Definition cfun_unit := [pred phi : classfun | [forall x in G, phi x != 0]].
Definition cfun_inv phi :=
if phi \in cfun_unit then cfun_comp (invr0 _) phi else phi.
Definition cfun_scale a := cfun_comp (mulr0 a).
Fact cfun_addA : associative cfun_add.
Fact cfun_mulD : left_distributive cfun_mul cfun_add.
Definition cfun_ringMixin :=
ComRingMixin cfun_mulA cfun_mulC cfun_mul1 cfun_mulD cfun_nz1.
Canonical cfun_ringType := RingType classfun cfun_ringMixin.
Canonical cfun_comRingType := ComRingType classfun cfun_mulC.

Lemma expS_cfunE phi n x : (phi ^+ n.+1) x = phi x ^+ n.+1.
Fact cfun_unitP phi psi : psi * phi = 1 -> phi \in cfun_unit.