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diff --git a/theories7/Relations/Rstar.v b/theories7/Relations/Rstar.v new file mode 100755 index 00000000..3747b45e --- /dev/null +++ b/theories7/Relations/Rstar.v @@ -0,0 +1,78 @@ +(************************************************************************) +(* 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 1.1.2.1 2004/07/16 19:31:38 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](P:A->A->Prop) + ((u:A)(P u u))->((u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)) -> (P x y). + +Theorem Rstar_reflexive: (x:A)(Rstar x x). + Proof [x:A][P:A->A->Prop] + [h1:(u:A)(P u u)][h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)] + (h1 x). + +Theorem Rstar_R: (x:A)(y:A)(z:A)(R x y)->(Rstar y z)->(Rstar x z). + Proof [x:A][y:A][z:A][t1:(R x y)][t2:(Rstar y z)] + [P:A->A->Prop] + [h1:(u:A)(P u u)][h2:(u:A)(v:A)(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: (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z). + Proof [x:A][y:A][z:A][h:(Rstar x y)] + (h ([u:A][v:A](Rstar v z)->(Rstar u z)) + ([u:A][t:(Rstar u z)]t) + ([u:A][v:A][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:A][y:A](P:A->A->Prop) + ((P x x))->((u:A)(R x u)->(Rstar u y)->(P x y)) -> (P x y). + +Theorem Rstar'_reflexive: (x:A)(Rstar' x x). + Proof [x:A][P:A->A->Prop][h:(P x x)][h':(u:A)(R x u)->(Rstar u x)->(P x x)]h. + +Theorem Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x y). + Proof [x:A][y:A][z:A][t1:(R x z)][t2:(Rstar z y)] + [P:A->A->Prop][h1:(P x x)] + [h2:(u:A)(R x u)->(Rstar u y)->(P x y)](h2 z t1 t2). + +(** Equivalence of the two definitions: *) + +Theorem Rstar'_Rstar: (x:A)(y:A)(Rstar' x y)->(Rstar x y). + Proof [x:A][y:A][h:(Rstar' x y)] + (h Rstar (Rstar_reflexive x) ([u:A](Rstar_R x u y))). + +Theorem Rstar_Rstar': (x:A)(y:A)(Rstar x y)->(Rstar' x y). + Proof [x:A][y:A][h:(Rstar x y)](h Rstar' ([u:A](Rstar'_reflexive u)) + ([u:A][v:A][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] + (x,y:A)(R1 y x)->(z:A)(R2 z y) + ->(EX y':A |(R2 y' x) & (R1 z y')). + + +End Rstar. + |