%   ORIGINAL: 'h4/thm/list/FLAT_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: !ls. 'h4/const/list/FLAT' ls = 'h4/const/list/NIL' <=> 'h4/const/list/EVERY' ($equals 'h4/const/list/NIL') ls
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !ls. 'h4/const/list/FLAT' ls = 'h4/const/list/NIL' <=> 'h4/const/list/EVERY' ($equals 'h4/const/list/NIL') ls
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_3f12144,V_3f12140]: ![F, G]: (![X]: s(V_3f12140,happ(s(fun(V_3f12144,V_3f12140),F),s(V_3f12144,X))) = s(V_3f12140,happ(s(fun(V_3f12144,V_3f12140),G),s(V_3f12144,X))) => s(fun(V_3f12144,V_3f12140),F) = s(fun(V_3f12144,V_3f12140),G))).
fof('h4/thm/list/FLAT_EQ_NIL_', conjecture, ![A]: ![Ls]: (s('h4/type/list/list'(A),'h4/const/list/FLAT'(s('h4/type/list/list'('h4/type/list/list'(A)),Ls))) = s('h4/type/list/list'(A),'h4/const/list/NIL') <=> p(s(bool,'h4/const/list/EVERY'(s(fun('h4/type/list/list'(A),bool),'$equals'(s('h4/type/list/list'(A),'h4/const/list/NIL'))),s('h4/type/list/list'('h4/type/list/list'(A)),Ls)))))).
