%   ORIGINAL: h4/lebesgue/max__fn__seq__mono
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/extreal/le__refl: !x. h4/extreal/extreal__le x x
% Assm: h4/extreal/extreal__max__def: !y x. h4/extreal/extreal__max x y = h4/bool/COND (h4/extreal/extreal__le x y) y x
% Assm: h4/lebesgue/max__fn__seq__def_c1: !x n g. h4/lebesgue/max__fn__seq g (h4/num/SUC n) x = h4/extreal/extreal__max (h4/lebesgue/max__fn__seq g n x) (g (h4/num/SUC n) x)
% Goal: !x n g. h4/extreal/extreal__le (h4/lebesgue/max__fn__seq g n x) (h4/lebesgue/max__fn__seq g (h4/num/SUC n) x)
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_extreals_leu_u_refl]: !x. h4/extreal/extreal__le x x
% Assm [h4s_extreals_extrealu_u_maxu_u_def]: !y x. h4/extreal/extreal__max x y = h4/bool/COND (h4/extreal/extreal__le x y) y x
% Assm [h4s_lebesgues_maxu_u_fnu_u_sequ_u_defu_c1]: !x n g. h4/lebesgue/max__fn__seq g (h4/num/SUC n) x = h4/extreal/extreal__max (h4/lebesgue/max__fn__seq g n x) (happ (happ g (h4/num/SUC n)) x)
% Goal: !x n g. h4/extreal/extreal__le (h4/lebesgue/max__fn__seq g n x) (h4/lebesgue/max__fn__seq g (h4/num/SUC n) x)
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_Q269770,TV_Q269766]: ![V_f, V_g]: (![V_x]: s(TV_Q269766,happ(s(t_fun(TV_Q269770,TV_Q269766),V_f),s(TV_Q269770,V_x))) = s(TV_Q269766,happ(s(t_fun(TV_Q269770,TV_Q269766),V_g),s(TV_Q269770,V_x))) => s(t_fun(TV_Q269770,TV_Q269766),V_f) = s(t_fun(TV_Q269770,TV_Q269766),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_extreals_leu_u_refl, axiom, ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_x))))).
fof(ah4s_extreals_extrealu_u_maxu_u_def, axiom, ![V_y, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_max(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y))),s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x)))).
fof(ah4s_lebesgues_maxu_u_fnu_u_sequ_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_g]: s(t_h4s_extreals_extreal,h4s_lebesgues_maxu_u_fnu_u_seq(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_max(s(t_h4s_extreals_extreal,h4s_lebesgues_maxu_u_fnu_u_seq(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_g),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(TV_u_27a,V_x)))))).
fof(ch4s_lebesgues_maxu_u_fnu_u_sequ_u_mono, conjecture, ![TV_u_27a]: ![V_x, V_n, V_g]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_lebesgues_maxu_u_fnu_u_seq(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_g),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,h4s_lebesgues_maxu_u_fnu_u_seq(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))))))).
