Require Export heap_lang.heap_lang heap_lang.lifting. Import uPred. Import heap_lang. (** Define some syntactic sugar. LitTrue and LitFalse are defined in heap_lang.v. *) Definition Lam (e : {bind expr}) := Rec e.[ren(+1)]. Definition Let (e1 : expr) (e2: {bind expr}) := App (Lam e2) e1. Definition Seq (e1 e2 : expr) := Let e1 e2.[ren(+1)]. Definition If (e0 e1 e2 : expr) := Case e0 e1.[ren(+1)] e2.[ren(+1)]. Definition Lt e1 e2 := Le (Plus e1 \$ LitNat 1) e2. Definition Eq e1 e2 := Let e1 (Let e2.[ren(+1)] (If (Le (Var 0) (Var 1)) (Le (Var 1) (Var 0)) LitFalse)). Definition LamV (e : {bind expr}) := RecV e.[ren(+1)]. Definition LetCtx (e2 : {bind expr}) := AppRCtx (LamV e2). Definition SeqCtx (e2 : expr) := LetCtx (e2.[ren(+1)]). Delimit Scope lang_scope with L. Bind Scope lang_scope with expr. Arguments wp {_ _} _ _%L _. (* TODO: The levels are all random. Also maybe we should not make 'new' a keyword. What about Arguments for hoare triples? Also find better notation for function application. Or maybe we can make "App" a coercion from expr to (expr → expr)? *) (* The colons indicate binders. *) Notation "'rec::' e" := (Rec e) (at level 100) : lang_scope. Notation "'λ:' e" := (Lam e) (at level 100) : lang_scope. Infix "\$" := App : lang_scope. Notation "'let:' e1 'in' e2" := (Let e1 e2) (at level 70) : lang_scope. Notation "e1 ';' e2" := (Seq e1 e2) (at level 70) : lang_scope. Notation "'if' e1 'then' e2 'else' e3" := (If e1 e2 e3) : lang_scope. Notation "'#0'" := (Var 0) (at level 10) : lang_scope. Notation "'#1'" := (Var 1) (at level 10) : lang_scope. Notation "'#2'" := (Var 2) (at level 10) : lang_scope. Notation "'#3'" := (Var 3) (at level 10) : lang_scope. Notation "'#4'" := (Var 4) (at level 10) : lang_scope. Notation "'#5'" := (Var 5) (at level 10) : lang_scope. Notation "'#6'" := (Var 6) (at level 10) : lang_scope. Notation "'#7'" := (Var 7) (at level 10) : lang_scope. Notation "'#8'" := (Var 8) (at level 10) : lang_scope. Notation "'#9'" := (Var 9) (at level 10) : lang_scope. Notation "'★' e" := (Load e) (at level 30) : lang_scope. Notation "e1 '<-' e2" := (Store e1 e2) (at level 60) : lang_scope. Notation "'new' e" := (Alloc e) (at level 60) : lang_scope. Notation "e1 '+' e2" := (Plus e1 e2) : lang_scope. Notation "e1 '≤' e2" := (Le e1 e2) : lang_scope. Notation "e1 '<' e2" := (Lt e1 e2) : lang_scope. Coercion LitNat : nat >-> expr. Coercion Loc : loc >-> expr. Section suger. Context {Σ : iFunctor}. Implicit Types P : iProp heap_lang Σ. Implicit Types Q : val → iProp heap_lang Σ. (** Proof rules for the sugar *) Lemma wp_lam E ef e v Q : to_val e = Some v → ▷ wp E ef.[e/] Q ⊑ wp E (App (Lam ef) e) Q. Proof. intros Hv. rewrite -wp_rec; last eassumption. (* RJ: This pulls in functional extensionality. If that bothers us, we have to talk to the Autosubst guys. *) by asimpl. Qed. Lemma wp_let E e1 e2 Q : wp E e1 (λ v, ▷wp E (e2.[of_val v/]) Q) ⊑ wp E (Let e1 e2) Q. Proof. rewrite -(wp_bind [LetCtx e2]). apply wp_mono=>v. by rewrite -wp_lam //= to_of_val. Qed. Lemma wp_if_true E e1 e2 Q : ▷ wp E e1 Q ⊑ wp E (If LitTrue e1 e2) Q. Proof. rewrite -wp_case_inl //. by asimpl. Qed. Lemma wp_if_false E e1 e2 Q : ▷ wp E e2 Q ⊑ wp E (If LitFalse e1 e2) Q. Proof. rewrite -wp_case_inr //. by asimpl. Qed. Lemma wp_lt E n1 n2 P Q : (n1 < n2 → P ⊑ ▷ Q LitTrueV) → (n1 ≥ n2 → P ⊑ ▷ Q LitFalseV) → P ⊑ wp E (Lt (LitNat n1) (LitNat n2)) Q. Proof. intros; rewrite -(wp_bind [LeLCtx _]) -wp_plus -later_intro /=. auto using wp_le with lia. Qed. Lemma wp_eq E n1 n2 P Q : (n1 = n2 → P ⊑ ▷ Q LitTrueV) → (n1 ≠ n2 → P ⊑ ▷ Q LitFalseV) → P ⊑ wp E (Eq (LitNat n1) (LitNat n2)) Q. Proof. intros HPeq HPne. rewrite -wp_let -wp_value' // -later_intro; asimpl. rewrite -wp_rec //; asimpl. rewrite -(wp_bind [CaseCtx _ _]) -later_intro; asimpl. apply wp_le; intros; asimpl. * rewrite -wp_case_inl // -!later_intro. apply wp_le; auto with lia. * rewrite -wp_case_inr // -later_intro -wp_value' //; auto with lia. Qed. End suger.