(*************************************************************************** * Safety for STLC with References - Infrastructure * * Extented from "Type Safety for STLC" * * Arthur Charguéraud, July 2007, Coq v8.1 * ***************************************************************************) Set Implicit Arguments. Require Import Metatheory STLC_Ref_Definitions. (* ********************************************************************** *) (** ** Additional Definitions used in the Proofs *) (** Computing free variables of a term. *) Fixpoint fv (t : trm) {struct t} : vars := match t with | trm_bvar i => {} | trm_fvar x => {{x}} | trm_abs t1 => (fv t1) | trm_app t1 t2 => (fv t1) \u (fv t2) | trm_unit => {} | trm_loc l => {} | trm_ref t1 => (fv t1) | trm_get t1 => (fv t1) | trm_set t1 t2 => (fv t1) \u (fv t2) end. (** Substitution for names *) Fixpoint subst (z : var) (u : trm) (t : trm) {struct t} : trm := match t with | trm_bvar i => trm_bvar i | trm_fvar x => if x == z then u else (trm_fvar x) | trm_abs t1 => trm_abs (subst z u t1) | trm_app t1 t2 => trm_app (subst z u t1) (subst z u t2) | trm_unit => trm_unit | trm_loc l => trm_loc l | trm_ref t1 => trm_ref (subst z u t1) | trm_get t1 => trm_get (subst z u t1) | trm_set t1 t2 => trm_set (subst z u t1) (subst z u t2) end. Notation "[ z ~> u ] t" := (subst z u t) (at level 68). (** Definition of the body of an abstraction *) Definition body t := exists L, forall x, x \notin L -> term (t ^ x). (* ********************************************************************** *) (** ** Instanciation of Tactics *) Ltac gather_vars := let A := gather_vars_with (fun x : vars => x) in let B := gather_vars_with (fun x : var => {{ x }}) in let C := gather_vars_with (fun x : env => dom x) in let D := gather_vars_with (fun x : trm => fv x) in constr:(A \u B \u C \u D). Ltac pick_fresh Y := let L := gather_vars in (pick_fresh_gen L Y). Tactic Notation "apply_fresh" constr(T) "as" ident(x) := apply_fresh_base T gather_vars x. Tactic Notation "apply_fresh" "*" constr(T) "as" ident(x) := apply_fresh T as x; auto*. Hint Constructors term sto_ok value red. (* ********************************************************************** *) (** ** Properties of substitution *) (** Substitution for a fresh name is identity. *) Lemma subst_fresh : forall x t u, x \notin fv t -> [x ~> u] t = t. Proof. intros. induction t; simpls; f_equal*. case_var*. notin_contradiction. Qed. (** Substitution distributes on the open operation. *) Lemma subst_open : forall x u t1 t2, term u -> [x ~> u] (t1 ^^ t2) = ([x ~> u]t1) ^^ ([x ~> u]t2). Proof. intros. unfold open. generalize 0. induction t1; intros; simpl; f_equal*. case_nat*. case_var*. apply* open_rec_term. Qed. (** Substitution and open_var for distinct names commute. *) Lemma subst_open_var : forall x y u t, y <> x -> term u -> ([x ~> u]t) ^ y = [x ~> u] (t ^ y). Proof. introv Neq Wu. rewrite* subst_open. simpl. case_var*. Qed. (** Opening up an abstraction of body t with a term u is the same as opening up the abstraction with a fresh name x and then substituting u for x. *) Lemma subst_intro : forall x t u, x \notin (fv t) -> term u -> t ^^ u = [x ~> u](t ^ x). Proof. introv Fr Wu. rewrite* subst_open. rewrite* subst_fresh. simpl. case_var*. Qed. (* ********************************************************************** *) (** ** Terms are stable through substitutions *) (** Terms are stable by substitution *) Lemma subst_term : forall t z u, term u -> term t -> term ([z ~> u]t). Proof. induction 2; simpl*. case_var*. apply_fresh term_abs as y. rewrite* subst_open_var. Qed. Hint Resolve subst_term. (* ********************************************************************** *) (** ** Terms are stable through open *) (** Conversion from locally closed abstractions and bodies *) Lemma term_abs_to_body : forall t1, term (trm_abs t1) -> body t1. Proof. intros. unfold body. inversion* H. Qed. Lemma body_to_term_abs : forall t1, body t1 -> term (trm_abs t1). Proof. intros. inversion* H. Qed. Hint Resolve term_abs_to_body body_to_term_abs. (** ** Opening a body with a term gives a term *) Lemma open_term : forall t u, body t -> term u -> term (t ^^ u). Proof. intros. destruct H. pick_fresh y. rewrite* (@subst_intro y). Qed. Hint Resolve open_term. (* ********************************************************************** *) (** ** Well-formedness of store *) Lemma sto_ok_binds : forall mu l t, sto_ok mu -> binds l t mu -> term t. Proof. unfolds binds. introv. induction mu as [|(l1,t1)]; simpl; env_fix; introv Ok Binds. contradictions. inversions Ok. case_var. inversions* Binds. auto. Qed. Hint Resolve sto_ok_binds. (* ********************************************************************** *) (** ** Regularity of relations *) (** A typing relation holds only if the environment has no duplicated keys and the pre-term is locally-closed. *) Lemma typing_regular : forall E P e T, typing E P e T -> ok E /\ term e. Proof. split; induction H; auto*. pick_fresh y; forward~ (H0 y) as K. inversions* K. Qed. (** The value predicate only holds on locally-closed terms. *) Lemma value_regular : forall e, value e -> term e. Proof. induction 1; auto*. Qed. (** A reduction relation only holds only on well-formed objects. *) Lemma red_regular : forall c c', red c c' -> (term (fst c) /\ term (fst c')) /\ (sto_ok (snd c) /\ sto_ok (snd c')). Proof. induction 1; simpl; env_fix; use value_regular. Qed. (** Automation for reasoning on well-formedness. *) Hint Extern 1 (ok ?E) => match goal with | H: typing E _ _ _ |- _ => apply (proj1 (typing_regular H)) end. Hint Extern 1 (term ?t) => match goal with | H: typing _ _ t _ |- _ => apply (proj2 (typing_regular H)) | H: red (t,_) _ |- _ => apply (proj1 (proj1 (red_regular H))) | H: red _ (t,_) |- _ => apply (proj2 (proj1 (red_regular H))) | H: value t |- _ => apply (value_regular H) end. Hint Extern 1 (sto_ok ?mu) => match goal with | H: red (_,mu) _ |- _ => apply (proj1 (proj2 (red_regular H))) | H: red _ (_,mu) |- _ => apply (proj2 (proj2 (red_regular H))) | H: sto_typing _ mu |- _ => apply (proj1 H) end.