constructor.
- reflexivity.
- reflexivity.
- exact Nwadd_ok.
- intros.
unfold Nwsub.
rewrite Nwadd_ok.
rewrite Nwopp_ok.
norm.
- exact Nwmul_ok.
- exact Nwopp_ok.
- exact gen_phiNword_ok.
Qed.
End NWORDMORPHISM.
Section GEN_DIV.
Variables  (R : Type) (rO : R) (rI : R) (radd : R -> R -> R)
(rmul : R -> R -> R) (rsub : R -> R -> R) (ropp : R -> R)
(req : R -> R -> Prop) (C : Type) (cO : C) (cI : C)
(cadd : C -> C -> C) (cmul : C -> C -> C) (csub : C -> C -> C)
(copp : C -> C) (ceqb : C -> C -> bool) (phi : C -> R).
Variable Rsth : Setoid_Theory R req.
Variable Reqe : ring_eq_ext radd rmul ropp req.
Variable ARth : almost_ring_theory rO rI radd rmul rsub ropp req.
Variable morph : ring_morph rO rI radd rmul rsub ropp req cO cI cadd cmul csub copp ceqb phi.
(* Useful tactics *)
Add Parametric Relation : R req
reflexivity  proved by (@Equivalence_Reflexive _ _ Rsth)
symmetry     proved by (@Equivalence_Symmetric _ _ Rsth)
transitivity proved by (@Equivalence_Transitive _ _ Rsth)
as R_set1.
Ltac rrefl := gen_reflexivity Rsth.
Add Morphism radd with signature (req ==> req ==> req) as radd_ext.
Variable zphi : Z -> R.

(* Specifications: *)
Lemma Ztriv_div_th : div_theory req Z.add Z.mul zphi Z.quotrem.
Variable nphi : N -> R.
Lemma Ntriv_div_th : div_theory req N.add N.mul nphi N.div_eucl.
constructor.