Definition p_compatible_QI qic qif:= forall f lp t, forall s:variable -> value,
let ru := rule_intro f lp t in
(In ru prog) ->
p_term_assignment qic qif (subst s t) ≤p
p_term_assignment qic qif (subst s (lhs_of_rule ru)).

Lemma p_term_assignment_ext qic f f':
(forall x, f x = f' x) ->
forall t, p_term_assignment qic f t= p_term_assignment qic f' t.