```(*************************************************************************** * Church-Rosser Property in Pure Lambda-Calculus - Infrastructure * * Arthur Charguéraud, March 2007, Coq v8.1 * ***************************************************************************) Set Implicit Arguments. Require Import Metatheory Lambda_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_app t1 t2 => (fv t1) \u (fv t2) | trm_abs t1 => (fv t1) end. (* ********************************************************************** *) (** Abstracting a name out of a term *) Fixpoint close_var_rec (k : nat) (z : var) (t : trm) {struct t} : trm := match t with | trm_bvar i => trm_bvar i | trm_fvar x => if x == z then (trm_bvar k) else (trm_fvar x) | trm_app t1 t2 => trm_app (close_var_rec k z t1) (close_var_rec k z t2) | trm_abs t1 => trm_abs (close_var_rec (S k) z t1) end. Definition close_var z t := close_var_rec 0 z t. (* ********************************************************************** *) (** Substitution for a name *) 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_app t1 t2 => trm_app (subst z u t1) (subst z u t2) | trm_abs t1 => trm_abs (subst z u t1) end. Notation "[ z ~> u ] t" := (subst z u t) (at level 68). (* ********************************************************************** *) (** Definition of parallel relation *) Inductive para : relation := | para_red : forall L t1 t1' t2 t2', (forall x, x \notin L -> para (t1 ^ x) (t1' ^ x) ) -> para t2 t2' -> para (trm_app (trm_abs t1) t2) (t1' ^^ t2') | para_var : forall x, para (trm_fvar x) (trm_fvar x) | para_app : forall t1 t1' t2 t2', para t1 t1' -> para t2 t2' -> para (trm_app t1 t2) (trm_app t1' t2') | para_abs : forall L t1 t1', (forall x, x \notin L -> para (t1 ^ x) (t1' ^ x) ) -> para (trm_abs t1) (trm_abs t1'). (* ********************************************************************** *) (** Definition of the transitive closure of a relation *) Inductive iter_ (R : relation) : relation := | iter_trans : forall t2 t1 t3, iter_ R t1 t2 -> iter_ R t2 t3 -> iter_ R t1 t3 | iter_step : forall t t', R t t' -> iter_ R t t'. Notation "R 'iter'" := (iter_ R) (at level 69). (* ********************************************************************** *) (** Inclusion between relations (simulation or a relation by another) *) Definition simulated (R1 R2 : relation) := forall (t t' : trm), R1 t t' -> R2 t t'. Infix "simulated_by" := simulated (at level 69). (* ********************************************************************** *) (** Properties of relations *) Definition red_regular (R : relation) := forall t t', R t t' -> term t /\ term t'. Definition red_refl (R : relation) := forall t, term t -> R t t. Definition red_in (R : relation) := forall t x u u', term t -> R u u' -> R ([x ~> u]t) ([x ~> u']t). Definition red_all (R : relation) := forall x t t', R t t' -> forall u u', R u u' -> R ([x~>u]t) ([x~>u']t'). Definition red_out (R : relation) := forall x u t t', term u -> R t t' -> R ([x~>u]t) ([x~>u]t'). Definition red_rename (R : relation) := forall x t t' y, x \notin (fv t) -> x \notin (fv t') -> R (t ^ x) (t' ^ x) -> R (t ^ y) (t' ^ y). Definition red_through (R : relation) := forall x t1 t2 u1 u2, x \notin (fv t1) -> x \notin (fv u1) -> R (t1 ^ x) (u1 ^ x) -> R t2 u2 -> R (t1 ^^ t2) (u1 ^^ u2). (* ********************************************************************** *) (** ** 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 : trm => fv x) in constr:(A \u B \u C). Ltac pick_fresh X := let L := gather_vars in (pick_fresh_gen L X). 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. (* ********************************************************************** *) (** ** Lemmas *) (* ********************************************************************** *) (** Properties of substitutions *) (** 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 e, y <> x -> term u -> ([x ~> u]e) ^ y = [x ~> u] (e ^ 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. (** Tactic to permute subst and open var *) Ltac cross := rewrite subst_open_var; try cross. Tactic Notation "cross" "*" := cross; auto*. (* ********************************************************************** *) (** ** 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. Lemma subst_body : forall z u t, body t -> term u -> body ([z ~> u]t). unfold body. introv [L H]. exists (L \u {{z}}). intros. rewrite~ subst_open_var. use subst_term. Qed. Hint Resolve subst_term subst_body. (* ********************************************************************** *) (** ** 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. (* ********************************************************************** *) (** ** Additional results on primitive operations *) (** Open_var with fresh names is an injective operation *) Lemma open_var_inj : forall x t1 t2, x \notin (fv t1) -> x \notin (fv t2) -> (t1 ^ x = t2 ^ x) -> (t1 = t2). Proof. intros x t1. unfold open. generalize 0. induction t1; intro k; destruct t2; simpl; intros; inversion H1; try solve [ f_equal* | do 2 try case_nat; inversions* H1; try notin_contradiction ]. Qed. (** Close var is an operation returning a body of an abstraction *) Lemma close_var_fresh : forall x t, x \notin fv (close_var x t). Proof. introv. unfold close_var. generalize 0. induction t; intros k; simpls; notin_simpl; auto. case_var; simpl*. Qed. (** Close var is an operation returning a body of an abstraction *) Lemma close_var_body : forall x t, term t -> body (close_var x t). Proof. introv W. exists {{x}}. intros y Fr. unfold open, close_var. generalize 0. gen y. induction W; intros y Fr k; simpls. case_var; simpl*. case_nat*. auto*. apply_fresh* term_abs as z. unfolds open. rewrite* close_var_rec_open. Qed. (** Close var is the right inverse of open_var *) Lemma close_var_open : forall x t, term t -> t = (close_var x t) ^ x. Proof. introv W. unfold close_var, open. generalize 0. induction W; intros k; simpls; f_equal*. case_var*. simpl. case_nat*. let L := gather_vars in match goal with |- _ = ?t => destruct (var_fresh (L \u fv t)) as [y Fr] end. apply* (@open_var_inj y). unfolds open. rewrite* close_var_rec_open. Qed. (** An abstract specification of close_var, which packages the result of the operation and all the properties about it. *) Lemma close_var_spec : forall t x, term t -> exists u, t = u ^ x /\ body u /\ x \notin (fv u). Proof. intros. exists (close_var x t). split3. apply* close_var_open. apply* close_var_body. apply* close_var_fresh. Qed. (* ********************************************************************** *) (** Regularity: relations only hold on well-formed terms *) Section TermRelations. Hint Extern 1 (term (trm_abs ?t)) => match goal with H: context [term (t ^ _) ] |- _ => let y := fresh in let K := fresh in apply_fresh term_abs as y; inst_notin H y as K; destruct K; auto end. Lemma red_regular_beta : red_regular beta. Proof. introz. induction* H. Qed. Lemma red_regular_beta_star : red_regular (beta star). Proof. introz. induction* H. apply* red_regular_beta. Qed. Lemma red_regular_para : red_regular para. Proof. introz. induction* H. Qed. Lemma red_regular_para_iter : red_regular (para iter). Proof. introz. induction* H. apply* red_regular_para. Qed. End TermRelations. Hint Resolve red_regular_beta red_regular_beta_star red_regular_para red_regular_para_iter. Hint Extern 1 (term ?t) => match goal with | H: beta t _ |- _ => apply (proj1 (red_regular_beta H)) | H: beta _ t |- _ => apply (proj2 (red_regular_beta H)) | H: para t _ |- _ => apply (proj1 (red_regular_para H)) | H: para _ t |- _ => apply (proj2 (red_regular_para H)) | H: (beta star) t _ |- _ => apply (proj1 (red_regular_beta_star H)) | H: (beta star) _ t |- _ => apply (proj2 (red_regular_beta_star H)) | H: (para iter) t _ |- _ => apply (proj1 (red_regular_para_iter)) | H: (para iter) _ t |- _ => apply (proj2 (red_regular_para_iter)) end. (* A last helper to solve the case where we have to prove [body t] and that we only know that [t ^ x] is a term because it appears as an argument to some reduction R. *) Hint Extern 1 (body ?t) => match goal with H: context [?R (t ^ _) _ ] |- _ => let y := fresh in apply term_abs_to_body; apply_fresh term_abs as y; inst_notin H y as K; clear H end. ```