Section NormedZmodule.
Variables (R : numDomainType) (V : normedZmodType R).
Implicit Types (l : R) (x y : V).
(* Closure properties of the real predicates. *)
Fact pos_divr_closed : divr_closed (@pos R).
Canonical pos_mulrPred := MulrPred pos_divr_closed.
Canonical pos_divrPred := DivrPred pos_divr_closed.
Fact nneg_divr_closed : divr_closed (@nneg R).
Canonical nneg_mulrPred := MulrPred nneg_divr_closed.
Canonical nneg_divrPred := DivrPred nneg_divr_closed.
Fact nneg_addr_closed : addr_closed (@nneg R).
Canonical real_addrPred := AddrPred real_addr_closed.
Canonical real_zmodPred := ZmodPred real_oppr_closed.
Fact real_divr_closed : divr_closed (@real R).
Canonical real_mulrPred := MulrPred real_divr_closed.
Canonical real_smulrPred := SmulrPred real_divr_closed.
Canonical real_divrPred := DivrPred real_divr_closed.
Canonical real_sdivrPred := SdivrPred real_divr_closed.
Canonical real_semiringPred := SemiringPred real_divr_closed.
Canonical real_subringPred := SubringPred real_divr_closed.
Canonical real_divringPred := DivringPred real_divr_closed.
End NumDomain.

Lemma num_real (R : realDomainType) (x : R) : x \is real.