Section AlgIntSubring.
Canonical Aint_opprPred := OpprPred Aint_subring.
Canonical Aint_addrPred := AddrPred Aint_subring.
Canonical Aint_mulrPred := MulrPred Aint_subring.
Canonical Aint_zmodPred := ZmodPred Aint_subring.
Canonical Aint_semiringPred := SemiringPred Aint_subring.
Canonical Aint_smulrPred := SmulrPred Aint_subring.
Canonical Aint_subringPred := SubringPred Aint_subring.
End AlgIntSubring.
Canonical dvdA_opprPred e := OpprPred (dvdA_zmod_closed e).
Canonical dvdA_addrPred e := AddrPred (dvdA_zmod_closed e).
Canonical dvdA_zmodPred e := ZmodPred (dvdA_zmod_closed e).
Definition eqAmod (e x y : Algebraics.divisor) := (e %| x - y)%A.
Notation "x == y %[mod e ]" := (eqAmod e x y) : algC_scope.
Notation "x != y %[mod e ]" := (~~ (eqAmod e x y)) : algC_scope.

Lemma eqAmod0_rat : {in Crat &, forall e n, (n == 0 %[mod e])%A = (e %| n)%C}.