%   ORIGINAL: h4/words/bit__count__upto__SUC
% 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/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/sum__num/SUM__def_c1: !m f. h4/sum__num/SUM (h4/num/SUC m) f = h4/arithmetic/_2B (h4/sum__num/SUM m f) (f m)
% Assm: h4/words/bit__count__upto__def: !w n. h4/words/bit__count__upto n w = h4/sum__num/SUM n (\i. h4/bool/COND (h4/fcp/fcp__index w i) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0)
% Goal: !w n. h4/words/bit__count__upto (h4/num/SUC n) w = h4/arithmetic/_2B (h4/bool/COND (h4/fcp/fcp__index w n) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0) (h4/words/bit__count__upto n w)
%   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_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_sumu_u_nums_SUMu_u_defu_c1]: !m f. h4/sum__num/SUM (h4/num/SUC m) f = h4/arithmetic/_2B (h4/sum__num/SUM m f) (happ f m)
% Assm [h4s_wordss_bitu_u_countu_u_uptou_u_def]: !_0. (!w i. happ (happ _0 w) i = h4/bool/COND (h4/fcp/fcp__index w i) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0) ==> (!w n. h4/words/bit__count__upto n w = h4/sum__num/SUM n (happ _0 w))
% Goal: !w n. h4/words/bit__count__upto (h4/num/SUC n) w = h4/arithmetic/_2B (h4/bool/COND (h4/fcp/fcp__index w n) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0) (h4/words/bit__count__upto n w)
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_Q130287,TV_Q130283]: ![V_f, V_g]: (![V_x]: s(TV_Q130283,happ(s(t_fun(TV_Q130287,TV_Q130283),V_f),s(TV_Q130287,V_x))) = s(TV_Q130283,happ(s(t_fun(TV_Q130287,TV_Q130283),V_g),s(TV_Q130287,V_x))) => s(t_fun(TV_Q130287,TV_Q130283),V_f) = s(t_fun(TV_Q130287,TV_Q130283),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_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_sumu_u_nums_SUMu_u_defu_c1, axiom, ![V_m, V_f]: s(t_h4s_nums_num,h4s_sumu_u_nums_sum(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_sumu_u_nums_sum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_wordss_bitu_u_countu_u_uptou_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_w, V_i]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_0),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,V_i))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_fcps_fcpu_u_index(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_nums_0))) => ![V_w, V_n]: s(t_h4s_nums_num,h4s_wordss_bitu_u_countu_u_upto(s(t_h4s_nums_num,V_n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_nums_num,h4s_sumu_u_nums_sum(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_0),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))))))).
fof(ch4s_wordss_bitu_u_countu_u_uptou_u_SUC, conjecture, ![TV_u_27a]: ![V_w, V_n]: s(t_h4s_nums_num,h4s_wordss_bitu_u_countu_u_upto(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_fcps_fcpu_u_index(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,h4s_wordss_bitu_u_countu_u_upto(s(t_h4s_nums_num,V_n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)))))).
