Thm* R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop). (Sym x,y:T. x R y) ![](FONT/eq.png) (Sym x,y:T. x (R^*) y) | [rel_star_symmetric] |
Thm* R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop), x,y:T. (x R^*^-1 y) ![](FONT/if_big.png) (x (R^-1^*) y) | [rel_inverse_star] |
Thm* x,y:T, R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop). x = y ![](FONT/eq.png) (x (R^*) y) | [rel_star_weakening] |
Thm* R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop), x,y,z:T. (x R y) ![](FONT/eq.png) (y (R^*) z) ![](FONT/eq.png) (x (R^*) z) | [rel_star_trans] |
Thm* R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop), x,y:T. (x R y) ![](FONT/eq.png) (x (R^*) y) | [rel_rel_star] |
Thm* E:(T![](FONT/dash.png) T![](FONT/dash.png) Prop), x,y:T. EquivRel(T)(_1 E _2) ![](FONT/eq.png) (x (E^*) y) ![](FONT/eq.png) (x E y) | [rel_star_of_equiv] |
Thm* R,R2:(T![](FONT/dash.png) T![](FONT/dash.png) Prop).
Thm* Trans(T)(R2(_1,_2))
Thm* ![](FONT/eq.png)
Thm* ( x,y:T. (x R y) ![](FONT/eq.png) (x R2 y)) ![](FONT/eq.png) ( x,y:T. (x (R^*) y) ![](FONT/eq.png) (x R2 y) x = y) | [rel_star_closure] |
Thm* P:(T![](FONT/dash.png) Prop), R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop). R preserves P ![](FONT/eq.png) R^* preserves P | [preserved_by_star] |
Thm* R1,R2:(T![](FONT/dash.png) T![](FONT/dash.png) Prop), x,y:T. R1 => R2 ![](FONT/eq.png) (x (R1^*) y) ![](FONT/eq.png) (x (R2^*) y) | [rel_star_monotonic] |
Thm* R:(T![](FONT/dash.png) T![](FONT/dash.png) Prop), x,y,z:T. (x (R^*) y) ![](FONT/eq.png) (y (R^*) z) ![](FONT/eq.png) (x (R^*) z) | [rel_star_transitivity] |
Thm* R1,R2:(T![](FONT/dash.png) T![](FONT/dash.png) Prop). R1 => R2 ![](FONT/eq.png) R1^* => R2^* | [rel_star_monotone] |