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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(****************************************************************************)
(*                               Reduction.v                                *)
(*                                                                          *)
(*                                Gerard Huet                               *)
(*                                                                          *)
(* Developed in V5.8  January 1993                                          *)
(* Ported    to V5.10 January 1995                                          *)
(****************************************************************************)
(* Beta reduction *)
Inductive red1 : lambda -> lambda -> Prop :=
| beta : forall M N : lambda, red1 (App (Abs M) N) (subst N M)
| abs_red : forall M N : lambda, red1 M N -> red1 (Abs M) (Abs N)
| app_red_l :
forall M1 N1 : lambda,
red1 M1 N1 -> forall M2 : lambda, red1 (App M1 M2) (App N1 M2)
| app_red_r :
forall M2 N2 : lambda,
red1 M2 N2 -> forall M1 : lambda, red1 (App M1 M2) (App M1 N2).
(* Transitive closure of red1 *)
Inductive red : lambda -> lambda -> Prop :=
| one_step_red : forall M N : lambda, red1 M N -> red M N
| refl_red : forall M : lambda, red M M
| trans_red : forall M N P : lambda, red M N -> red N P -> red M P.

(* Specifications: *)
Lemma red_abs : forall M M' : lambda, red M M' -> red (Abs M) (Abs M').
Lemma red_appl :
forall M M' : lambda,
red M M' -> forall N : lambda, red (App M N) (App M' N).
Lemma red_appr :
forall M M' : lambda,
red M M' -> forall N : lambda, red (App N M) (App N M').
Lemma red_app :
forall M M' N N' : lambda, red M M' -> red N N' -> red (App M N) (App M' N').
Lemma red_beta :
forall M M' N N' : lambda,
red M M' -> red N N' -> red (App (Abs M) N) (subst N' M').