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Diffstat (limited to 'theories/Relations/Rstar.v')
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diff --git a/theories/Relations/Rstar.v b/theories/Relations/Rstar.v new file mode 100755 index 00000000..7bb3ee93 --- /dev/null +++ b/theories/Relations/Rstar.v @@ -0,0 +1,87 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rstar.v,v 1.8.2.1 2004/07/16 19:31:16 herbelin Exp $ i*) + +(** Properties of a binary relation [R] on type [A] *) + +Section Rstar. + +Variable A : Type. +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)). + + +(** Property of Commutativity of two relations *) + +Definition commut (A:Set) (R1 R2:A -> A -> Prop) := + forall x y:A, + R1 y x -> forall z:A, R2 z y -> exists2 y' : A, R2 y' x & R1 z y'. + + +End Rstar. |