%   ORIGINAL: 'h4/thm/bool/BOOL_FUN_INDUCT_'
% 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. P (\b. T) /\ P (\b. F) /\ P (\b. b) /\ P (\b. ~b) ==> (!f. P f)
%   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: !_3. (!b. happ _3 b <=> ~b) ==> (!_2. (!b. happ _2 b <=> b) ==> (!_1. (!b. happ _1 b <=> F) ==> (!_0. (!b. happ _0 b <=> T) ==> (!P. happ P _0 /\ happ P _1 /\ happ P _2 /\ happ P _3 ==> (!f. happ P f)))))
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_3f35353,V_3f35349]: ![F, G]: (![X]: s(V_3f35349,happ(s(fun(V_3f35353,V_3f35349),F),s(V_3f35353,X))) = s(V_3f35349,happ(s(fun(V_3f35353,V_3f35349),G),s(V_3f35353,X))) => s(fun(V_3f35353,V_3f35349),F) = s(fun(V_3f35353,V_3f35349),G))).
fof('h4/thm/bool/BOOL_FUN_INDUCT_', conjecture, ![V__3]: (![B]: (p(s(bool,happ(s(fun(bool,bool),V__3),s(bool,B)))) <=> ~ (p(s(bool,B)))) => ![V__2]: (![B]: s(bool,happ(s(fun(bool,bool),V__2),s(bool,B))) = s(bool,B) => ![V__1]: (![B]: s(bool,happ(s(fun(bool,bool),V__1),s(bool,B))) = s(bool,'F') => ![V__0]: (![B]: s(bool,happ(s(fun(bool,bool),V__0),s(bool,B))) = s(bool,'T') => ![P]: ((p(s(bool,happ(s(fun(fun(bool,bool),bool),P),s(fun(bool,bool),V__0)))) & (p(s(bool,happ(s(fun(fun(bool,bool),bool),P),s(fun(bool,bool),V__1)))) & (p(s(bool,happ(s(fun(fun(bool,bool),bool),P),s(fun(bool,bool),V__2)))) & p(s(bool,happ(s(fun(fun(bool,bool),bool),P),s(fun(bool,bool),V__3))))))) => ![F]: p(s(bool,happ(s(fun(fun(bool,bool),bool),P),s(fun(bool,bool),F)))))))))).
