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).

Lemma all_supp_insP k v : p k -> all p (supp f) -> all p (supp (ins k v f)).