%   ORIGINAL: h4/llist/LNTH__THM_c1
% 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/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/llist/LHD__THM_c1: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm: h4/llist/LNTH0_c0: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Goal: !t h. h4/llist/LNTH h4/num/0 (h4/llist/LCONS h t) = h4/option/SOME h
%   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_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_llists_LHDu_u_THMu_c1]: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm [h4s_llists_LNTH0u_c0]: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Goal: !t h. h4/llist/LNTH h4/num/0 (h4/llist/LCONS h t) = h4/option/SOME h
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_Q169271,TV_Q169267]: ![V_f, V_g]: (![V_x]: s(TV_Q169267,happ(s(t_fun(TV_Q169271,TV_Q169267),V_f),s(TV_Q169271,V_x))) = s(TV_Q169267,happ(s(t_fun(TV_Q169271,TV_Q169267),V_g),s(TV_Q169271,V_x))) => s(t_fun(TV_Q169271,TV_Q169267),V_f) = s(t_fun(TV_Q169271,TV_Q169267),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_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_llists_LHDu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(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(TV_u_27b),h4s_options_some(s(TV_u_27b,V_h)))).
fof(ah4s_llists_LNTH0u_c0, axiom, ![TV_u_27a]: ![V_ll]: s(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(ch4s_llists_LNTHu_u_THMu_c1, conjecture, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lnth(s(t_h4s_nums_num,h4s_nums_0),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(TV_u_27b),h4s_options_some(s(TV_u_27b,V_h)))).
