Section ALMOST_RING.
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.
Infix "==" := req. Infix "+" := radd.  Infix "* " := rmul.
(* a semi_morph can be extended to a ring_morph for the almost_ring derived from a semi_ring, provided the ring is a setoid (we only need reflexivity) *)
Variable C : Type.
Variable (cO cI : C) (cadd cmul: C->C->C).
Variable (ceqb : C -> C -> bool).
Variable phi : C -> R.
Variable Smorph : semi_morph rO rI radd rmul req cO cI cadd cmul ceqb phi.
End  SEMI_RING.
Infix "-" := rsub.
Notation "- x" := (ropp x).
Variable Reqe : ring_eq_ext radd rmul ropp req.
Add Morphism radd with signature (req ==> req ==> req) as radd_ext2.

Lemma  Rth_ARth : almost_ring_theory 0 1 radd rmul rsub ropp req.