%   ORIGINAL: h4/bag/BAG__IMAGE__EMPTY
% 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/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bag/EMPTY__BAG__alt: h4/bag/EMPTY__BAG = (\x. h4/num/0)
% Assm: h4/bag/FINITE__BAG__THM_c0: h4/bag/FINITE__BAG h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__CARD__EMPTY: h4/bag/BAG__CARD h4/bag/EMPTY__BAG = h4/num/0
% Assm: h4/bag/BAG__FILTER__EMPTY: !P. h4/bag/BAG__FILTER P h4/bag/EMPTY__BAG = h4/bag/EMPTY__BAG
% Assm: h4/bag/FINITE__BAG__FILTER: !b P. h4/bag/FINITE__BAG b ==> h4/bag/FINITE__BAG (h4/bag/BAG__FILTER P b)
% Assm: h4/bag/BAG__IMAGE__DEF: !f b. h4/bag/BAG__IMAGE f b = (\e. h4/bool/LET (\sb. h4/bool/COND (h4/bag/FINITE__BAG sb) (h4/bag/BAG__CARD sb) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/bag/BAG__FILTER (\e0. f e0 = e) b))
% Goal: !f. h4/bag/BAG__IMAGE f h4/bag/EMPTY__BAG = h4/bag/EMPTY__BAG
%   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_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% 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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bags_EMPTYu_u_BAGu_u_alt]: !x. happ h4/bag/EMPTY__BAG x = h4/num/0
% Assm [h4s_bags_FINITEu_u_BAGu_u_THMu_c0]: h4/bag/FINITE__BAG h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_CARDu_u_EMPTY]: h4/bag/BAG__CARD h4/bag/EMPTY__BAG = h4/num/0
% Assm [h4s_bags_BAGu_u_FILTERu_u_EMPTY]: !P. h4/bag/BAG__FILTER P h4/bag/EMPTY__BAG = h4/bag/EMPTY__BAG
% Assm [h4s_bags_FINITEu_u_BAGu_u_FILTER]: !b P. h4/bag/FINITE__BAG b ==> h4/bag/FINITE__BAG (h4/bag/BAG__FILTER P b)
% Assm [h4s_bags_BAGu_u_IMAGEu_u_DEF]: !_1. (!f x e0. happ (happ (happ _1 f) x) e0 <=> happ f e0 = x) ==> (!_0. (!sb. happ _0 sb = h4/bool/COND (h4/bag/FINITE__BAG sb) (h4/bag/BAG__CARD sb) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> (!f b x. happ (h4/bag/BAG__IMAGE f b) x = h4/bool/LET _0 (h4/bag/BAG__FILTER (happ (happ _1 f) x) b)))
% Goal: !f. h4/bag/BAG__IMAGE f h4/bag/EMPTY__BAG = h4/bag/EMPTY__BAG
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q231186,TV_Q231182]: ![V_f, V_g]: (![V_x]: s(TV_Q231182,happ(s(t_fun(TV_Q231186,TV_Q231182),V_f),s(TV_Q231186,V_x))) = s(TV_Q231182,happ(s(t_fun(TV_Q231186,TV_Q231182),V_g),s(TV_Q231186,V_x))) => s(t_fun(TV_Q231186,TV_Q231182),V_f) = s(t_fun(TV_Q231186,TV_Q231182),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bags_EMPTYu_u_BAGu_u_alt, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bags_FINITEu_u_BAGu_u_THMu_c0, axiom, ![TV_u_27a]: p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))).
fof(ah4s_bags_BAGu_u_CARDu_u_EMPTY, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_bags_bagu_u_card(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bags_BAGu_u_FILTERu_u_EMPTY, axiom, ![TV_u_27a]: ![V_P]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_FINITEu_u_BAGu_u_FILTER, axiom, ![TV_u_27a]: ![V_b, V_P]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))))).
fof(ah4s_bags_BAGu_u_IMAGEu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_f, V_x, V_e0]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(TV_u_27a,V_x))),s(TV_u_27b,V_e0)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_e0))) = s(TV_u_27a,V_x)) => ![V_uu_0]: (![V_sb]: s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27b,t_h4s_nums_num),t_h4s_nums_num),V_uu_0),s(t_fun(TV_u_27b,t_h4s_nums_num),V_sb))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27b,t_h4s_nums_num),V_sb))),s(t_h4s_nums_num,h4s_bags_bagu_u_card(s(t_fun(TV_u_27b,t_h4s_nums_num),V_sb))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) => ![V_f, V_b, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_image(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_fun(TV_u_27b,t_h4s_nums_num),V_b))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_bools_let(s(t_fun(t_fun(TV_u_27b,t_h4s_nums_num),t_h4s_nums_num),V_uu_0),s(t_fun(TV_u_27b,t_h4s_nums_num),h4s_bags_bagu_u_filter(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_h4s_nums_num),V_b)))))))).
fof(ch4s_bags_BAGu_u_IMAGEu_u_EMPTY, conjecture, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27b,t_h4s_nums_num),h4s_bags_bagu_u_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) = s(t_fun(TV_u_27b,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
