From iris.heap_lang Require Export lifting. Import uPred. (** Define some derived forms, and derived lemmas about them. *) 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. (** Proof rules for the sugar *) Lemma wp_lam E x ef e Φ : is_Some (to_val e) → Closed (x :b: []) ef → ▷ 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 → ▷ 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 → ▷ WP e2 @ E {{ Φ }} ⊢ WP Seq e1 e2 @ E {{ Φ }}. Proof. intros ??. by rewrite -wp_let. Qed. Lemma wp_skip E Φ : ▷ 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 → ▷ 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 → ▷ 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.