%   ORIGINAL: h4/bag/BAG__CHOICE__SING
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bag/BAG__IN__BAG__INSERT: !e2 e1 b. h4/bag/BAG__IN e1 (h4/bag/BAG__INSERT e2 b) <=> e1 = e2 \/ h4/bag/BAG__IN e1 b
% Assm: h4/bag/BAG__INSERT__NOT__EMPTY: !x b. ~(h4/bag/BAG__INSERT x b = h4/bag/EMPTY__BAG)
% Assm: h4/bag/NOT__IN__EMPTY__BAG: !x. ~h4/bag/BAG__IN x h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__CHOICE__DEF: !b. ~(b = h4/bag/EMPTY__BAG) ==> h4/bag/BAG__IN (h4/bag/BAG__CHOICE b) b
% Goal: !x. h4/bag/BAG__CHOICE (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) = x
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bags_BAGu_u_INu_u_BAGu_u_INSERT]: !e2 e1 b. h4/bag/BAG__IN e1 (h4/bag/BAG__INSERT e2 b) <=> e1 = e2 \/ h4/bag/BAG__IN e1 b
% Assm [h4s_bags_BAGu_u_INSERTu_u_NOTu_u_EMPTY]: !x b. ~(h4/bag/BAG__INSERT x b = h4/bag/EMPTY__BAG)
% Assm [h4s_bags_NOTu_u_INu_u_EMPTYu_u_BAG]: !x. ~h4/bag/BAG__IN x h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_CHOICEu_u_DEF]: !b. ~(b = h4/bag/EMPTY__BAG) ==> h4/bag/BAG__IN (h4/bag/BAG__CHOICE b) b
% Goal: !x. h4/bag/BAG__CHOICE (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) = x
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_Q231413,TV_Q231409]: ![V_f, V_g]: (![V_x]: s(TV_Q231409,happ(s(t_fun(TV_Q231413,TV_Q231409),V_f),s(TV_Q231413,V_x))) = s(TV_Q231409,happ(s(t_fun(TV_Q231413,TV_Q231409),V_g),s(TV_Q231413,V_x))) => s(t_fun(TV_Q231413,TV_Q231409),V_f) = s(t_fun(TV_Q231413,TV_Q231409),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bags_BAGu_u_INu_u_BAGu_u_INSERT, axiom, ![TV_u_27a]: ![V_e2, V_e1, V_b]: (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_e1),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e2),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))) <=> (s(TV_u_27a,V_e1) = s(TV_u_27a,V_e2) | p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_e1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))))).
fof(ah4s_bags_BAGu_u_INSERTu_u_NOTu_u_EMPTY, axiom, ![TV_u_27a]: ![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_emptyu_u_bag))).
fof(ah4s_bags_NOTu_u_INu_u_EMPTYu_u_BAG, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))))).
fof(ah4s_bags_BAGu_u_CHOICEu_u_DEF, axiom, ![TV_u_27a]: ![V_b]: (~ (s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)) => p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))).
fof(ch4s_bags_BAGu_u_CHOICEu_u_SING, conjecture, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_bags_bagu_u_choice(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),h4s_bags_emptyu_u_bag))))) = s(TV_u_27a,V_x)).
