%   ORIGINAL: h4/rich__list/MEM__FOLDR__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
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/rich__list/FOLDR__MAP: !l g f e. h4/list/FOLDR f e (h4/list/MAP g l) = h4/list/FOLDR (\x y. f (g x) y) e l
% Assm: h4/rich__list/MEM__FOLDR: !y l. h4/bool/IN y (h4/list/LIST__TO__SET l) <=> h4/list/FOLDR (\x l_27. y = x \/ l_27) F l
% Goal: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> h4/list/FOLDR $or F (h4/list/MAP ($equals x) l)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_richu_u_lists_FOLDRu_u_MAP]: !_1. (!f g x y. happ (happ (happ (happ _1 f) g) x) y = happ (happ f (happ g x)) y) ==> (!_0. (!f g x. happ (happ (happ _0 f) g) x = happ (happ (happ _1 f) g) x) ==> (!l g f e. h4/list/FOLDR f e (h4/list/MAP g l) = h4/list/FOLDR (happ (happ _0 f) g) e l))
% Assm [h4s_richu_u_lists_MEMu_u_FOLDR]: !_1. (!y x l_27. happ (happ (happ _1 y) x) l_27 <=> y = x \/ l_27) ==> (!_0. (!y x. happ (happ _0 y) x = happ (happ _1 y) x) ==> (!y l. h4/bool/IN y (h4/list/LIST__TO__SET l) <=> h4/list/FOLDR (happ _0 y) F l))
% Goal: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> h4/list/FOLDR $or F (h4/list/MAP ($equals x) l)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q161583,TV_Q161579]: ![V_f, V_g]: (![V_x]: s(TV_Q161579,happ(s(t_fun(TV_Q161583,TV_Q161579),V_f),s(TV_Q161583,V_x))) = s(TV_Q161579,happ(s(t_fun(TV_Q161583,TV_Q161579),V_g),s(TV_Q161583,V_x))) => s(t_fun(TV_Q161583,TV_Q161579),V_f) = s(t_fun(TV_Q161583,TV_Q161579),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_richu_u_lists_FOLDRu_u_MAP, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_1]: (![V_f, V_g, V_x, V_y]: s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x))))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_f, V_g, V_x]: s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) => ![V_l, V_g, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(t_h4s_lists_list(TV_u_27c),V_l))))) = s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27c),V_l)))))).
fof(ah4s_richu_u_lists_MEMu_u_FOLDR, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_y, V_x, V_lu_27]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_1),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))),s(t_bool,V_lu_27)))) <=> (s(TV_u_27a,V_y) = s(TV_u_27a,V_x) | p(s(t_bool,V_lu_27)))) => ![V_uu_0]: (![V_y, V_x]: s(t_fun(t_bool,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_0),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))) = s(t_fun(t_bool,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_1),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))) => ![V_y, V_l]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_bool,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_0),s(TV_u_27a,V_y))),s(t_bool,f),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ch4s_richu_u_lists_MEMu_u_FOLDRu_u_MAP, conjecture, ![TV_u_27a]: ![V_x, V_l]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_bool,h4s_lists_foldr(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_or),s(t_bool,f),s(t_h4s_lists_list(t_bool),h4s_lists_map(s(t_fun(TV_u_27a,t_bool),d_equals(s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
