lifting.v 4.49 KB
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From iris.program_logic Require Export weakestpre.
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From iris.proofmode Require Import tactics.
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Set Default Proof Using "Type".
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Section lifting.
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Context `{irisG Λ Σ}.
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Implicit Types s : stuckness.
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Implicit Types v : val Λ.
Implicit Types e : expr Λ.
Implicit Types σ : state Λ.
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Implicit Types P Q : iProp Σ.
Implicit Types Φ : val Λ  iProp Σ.
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Lemma wp_lift_step s E Φ e1 :
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  to_val e1 = None 
  ( σ1, state_interp σ1 ={E,}=
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    if s is NotStuck then reducible e1 σ1 else True 
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      e2 σ2 efs, prim_step e1 σ1 e2 σ2 efs ={,E}=
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      state_interp σ2  WP e2 @ s; E {{ Φ }}  [ list] ef  efs, WP ef @ s;  {{ _, True }})
   WP e1 @ s; E {{ Φ }}.
Proof.
  rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1) "Hσ".
  iMod ("H" with "Hσ") as "(%&?)". iModIntro. iSplit. by destruct s. done.
Qed.

Lemma wp_lift_stuck E Φ e :
  to_val e = None 
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  ( σ, state_interp σ ={E,}= stuck e σ⌝)
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   WP e @ E ?{{ Φ }}.
Proof.
  rewrite wp_unfold /wp_pre=>->. iIntros "H" (σ1) "Hσ".
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  iMod ("H" with "Hσ") as %[? Hirr]. iModIntro. iSplit; first done.
  iIntros "!>" (e2 σ2 efs) "%". by case: (Hirr e2 σ2 efs).
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Qed.
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(** Derived lifting lemmas. *)
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Lemma wp_lift_pure_step `{Inhabited (state Λ)} s E E' Φ e1 :
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  ( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None) 
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  ( σ1 e2 σ2 efs, prim_step e1 σ1 e2 σ2 efs  σ1 = σ2) 
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  (|={E,E'}=>  e2 efs σ, prim_step e1 σ e2 σ efs 
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    WP e2 @ s; E {{ Φ }}  [ list] ef  efs, WP ef @ s;  {{ _, True }})
   WP e1 @ s; E {{ Φ }}.
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Proof.
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  iIntros (Hsafe Hstep) "H". iApply wp_lift_step.
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  { specialize (Hsafe inhabitant). destruct s; last done.
      by eapply reducible_not_val. }
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  iIntros (σ1) "Hσ". iMod "H".
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  iMod fupd_intro_mask' as "Hclose"; last iModIntro; first by set_solver. iSplit.
  { iPureIntro. destruct s; done. }
  iNext. iIntros (e2 σ2 efs ?).
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  destruct (Hstep σ1 e2 σ2 efs); auto; subst.
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  iMod "Hclose" as "_". iFrame "Hσ". iMod "H". iApply "H"; auto.
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Qed.
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Lemma wp_lift_pure_stuck `{Inhabited (state Λ)} E Φ e :
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  ( σ, stuck e σ) 
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  True  WP e @ E ?{{ Φ }}.
Proof.
  iIntros (Hstuck) "_". iApply wp_lift_stuck.
  - destruct(to_val e) as [v|] eqn:He; last done.
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    rewrite -He. by case: (Hstuck inhabitant).
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  - iIntros (σ) "_". iMod (fupd_intro_mask' E ) as "_".
    by set_solver. by auto.
Qed.

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(* Atomic steps don't need any mask-changing business here, one can
   use the generic lemmas here. *)
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Lemma wp_lift_atomic_step {s E Φ} e1 :
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  to_val e1 = None 
  ( σ1, state_interp σ1 ={E}=
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    if s is NotStuck then reducible e1 σ1 else True 
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      e2 σ2 efs, prim_step e1 σ1 e2 σ2 efs ={E}=
      state_interp σ2 
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      default False (to_val e2) Φ  [ list] ef  efs, WP ef @ s;  {{ _, True }})
   WP e1 @ s; E {{ Φ }}.
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Proof.
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  iIntros (?) "H". iApply (wp_lift_step s E _ e1)=>//; iIntros (σ1) "Hσ1".
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  iMod ("H" $! σ1 with "Hσ1") as "[$ H]".
  iMod (fupd_intro_mask' E ) as "Hclose"; first set_solver.
  iModIntro; iNext; iIntros (e2 σ2 efs) "%". iMod "Hclose" as "_".
  iMod ("H" $! e2 σ2 efs with "[#]") as "($ & HΦ & $)"; first by eauto.
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  destruct (to_val e2) eqn:?; last by iExFalso.
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  by iApply wp_value.
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Qed.

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Lemma wp_lift_pure_det_step `{Inhabited (state Λ)} {s E E' Φ} e1 e2 efs :
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  ( σ1, if s is NotStuck then reducible e1 σ1 else to_val e1 = None) 
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  ( σ1 e2' σ2 efs', prim_step e1 σ1 e2' σ2 efs'  σ1 = σ2  e2 = e2'  efs = efs')
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  (|={E,E'}=> WP e2 @ s; E {{ Φ }}  [ list] ef  efs, WP ef @ s;  {{ _, True }})
   WP e1 @ s; E {{ Φ }}.
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Proof.
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  iIntros (? Hpuredet) "H". iApply (wp_lift_pure_step s E E'); try done.
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  { by intros; eapply Hpuredet. }
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  iApply (step_fupd_wand with "H"); iIntros "H".
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  by iIntros (e' efs' σ (_&->&->)%Hpuredet).
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Qed.
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Lemma wp_pure_step_fupd `{Inhabited (state Λ)} s E E' e1 e2 φ Φ :
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  PureExec φ e1 e2 
  φ 
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  (|={E,E'}=> WP e2 @ s; E {{ Φ }})  WP e1 @ s; E {{ Φ }}.
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Proof.
  iIntros ([??] Hφ) "HWP".
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  iApply (wp_lift_pure_det_step with "[HWP]").
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  - intros σ. specialize (pure_exec_safe σ). destruct s; eauto using reducible_not_val.
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  - destruct s; naive_solver.
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  - by rewrite big_sepL_nil right_id.
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Qed.

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Lemma wp_pure_step_later `{Inhabited (state Λ)} s E e1 e2 φ Φ :
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  PureExec φ e1 e2 
  φ 
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   WP e2 @ s; E {{ Φ }}  WP e1 @ s; E {{ Φ }}.
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Proof.
  intros ??. rewrite -wp_pure_step_fupd //. rewrite -step_fupd_intro //.
Qed.
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End lifting.