Inductive base := 
  Undef | Def (f : {finMap K -> V}) of all p (supp f).Inductive base := 
  Undef | Def (f : {finMap K -> V}) of all p (supp f).Inductive base := 
  Undef | Def (f : {finMap K -> V}) of all p (supp f).

(* Specifications: *)
Lemma all_supp_insP k v : p k -> all p (supp f) -> all p (supp (ins k v f)).
Lemma all_supp_remP k : all p (supp f) -> all p (supp (rem k f)).
Lemma all_supp_fcatP : all p (supp f) -> all p (supp g) -> all p (supp (fcat f g)).
Lemma all_supp_kfilterP q : all p (supp f) -> all p (supp (kfilter q f)).