%   ORIGINAL: h4/llist/LNTH__THM_c2
% 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/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/OPTION__JOIN__DEF_c1: !x. h4/option/OPTION__JOIN (h4/option/SOME x) = x
% Assm: h4/llist/LTL__THM_c1: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% 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: !t n h. h4/llist/LNTH (h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LNTH n t
%   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_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_OPTIONu_u_JOINu_u_DEFu_c1]: !x. h4/option/OPTION__JOIN (h4/option/SOME x) = x
% Assm [h4s_llists_LTLu_u_THMu_c1]: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% 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: !t n h. happ (h4/llist/LNTH (h4/num/SUC n)) (h4/llist/LCONS h t) = happ (h4/llist/LNTH n) t
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q169296,TV_Q169292]: ![V_f, V_g]: (![V_x]: s(TV_Q169292,happ(s(t_fun(TV_Q169296,TV_Q169292),V_f),s(TV_Q169296,V_x))) = s(TV_Q169292,happ(s(t_fun(TV_Q169296,TV_Q169292),V_g),s(TV_Q169296,V_x))) => s(t_fun(TV_Q169296,TV_Q169292),V_f) = s(t_fun(TV_Q169296,TV_Q169292),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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,t0)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, 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_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_OPTIONu_u_JOINu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_x]: 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_some(s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_llists_LTLu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_options_some(s(t_h4s_llists_llist(TV_u_27b),V_t)))).
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_c2, conjecture, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(t_h4s_llists_llist(TV_u_27c),t_h4s_options_option(TV_u_27c)),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_27c),h4s_llists_lcons(s(TV_u_27c,V_h),s(t_h4s_llists_llist(TV_u_27c),V_t))))) = s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(t_h4s_llists_llist(TV_u_27c),t_h4s_options_option(TV_u_27c)),h4s_llists_lnth(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27c),V_t)))).
