%   ORIGINAL: 'h4/thm/list/EVERY_MAP_'
% 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 f l. 'h4/const/list/EVERY' P ('h4/const/list/MAP' f l) <=> 'h4/const/list/EVERY' (\x. P (f 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 f x. happ (happ (happ _0 P) f) x <=> happ P (happ f x)) ==> (!P f l. 'h4/const/list/EVERY' P ('h4/const/list/MAP' f l) <=> 'h4/const/list/EVERY' (happ (happ _0 P) f) 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_3f3538,V_3f3534]: ![F, G]: (![X]: s(V_3f3534,happ(s(fun(V_3f3538,V_3f3534),F),s(V_3f3538,X))) = s(V_3f3534,happ(s(fun(V_3f3538,V_3f3534),G),s(V_3f3538,X))) => s(fun(V_3f3538,V_3f3534),F) = s(fun(V_3f3538,V_3f3534),G))).
fof('h4/thm/list/EVERY_MAP_', conjecture, ![B,A]: ![V__0]: (![P, F, X]: s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,B),fun(A,bool)),happ(s(fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),V__0),s(fun(B,bool),P))),s(fun(A,B),F))),s(A,X))) = s(bool,happ(s(fun(B,bool),P),s(B,happ(s(fun(A,B),F),s(A,X))))) => ![P, F, L]: s(bool,'h4/const/list/EVERY'(s(fun(B,bool),P),s('h4/type/list/list'(B),'h4/const/list/MAP'(s(fun(A,B),F),s('h4/type/list/list'(A),L))))) = s(bool,'h4/const/list/EVERY'(s(fun(A,bool),happ(s(fun(fun(A,B),fun(A,bool)),happ(s(fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),V__0),s(fun(B,bool),P))),s(fun(A,B),F))),s('h4/type/list/list'(A),L))))).
