diff options
Diffstat (limited to 'theories/Relations/Rstar.v')
| -rwxr-xr-x | theories/Relations/Rstar.v | 17 |
1 files changed, 8 insertions, 9 deletions
diff --git a/theories/Relations/Rstar.v b/theories/Relations/Rstar.v index ff2e02a4b..90ab6d6c2 100755 --- a/theories/Relations/Rstar.v +++ b/theories/Relations/Rstar.v @@ -8,15 +8,15 @@ (*i $Id$ i*) -(* Properties of a binary relation R on type A *) +(** 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 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). @@ -32,7 +32,7 @@ Theorem Rstar_R: (x:A)(y:A)(z:A)(R x y)->(Rstar y z)->(Rstar x z). [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 : *) +(** 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)] @@ -41,8 +41,8 @@ Theorem Rstar_transitive: (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z) ([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* *) +(** 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). @@ -55,7 +55,7 @@ Theorem Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x 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: *) +(** 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)] @@ -67,8 +67,7 @@ Theorem Rstar_Rstar': (x:A)(y:A)(Rstar x y)->(Rstar' x y). (Rstar'_R u w v h1 (Rstar'_Rstar v w h2)))). - -(* Property of Commutativity of two relations *) +(** 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) |