Commit ebf06f91 authored by Robbert Krebbers's avatar Robbert Krebbers

Fine-grained post-conditions for forked-off threads.

This commit extends the state interpretation with an additional parameter to
talk about the number of forked-off threads, and a fixed postcondition for each
forked-off thread:

    state_interp : Λstate → list Λobservation → nat → iProp Σ;
    fork_post : iProp Σ;

This way, instead of having `True` as the post-condition of `Fork`, one can
have any post-condition, which is then recorded in the state interpretation.
The point of keeping track of the postconditions of forked-off threads, is that
we get an (additional) stronger adequacy theorem:

    Theorem wp_strong_all_adequacy Σ Λ `{invPreG Σ} s e σ1 v vs σ2 φ :
       (∀ `{Hinv : invG Σ} κs,
         (|={⊤}=> ∃
             (stateI : state Λ → list (observation Λ) → nat → iProp Σ)
             (fork_post : iProp Σ),
           let _ : irisG Λ Σ := IrisG _ _ _ Hinv stateI fork_post in
           stateI σ1 κs 0 ∗ WP e @ s; ⊤ {{ v,
             let m := length vs in
             stateI σ2 [] m -∗ [∗] replicate m fork_post ={⊤,∅}=∗ ⌜ φ v ⌝ }})%I) →
      rtc erased_step ([e], σ1) (of_val <$> v :: vs, σ2) →
      φ v.

The difference with the ordinary adequacy theorem is that this one only applies
once all threads terminated. In this case, one gets back the post-conditions
`[∗] replicate m fork_post` of all forked-off threads.

In Iron we showed that we can use this mechanism to make sure that all
resources are disposed of properly in the presence of fork-based concurrency.
parent b0e4b6fa
......@@ -16,8 +16,9 @@ Class heapG Σ := HeapG {
Instance heapG_irisG `{heapG Σ} : irisG heap_lang Σ := {
iris_invG := heapG_invG;
state_interp σ κs :=
(gen_heap_ctx σ.(heap) proph_map_ctx κs σ.(used_proph_id))%I
state_interp σ κs _ :=
(gen_heap_ctx σ.(heap) proph_map_ctx κs σ.(used_proph_id))%I;
fork_post := True%I;
}.
(** Override the notations so that scopes and coercions work out *)
......@@ -162,7 +163,7 @@ Lemma wp_fork s E e Φ :
WP e @ s; {{ _, True }} - Φ (LitV LitUnit) - WP Fork e @ s; E {{ Φ }}.
Proof.
iIntros "He HΦ".
iApply wp_lift_pure_det_head_step; [by eauto|intros; inv_head_step; by eauto|].
iApply wp_lift_pure_det_head_step; [done|auto|intros; inv_head_step; eauto|].
iModIntro; iNext; iIntros "!> /= {$He}". by iApply wp_value.
Qed.
......@@ -170,7 +171,7 @@ Lemma twp_fork s E e Φ :
WP e @ s; [{ _, True }] - Φ (LitV LitUnit) - WP Fork e @ s; E [{ Φ }].
Proof.
iIntros "He HΦ".
iApply twp_lift_pure_det_head_step; [eauto|intros; inv_head_step; eauto|].
iApply twp_lift_pure_det_head_step; [done|auto|intros; inv_head_step; eauto|].
iIntros "!> /= {$He}". by iApply twp_value.
Qed.
......@@ -179,7 +180,7 @@ Lemma wp_alloc s E v :
{{{ True }}} Alloc (Val v) @ s; E {{{ l, RET LitV (LitLoc l); l v }}}.
Proof.
iIntros (Φ) "_ HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ Hκs] !>"; iSplit; first by auto.
iIntros (σ1 κ κs n) "[Hσ Hκs] !>"; iSplit; first by auto.
iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_alloc with "Hσ") as "[Hσ Hl]"; first done.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
......@@ -188,7 +189,7 @@ Lemma twp_alloc s E v :
[[{ True }]] Alloc (Val v) @ s; E [[{ l, RET LitV (LitLoc l); l v }]].
Proof.
iIntros (Φ) "_ HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κs) "[Hσ Hκs] !>"; iSplit; first by eauto.
iIntros (σ1 κs n) "[Hσ Hκs] !>"; iSplit; first by auto.
iIntros (κ v2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_alloc with "Hσ") as "[Hσ Hl]"; first done.
iModIntro; iSplit=> //. iSplit; first done. iFrame. by iApply "HΦ".
......@@ -198,7 +199,7 @@ Lemma wp_load s E l q v :
{{{ l {q} v }}} Load (Val $ LitV $ LitLoc l) @ s; E {{{ RET v; l {q} v }}}.
Proof.
iIntros (Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κ κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
......@@ -206,7 +207,7 @@ Lemma twp_load s E l q v :
[[{ l {q} v }]] Load (Val $ LitV $ LitLoc l) @ s; E [[{ RET v; l {q} v }]].
Proof.
iIntros (Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (κ v2 σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iSplit; first done. iFrame. by iApply "HΦ".
Qed.
......@@ -217,7 +218,7 @@ Lemma wp_store s E l v' v :
Proof.
iIntros (Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κ κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. iFrame. by iApply "HΦ".
......@@ -228,7 +229,7 @@ Lemma twp_store s E l v' v :
Proof.
iIntros (Φ) "Hl HΦ".
iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (κ v2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. iSplit; first done. iFrame. by iApply "HΦ".
......@@ -240,7 +241,7 @@ Lemma wp_cas_fail s E l q v' v1 v2 :
{{{ RET LitV (LitBool false); l {q} v' }}}.
Proof.
iIntros (?? Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κ κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
......@@ -250,7 +251,7 @@ Lemma twp_cas_fail s E l q v' v1 v2 :
[[{ RET LitV (LitBool false); l {q} v' }]].
Proof.
iIntros (?? Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (κ v2' σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iSplit; first done. iFrame. by iApply "HΦ".
Qed.
......@@ -261,7 +262,7 @@ Lemma wp_cas_suc s E l v1 v2 :
{{{ RET LitV (LitBool true); l v2 }}}.
Proof.
iIntros (? Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κ κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. iFrame. by iApply "HΦ".
......@@ -272,7 +273,7 @@ Lemma twp_cas_suc s E l v1 v2 :
[[{ RET LitV (LitBool true); l v2 }]].
Proof.
iIntros (? Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (κ v2' σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. iSplit; first done. iFrame. by iApply "HΦ".
......@@ -283,7 +284,7 @@ Lemma wp_faa s E l i1 i2 :
{{{ RET LitV (LitInt i1); l LitV (LitInt (i1 + i2)) }}}.
Proof.
iIntros (Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κ κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. iFrame. by iApply "HΦ".
......@@ -293,7 +294,7 @@ Lemma twp_faa s E l i1 i2 :
[[{ RET LitV (LitInt i1); l LitV (LitInt (i1 + i2)) }]].
Proof.
iIntros (Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iIntros (σ1 κs n) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (κ e2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. iSplit; first done. iFrame. by iApply "HΦ".
......@@ -304,7 +305,7 @@ Lemma wp_new_proph :
{{{ True }}} NewProph {{{ v (p : proph_id), RET (LitV (LitProphecy p)); proph p v }}}.
Proof.
iIntros (Φ) "_ HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
iIntros (σ1 κ κs n) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
iSplit; first by eauto.
iNext; iIntros (v2 σ2 efs Hstep). inv_head_step.
iMod (@proph_map_alloc with "HR") as "[HR Hp]".
......@@ -323,7 +324,7 @@ Lemma wp_resolve_proph p v w:
{{{ RET (LitV LitUnit); v = Some w }}}.
Proof.
iIntros (Φ) "Hp HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 κ κs) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
iIntros (σ1 κ κs n) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
iDestruct (@proph_map_valid with "HR Hp") as %Hlookup.
iSplit; first by eauto.
iNext; iIntros (v2 σ2 efs Hstep); inv_head_step. iApply fupd_frame_l.
......@@ -331,8 +332,7 @@ Proof.
iMod (@proph_map_remove with "HR Hp") as "Hp". iModIntro.
iSplitR "HΦ".
- iExists _. iFrame. iPureIntro. split; first by eapply first_resolve_delete.
rewrite dom_delete. rewrite <- difference_empty_L. by eapply difference_mono.
rewrite dom_delete. set_solver.
- iApply "HΦ". iPureIntro. by eapply first_resolve_eq.
Qed.
End lifting.
......@@ -12,6 +12,8 @@ Proof.
iMod (gen_heap_init σ.(heap)) as (?) "Hh".
iMod (proph_map_init [] σ.(used_proph_id)) as (?) "Hp".
iModIntro.
iExists (λ σ κs, (gen_heap_ctx σ.(heap) proph_map_ctx κs σ.(used_proph_id))%I). iFrame.
iExists
(λ σ κs _, (gen_heap_ctx σ.(heap) proph_map_ctx κs σ.(used_proph_id))%I),
True%I; iFrame.
iApply (Hwp (HeapG _ _ _ _)).
Qed.
This diff is collapsed.
......@@ -16,48 +16,53 @@ Hint Resolve head_stuck_stuck.
Lemma wp_lift_head_step_fupd {s E Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E,}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E,}=
head_reducible e1 σ1
e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={,,E}=
state_interp σ2 κs WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
WP e2 @ s; E {{ Φ }}
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_step_fupd=>//. iIntros (σ1 κ κs) "Hσ".
iIntros (?) "H". iApply wp_lift_step_fupd=>//. iIntros (σ1 κ κs Qs) "Hσ".
iMod ("H" with "Hσ") as "[% H]"; iModIntro.
iSplit; first by destruct s; eauto. iIntros (e2 σ2 efs Hstep).
iSplit; first by destruct s; eauto. iIntros (e2 σ2 efs ?).
iApply "H"; eauto.
Qed.
Lemma wp_lift_head_step {s E Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E,}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E,}=
head_reducible e1 σ1
e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={,E}=
state_interp σ2 κs WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
WP e2 @ s; E {{ Φ }}
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_head_step_fupd; [done|]. iIntros (???) "?".
iIntros (?) "H". iApply wp_lift_head_step_fupd; [done|]. iIntros (????) "?".
iMod ("H" with "[$]") as "[$ H]". iIntros "!>" (e2 σ2 efs ?) "!> !>". by iApply "H".
Qed.
Lemma wp_lift_head_stuck E Φ e :
to_val e = None
sub_redexes_are_values e
( σ κs, state_interp σ κs ={E,}= head_stuck e σ⌝)
( σ κs n, state_interp σ κs n ={E,}= head_stuck e σ⌝)
WP e @ E ?{{ Φ }}.
Proof.
iIntros (??) "H". iApply wp_lift_stuck; first done.
iIntros (σ κs) "Hσ". iMod ("H" with "Hσ") as "%". by auto.
iIntros (σ κs n) "Hσ". iMod ("H" with "Hσ") as "%". by auto.
Qed.
Lemma wp_lift_pure_head_step {s E E' Φ} e1 :
state_interp_fork_indep
( σ1, head_reducible e1 σ1)
( σ1 κ e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs κ = [] σ1 = σ2)
(|={E,E'}=> κ e2 efs σ, head_step e1 σ κ e2 σ efs
WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof using Hinh.
iIntros (??) "H". iApply wp_lift_pure_step; [|by eauto|].
iIntros (???) "H". iApply wp_lift_pure_step; [done| |by eauto|].
{ by destruct s; auto. }
iApply (step_fupd_wand with "H"); iIntros "H".
iIntros (?????). iApply "H"; eauto.
......@@ -70,74 +75,77 @@ Lemma wp_lift_pure_head_stuck E Φ e :
WP e @ E ?{{ Φ }}%I.
Proof using Hinh.
iIntros (?? Hstuck). iApply wp_lift_head_stuck; [done|done|].
iIntros (σ κs) "_". iMod (fupd_intro_mask' E ) as "_"; first set_solver.
iIntros (σ κs n) "_". iMod (fupd_intro_mask' E ) as "_"; first set_solver.
by auto.
Qed.
Lemma wp_lift_atomic_head_step_fupd {s E1 E2 Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E1}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E1}=
head_reducible e1 σ1
e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={E1,E2}=
state_interp σ2 κs
from_option Φ False (to_val e2) [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
from_option Φ False (to_val e2)
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E1 {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_atomic_step_fupd; [done|].
iIntros (σ1 κ κs) "Hσ1". iMod ("H" with "Hσ1") as "[% H]"; iModIntro.
iIntros (σ1 κ κs Qs) "Hσ1". iMod ("H" with "Hσ1") as "[% H]"; iModIntro.
iSplit; first by destruct s; auto. iIntros (e2 σ2 efs Hstep).
iApply "H"; eauto.
Qed.
Lemma wp_lift_atomic_head_step {s E Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E}=
head_reducible e1 σ1
e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={E}=
state_interp σ2 κs
from_option Φ False (to_val e2) [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
from_option Φ False (to_val e2)
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_atomic_step; eauto.
iIntros (σ1 κ κs) "Hσ1". iMod ("H" with "Hσ1") as "[% H]"; iModIntro.
iIntros (σ1 κ κs Qs) "Hσ1". iMod ("H" with "Hσ1") as "[% H]"; iModIntro.
iSplit; first by destruct s; auto. iNext. iIntros (e2 σ2 efs Hstep).
iApply "H"; eauto.
Qed.
Lemma wp_lift_atomic__head_step_no_fork_fupd {s E1 E2 Φ} e1 :
Lemma wp_lift_atomic_head_step_no_fork_fupd {s E1 E2 Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E1}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E1}=
head_reducible e1 σ1
e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={E1,E2}=
efs = [] state_interp σ2 κs from_option Φ False (to_val e2))
efs = [] state_interp σ2 κs n from_option Φ False (to_val e2))
WP e1 @ s; E1 {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_atomic_head_step_fupd; [done|].
iIntros (σ1 κ κs) "Hσ1". iMod ("H" $! σ1 with "Hσ1") as "[$ H]"; iModIntro.
iIntros (σ1 κ κs Qs) "Hσ1". iMod ("H" $! σ1 with "Hσ1") as "[$ H]"; iModIntro.
iIntros (v2 σ2 efs Hstep).
iMod ("H" $! v2 σ2 efs with "[# //]") as "H".
iIntros "!> !>". iMod "H" as "(% & $ & $)"; subst; auto.
iIntros "!> !>". iMod "H" as "(-> & ? & ?) /=". by iFrame.
Qed.
Lemma wp_lift_atomic_head_step_no_fork {s E Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E}=
head_reducible e1 σ1
e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={E}=
efs = [] state_interp σ2 κs from_option Φ False (to_val e2))
efs = [] state_interp σ2 κs n from_option Φ False (to_val e2))
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_atomic_head_step; eauto.
iIntros (σ1 κ κs) "Hσ1". iMod ("H" $! σ1 with "Hσ1") as "[$ H]"; iModIntro.
iIntros (σ1 κ κs Qs) "Hσ1". iMod ("H" $! σ1 with "Hσ1") as "[$ H]"; iModIntro.
iNext; iIntros (v2 σ2 efs Hstep).
iMod ("H" $! v2 σ2 efs with "[# //]") as "(% & $ & $)". subst; auto.
iMod ("H" $! v2 σ2 efs with "[//]") as "(-> & ? & ?) /=". by iFrame.
Qed.
Lemma wp_lift_pure_det_head_step {s E E' Φ} e1 e2 efs :
state_interp_fork_indep
( σ1, head_reducible e1 σ1)
( σ1 κ e2' σ2 efs',
head_step e1 σ1 κ e2' σ2 efs' κ = [] σ1 = σ2 e2 = e2' efs = efs')
(|={E,E'}=> WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
(|={E,E'}=> WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof using Hinh.
intros. rewrite -(wp_lift_pure_det_step e1 e2 efs); eauto.
......@@ -148,10 +156,10 @@ Lemma wp_lift_pure_det_head_step_no_fork {s E E' Φ} e1 e2 :
to_val e1 = None
( σ1, head_reducible e1 σ1)
( σ1 κ e2' σ2 efs',
head_step e1 σ1 κ e2' σ2 efs' κ = [] σ1 = σ2 e2 = e2' [] = efs')
head_step e1 σ1 κ e2' σ2 efs' κ = [] σ1 = σ2 e2 = e2' efs' = [])
(|={E,E'}=> WP e2 @ s; E {{ Φ }}) WP e1 @ s; E {{ Φ }}.
Proof using Hinh.
intros. rewrite -(wp_lift_pure_det_step e1 e2 []) /= ?right_id; eauto.
intros. rewrite -(wp_lift_pure_det_step_no_fork e1 e2); eauto.
destruct s; by auto.
Qed.
......@@ -159,7 +167,7 @@ Lemma wp_lift_pure_det_head_step_no_fork' {s E Φ} e1 e2 :
to_val e1 = None
( σ1, head_reducible e1 σ1)
( σ1 κ e2' σ2 efs',
head_step e1 σ1 κ e2' σ2 efs' κ = [] σ1 = σ2 e2 = e2' [] = efs')
head_step e1 σ1 κ e2' σ2 efs' κ = [] σ1 = σ2 e2 = e2' efs' = [])
WP e2 @ s; E {{ Φ }} WP e1 @ s; E {{ Φ }}.
Proof using Hinh.
intros. rewrite -[(WP e1 @ s; _ {{ _ }})%I]wp_lift_pure_det_head_step_no_fork //.
......
......@@ -15,23 +15,25 @@ Hint Resolve reducible_no_obs_reducible.
Lemma wp_lift_step_fupd s E Φ e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E,}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E,}=
if s is NotStuck then reducible e1 σ1 else True
e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={,,E}=
state_interp σ2 κs WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
WP e2 @ s; E {{ Φ }}
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1 κ κs) "Hσ".
rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1 κ κs n) "Hσ".
iMod ("H" with "Hσ") as "(%&H)". iModIntro. iSplit. by destruct s.
iIntros (????). iApply "H". eauto.
Qed.
Lemma wp_lift_stuck E Φ e :
to_val e = None
( σ κs, state_interp σ κs ={E,}= stuck e σ⌝)
( σ κs n, state_interp σ κs n ={E,}= stuck e σ⌝)
WP e @ E ?{{ Φ }}.
Proof.
rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1 κ κs) "Hσ".
rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1 κ κs n) "Hσ".
iMod ("H" with "Hσ") as %[? Hirr]. iModIntro. iSplit; first done.
iIntros (e2 σ2 efs ?). by case: (Hirr κ e2 σ2 efs).
Qed.
......@@ -39,32 +41,53 @@ Qed.
(** Derived lifting lemmas. *)
Lemma wp_lift_step s E Φ e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E,}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E,}=
if s is NotStuck then reducible e1 σ1 else True
e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={,E}=
state_interp σ2 κs WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
WP e2 @ s; E {{ Φ }}
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_step_fupd; [done|]. iIntros (???) "Hσ".
iMod ("H" with "Hσ") as "[$ H]". iIntros "!> * % !>". by iApply "H".
iIntros (?) "H". iApply wp_lift_step_fupd; [done|]. iIntros (????) "Hσ".
iMod ("H" with "Hσ") as "[$ H]". iIntros "!> * % !> !>". by iApply "H".
Qed.
Lemma wp_lift_pure_step `{Inhabited (state Λ)} s E E' Φ e1 :
state_interp_fork_indep
( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None)
( κ σ1 e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs κ = [] σ1 = σ2)
(|={E,E'}=> κ e2 efs σ, prim_step e1 σ κ e2 σ efs
WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (Hsafe Hstep) "H". iApply wp_lift_step.
iIntros (Hfork Hsafe Hstep) "H". iApply wp_lift_step.
{ specialize (Hsafe inhabitant). destruct s; last done.
by eapply reducible_not_val. }
iIntros (σ1 κ κs) "Hσ". iMod "H".
iIntros (σ1 κ κs n) "Hσ". iMod "H".
iMod fupd_intro_mask' as "Hclose"; last iModIntro; first by set_solver. iSplit.
{ iPureIntro. destruct s; done. }
iNext. iIntros (e2 σ2 efs Hstep').
destruct (Hstep κ σ1 e2 σ2 efs); auto; subst; clear Hstep.
iMod "Hclose" as "_". iFrame "Hσ". iMod "H". iApply "H"; auto.
iMod "Hclose" as "_". iMod "H". iModIntro.
rewrite (Hfork _ _ _ n). iFrame "Hσ". iApply "H"; auto.
Qed.
Lemma wp_lift_pure_step_no_fork `{Inhabited (state Λ)} s E E' Φ e1 :
( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None)
( κ σ1 e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs κ = [] σ1 = σ2 efs = [])
(|={E,E'}=> κ e2 efs σ, prim_step e1 σ κ e2 σ efs WP e2 @ s; E {{ Φ }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (Hsafe Hstep) "H". iApply wp_lift_step.
{ specialize (Hsafe inhabitant). destruct s; eauto using reducible_not_val. }
iIntros (σ1 κ κs n) "Hσ". iMod "H".
iMod fupd_intro_mask' as "Hclose"; last iModIntro; first by set_solver. iSplit.
{ iPureIntro. destruct s; done. }
iNext. iIntros (e2 σ2 efs ?).
destruct (Hstep κ σ1 e2 σ2 efs) as (-> & <- & ->); auto.
iMod "Hclose" as "_". iMod "H". iModIntro.
iDestruct ("H" with "[//]") as "H". simpl. iFrame.
Qed.
Lemma wp_lift_pure_stuck `{Inhabited (state Λ)} E Φ e :
......@@ -74,59 +97,75 @@ Proof.
iIntros (Hstuck) "_". iApply wp_lift_stuck.
- destruct(to_val e) as [v|] eqn:He; last done.
rewrite -He. by case: (Hstuck inhabitant).
- iIntros (σ κs) "_". iMod (fupd_intro_mask' E ) as "_".
by set_solver. by auto.
- iIntros (σ κs n) "_". by iMod (fupd_intro_mask' E ) as "_"; first set_solver.
Qed.
(* Atomic steps don't need any mask-changing business here, one can
use the generic lemmas here. *)
Lemma wp_lift_atomic_step_fupd {s E1 E2 Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E1}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E1}=
if s is NotStuck then reducible e1 σ1 else True
e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={E1,E2}=
state_interp σ2 κs
from_option Φ False (to_val e2) [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
from_option Φ False (to_val e2)
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E1 {{ Φ }}.
Proof.
iIntros (?) "H". iApply (wp_lift_step_fupd s E1 _ e1)=>//; iIntros (σ1 κ κs) "Hσ1".
iIntros (?) "H".
iApply (wp_lift_step_fupd s E1 _ e1)=>//; iIntros (σ1 κ κs n) "Hσ1".
iMod ("H" $! σ1 with "Hσ1") as "[$ H]".
iMod (fupd_intro_mask' E1 ) as "Hclose"; first set_solver.
iIntros "!>" (e2 σ2 efs ?). iMod "Hclose" as "_".
iMod ("H" $! e2 σ2 efs with "[#]") as "H"; [done|].
iMod (fupd_intro_mask' E2 ) as "Hclose"; [set_solver|]. iIntros "!> !>".
iMod "Hclose" as "_". iMod "H" as "($ & HΦ & $)".
iMod "Hclose" as "_". iMod "H" as "($ & HQ & $)".
destruct (to_val e2) eqn:?; last by iExFalso.
iApply wp_value; last done. by apply of_to_val.
Qed.
Lemma wp_lift_atomic_step {s E Φ} e1 :
to_val e1 = None
( σ1 κ κs, state_interp σ1 (κ ++ κs) ={E}=
( σ1 κ κs n, state_interp σ1 (κ ++ κs) n ={E}=
if s is NotStuck then reducible e1 σ1 else True
e2 σ2 efs, prim_step e1 σ1 κ e2 σ2 efs ={E}=
state_interp σ2 κs
from_option Φ False (to_val e2) [ list] ef efs, WP ef @ s; {{ _, True }})
state_interp σ2 κs (length efs + n)
from_option Φ False (to_val e2)
[ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (?) "H". iApply wp_lift_atomic_step_fupd; [done|].
iIntros (???) "?". iMod ("H" with "[$]") as "[$ H]".
iIntros (????) "?". iMod ("H" with "[$]") as "[$ H]".
iIntros "!> *". iIntros (Hstep) "!> !>".
by iApply "H".
Qed.
Lemma wp_lift_pure_det_step `{Inhabited (state Λ)} {s E E' Φ} e1 e2 efs :
state_interp_fork_indep
( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None)
( σ1 κ e2' σ2 efs', prim_step e1 σ1 κ e2' σ2 efs' κ = [] σ1 = σ2 e2 = e2' efs = efs')
(|={E,E'}=> WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, True }})
( σ1 κ e2' σ2 efs', prim_step e1 σ1 κ e2' σ2 efs'
κ = [] σ1 = σ2 e2 = e2' efs = efs')
(|={E,E'}=> WP e2 @ s; E {{ Φ }} [ list] ef efs, WP ef @ s; {{ _, fork_post }})
WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (? Hpuredet) "H". iApply (wp_lift_pure_step s E E'); try done.
iIntros (?? Hpuredet) "H". iApply (wp_lift_pure_step s E E'); try done.
{ by naive_solver. }
iApply (step_fupd_wand with "H"); iIntros "H".
by iIntros (κ e' efs' σ (->&_&->&->)%Hpuredet).
Qed.
Lemma wp_lift_pure_det_step_no_fork `{Inhabited (state Λ)} {s E E' Φ} e1 e2 :
( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None)
( σ1 κ e2' σ2 efs', prim_step e1 σ1 κ e2' σ2 efs'
κ = [] σ1 = σ2 e2 = e2' efs' = [])
(|={E,E'}=> WP e2 @ s; E {{ Φ }}) WP e1 @ s; E {{ Φ }}.
Proof.
iIntros (? Hpuredet) "H". iApply (wp_lift_pure_step_no_fork s E E'); try done.
{ naive_solver. }
iApply (step_fupd_wand with "H"); iIntros "H".
iIntros (κ e' efs' σ (_&?&->&?)%Hpuredet); auto.
Qed.
Lemma wp_pure_step_fupd `{Inhabited (state Λ)} s E E' e1 e2 φ n Φ :
PureExec φ n e1 e2