```(*************************************************************************** * Church-Rosser Property in Pure Lambda-Calculus - Definitions * * Arthur Charguéraud, March 2007, Coq v8.1 * ***************************************************************************) Set Implicit Arguments. Require Import Metatheory. (* ********************************************************************** *) (** ** Description of the Task (only part which has to be trusted) *) (* ********************************************************************** *) (** Grammar of pre-terms *) Inductive trm : Set := | trm_bvar : nat -> trm | trm_fvar : var -> trm | trm_app : trm -> trm -> trm | trm_abs : trm -> trm. (* ********************************************************************** *) (** Operation to open up abstractions. *) Fixpoint open_rec (k : nat) (u : trm) (t : trm) {struct t} : trm := match t with | trm_bvar i => if k === i then u else (trm_bvar i) | trm_fvar x => trm_fvar x | trm_app t1 t2 => trm_app (open_rec k u t1) (open_rec k u t2) | trm_abs t1 => trm_abs (open_rec (S k) u t1) end. Definition open t u := open_rec 0 u t. Notation "{ k ~> u } t" := (open_rec k u t) (at level 67). Notation "t ^^ u" := (open t u) (at level 67). Notation "t ^ x" := (open t (trm_fvar x)). (* ********************************************************************** *) (** Definition of well-formedness of a term *) Inductive term : trm -> Prop := | term_var : forall x, term (trm_fvar x) | term_app : forall t1 t2, term t1 -> term t2 -> term (trm_app t1 t2) | term_abs : forall L t1, (forall x, x \notin L -> term (t1 ^ x)) -> term (trm_abs t1). (* ********************************************************************** *) (** Definition of the body of an abstraction *) Definition body t := exists L, forall x, x \notin L -> term (t ^ x). (* ********************************************************************** *) (** Definition of the beta relation *) Definition relation := trm -> trm -> Prop. Inductive beta : relation := | beta_red : forall t1 t2, body t1 -> term t2 -> beta (trm_app (trm_abs t1) t2) (t1 ^^ t2) | beta_app1 : forall t1 t1' t2, term t2 -> beta t1 t1' -> beta (trm_app t1 t2) (trm_app t1' t2) | beta_app2 : forall t1 t2 t2', term t1 -> beta t2 t2' -> beta (trm_app t1 t2) (trm_app t1 t2') | beta_abs : forall L t1 t1', (forall x, x \notin L -> beta (t1 ^ x) (t1' ^ x)) -> beta (trm_abs t1) (trm_abs t1'). (* ********************************************************************** *) (** Definition of the reflexive-transitive closure of a relation *) Inductive star_ (R : relation) : relation := | star_refl : forall t, term t -> star_ R t t | star_trans : forall t2 t1 t3, star_ R t1 t2 -> star_ R t2 t3 -> star_ R t1 t3 | star_step : forall t t', R t t' -> star_ R t t'. Notation "R 'star'" := (star_ R) (at level 69). (* ********************************************************************** *) (** Definition of the reflexive-symmetric-transitive closure of a relation *) Inductive equiv_ (R : relation) : relation := | equiv_refl : forall t, term t -> equiv_ R t t | equiv_sym: forall t t', equiv_ R t t' -> equiv_ R t' t | equiv_trans : forall t2 t1 t3, equiv_ R t1 t2 -> equiv_ R t2 t3 -> equiv_ R t1 t3 | equiv_step : forall t t', R t t' -> equiv_ R t t'. Notation "R 'equiv'" := (equiv_ R) (at level 69). (* ********************************************************************** *) (** Definition of confluence and of the Church-Rosser property (Our goal is to prove the Church-Rosser Property for beta relation) *) Definition confluence (R : relation) := forall M S T, R M S -> R M T -> exists P : trm, R S P /\ R T P. Definition church_rosser (R : relation) := forall t1 t2, (R equiv) t1 t2 -> exists t, (R star) t1 t /\ (R star) t2 t. ```