%   ORIGINAL: h4/llist/LNTH__THM_c0
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__JOIN__DEF_c0: h4/option/OPTION__JOIN h4/option/NONE = h4/option/NONE
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/llist/LHD__THM_c0: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LTL__THM_c0: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LNTH0_c0: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Assm: h4/llist/LNTH0_c1: !n ll. h4/llist/LNTH (h4/num/SUC n) ll = h4/option/OPTION__JOIN (h4/option/OPTION__MAP (h4/llist/LNTH n) (h4/llist/LTL ll))
% Goal: !n. h4/llist/LNTH n h4/llist/LNIL = h4/option/NONE
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_JOINu_u_DEFu_c0]: h4/option/OPTION__JOIN h4/option/NONE = h4/option/NONE
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_llists_LHDu_u_THMu_c0]: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LTLu_u_THMu_c0]: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LNTH0u_c0]: !ll. happ (h4/llist/LNTH h4/num/0) ll = h4/llist/LHD ll
% Assm [h4s_llists_LNTH0u_c1]: !n ll. happ (h4/llist/LNTH (h4/num/SUC n)) ll = h4/option/OPTION__JOIN (h4/option/OPTION__MAP (h4/llist/LNTH n) (h4/llist/LTL ll))
% Goal: !n. happ (h4/llist/LNTH n) h4/llist/LNIL = h4/option/NONE
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_Q169246,TV_Q169242]: ![V_f, V_g]: (![V_x]: s(TV_Q169242,happ(s(t_fun(TV_Q169246,TV_Q169242),V_f),s(TV_Q169246,V_x))) = s(TV_Q169242,happ(s(t_fun(TV_Q169246,TV_Q169242),V_g),s(TV_Q169246,V_x))) => s(t_fun(TV_Q169246,TV_Q169242),V_f) = s(t_fun(TV_Q169246,TV_Q169242),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_JOINu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_join(s(t_h4s_options_option(t_h4s_options_option(TV_u_27a)),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_llists_LHDu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_llists_LTLu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_options_none)).
fof(ah4s_llists_LNTH0u_c0, axiom, ![TV_u_27a]: ![V_ll]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_llists_lnth(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll)))).
fof(ah4s_llists_LNTH0u_c1, axiom, ![TV_u_27a]: ![V_n, V_ll]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_llists_lnth(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_join(s(t_h4s_options_option(t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_llists_lnth(s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),V_ll)))))))).
fof(ch4s_llists_LNTHu_u_THMu_c0, conjecture, ![TV_u_27a]: ![V_n]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_llists_lnth(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
