Library iris.heap_lang.lifting

From iris.program_logic Require Export weakestpre stepshifts.
From iris.program_logic Require Import ownership ectx_lifting. From iris.heap_lang Require Export lang.
From iris.heap_lang Require Import tactics.
From iris.proofmode Require Import weakestpre pstepshifts.
Import uPred.
Local Hint Extern 0 (head_reducible _ _) ⇒ do_head_step eauto 2.

Section lifting.
Context {Σ : iFunctor}.
Implicit Types P Q : iProp heap_lang Σ.
Implicit Types Φ : val iProp heap_lang Σ.
Implicit Types ef : option expr.

Bind. This bundles some arguments that wp_ectx_bind leaves as indices.
Lemma wp_bind {E e} K Φ :
  WP e @ E {{ v, WP fill K (of_val v) @ E {{ Φ }} }} WP fill K e @ E {{ Φ }}.
Proof. exact: wp_ectx_bind. Qed.

Lemma wp_bindi {E e} Ki Φ :
  WP e @ E {{ v, WP fill_item Ki (of_val v) @ E {{ Φ }} }}
     WP fill_item Ki e @ E {{ Φ }}.
Proof. exact: weakestpre.wp_bind. Qed.

Base axioms for core primitives of the language: Stateful reductions.
Lemma wp_alloc_pst E σ v Φ :
  ( ownP σ ( l, (σ !! l = None) ownP (<[l:=v]>σ) -★ |={E}=>> Φ (LitV (LitLoc l))))
   WP Alloc (of_val v) @ E {{ Φ }}.
Proof.
  iIntros "[HP HΦ]".
  iApply (wp_lift_atomic_head_step (Alloc (of_val v)) σ); eauto with fsaV.
  iFrame "HP". iNext. iIntros (v2 σ2 ef). rewrite affine_and_r. iIntros "[% HP]". inv_head_step.
  set (l := fresh (dom (gset positive) σ)).
  iSpecialize ("HΦ" $! l).
  rewrite pure_equiv //=; last first.
  { apply (not_elem_of_dom (D:= gset _)), is_fresh. }
  rewrite -emp_True left_id. rewrite /ownP.
  iPsvs ("HΦ" with "HP").
  rewrite ?right_id.
  match goal with H: _ = of_val v2 |- _apply (inj of_val (LitV _)) in H end.
  iPvsIntro. by subst v2.
Qed.

Lemma wp_alloc_pst' E σ v Φ :
  ( ownP σ ▷(ownP (<[fresh (dom _ σ) :=v]>σ) -★ |={E}=>> Φ (LitV (LitLoc (fresh (dom _ σ))))))
   WP Alloc (of_val v) @ E {{ Φ }}.
Proof.
  iIntros "[HP HΦ]".
  iApply (wp_lift_atomic_head_step (Alloc (of_val v)) σ); eauto with fsaV.
  iFrame "HP". iNext. iIntros (v2 σ2 ef). rewrite affine_and_r. iIntros "[% HP]". inv_head_step.
  set (l := fresh (dom (gset positive) σ)).
  rewrite /ownP. iPsvs ("HΦ" with "HP").
  rewrite ?right_id.
  match goal with H: _ = of_val v2 |- _apply (inj of_val (LitV _)) in H end.
  iPvsIntro. by subst v2.
Qed.

Lemma wp_load_pst E σ l v Φ :
  σ !! l = Some v
  ( ownP σ (ownP σ -★ |={E}=>> Φ v)) WP Load (Lit (LitLoc l)) @ E {{ Φ }}.
Proof.
  intros. rewrite -(wp_lift_atomic_det_head_step σ v σ None) ?right_id //;
    last (by intros; inv_head_step; eauto using to_of_val). solve_atomic.
Qed.

Lemma wp_store_pst E σ l v v' Φ :
  σ !! l = Some v'
  ( ownP σ (ownP (<[l:=v]>σ) -★ |={E}=>> Φ (LitV LitUnit)))
   WP Store (Lit (LitLoc l)) (of_val v) @ E {{ Φ }}.
Proof.
  intros ?.
  rewrite-(wp_lift_atomic_det_head_step σ (LitV LitUnit) (<[l:=v]>σ) None)
    ?right_id //; last (by intros; inv_head_step; eauto). solve_atomic.
Qed.

Lemma wp_cas_fail_pst E σ l v1 v2 v' Φ :
  σ !! l = Some v' v' v1
  ( ownP σ (ownP σ -★ |={E}=>> Φ (LitV $ LitBool false)))
   WP CAS (Lit (LitLoc l)) (of_val v1) (of_val v2) @ E {{ Φ }}.
Proof.
  intros ??.
  rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool false) σ None)
    ?right_id //; last (by intros; inv_head_step; eauto). solve_atomic.
Qed.

Lemma wp_cas_suc_pst E σ l v1 v2 Φ :
  σ !! l = Some v1
  ( ownP σ (ownP (<[l:=v2]>σ) -★ |={E}=>> Φ (LitV $ LitBool true)))
   WP CAS (Lit (LitLoc l)) (of_val v1) (of_val v2) @ E {{ Φ }}.
Proof.
  intros ?. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool true)
    (<[l:=v2]>σ) None) ?right_id //; last (by intros; inv_head_step; eauto).
  solve_atomic.
Qed.

Lemma wp_swap_pst E σ l v v' Φ :
  σ !! l = Some v'
  ( ownP σ (ownP (<[l:=v]>σ) -★ |={E}=>> Φ v'))
   WP Swap (Lit (LitLoc l)) (of_val v) @ E {{ Φ }}.
Proof.
  intros ?.
  rewrite-(wp_lift_atomic_det_head_step σ v' (<[l:=v]>σ) None)
    ?right_id //; last (by intros; inv_head_step; eauto using to_of_val). solve_atomic.
Qed.

Lemma wp_fai_pst E σ l k Φ :
  σ !! l = Some (LitV $ LitInt k)
  ( ownP σ (ownP (<[l:=LitV $ LitInt (k+1)]>σ) -★ |={E}=>> Φ (LitV $ LitInt k)))
   WP FAI (Lit (LitLoc l)) @ E {{ Φ }}.
Proof.
  intros ?.
  rewrite-(wp_lift_atomic_det_head_step σ (LitV $ LitInt k) (<[l:=LitV $ LitInt (k+1)]>σ) None)
    ?right_id //; last (by intros; inv_head_step; eauto). solve_atomic.
Qed.

Base axioms for core primitives of the language: Stateless reductions
Lemma wp_fork E e Φ :
   (|={E}=>> Φ (LitV LitUnit) WP e {{ _, uPred_stopped }}) WP Fork e @ E {{ Φ }}.
Proof.
  rewrite -(wp_lift_pure_det_head_step (Fork e) (Lit LitUnit) (Some e)) //=;
    last by intros; inv_head_step; eauto.
  rewrite -(wp_value _ _ (Lit _)) //.
Qed.

Lemma wp_rec E f x erec e1 e2 Φ :
  e1 = Rec f x erec
  is_Some (to_val e2)
  Closed (f :b: x :b: []) erec
   (|={E}=>> WP subst' x e2 (subst' f e1 erec) @ E {{ Φ }})
   WP App e1 e2 @ E {{ Φ }}.
Proof.
  intros → [v2 ?] ?. rewrite -(wp_lift_pure_det_head_step (App _ _)
    (subst' x e2 (subst' f (Rec f x erec) erec)) None) //= ?right_id;
    intros; inv_head_step; eauto.
Qed.

Lemma wp_un_op E op l l' Φ :
  un_op_eval op l = Some l'
   (|={E}=>> Φ (LitV l')) WP UnOp op (Lit l) @ E {{ Φ }}.
Proof.
  intros. rewrite -(wp_lift_pure_det_head_step (UnOp op _) (Lit l') None)
    ?right_id -?wp_value_pvs //; intros; inv_head_step; eauto.
  iIntros "HP". iNext. iPsvs "HP". by do 2 iPvsIntro.
Qed.

Lemma wp_bin_op E op l1 l2 l' Φ :
  bin_op_eval op l1 l2 = Some l'
   (|={E}=>> Φ (LitV l')) WP BinOp op (Lit l1) (Lit l2) @ E {{ Φ }}.
Proof.
  intros Heval. rewrite -(wp_lift_pure_det_head_step (BinOp op _ _) (Lit l') None)
    ?right_id -?wp_value_pvs //; intros; inv_head_step; eauto.
  iIntros "HP". iNext. iPsvs "HP". by do 2 iPvsIntro.
Qed.

Lemma wp_if_true E e1 e2 Φ :
   (|={E}=>> WP e1 @ E {{ Φ }}) WP If (Lit (LitBool true)) e1 e2 @ E {{ Φ }}.
Proof.
  rewrite -(wp_lift_pure_det_head_step (If _ _ _) e1 None)
    ?right_id //; intros; inv_head_step; eauto.
Qed.

Lemma wp_if_false E e1 e2 Φ :
   (|={E}=>> WP e2 @ E {{ Φ }}) WP If (Lit (LitBool false)) e1 e2 @ E {{ Φ }}.
Proof.
  rewrite -(wp_lift_pure_det_head_step (If _ _ _) e2 None)
    ?right_id //; intros; inv_head_step; eauto.
Qed.

Lemma wp_fst E e1 v1 e2 Φ :
  to_val e1 = Some v1 is_Some (to_val e2)
   (|={E}=>> Φ v1) WP Fst (Pair e1 e2) @ E {{ Φ }}.
Proof.
  intros ? [v2 ?]. rewrite -(wp_lift_pure_det_head_step (Fst _) e1 None)
    ?right_id -?wp_value_pvs //; intros; inv_head_step; eauto.
  iIntros "HP". iNext. iPsvs "HP". by do 2 iPvsIntro.
Qed.

Lemma wp_snd E e1 e2 v2 Φ :
  is_Some (to_val e1) to_val e2 = Some v2
   (|={E}=>> Φ v2) WP Snd (Pair e1 e2) @ E {{ Φ }}.
Proof.
  intros [v1 ?] ?. rewrite -(wp_lift_pure_det_head_step (Snd _) e2 None)
    ?right_id -?wp_value_pvs //; intros; inv_head_step; eauto.
  iIntros "HP". iNext. iPsvs "HP". by do 2 iPvsIntro.
Qed.

Lemma wp_letp E x y e1 e2 eb Φ :
  is_Some (to_val e1) is_Some (to_val e2)
  Closed (x :b: y :b: []) eb
   (|={E}=>> WP subst' y e2 (subst' x e1 eb) @ E {{ Φ }})
   WP Letp x y (Pair e1 e2) eb @ E {{ Φ }}.
Proof.
  intros [v1 ?] [V2 ?] Hclo. rewrite -(wp_lift_pure_det_head_step (Letp _ _ _ _)
    (subst' y e2 (subst' x e1 eb)) None) //= ?right_id;
    intros; inv_head_step; eauto.
Qed.

Lemma wp_case_inl E e0 e1 e2 Φ :
  is_Some (to_val e0)
   (|={E}=>> WP App e1 e0 @ E {{ Φ }}) WP Case (InjL e0) e1 e2 @ E {{ Φ }}.
Proof.
  intros [v0 ?]. rewrite -(wp_lift_pure_det_head_step (Case _ _ _)
    (App e1 e0) None) ?right_id //; intros; inv_head_step; eauto.
Qed.

Lemma wp_case_inr E e0 e1 e2 Φ :
  is_Some (to_val e0)
   (|={E}=>> WP App e2 e0 @ E {{ Φ }}) WP Case (InjR e0) e1 e2 @ E {{ Φ }}.
Proof.
  intros [v0 ?]. rewrite -(wp_lift_pure_det_head_step (Case _ _ _)
    (App e2 e0) None) ?right_id //; intros; inv_head_step; eauto.
Qed.
End lifting.