(* Definition of integral domains: commutative ring without zero divisor *)

Class Integral_domain {R : Type}`{Rcr:Cring R} := {
 integral_domain_product:
   forall x y, x * y == 0 -> x == 0 \/ y == 0;
 integral_domain_one_zero: not (1 == 0)}.(* Definition of integral domains: commutative ring without zero divisor *)

Class Integral_domain {R : Type}`{Rcr:Cring R} := {
 integral_domain_product:
   forall x y, x * y == 0 -> x == 0 \/ y == 0;
 integral_domain_one_zero: not (1 == 0)}.(* Definition of integral domains: commutative ring without zero divisor *)

Class Integral_domain {R : Type}`{Rcr:Cring R} := {
 integral_domain_product:
   forall x y, x * y == 0 -> x == 0 \/ y == 0;
 integral_domain_one_zero: not (1 == 0)}.

Lemma integral_domain_minus_one_zero: ~ - (1:R) == 0.