include "logic/pts.ma".
-ndefinition reflexive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x.P x x.
-ndefinition symmetric1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x,y.P x y → P y x.
-ndefinition transitive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀z,x,y. P x z → P z y → P x y.
+definition reflexive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x.P x x.
+definition symmetric1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x,y.P x y → P y x.
+definition transitive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀z,x,y. P x z → P z y → P x y.
-nrecord equivalence_relation1 (A:Type[1]) : Type[2] ≝
+record equivalence_relation1 (A:Type[1]) : Type[2] ≝
{ eq_rel1:2> A → A → CProp[1];
- refl1: reflexive1 ? eq_rel1;
- sym1: symmetric1 ? eq_rel1;
- trans1: transitive1 ? eq_rel1
+ refl1: <A href="cic:/matita/ng/properties/relations1/reflexive1.def(1)">reflexive1</A> ? eq_rel1;
+ sym1: <A href="cic:/matita/ng/properties/relations1/symmetric1.def(1)">symmetric1</A> ? eq_rel1;
+ trans1: <A href="cic:/matita/ng/properties/relations1/transitive1.def(1)">transitive1</A> ? eq_rel1
}.