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Relations.v
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(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 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 *)
(* *)
(****************************************************************************)
(* Relations.v *)
(****************************************************************************)
(* Formal Language Theory *)
(* *)
(* Judicael Courant - Jean-Christophe Filliatre *)
(* *)
(* Developped in V5.8 June-July 1993 *)
(* Ported to V5.10 October 1994 *)
(****************************************************************************)
(*Resultats sur les relations sur un Set,*)
(*copie de THEORIES/RELATIONS/Relations.v*)
(*ou l'on a remplace Type par Set et ou l'on definit le predicat Rstar_inv *)
Section Relations.
(* Properties of a binary relation R on type A *)
Variable A : Set.
Variable R : A -> A -> Prop.
(* Definition of the reflexive-transitive closure R* of R *)
(* Smallest reflexive P containing R o P *)
Definition Rstar (x y : A) :=
forall P : A -> A -> Prop,
(forall u : A, P u u) ->
(forall u v w : A, R u v -> P v w -> P u w) -> P x y.
Theorem Rstar_reflexive : forall x : A, Rstar x x.
Proof
fun (x : A) (P : A -> A -> Prop) (h1 : forall u : A, P u u)
(h2 : forall u v w : A, R u v -> P v w -> P u w) =>
h1 x.
Theorem Rstar_R : forall x y z : A, R x y -> Rstar y z -> Rstar x z.
Proof
fun (x y z : A) (t1 : R x y) (t2 : Rstar y z) (P : A -> A -> Prop)
(h1 : forall u : A, P u u)
(h2 : forall u v w : A, R u v -> P v w -> P u w) =>
h2 x y z t1 (t2 P h1 h2).
(* We conclude with transitivity of Rstar : *)
Theorem Rstar_transitive :
forall x y z : A, Rstar x y -> Rstar y z -> Rstar x z.
Proof
fun (x y z : A) (h : Rstar x y) =>
h (fun u v : A => Rstar v z -> Rstar u z)
(fun (u : A) (t : Rstar u z) => t)
(fun (u v w : A) (t1 : R u v) (t2 : Rstar w z -> Rstar v z)
(t3 : Rstar w z) => Rstar_R u v z t1 (t2 t3)).
(* Another characterization of R* *)
(* Smallest reflexive P containing R o R* *)
Definition Rstar' (x y : A) :=
forall P : A -> A -> Prop,
P x x -> (forall u : A, R x u -> Rstar u y -> P x y) -> P x y.
Theorem Rstar'_reflexive : forall x : A, Rstar' x x.
Proof
fun (x : A) (P : A -> A -> Prop) (h : P x x)
(h' : forall u : A, R x u -> Rstar u x -> P x x) => h.
Theorem Rstar'_R : forall x y z : A, R x z -> Rstar z y -> Rstar' x y.
Proof
fun (x y z : A) (t1 : R x z) (t2 : Rstar z y) (P : A -> A -> Prop)
(h1 : P x x) (h2 : forall u : A, R x u -> Rstar u y -> P x y) =>
h2 z t1 t2.
(* Equivalence of the two definitions: *)
Theorem Rstar'_Rstar : forall x y : A, Rstar' x y -> Rstar x y.
Proof
fun (x y : A) (h : Rstar' x y) =>
h Rstar (Rstar_reflexive x) (fun u : A => Rstar_R x u y).
Theorem Rstar_Rstar' : forall x y : A, Rstar x y -> Rstar' x y.
Proof
fun (x y : A) (h : Rstar x y) =>
h Rstar' (fun u : A => Rstar'_reflexive u)
(fun (u v w : A) (h1 : R u v) (h2 : Rstar' v w) =>
Rstar'_R u w v h1 (Rstar'_Rstar v w h2)).
(* inversion de Rstar*)
Lemma Rstar_inv :
forall x y : A,
Rstar x y -> x = y \/ ex2 (fun z : A => R x z) (fun z : A => Rstar z y).
intros x y Rstar_x_y.
pattern x, y in |- *.
apply Rstar_x_y.
auto.
intros u v w R_u_v Hyp.
apply or_intror.
exists v.
assumption.
elim Hyp.
intro Rew.
rewrite Rew.
apply Rstar_reflexive.
intro temp; elim temp; clear temp.
intros z R_v_z Rstar_z_w.
apply Rstar_R with z; assumption.
Qed.
End Relations.
Hint Resolve Rstar_reflexive.