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 Robbert Krebbers committed Feb 13, 2016 1 ``````From heap_lang Require Export lifting. `````` Ralf Jung committed Feb 11, 2016 2 3 4 ``````Import uPred. (** Define some derived forms, and derived lemmas about them. *) `````` Ralf Jung committed Mar 02, 2016 5 ``````Notation Lam x e := (Rec BAnon x e). `````` Ralf Jung committed Feb 11, 2016 6 ``````Notation Let x e1 e2 := (App (Lam x e2) e1). `````` Ralf Jung committed Mar 02, 2016 7 8 ``````Notation Seq e1 e2 := (Let BAnon e1 e2). Notation LamV x e := (RecV BAnon x e). `````` Ralf Jung committed Feb 11, 2016 9 ``````Notation LetCtx x e2 := (AppRCtx (LamV x e2)). `````` Ralf Jung committed Mar 02, 2016 10 ``````Notation SeqCtx e2 := (LetCtx BAnon e2). `````` Ralf Jung committed Feb 13, 2016 11 ``````Notation Skip := (Seq (Lit LitUnit) (Lit LitUnit)). `````` Ralf Jung committed Mar 02, 2016 12 ``````Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)). `````` Ralf Jung committed Feb 11, 2016 13 14 `````` Section derived. `````` Robbert Krebbers committed Mar 02, 2016 15 ``````Context {Σ : rFunctor}. `````` Robbert Krebbers committed Feb 18, 2016 16 17 ``````Implicit Types P Q : iProp heap_lang Σ. Implicit Types Φ : val → iProp heap_lang Σ. `````` Ralf Jung committed Feb 11, 2016 18 19 `````` (** Proof rules for the sugar *) `````` Robbert Krebbers committed Feb 27, 2016 20 ``````Lemma wp_lam E x ef e v Φ : `````` Robbert Krebbers committed Feb 19, 2016 21 `````` to_val e = Some v → `````` Robbert Krebbers committed Mar 02, 2016 22 23 `````` ▷ || subst' ef x v @ E {{ Φ }} ⊑ || App (Lam x ef) e @ E {{ Φ }}. Proof. intros. by rewrite -wp_rec. Qed. `````` Ralf Jung committed Feb 11, 2016 24 `````` `````` Robbert Krebbers committed Feb 27, 2016 25 ``````Lemma wp_let E x e1 e2 v Φ : `````` Robbert Krebbers committed Feb 19, 2016 26 `````` to_val e1 = Some v → `````` Robbert Krebbers committed Mar 02, 2016 27 `````` ▷ || subst' e2 x v @ E {{ Φ }} ⊑ || Let x e1 e2 @ E {{ Φ }}. `````` Robbert Krebbers committed Feb 27, 2016 28 ``````Proof. apply wp_lam. Qed. `````` Ralf Jung committed Feb 11, 2016 29 `````` `````` 30 31 32 ``````Lemma wp_seq E e1 e2 v Φ : to_val e1 = Some v → ▷ || e2 @ E {{ Φ }} ⊑ || Seq e1 e2 @ E {{ Φ }}. `````` Robbert Krebbers committed Mar 02, 2016 33 ``````Proof. intros ?. by rewrite -wp_let. Qed. `````` Ralf Jung committed Feb 11, 2016 34 `````` `````` Robbert Krebbers committed Feb 19, 2016 35 ``````Lemma wp_skip E Φ : ▷ Φ (LitV LitUnit) ⊑ || Skip @ E {{ Φ }}. `````` 36 ``````Proof. rewrite -wp_seq // -wp_value //. Qed. `````` Ralf Jung committed Feb 13, 2016 37 `````` `````` Ralf Jung committed Mar 02, 2016 38 39 40 41 42 43 44 45 46 47 48 49 50 51 ``````Lemma wp_match_inl E e0 v0 x1 e1 x2 e2 Φ : to_val e0 = Some v0 → ▷ || subst' e1 x1 v0 @ E {{ Φ }} ⊑ || Match (InjL e0) x1 e1 x2 e2 @ E {{ Φ }}. Proof. intros. rewrite -wp_case_inl // -[X in _ ⊑ X]later_intro. by apply wp_let. Qed. Lemma wp_match_inr E e0 v0 x1 e1 x2 e2 Φ : to_val e0 = Some v0 → ▷ || subst' e2 x2 v0 @ E {{ Φ }} ⊑ || Match (InjR e0) x1 e1 x2 e2 @ E {{ Φ }}. Proof. intros. rewrite -wp_case_inr // -[X in _ ⊑ X]later_intro. by apply wp_let. Qed. `````` Robbert Krebbers committed Feb 18, 2016 52 ``````Lemma wp_le E (n1 n2 : Z) P Φ : `````` Robbert Krebbers committed Feb 19, 2016 53 54 55 `````` (n1 ≤ n2 → P ⊑ ▷ Φ (LitV (LitBool true))) → (n2 < n1 → P ⊑ ▷ Φ (LitV (LitBool false))) → P ⊑ || BinOp LeOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}. `````` Ralf Jung committed Feb 11, 2016 56 57 58 59 60 ``````Proof. intros. rewrite -wp_bin_op //; []. destruct (bool_decide_reflect (n1 ≤ n2)); by eauto with omega. Qed. `````` Robbert Krebbers committed Feb 18, 2016 61 ``````Lemma wp_lt E (n1 n2 : Z) P Φ : `````` Robbert Krebbers committed Feb 19, 2016 62 63 64 `````` (n1 < n2 → P ⊑ ▷ Φ (LitV (LitBool true))) → (n2 ≤ n1 → P ⊑ ▷ Φ (LitV (LitBool false))) → P ⊑ || BinOp LtOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}. `````` Ralf Jung committed Feb 11, 2016 65 66 67 68 69 ``````Proof. intros. rewrite -wp_bin_op //; []. destruct (bool_decide_reflect (n1 < n2)); by eauto with omega. Qed. `````` Robbert Krebbers committed Feb 18, 2016 70 ``````Lemma wp_eq E (n1 n2 : Z) P Φ : `````` Robbert Krebbers committed Feb 19, 2016 71 72 73 `````` (n1 = n2 → P ⊑ ▷ Φ (LitV (LitBool true))) → (n1 ≠ n2 → P ⊑ ▷ Φ (LitV (LitBool false))) → P ⊑ || BinOp EqOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}. `````` Ralf Jung committed Feb 11, 2016 74 75 76 77 78 ``````Proof. intros. rewrite -wp_bin_op //; []. destruct (bool_decide_reflect (n1 = n2)); by eauto with omega. Qed. End derived.``````