Definition separable U W : bool :=
all (separable_element U) (vbasis W).
Definition purely_inseparable U W : bool :=
all (purely_inseparable_element U) (vbasis W).

Lemma adjoin_separable_eq K x : separable_element K x = separable K <<K; x>>%VS.
Definition separable_generator K E : L :=
s2val (locked (separable_inseparable_decomposition E K)).