(** Two generic morphisms from Z to (abrbitrary) rings, *)
(**second one is more convenient for proofs but they are ext. equal*)
Variable R : Type.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R).
Variable req : R -> R -> Prop.
Notation "0" := rO.  Notation "1" := rI.
Notation "x + y" := (radd x y).  Notation "x * y " := (rmul x y).
Notation "x - y " := (rsub x y).  Notation "- x" := (ropp x).
Notation "x == y" := (req x y).
Variable Rsth : Setoid_Theory R req.
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_setoid3.
Ltac rrefl := gen_reflexivity Rsth.
Variable Reqe : ring_eq_ext radd rmul ropp req.
Add Morphism radd with signature (req ==> req ==> req) as radd_ext3.
Section ALMOST_RING.
Variable ARth : almost_ring_theory 0 1 radd rmul rsub ropp req.
Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext3.

Lemma ARgen_phiPOS_add : forall x y,
gen_phiPOS1 (x + y) == (gen_phiPOS1 x) + (gen_phiPOS1 y).