%   ORIGINAL: h4/extreal/extreal__induction
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/extreal/extreal__repfns_c0: !a. h4/extreal/_20_40ind__typeextreal3 (h4/extreal/_20_40ind__typeextreal4 a) = a
% Assm: h4/extreal/extreal__repfns_c1: !r. (\a0. !_27extreal_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) a1 (\n. h4/ind__type/BOTTOM)) a) ==> _27extreal_27 a00) ==> _27extreal_27 a0) r <=> h4/extreal/_20_40ind__typeextreal4 (h4/extreal/_20_40ind__typeextreal3 r) = r
% Assm: h4/extreal/hidden____20__40ind____typeextreal0____def: h4/extreal/_20_40ind__typeextreal0 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/extreal/hidden____20__40ind____typeextreal1____def: h4/extreal/_20_40ind__typeextreal1 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/extreal/hidden____20__40ind____typeextreal2____def: h4/extreal/_20_40ind__typeextreal2 = (\a. h4/extreal/_20_40ind__typeextreal3 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) a0 (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/extreal/NegInf0: h4/extreal/NegInf = h4/extreal/_20_40ind__typeextreal0
% Assm: h4/extreal/PosInf0: h4/extreal/PosInf = h4/extreal/_20_40ind__typeextreal1
% Assm: h4/extreal/Normal0: h4/extreal/Normal = h4/extreal/_20_40ind__typeextreal2
% Goal: !P. P h4/extreal/NegInf /\ P h4/extreal/PosInf /\ (!r. P (h4/extreal/Normal r)) ==> (!e. P e)
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_extreals_extrealu_u_repfnsu_c0]: !a. h4/extreal/_20_40ind__typeextreal3 (h4/extreal/_20_40ind__typeextreal4 a) = a
% Assm [h4s_extreals_extrealu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27extreal_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0 \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) a _0) ==> happ _27extreal_27 a00) ==> happ _27extreal_27 r) <=> h4/extreal/_20_40ind__typeextreal4 (h4/extreal/_20_40ind__typeextreal3 r) = r)
% Assm [h4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/extreal/_20_40ind__typeextreal0 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/extreal/_20_40ind__typeextreal1 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0)
% Assm [h4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/extreal/_20_40ind__typeextreal2 x = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) x _0))
% Assm [h4s_extreals_NegInf0]: h4/extreal/NegInf = h4/extreal/_20_40ind__typeextreal0
% Assm [h4s_extreals_PosInf0]: h4/extreal/PosInf = h4/extreal/_20_40ind__typeextreal1
% Assm [h4s_extreals_Normal0]: h4/extreal/Normal = h4/extreal/_20_40ind__typeextreal2
% Goal: !P. happ P h4/extreal/NegInf /\ happ P h4/extreal/PosInf /\ (!r. happ P (happ h4/extreal/Normal r)) ==> (!e. happ P e)
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_Q172563,TV_Q172559]: ![V_f, V_g]: (![V_x]: s(TV_Q172559,happ(s(t_fun(TV_Q172563,TV_Q172559),V_f),s(TV_Q172563,V_x))) = s(TV_Q172559,happ(s(t_fun(TV_Q172563,TV_Q172559),V_g),s(TV_Q172563,V_x))) => s(t_fun(TV_Q172563,TV_Q172559),V_f) = s(t_fun(TV_Q172563,TV_Q172559),V_g))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_extreals_extrealu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_extreals_u_20u_40indu_u_typeextreal4(s(t_h4s_extreals_extreal,V_a))))) = s(t_h4s_extreals_extreal,V_a)).
fof(ah4s_extreals_extrealu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27extrealu_27]: (![V_a00]: ((s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | (s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | ?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_realaxs_real,V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27extrealu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27extrealu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_extreals_u_20u_40indu_u_typeextreal4(s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r)))).
fof(ah4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal0u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal0) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal1u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal1) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal2u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_u_20u_40indu_u_typeextreal2),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_realaxs_real,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_extreals_NegInf0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal0)).
fof(ah4s_extreals_PosInf0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_posinf) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal1)).
fof(ah4s_extreals_Normal0, axiom, s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal) = s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_u_20u_40indu_u_typeextreal2)).
fof(ch4s_extreals_extrealu_u_induction, conjecture, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_P),s(t_h4s_extreals_extreal,h4s_extreals_neginf)))) & (p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_P),s(t_h4s_extreals_extreal,h4s_extreals_posinf)))) & ![V_r]: p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_P),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_r)))))))) => ![V_e]: p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_P),s(t_h4s_extreals_extreal,V_e)))))).
