%   ORIGINAL: 'h4/thm/list/FILTER_EQ_NIL_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !P l. 'h4/const/list/FILTER' P l = 'h4/const/list/NIL' <=> 'h4/const/list/EVERY' (\x. ~P x) l
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!P l. 'h4/const/list/FILTER' P l = 'h4/const/list/NIL' <=> 'h4/const/list/EVERY' (happ _0 P) l)
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f4863,V_3f4859]: ![F, G]: (![X]: s(V_3f4859,happ(s(fun(V_3f4863,V_3f4859),F),s(V_3f4863,X))) = s(V_3f4859,happ(s(fun(V_3f4863,V_3f4859),G),s(V_3f4863,X))) => s(fun(V_3f4863,V_3f4859),F) = s(fun(V_3f4863,V_3f4859),G))).
fof('h4/thm/list/FILTER_EQ_NIL_', conjecture, ![A]: ![V__0]: (![P, X]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__0),s(fun(A,bool),P))),s(A,X)))) <=> ~ (p(s(bool,happ(s(fun(A,bool),P),s(A,X)))))) => ![P, L]: (s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P),s('h4/type/list/list'(A),L))) = s('h4/type/list/list'(A),'h4/const/list/NIL') <=> p(s(bool,'h4/const/list/EVERY'(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__0),s(fun(A,bool),P))),s('h4/type/list/list'(A),L))))))).
