Arguments cfun_onP {A phi}.

(* Specifications: *)
Lemma cfun_on0 A phi x : phi \in 'CF(G, A) -> x \notin A -> phi x = 0.
Lemma cfun_base_free A : free (cfun_base G A).
Lemma dim_cfun : \dim 'CF(G) = #|classes G|.
Lemma cfuni_on A : '1_A \in 'CF(G, A).
Lemma mul_cfuni_on A phi : phi * '1_A \in 'CF(G, A).
Lemma cfun_onE phi A : (phi \in 'CF(G, A)) = (support phi \subset A).
Lemma cfun_onG phi : phi \in 'CF(G, G).
Lemma cfConjCE phi x : (phi^*)%CF x = (phi x)^*.
Canonical cfker_group phi := Group (cfker_is_group phi).
Lemma cfker_sub phi : cfker phi \subset G.
Lemma cfker_normal phi : cfker phi <| G.
Lemma cfker1 phi x : x \in cfker phi -> phi x = phi 1%g.
Lemma cfker_add phi psi : cfker phi :&: cfker psi \subset cfker (phi + psi).
Lemma cfker_sum I r (P : pred I) (Phi : I -> 'CF(G)) :
G :&: \bigcap_(i <- r | P i) cfker (Phi i)
\subset cfker (\sum_(i <- r | P i) Phi i).
Lemma cfker_scale a phi : cfker phi \subset cfker (a *: phi).
Lemma cfker_scale_nz a phi : a != 0 -> cfker (a *: phi) = cfker phi.
Lemma cfker_opp phi : cfker (- phi) = cfker phi.
Lemma cfker_mul phi psi : cfker phi :&: cfker psi \subset cfker (phi * psi).
Lemma cfker_prod I r (P : pred I) (Phi : I -> 'CF(G)) :
G :&: \bigcap_(i <- r | P i) cfker (Phi i)
\subset cfker (\prod_(i <- r | P i) Phi i).
Lemma cfaithfulE phi : cfaithful phi = (cfker phi \subset [1]).
Lemma cfdotEl A phi psi :
phi \in 'CF(G, A) ->
'[phi, psi] = #|G|%:R^-1 * \sum_(x in A) phi x * (psi x)^*.
Lemma cfnormE A phi :
phi \in 'CF(G, A) -> '[phi] = #|G|%:R^-1 * (\sum_(x in A) `|phi x| ^+ 2).
Lemma cfdot_cfuni A B :
A <| G -> B <| G -> '['1_A, '1_B]_G = #|A :&: B|%:R / #|G|%:R.
Lemma cfnorm1 : '[1]_G = 1.
Canonical cfdotr_additive xi := Additive (cfdotr_is_linear xi).