Let gen_phiPOS_inject :=
gen_phiPOS_inj AFth ring_S_inj gen_phiPOS_not_0.

(* Specifications: *)
Lemma gen_phiPOS_discr_sgn x y :
~ gen_phiPOS rI radd rmul x == - gen_phiPOS rI radd rmul y.
Lemma gen_phiZ_inj x y :
gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y ->
x = y.
Lemma gen_phiZ_complete x y :
gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y ->
Zeq_bool x y = true.