Definition Pcond_simpl_complete :=
fcons_ok _ PFcons2_fcons_inv.
End Fcons_simpl.
End AlmostField.
Section FieldAndSemiField.
Record field_theory : Prop := mk_field {
F_R : ring_theory rO rI radd rmul rsub ropp req;
F_1_neq_0 : ~ 1 == 0;
Fdiv_def : forall p q, p / q == p * / q;
Finv_l : forall p, ~ p == 0 ->  / p * p == 1
}.
Definition F2AF f :=
mk_afield
(Rth_ARth Rsth Reqe (F_R f)) (F_1_neq_0 f) (Fdiv_def f) (Finv_l f).
Record semi_field_theory : Prop := mk_sfield {
SF_SR : semi_ring_theory rO rI radd rmul req;
SF_1_neq_0 : ~ 1 == 0;
SFdiv_def : forall p q, p / q == p * / q;
SFinv_l : forall p, ~ p == 0 ->  / p * p == 1
}.
End FieldAndSemiField.
End MakeFieldPol.
Definition SF2AF R (rO rI:R) radd rmul rdiv rinv req Rsth
(sf:semi_field_theory rO rI radd rmul rdiv rinv req)  :=
mk_afield _ _
(SRth_ARth Rsth (SF_SR sf))
(SF_1_neq_0 sf)
(SFdiv_def sf)
(SFinv_l sf).
Section Complete.
Variable R : Type.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R).
Variable (rdiv : R -> R ->  R) (rinv : 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 " := (rdiv x y).  Notation "/ x" := (rinv x).
Notation "x == y" := (req x y) (at level 70, no associativity).
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.
Variable Reqe : ring_eq_ext radd rmul ropp req.
Add Morphism radd with signature (req ==> req ==> req) as radd_ext3.

(* Specifications: *)
Lemma gen_phiPOS_inj x y :
gen_phiPOS rI radd rmul x == gen_phiPOS rI radd rmul y ->
x = y.
Lemma gen_phiN_inj x y :
gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y ->
x = y.
Lemma gen_phiN_complete x y :
gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y ->
N.eqb x y = true.