) ! HP".
-  iApply (wp_lift_step E1 E2 φ _ e1 σ1); auto.
-  iPvs ("Hvs" with "HP") as "[Hσ HP]"; first set_solver.
-  iPvsIntro. iNext. iSplitL "Hσ"; [done|iIntros {e2 σ2 ef} "[#Hφ Hown]"].
+  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.
+  iNext. iIntros {e2 σ2 ef} "[#Hφ Hown]".
   iSpecialize ("HΦ" $! e2 σ2 ef with "[-]"). by iFrame "Hφ HP Hown".
-  iPvs "HΦ" as "[H1 H2]"; first by set_solver.
-  iPvsIntro. iSplitL "H1".
+  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)).
 Qed.
@@ -51,11 +52,11 @@ 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);
-    try by (rewrite /φ'; eauto using atomic_not_val, atomic_step).
+    (λ e2 σ2 ef, ownP σ2 ★ ■ (φ' e2 σ2 ef))%I (λ e2 σ2 ef, ■ φ e2 σ2 ef ★ P)%I);
+    try by (eauto using atomic_not_val).
   repeat iSplit.
-  - by iIntros "! ?".
+  - 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 {e2 σ2 ef} "! [Hown #Hφ]"; iDestruct "Hφ" as %[[v2 <-%of_to_val] ?].

program_logic/lifting.v

 From iris.program_logic Require Export weakestpre.
 From iris.program_logic Require Import wsat ownership.
+From iris.proofmode Require Import pviewshifts.
 Local Hint Extern 10 (_ ≤ _) => omega.
 Local Hint Extern 100 (_ ⊥ _) => set_solver.
 Local Hint Extern 10 (✓{_} _) =>
@@ -18,18 +19,19 @@ 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 σ1 :
+    (φ : expr Λ → state Λ → option (expr Λ) → Prop) Φ e1 :
   E2 ⊆ E1 → to_val e1 = None →
-  reducible e1 σ1 →
-  (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
-  (|={E1,E2}=> ▷ 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 ∧
+      ■ (∀ 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)
   ⊢ WP e1 @ E1 {{ Φ }}.
 Proof.
-  intros ? He Hsafe Hstep. 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'&(r1&r2&?&?&Hwp)&Hws); auto; clear Hvs; cofe_subst r'.
+  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);
+    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. }
   { by rewrite assoc. }
@@ -64,24 +66,18 @@ Lemma wp_lift_atomic_step {E Φ} e1
     (φ : expr Λ → state Λ → option (expr Λ) → Prop) σ1 :
   atomic e1 →
   reducible e1 σ1 →
-  (∀ e2 σ2 ef,
-    prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
+  (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
   ▷ ownP σ1 ★ ▷ (∀ v2 σ2 ef,
     ■ φ (of_val v2) σ2 ef ∧ ownP σ2 -★ (|={E}=> Φ v2) ★ wp_fork ef)
   ⊢ WP e1 @ E {{ Φ }}.
 Proof.
-  intros. rewrite -(wp_lift_step E E (λ e2 σ2 ef,
-    is_Some (to_val e2) ∧ φ e2 σ2 ef) _ e1 σ1) //;
-    try by (eauto using atomic_not_val, atomic_step).
-  rewrite -pvs_intro. apply sep_mono, later_mono; first done.
-  apply forall_intro=>e2'; apply forall_intro=>σ2'.
-  apply forall_intro=>ef; apply wand_intro_l.
-  rewrite always_and_sep_l -assoc -always_and_sep_l.
-  apply pure_elim_l=>-[[v2 Hv] ?] /=.
-  rewrite -pvs_intro -wp_pvs.
-  rewrite (forall_elim v2) (forall_elim σ2') (forall_elim ef) pure_equiv //.
-  rewrite left_id wand_elim_r -(wp_value _ _ e2' v2) //.
-  by erewrite of_to_val.
+  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.
+  iDestruct ("Hwp" $! v2 σ2 ef with "[Hσ2]") as "[HΦ ?]". by eauto.
+  iFrame. iPvs "HΦ". iPvsIntro. iApply wp_value; auto using to_of_val.
 Qed.
 
 Lemma wp_lift_atomic_det_step {E Φ e1} σ1 v2 σ2 ef :
@@ -91,13 +87,10 @@ Lemma wp_lift_atomic_det_step {E Φ e1} σ1 v2 σ2 ef :
     σ2 = σ2' ∧ to_val e2' = Some v2 ∧ ef = ef') →
   ▷ ownP σ1 ★ ▷ (ownP σ2 -★ (|={E}=> Φ v2) ★ wp_fork ef) ⊢ WP e1 @ E {{ Φ }}.
 Proof.
-  intros. rewrite -(wp_lift_atomic_step _ (λ e2' σ2' ef',
-    σ2 = σ2' ∧ to_val e2' = Some v2 ∧ ef = ef') σ1) //.
-  apply sep_mono, later_mono; first done.
-  apply forall_intro=>e2'; apply forall_intro=>σ2'; apply forall_intro=>ef'.
-  apply wand_intro_l.
-  rewrite always_and_sep_l -assoc -always_and_sep_l to_of_val.
-  apply pure_elim_l=>-[-> [[->] ->]] /=. by rewrite wand_elim_r.
+  iIntros {???} "[Hσ1 Hσ2]". iApply (wp_lift_atomic_step _ (λ e2' σ2' ef',
+    σ2 = σ2' ∧ to_val e2' = Some v2 ∧ ef = ef') σ1); try done. iFrame. iNext.
+  iIntros {v2' σ2' ef'} "[#Hφ Hσ2']". rewrite to_of_val.
+  iDestruct "Hφ" as %(->&[= ->]&->). by iApply "Hσ2".
 Qed.
 
 Lemma wp_lift_pure_det_step {E Φ} e1 e2 ef :
@@ -106,9 +99,8 @@ Lemma wp_lift_pure_det_step {E Φ} e1 e2 ef :
   (∀ σ1 e2' σ2 ef', prim_step e1 σ1 e2' σ2 ef' → σ1 = σ2 ∧ e2 = e2' ∧ ef = ef')→
   ▷ (WP e2 @ E {{ Φ }} ★ wp_fork ef) ⊢ WP e1 @ E {{ Φ }}.
 Proof.
-  intros.
-  rewrite -(wp_lift_pure_step E (λ e2' ef', e2 = e2' ∧ ef = ef') _ e1) //=.
-  apply later_mono, forall_intro=>e'; apply forall_intro=>ef'.
-  by apply impl_intro_l, pure_elim_l=>-[-> ->].
+  iIntros {???} "?".
+  iApply (wp_lift_pure_step E (λ e2' ef', e2 = e2' ∧ ef = ef')); try done.
+  iNext. by iIntros {e' ef' [-> ->] }.
 Qed.
 End lifting.