%   ORIGINAL: h4/sorting/SUM__IMAGE__count__SUM__GENLIST
% 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/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/pred__set/FINITE__COUNT: !n. h4/pred__set/FINITE (h4/pred__set/count n)
% Assm: h4/list/MAP__GENLIST: !n g f. h4/list/MAP f (h4/list/GENLIST g n) = h4/list/GENLIST (h4/combin/o f g) n
% Assm: h4/list/SUM__IMAGE__eq__SUM__MAP__SET__TO__LIST: !s f. h4/pred__set/FINITE s ==> h4/pred__set/SUM__IMAGE f s = h4/list/SUM (h4/list/MAP f (h4/list/SET__TO__LIST s))
% Assm: h4/rich__list/COUNT__LIST__GENLIST: !n. h4/rich__list/COUNT__LIST n = h4/list/GENLIST h4/combin/I n
% Assm: h4/sorting/PERM__INTRO: !y x. x = y ==> h4/sorting/PERM x y
% Assm: h4/sorting/PERM__TRANS: !z y x. h4/sorting/PERM x y /\ h4/sorting/PERM y z ==> h4/sorting/PERM x z
% Assm: h4/sorting/PERM__MAP: !l2 l1 f. h4/sorting/PERM l1 l2 ==> h4/sorting/PERM (h4/list/MAP f l1) (h4/list/MAP f l2)
% Assm: h4/sorting/PERM__SUM: !l2 l1. h4/sorting/PERM l1 l2 ==> h4/list/SUM l1 = h4/list/SUM l2
% Assm: h4/sorting/PERM__SET__TO__LIST__count__COUNT__LIST: !n. h4/sorting/PERM (h4/list/SET__TO__LIST (h4/pred__set/count n)) (h4/rich__list/COUNT__LIST n)
% Goal: !n f. h4/pred__set/SUM__IMAGE f (h4/pred__set/count n) = h4/list/SUM (h4/list/GENLIST f n)
%   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_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_predu_u_sets_FINITEu_u_COUNT]: !n. h4/pred__set/FINITE (h4/pred__set/count n)
% Assm [h4s_lists_MAPu_u_GENLIST]: !n g f. h4/list/MAP f (h4/list/GENLIST g n) = h4/list/GENLIST (h4/combin/o f g) n
% Assm [h4s_lists_SUMu_u_IMAGEu_u_equ_u_SUMu_u_MAPu_u_SETu_u_TOu_u_LIST]: !s f. h4/pred__set/FINITE s ==> h4/pred__set/SUM__IMAGE f s = h4/list/SUM (h4/list/MAP f (h4/list/SET__TO__LIST s))
% Assm [h4s_richu_u_lists_COUNTu_u_LISTu_u_GENLIST]: !n. h4/rich__list/COUNT__LIST n = h4/list/GENLIST h4/combin/I n
% Assm [h4s_sortings_PERMu_u_INTRO]: !y x. x = y ==> h4/sorting/PERM x y
% Assm [h4s_sortings_PERMu_u_TRANS]: !z y x. h4/sorting/PERM x y /\ h4/sorting/PERM y z ==> h4/sorting/PERM x z
% Assm [h4s_sortings_PERMu_u_MAP]: !l2 l1 f. h4/sorting/PERM l1 l2 ==> h4/sorting/PERM (h4/list/MAP f l1) (h4/list/MAP f l2)
% Assm [h4s_sortings_PERMu_u_SUM]: !l2 l1. h4/sorting/PERM l1 l2 ==> h4/list/SUM l1 = h4/list/SUM l2
% Assm [h4s_sortings_PERMu_u_SETu_u_TOu_u_LISTu_u_countu_u_COUNTu_u_LIST]: !n. h4/sorting/PERM (h4/list/SET__TO__LIST (h4/pred__set/count n)) (h4/rich__list/COUNT__LIST n)
% Goal: !n f. h4/pred__set/SUM__IMAGE f (h4/pred__set/count n) = h4/list/SUM (h4/list/GENLIST f n)
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_Q276381,TV_Q276377]: ![V_f, V_g]: (![V_x]: s(TV_Q276377,happ(s(t_fun(TV_Q276381,TV_Q276377),V_f),s(TV_Q276381,V_x))) = s(TV_Q276377,happ(s(t_fun(TV_Q276381,TV_Q276377),V_g),s(TV_Q276381,V_x))) => s(t_fun(TV_Q276381,TV_Q276377),V_f) = s(t_fun(TV_Q276381,TV_Q276377),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_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_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_predu_u_sets_FINITEu_u_COUNT, axiom, ![V_n]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_lists_MAPu_u_GENLIST, axiom, ![TV_u_27b,TV_u_27a]: ![V_n, V_g, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,V_n))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_nums_num,TV_u_27a),V_g))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_lists_SUMu_u_IMAGEu_u_equ_u_SUMu_u_MAPu_u_SETu_u_TOu_u_LIST, axiom, ![TV_u_27a]: ![V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_map(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_richu_u_lists_COUNTu_u_LISTu_u_GENLIST, axiom, ![V_n]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i),s(t_h4s_nums_num,V_n)))).
fof(ah4s_sortings_PERMu_u_INTRO, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_lists_list(TV_u_27a),V_x) = s(t_h4s_lists_list(TV_u_27a),V_y) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(TV_u_27a),V_y)))))).
fof(ah4s_sortings_PERMu_u_TRANS, axiom, ![TV_u_27a]: ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(TV_u_27a),V_y)))) & p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_y),s(t_h4s_lists_list(TV_u_27a),V_z))))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(TV_u_27a),V_z)))))).
fof(ah4s_sortings_PERMu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_l2, V_l1, V_f]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l2)))))))).
fof(ah4s_sortings_PERMu_u_SUM, axiom, ![V_l2, V_l1]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(t_h4s_nums_num),V_l1),s(t_h4s_lists_list(t_h4s_nums_num),V_l2)))) => s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),V_l1))) = s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),V_l2))))).
fof(ah4s_sortings_PERMu_u_SETu_u_TOu_u_LISTu_u_countu_u_COUNTu_u_LIST, axiom, ![V_n]: p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))))).
fof(ch4s_sortings_SUMu_u_IMAGEu_u_countu_u_SUMu_u_GENLIST, conjecture, ![V_n, V_f]: s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_h4s_nums_num,V_n)))))).
