Definition psub  := psub Z0  Z.add Z.sub Z.opp Zeq_bool.
Declare Equivalent Keys psub RingMicromega.psub.
Definition popp  := popp Z.opp.
Declare Equivalent Keys popp RingMicromega.popp.
Definition padd  := padd Z0  Z.add Zeq_bool.
Declare Equivalent Keys padd RingMicromega.padd.
Definition pmul := pmul 0 1 Z.add Z.mul Zeq_bool.
Definition normZ  := norm 0 1 Z.add Z.mul Z.sub Z.opp Zeq_bool.
Declare Equivalent Keys normZ RingMicromega.norm.
Definition eval_pol := eval_pol Z.add Z.mul (fun x => x).
Declare Equivalent Keys eval_pol RingMicromega.eval_pol.

Lemma eval_pol_add : forall env lhs rhs, eval_pol env (padd  lhs rhs) = eval_pol env lhs + eval_pol env rhs.