%   ORIGINAL: h4/extreal/extreal__abs__def_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/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/extreal/extreal__case__def_c0: !v1 v f. h4/extreal/extreal__CASE h4/extreal/NegInf v v1 f = v
% Assm: h4/extreal/extreal__case__def_c1: !v1 v f. h4/extreal/extreal__CASE h4/extreal/PosInf v v1 f = v1
% Assm: h4/extreal/extreal__case__def_c2: !v1 v f a. h4/extreal/extreal__CASE (h4/extreal/Normal a) v v1 f = f a
% Assm: h4/extreal/extreal__abs__primitive__def: h4/extreal/extreal__abs = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\extreal__abs a. h4/extreal/extreal__CASE a (h4/combin/I h4/extreal/PosInf) (h4/combin/I h4/extreal/PosInf) (\x. h4/combin/I (h4/extreal/Normal (h4/real/abs x))))
% Goal: h4/extreal/extreal__abs h4/extreal/PosInf = h4/extreal/PosInf
%   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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c0]: !v1 v f. h4/extreal/extreal__CASE h4/extreal/NegInf v v1 f = v
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c1]: !v1 v f. h4/extreal/extreal__CASE h4/extreal/PosInf v v1 f = v1
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c2]: !v1 v f a. h4/extreal/extreal__CASE (h4/extreal/Normal a) v v1 f = happ f a
% Assm [h4s_extreals_extrealu_u_absu_u_primitiveu_u_def]: !_3. (!x. happ _3 x = h4/combin/I (h4/extreal/Normal (h4/real/abs x))) ==> (!_2. (!a. happ _2 a = h4/extreal/extreal__CASE a (h4/combin/I h4/extreal/PosInf) (h4/combin/I h4/extreal/PosInf) _3) ==> (!_1. (!extreal__abs. happ _1 extreal__abs = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/extreal/extreal__abs = h4/relation/WFREC (h4/min/_40 _0) _1)))
% Goal: happ h4/extreal/extreal__abs h4/extreal/PosInf = h4/extreal/PosInf
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_Q173605,TV_Q173601]: ![V_f, V_g]: (![V_x]: s(TV_Q173601,happ(s(t_fun(TV_Q173605,TV_Q173601),V_f),s(TV_Q173605,V_x))) = s(TV_Q173601,happ(s(t_fun(TV_Q173605,TV_Q173601),V_g),s(TV_Q173605,V_x))) => s(t_fun(TV_Q173605,TV_Q173601),V_f) = s(t_fun(TV_Q173605,TV_Q173601),V_g))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,V_v1)).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f, V_a]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(t_h4s_realaxs_real,V_a)))).
fof(ah4s_extreals_extrealu_u_absu_u_primitiveu_u_def, axiom, ![V_uu_3]: (![V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),V_uu_3),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))))))) => ![V_uu_2]: (![V_a]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),V_uu_2),s(t_h4s_extreals_extreal,V_a))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,V_a),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_posinf))),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_posinf))),s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),V_uu_3))) => ![V_uu_1]: (![V_extrealu_u_abs]: s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal)),V_uu_1),s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),V_extrealu_u_abs))) = s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),V_R))) => s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),h4s_extreals_extrealu_u_abs) = s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),h4s_relations_wfrec(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal)),V_uu_1)))))))).
fof(ch4s_extreals_extrealu_u_absu_u_defu_c2, conjecture, s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),h4s_extreals_extrealu_u_abs),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)).
