#[global]
Instance Zops:@Ring_ops Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp (@eq Z).#[global]
Instance Zr: (@Ring _ _ _ _ _ _ _ _ Zops).Ltac rrefl := reflexivity.Fixpoint gen_phiPOS1 (p:positive) : R :=
  match p with
  | xH => 1
  | xO p => (1 + 1) * (gen_phiPOS1 p)
  | xI p => 1 + ((1 + 1) * (gen_phiPOS1 p))
  end.Fixpoint gen_phiPOS (p:positive) : R :=
  match p with
  | xH => 1
  | xO xH => (1 + 1)
  | xO p => (1 + 1) * (gen_phiPOS p)
  | xI xH => 1 + (1 +1)
  | xI p => 1 + ((1 + 1) * (gen_phiPOS p))
  end.Definition gen_phiZ1 z :=
  match z with
  | Zpos p => gen_phiPOS1 p
  | Z0 => 0
  | Zneg p => -(gen_phiPOS1 p)
  end.Definition gen_phiZ z :=
  match z with
  | Zpos p => gen_phiPOS p
  | Z0 => 0
  | Zneg p => -(gen_phiPOS p)
  end.Notation "[ x ]" := (gen_phiZ x) : ZMORPHISM.Definition get_signZ z :=
  match z with
  | Zneg p => Some (Zpos p)
  | _ => None
  end.Ltac norm := gen_rewrite.Ltac add_push :=  Ncring.gen_add_push.Ltac rsimpl := simpl.

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