proposition: Lemma carryandzero : carry p ( isaprimetoneq0 is ) 0 = 0.
proof: Proof. intros. unfold carry. rewrite precarryandzero. assert ( forall n : nat, (fun n : nat => hzremaindermod p (isaprimetoneq0 is) ( @ringunel1 ( fpscommring hz ) n)) n = @ringunel1 ( fpscommring hz ) n ) as f. { intros n. rewrite hzqrand0r. unfold carry. change ( @ringunel1 ( fpscommring hz) n ) with 0%hz. apply idpath. } apply ( funextfun _ _ f ). Defined.
imports: UniMath.PAdics.lemmas UniMath.PAdics.fps UniMath.PAdics.frac UniMath.PAdics.z_mod_p UniMath.NumberSystems.Integers