Commit 745b4422 authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan

Merge branch 'jh_state_lifting'

parents 7e9c378e 72e8f63c
Pipeline #2029 passed with stage
......@@ -29,14 +29,10 @@ Lemma wp_alloc_pst E σ e v Φ :
WP Alloc e @ E {{ Φ }}.
Proof.
iIntros {?} "[HP HΦ]".
(* TODO: This works around ssreflect bug #22. *)
set (φ (e' : expr []) σ' ef := l,
ef = None e' = Lit (LitLoc l) σ' = <[l:=v]>σ σ !! l = None).
iApply (wp_lift_atomic_head_step (Alloc e) φ σ); try (by simpl; eauto);
[by intros; subst φ; inv_head_step; eauto 8|].
iFrame "HP". iNext. iIntros {v2 σ2 ef} "[Hφ HP]".
iDestruct "Hφ" as %(l & -> & [= <-]%of_to_val_flip & -> & ?); simpl.
iSplit; last done. iApply "HΦ"; by iSplit.
iApply (wp_lift_atomic_head_step (Alloc e) σ); try (by simpl; eauto).
iFrame "HP". iNext. iIntros {v2 σ2 ef} "[% HP]". inv_head_step.
match goal with H: _ = of_val v2 |- _ => apply (inj of_val (LitV _)) in H end.
subst v2. iSplit; last done. iApply "HΦ"; by iSplit.
Qed.
Lemma wp_load_pst E σ l v Φ :
......
(** 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}.
......@@ -17,32 +18,40 @@ 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 σ1 :
Lemma wp_lift_head_step E1 E2 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
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. eauto using wp_lift_step. Qed.
Proof.
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 :
Lemma wp_lift_pure_head_step E Φ e1 :
to_val e1 = None
( σ1, head_reducible e1 σ1)
( σ1 e2 σ2 ef, head_step e1 σ1 e2 σ2 ef σ1 = σ2 φ e2 ef)
( e2 ef, φ e2 ef WP e2 @ E {{ Φ }} wp_fork ef) WP e1 @ E {{ Φ }}.
Proof. eauto using wp_lift_pure_step. Qed.
( σ1 e2 σ2 ef, head_step e1 σ1 e2 σ2 ef σ1 = σ2)
( e2 ef σ, head_step e1 σ e2 σ ef WP e2 @ E {{ Φ }} wp_fork ef)
WP e1 @ E {{ Φ }}.
Proof.
iIntros {???} "H". iApply wp_lift_pure_step; eauto. iNext.
iIntros {????}. iApply "H". eauto.
Qed.
Lemma wp_lift_atomic_head_step {E Φ} e1
(φ : expr state option expr Prop) σ1 :
Lemma wp_lift_atomic_head_step {E Φ} e1 σ1 :
atomic e1
head_reducible e1 σ1
( e2 σ2 ef, head_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)
head_step e1 σ1 (of_val v2) σ2 ef ownP σ2 - (|={E}=> Φ v2) wp_fork ef)
WP e1 @ E {{ Φ }}.
Proof. eauto using wp_lift_atomic_step. Qed.
Proof.
iIntros {??} "[? H]". iApply wp_lift_atomic_step; eauto. iFrame. iNext.
iIntros {???} "[% ?]". iApply "H". eauto.
Qed.
Lemma wp_lift_atomic_det_head_step {E Φ e1} σ1 v2 σ2 ef :
atomic e1
......
......@@ -18,80 +18,74 @@ 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 σ1 :
Lemma ht_lift_step E1 E2 P Pσ1 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')
( 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,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 {?? 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]"].
iSpecialize ("HΦ" $! e2 σ2 ef with "[-]"). by iFrame "Hφ HP Hown".
iPvs "HΦ" as "[H1 H2]"; first by set_solver.
iPvsIntro. iSplitL "H1".
iIntros {??} "#(#Hvs&HΦ&He2&Hef) ! HP".
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} "[#Hstep Hown]".
iSpecialize ("HΦ" $! σ1 e2 σ2 ef with "[-]"). by iFrame "Hstep 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)).
Qed.
Lemma ht_lift_atomic_step
E (φ : expr Λ state Λ option (expr Λ) Prop) P e1 σ1 :
Lemma ht_lift_atomic_step E P e1 σ1 :
atomic e1
reducible e1 σ1
( e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef φ e2 σ2 ef)
( e2 σ2 ef, {{ φ e2 σ2 ef P }} ef ?@ {{ _, True }})
{{ ownP σ1 P }} e1 @ E {{ v, σ2 ef, ownP σ2 φ (of_val v) σ2 ef }}.
( e2 σ2 ef, {{ prim_step e1 σ1 e2 σ2 ef P }} ef ?@ {{ _, True }})
{{ ownP σ1 P }} e1 @ E {{ v, σ2 ef, ownP σ2
prim_step e1 σ1 (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);
try by (rewrite /φ'; eauto using atomic_not_val, atomic_step).
iIntros {? Hsafe} "#Hef".
iApply (ht_lift_step E E _ (λ σ1', P σ1 = σ1')%I
(λ e2 σ2 ef, ownP σ2 (is_Some (to_val e2) prim_step e1 σ1 e2 σ2 ef))%I
(λ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef P)%I);
try by (eauto using atomic_not_val).
repeat iSplit.
- by iIntros "! ?".
- 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.
Qed.
Lemma ht_lift_pure_step E (φ : expr Λ option (expr Λ) Prop) P P' Ψ e1 :
Lemma ht_lift_pure_step E P P' Ψ e1 :
to_val e1 = None
( σ1, reducible e1 σ1)
( σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef σ1 = σ2 φ e2 ef)
( e2 ef, {{ φ e2 ef P }} e2 @ E {{ Ψ }})
( e2 ef, {{ φ e2 ef P' }} ef ?@ {{ _, True }})
( σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef σ1 = σ2)
( e2 ef σ, {{ prim_step e1 σ e2 σ ef P }} e2 @ E {{ Ψ }})
( e2 ef σ, {{ prim_step e1 σ e2 σ ef P' }} ef ?@ {{ _, True }})
{{ (P P') }} e1 @ E {{ Ψ }}.
Proof.
iIntros {? Hsafe Hstep} "[#He2 #Hef] ! HP".
iApply (wp_lift_pure_step E φ _ e1); auto.
iNext; iIntros {e2 ef Hφ}. iDestruct "HP" as "[HP HP']"; iSplitL "HP".
iIntros {? Hsafe Hpure} "[#He2 #Hef] ! HP". iApply wp_lift_pure_step; auto.
iNext; iIntros {e2 ef σ Hstep}. iDestruct "HP" as "[HP HP']"; iSplitL "HP".
- iApply "He2"; by iSplit.
- destruct ef as [e|]; last done.
iApply ("Hef" $! _ (Some e)); by iSplit.
- destruct ef as [e|]; last done. iApply ("Hef" $! _ (Some e)); by iSplit.
Qed.
Lemma ht_lift_pure_det_step
E (φ : expr Λ option (expr Λ) Prop) P P' Ψ e1 e2 ef :
Lemma ht_lift_pure_det_step E P P' Ψ e1 e2 ef :
to_val e1 = None
( σ1, reducible e1 σ1)
( σ1 e2' σ2 ef', prim_step e1 σ1 e2' σ2 ef' σ1 = σ2 e2 = e2' ef = ef')
{{ P }} e2 @ E {{ Ψ }} {{ P' }} ef ?@ {{ _, True }}
{{ (P P') }} e1 @ E {{ Ψ }}.
Proof.
iIntros {? Hsafe Hdet} "[#He2 #Hef]".
iApply (ht_lift_pure_step _ (λ e2' ef', e2 = e2' ef = ef')); eauto.
iSplit; iIntros {e2' ef'}.
- iIntros "! [#He ?]"; iDestruct "He" as %[-> ->]. by iApply "He2".
iIntros {? Hsafe Hpuredet} "[#He2 #Hef]".
iApply ht_lift_pure_step; eauto. by intros; eapply Hpuredet.
iSplit; iIntros {e2' ef' σ}.
- iIntros "! [% ?]". edestruct Hpuredet as (_&->&->). done. by iApply "He2".
- destruct ef' as [e'|]; last done.
iIntros "! [#He ?]"; iDestruct "He" as %[-> ->]. by iApply "Hef".
iIntros "! [% ?]". edestruct Hpuredet as (_&->&->). done. by iApply "Hef".
Qed.
End lifting.
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 ({_} _) =>
......@@ -17,41 +18,40 @@ 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 :
Lemma wp_lift_step E1 E2 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 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 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&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. }
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.
Lemma wp_lift_pure_step E (φ : expr Λ option (expr Λ) Prop) Φ e1 :
Lemma wp_lift_pure_step E Φ e1 :
to_val e1 = None
( σ1, reducible e1 σ1)
( σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef σ1 = σ2 φ e2 ef)
( e2 ef, φ e2 ef WP e2 @ E {{ Φ }} wp_fork ef) WP e1 @ E {{ Φ }}.
( σ1 e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef σ1 = σ2)
( e2 ef σ, prim_step e1 σ e2 σ ef WP e2 @ E {{ Φ }} wp_fork ef)
WP e1 @ E {{ Φ }}.
Proof.
intros He Hsafe Hstep; rewrite wp_eq; uPred.unseal.
split=> n r ? Hwp; constructor; auto.
intros k Ef σ1 rf ???; split; [done|]. destruct n as [|n]; first lia.
intros e2 σ2 ef ?; destruct (Hstep σ1 e2 σ2 ef); auto; subst.
destruct (Hwp e2 ef k r) as (r1&r2&Hr&?&?); auto.
destruct (Hwp e2 ef σ1 k r) as (r1&r2&Hr&?&?); auto.
exists r1,r2; split_and?; try done.
- rewrite -Hr; eauto using wsat_le.
- uPred.unseal; by intros ? ->.
......@@ -60,44 +60,31 @@ Qed.
(** Derived lifting lemmas. *)
Import uPred.
Lemma wp_lift_atomic_step {E Φ} e1
(φ : expr Λ state Λ option (expr Λ) Prop) σ1 :
Lemma wp_lift_atomic_step {E Φ} e1 σ1 :
atomic e1
reducible e1 σ1
( 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)
prim_step e1 σ1 (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 _ 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.
Lemma wp_lift_atomic_det_step {E Φ e1} σ1 v2 σ2 ef :
atomic e1
reducible e1 σ1
( e2' σ2' ef', prim_step e1 σ1 e2' σ2' ef'
σ2 = σ2' to_val e2' = Some v2 ef = 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 {?? Hdet} "[Hσ1 Hσ2]". iApply (wp_lift_atomic_step _ σ1); try done.
iFrame. iNext. iIntros {v2' σ2' ef'} "[% Hσ2']".
edestruct Hdet as (->&->%of_to_val%(inj of_val)&->). done. by iApply "Hσ2".
Qed.
Lemma wp_lift_pure_det_step {E Φ} e1 e2 ef :
......@@ -106,9 +93,7 @@ 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 {?? Hpuredet} "?". iApply (wp_lift_pure_step E); try done.
by intros; eapply Hpuredet. iNext. by iIntros {e' ef' σ (_&->&->)%Hpuredet}.
Qed.
End lifting.
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