%   ORIGINAL: h4/bag/BAG__INSERT__UNION
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bag/BAG__UNION__INSERT_c0: !e b2 b1. h4/bag/BAG__UNION (h4/bag/BAG__INSERT e b1) b2 = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm: h4/bag/C__BAG__INSERT__ONE__ONE: !y x b. h4/bag/BAG__INSERT x b = h4/bag/BAG__INSERT y b <=> x = y
% Assm: h4/bag/BAG__UNION__EMPTY_c1: !b. h4/bag/BAG__UNION h4/bag/EMPTY__BAG b = b
% Assm: h4/bag/EL__BAG0: !e. h4/bag/EL__BAG e = h4/bag/BAG__INSERT e h4/bag/EMPTY__BAG
% Goal: !e b. h4/bag/BAG__INSERT e b = h4/bag/BAG__UNION (h4/bag/EL__BAG e) b
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bags_BAGu_u_UNIONu_u_INSERTu_c0]: !e b2 b1. h4/bag/BAG__UNION (h4/bag/BAG__INSERT e b1) b2 = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm [h4s_bags_Cu_u_BAGu_u_INSERTu_u_ONEu_u_ONE]: !y x b. h4/bag/BAG__INSERT x b = h4/bag/BAG__INSERT y b <=> x = y
% Assm [h4s_bags_BAGu_u_UNIONu_u_EMPTYu_c1]: !b. h4/bag/BAG__UNION h4/bag/EMPTY__BAG b = b
% Assm [h4s_bags_ELu_u_BAG0]: !e. h4/bag/EL__BAG e = h4/bag/BAG__INSERT e h4/bag/EMPTY__BAG
% Goal: !e b. h4/bag/BAG__INSERT e b = h4/bag/BAG__UNION (h4/bag/EL__BAG e) b
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_Q227923,TV_Q227919]: ![V_f, V_g]: (![V_x]: s(TV_Q227919,happ(s(t_fun(TV_Q227923,TV_Q227919),V_f),s(TV_Q227923,V_x))) = s(TV_Q227919,happ(s(t_fun(TV_Q227923,TV_Q227919),V_g),s(TV_Q227923,V_x))) => s(t_fun(TV_Q227923,TV_Q227919),V_f) = s(t_fun(TV_Q227923,TV_Q227919),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_bags_BAGu_u_UNIONu_u_INSERTu_c0, axiom, ![TV_u_27a]: ![V_e, V_b2, V_b1]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))).
fof(ah4s_bags_Cu_u_BAGu_u_INSERTu_u_ONEu_u_ONE, axiom, ![TV_u_27a]: ![V_y, V_x, V_b]: (s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bags_BAGu_u_UNIONu_u_EMPTYu_c1, axiom, ![TV_u_27b]: ![V_b]: s(t_fun(TV_u_27b,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27b,t_h4s_nums_num),h4s_bags_emptyu_u_bag),s(t_fun(TV_u_27b,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27b,t_h4s_nums_num),V_b)).
fof(ah4s_bags_ELu_u_BAG0, axiom, ![TV_u_27a]: ![V_e]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_elu_u_bag(s(TV_u_27a,V_e))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))).
fof(ch4s_bags_BAGu_u_INSERTu_u_UNION, conjecture, ![TV_u_27a]: ![V_e, V_b]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_elu_u_bag(s(TV_u_27a,V_e))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))).
