%   ORIGINAL: h4/rich__list/MEM__FOLDR
% 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/MEM__EXISTS: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> h4/list/EXISTS ($equals x) l
% Assm: h4/rich__list/EXISTS__FOLDR: !l P. h4/list/EXISTS P l <=> h4/list/FOLDR (\x l_27. P x \/ l_27) F l
% Goal: !y l. h4/bool/IN y (h4/list/LIST__TO__SET l) <=> h4/list/FOLDR (\x l_27. y = x \/ l_27) F l
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_richu_u_lists_MEMu_u_EXISTS]: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> h4/list/EXISTS ($equals x) l
% Assm [h4s_richu_u_lists_EXISTSu_u_FOLDR]: !_1. (!P x l_27. happ (happ (happ _1 P) x) l_27 <=> happ P x \/ l_27) ==> (!_0. (!P x. happ (happ _0 P) x = happ (happ _1 P) x) ==> (!l P. h4/list/EXISTS P l <=> h4/list/FOLDR (happ _0 P) F l))
% Goal: !_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))
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_Q160108,TV_Q160104]: ![V_f, V_g]: (![V_x]: s(TV_Q160104,happ(s(t_fun(TV_Q160108,TV_Q160104),V_f),s(TV_Q160108,V_x))) = s(TV_Q160104,happ(s(t_fun(TV_Q160108,TV_Q160104),V_g),s(TV_Q160108,V_x))) => s(t_fun(TV_Q160108,TV_Q160104),V_f) = s(t_fun(TV_Q160108,TV_Q160104),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_MEMu_u_EXISTS, axiom, ![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_exists(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)))).
fof(ah4s_richu_u_lists_EXISTSu_u_FOLDR, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_P, 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(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))),s(t_bool,V_lu_27)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_lu_27)))) => ![V_uu_0]: (![V_P, 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(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),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(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) => ![V_l, V_P]: s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),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(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_bool,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_bool,f),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ch4s_richu_u_lists_MEMu_u_FOLDR, conjecture, ![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)))))).
