%   ORIGINAL: h4/patricia/SIZE__PTREE__OF__NUMSET__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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> 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/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/ALL__DISTINCT__SET__TO__LIST: !s. h4/pred__set/FINITE s ==> h4/list/ALL__DISTINCT (h4/list/SET__TO__LIST s)
% Assm: h4/patricia/TRAVERSE__def_c0: h4/patricia/TRAVERSE h4/patricia/Empty = h4/list/NIL
% Assm: h4/patricia/EMPTY__IS__PTREE: h4/patricia/IS__PTREE h4/patricia/Empty
% Assm: h4/patricia/SIZE0_c0: h4/patricia/SIZE h4/patricia/Empty = h4/num/0
% Assm: h4/patricia/SIZE__PTREE__OF__NUMSET: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t /\ h4/list/ALL__DISTINCT (h4/list/APPEND (h4/patricia/TRAVERSE t) (h4/list/SET__TO__LIST s)) ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET t s) = h4/arithmetic/_2B (h4/patricia/SIZE t) (h4/pred__set/CARD s)
% Goal: !s. h4/pred__set/FINITE s ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s) = h4/pred__set/CARD s
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> 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_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_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_SETu_u_TOu_u_LIST]: !s. h4/pred__set/FINITE s ==> h4/list/ALL__DISTINCT (h4/list/SET__TO__LIST s)
% Assm [h4s_patricias_TRAVERSEu_u_defu_c0]: h4/patricia/TRAVERSE h4/patricia/Empty = h4/list/NIL
% Assm [h4s_patricias_EMPTYu_u_ISu_u_PTREE]: h4/patricia/IS__PTREE h4/patricia/Empty
% Assm [h4s_patricias_SIZE0u_c0]: h4/patricia/SIZE h4/patricia/Empty = h4/num/0
% Assm [h4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSET]: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t /\ h4/list/ALL__DISTINCT (h4/list/APPEND (h4/patricia/TRAVERSE t) (h4/list/SET__TO__LIST s)) ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET t s) = h4/arithmetic/_2B (h4/patricia/SIZE t) (h4/pred__set/CARD s)
% Goal: !s. h4/pred__set/FINITE s ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s) = h4/pred__set/CARD s
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_Q288106,TV_Q288102]: ![V_f, V_g]: (![V_x]: s(TV_Q288102,happ(s(t_fun(TV_Q288106,TV_Q288102),V_f),s(TV_Q288106,V_x))) = s(TV_Q288102,happ(s(t_fun(TV_Q288106,TV_Q288102),V_g),s(TV_Q288106,V_x))) => s(t_fun(TV_Q288106,TV_Q288102),V_f) = s(t_fun(TV_Q288106,TV_Q288102),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_ANDu_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_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_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_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_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, 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_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_SETu_u_TOu_u_LIST, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_lists_allu_u_distinct(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_patricias_TRAVERSEu_u_defu_c0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_nil)).
fof(ah4s_patricias_EMPTYu_u_ISu_u_PTREE, axiom, ![TV_u_27a]: p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))))).
fof(ah4s_patricias_SIZE0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSET, axiom, ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_append(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),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),V_s))))))))) => s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
fof(ch4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSETu_u_EMPTY, conjecture, ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(t_h4s_nums_num,t_bool),V_s))))).
