Library iris.array_lang.derived

From iris.array_lang Require Export lifting.
Import uPred.

Notation Lam x e := (Rec BAnon x e).
Notation Let x e1 e2 := (App (Lam x e2) e1).
Notation Seq e1 e2 := (Let BAnon e1 e2).
Notation LamV x e := (RecV BAnon x e).
Notation LetCtx x e2 := (AppRCtx (LamV x e2)).
Notation SeqCtx e2 := (LetCtx BAnon e2).
Notation Skip := (Seq (Lit LitUnit) (Lit LitUnit)).
Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)).

Section derived.
Context {Σ : iFunctor}.
Implicit Types P Q : iProp heap_lang Σ.
Implicit Types Φ : val → iProp heap_lang Σ.

Lemma wp_lam E x ef e Φ :
  is_Some (to_val e) → Closed (x :b: []) ef →
  ▷ (|={E}=>> WP subst' x e ef @ E {{ Φ }}) ⊢ WP App (Lam x ef) e @ E {{ Φ }}.
Proof. intros. by rewrite -(wp_rec _ BAnon) //. Qed.

Lemma wp_let E x e1 e2 Φ :
  is_Some (to_val e1) → Closed (x :b: []) e2 →
  ▷ (|={E}=>> WP subst' x e1 e2 @ E {{ Φ }}) ⊢ WP Let x e1 e2 @ E {{ Φ }}.
Proof. apply wp_lam. Qed.

Lemma wp_seq E e1 e2 Φ :
  is_Some (to_val e1) → Closed [] e2 →
  ▷ (|={E}=>> WP e2 @ E {{ Φ }}) ⊢ WP Seq e1 e2 @ E {{ Φ }}.
Proof. intros ??. by rewrite -wp_let. Qed.

Lemma wp_skip E Φ : ▷ (|={E}=>> Φ (LitV LitUnit)) ⊢ WP Skip @ E {{ Φ }}.
Proof. rewrite -wp_seq; last eauto. by rewrite -wp_value. Qed.

Lemma wp_match_inl E e0 x1 e1 x2 e2 Φ :
  is_Some (to_val e0) → Closed (x1 :b: []) e1 →
  (|={E}=>> ▷ (|={E}=>> WP subst' x1 e0 e1 @ E {{ Φ }}))
    ⊢ WP Match (InjL e0) x1 e1 x2 e2 @ E {{ Φ }}.
Proof. intros. by rewrite -wp_case_inl // -[X in _ ⊢ X]later_intro -wp_let. Qed.

Lemma wp_match_inr E e0 x1 e1 x2 e2 Φ :
  is_Some (to_val e0) → Closed (x2 :b: []) e2 →
  (|={E}=>> ▷ (|={E}=>> WP subst' x2 e0 e2 @ E {{ Φ }}))
    ⊢ WP Match (InjR e0) x1 e1 x2 e2 @ E {{ Φ }}.
Proof. intros. by rewrite -wp_case_inr // -[X in _ ⊢ X]later_intro -wp_let. Qed.

Lemma wp_le E (n1 n2 : Z) P Φ :
  (n1 ≤ n2 → P ⊢ ▷ |={E}=>> Φ (LitV (LitBool true))) →
  (n2 < n1 → P ⊢ ▷ |={E}=>> Φ (LitV (LitBool false))) →
  P ⊢ WP BinOp LeOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
Proof.
  intros. rewrite -wp_bin_op //; [].
  destruct (bool_decide_reflect (n1 ≤ n2)); by eauto with omega.
Qed.

Lemma wp_lt E (n1 n2 : Z) P Φ :
  (n1 < n2 → P ⊢ ▷ |={E}=>> Φ (LitV (LitBool true))) →
  (n2 ≤ n1 → P ⊢ ▷ |={E}=>> Φ (LitV (LitBool false))) →
  P ⊢ WP BinOp LtOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
Proof.
  intros. rewrite -wp_bin_op //; [].
  destruct (bool_decide_reflect (n1 < n2)); by eauto with omega.
Qed.

Lemma wp_eq E (n1 n2 : Z) P Φ :
  (n1 = n2 → P ⊢ ▷ |={E}=>> Φ (LitV (LitBool true))) →
  (n1 ≠ n2 → P ⊢ ▷ |={E}=>> Φ (LitV (LitBool false))) →
  P ⊢ WP BinOp EqOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
Proof.
  intros. rewrite -wp_bin_op //; [].
  destruct (bool_decide_reflect (n1 = n2)); by eauto with omega.
Qed.
End derived.