Record div_theory : Prop := mkdiv_th {
    div_eucl_th : forall a b, let (q,r) := cdiv a b in [a] == [b *! q +! r]
  }.Record div_theory : Prop := mkdiv_th {
    div_eucl_th : forall a b, let (q,r) := cdiv a b in [a] == [b *! q +! r]
  }.Record div_theory : Prop := mkdiv_th {
    div_eucl_th : forall a b, let (q,r) := cdiv a b in [a] == [b *! q +! r]
  }.Record div_theory : Prop := mkdiv_th {
    div_eucl_th : forall a b, let (q,r) := cdiv a b in [a] == [b *! q +! r]
  }.Record div_theory : Prop := mkdiv_th {
    div_eucl_th : forall a b, let (q,r) := cdiv a b in [a] == [b *! q +! r]
  }.Definition IDphi (x:R) := x.

Lemma IDmorph : ring_morph rO rI radd rmul rsub ropp reqb IDphi.