Library iris.program_logic.total_ectx_lifting

Some derived lemmas for ectx-based languages
From iris.proofmode Require Import tactics.
From iris.program_logic Require Export ectx_language total_weakestpre total_lifting.
Set Default Proof Using "Type".

Section wp.
Context {Λ : ectxLanguage} `{!irisG Λ Σ} {Hinh : Inhabited (state Λ)}.
Implicit Types P : iProp Σ.
Implicit Types Φ : val Λ iProp Σ.
Implicit Types v : val Λ.
Implicit Types e : expr Λ.
Hint Resolve head_prim_reducible_no_obs head_reducible_prim_step
  head_reducible_no_obs_reducible : core.

Lemma twp_lift_head_step {s E Φ} e1 :
  to_val e1 = None
  ( σ1 κs n, state_interp σ1 κs n ={E,}=∗
    head_reducible_no_obs e1 σ1
     κ e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={,E}=∗
      κ = []
      state_interp σ2 κs (length efs + n)
      WP e2 @ s; E [{ Φ }]
      [∗ list] i ef efs, WP ef @ s; [{ fork_post }])
   WP e1 @ s; E [{ Φ }].
Proof.
  iIntros (?) "H".
  iApply (twp_lift_step _ E)=>//. iIntros (σ1 κs n) "Hσ".
  iMod ("H" $! σ1 with "Hσ") as "[% H]"; iModIntro.
  iSplit; [destruct s; auto|]. iIntros (κ e2 σ2 efs Hstep).
  iApply "H". by eauto.
Qed.

Lemma twp_lift_pure_head_step_no_fork {s E Φ} e1 :
  ( σ1, head_reducible_no_obs e1 σ1)
  ( σ1 κ e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs κ = [] σ2 = σ1 efs = [])
  (|={E}=> κ e2 efs σ, head_step e1 σ κ e2 σ efs WP e2 @ s; E [{ Φ }] )
   WP e1 @ s; E [{ Φ }].
Proof using Hinh.
  iIntros (??) ">H". iApply twp_lift_pure_step_no_fork; eauto.
  iIntros "!>" (?????). iApply "H"; eauto.
Qed.

Lemma twp_lift_atomic_head_step {s E Φ} e1 :
  to_val e1 = None
  ( σ1 κs n, state_interp σ1 κs n ={E}=∗
    head_reducible_no_obs e1 σ1
     κ e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={E}=∗
      κ = []
      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 twp_lift_atomic_step; eauto.
  iIntros (σ1 κs n) "Hσ1". iMod ("H" $! σ1 with "Hσ1") as "[% H]"; iModIntro.
  iSplit; first by destruct s; auto. iIntros (κ e2 σ2 efs Hstep). iApply "H"; eauto.
Qed.

Lemma twp_lift_atomic_head_step_no_fork {s E Φ} e1 :
  to_val e1 = None
  ( σ1 κs n, state_interp σ1 κs n ={E}=∗
    head_reducible_no_obs e1 σ1
     κ e2 σ2 efs, head_step e1 σ1 κ e2 σ2 efs ={E}=∗
      κ = [] efs = [] state_interp σ2 κs n from_option Φ False (to_val e2))
   WP e1 @ s; E [{ Φ }].
Proof.
  iIntros (?) "H". iApply twp_lift_atomic_head_step; eauto.
  iIntros (σ1 κs n) "Hσ1". iMod ("H" $! σ1 with "Hσ1") as "[$ H]"; iModIntro.
  iIntros (κ v2 σ2 efs Hstep).
  iMod ("H" with "[# //]") as "(-> & -> & ? & $) /=". by iFrame.
Qed.

Lemma twp_lift_pure_det_head_step_no_fork {s E Φ} e1 e2 :
  to_val e1 = None
  ( σ1, head_reducible_no_obs e1 σ1)
  ( σ1 κ e2' σ2 efs',
    head_step e1 σ1 κ e2' σ2 efs' κ = [] σ2 = σ1 e2' = e2 efs' = [])
  WP e2 @ s; E [{ Φ }] WP e1 @ s; E [{ Φ }].
Proof using Hinh.
  intros. rewrite -(twp_lift_pure_det_step_no_fork e1 e2); eauto.
Qed.
End wp.