include "logic/pts.ma".
-ndefinition reflexive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x.P x x.
-ndefinition symmetric2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x,y.P x y → P y x.
-ndefinition transitive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀z,x,y. P x z → P z y → P x y.
+definition reflexive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x.P x x.
+definition symmetric2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x,y.P x y → P y x.
+definition transitive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀z,x,y. P x z → P z y → P x y.
-nrecord equivalence_relation2 (A:Type[2]) : Type[3] ≝
+record equivalence_relation2 (A:Type[2]) : Type[3] ≝
{ eq_rel2:2> A → A → CProp[2];
- refl2: reflexive2 ? eq_rel2;
- sym2: symmetric2 ? eq_rel2;
- trans2: transitive2 ? eq_rel2
+ refl2: <A href="cic:/matita/ng/properties/relations2/reflexive2.def(1)">reflexive2</A> ? eq_rel2;
+ sym2: <A href="cic:/matita/ng/properties/relations2/symmetric2.def(1)">symmetric2</A> ? eq_rel2;
+ trans2: <A href="cic:/matita/ng/properties/relations2/transitive2.def(1)">transitive2</A> ? eq_rel2
}.