Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
Add Morphism rtimes with signature req ==> req ==> req as rtimes_morph.
Add Morphism ropp with signature req ==> req as ropp_morph.
Add Morphism rle with signature req ==> req ==> iff as rle_morph.
Add Morphism rlt with signature req ==> req ==> iff as rlt_morph.
Add Morphism rminus with signature req ==> req ==> req as rminus_morph.
Ltac le_less := rewrite (Rle_lt_eq sor); left; try assumption.
Ltac le_equal := rewrite (Rle_lt_eq sor); right; try reflexivity; try assumption.
Definition gen_order_phi_Z : Z -> R := gen_phiZ 0 1 rplus rtimes ropp.
Declare Equivalent Keys gen_order_phi_Z gen_phiZ.
Notation phi_pos := (gen_phiPOS 1 rplus rtimes).
Notation phi_pos1 := (gen_phiPOS1 1 rplus rtimes).
Notation "[ x ]" := (gen_order_phi_Z x).

Lemma phi_pos1_pos : forall x : positive, 0 < phi_pos1 x.