%   ORIGINAL: h4/float/ERROR__AT__WORST__LEMMA
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/ieee/Val0: !a. h4/ieee/Val a = h4/ieee/valof h4/ieee/float__format (h4/ieee/defloat a)
% Assm: h4/float/DEFLOAT__FLOAT__ROUND: !x. h4/ieee/defloat (h4/ieee/float0 (h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x)) = h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x
% Assm: h4/float/ISFINITE: !a. h4/ieee/Finite a <=> h4/ieee/is__finite h4/ieee/float__format (h4/ieee/defloat a)
% Assm: h4/float/error__def: !x. h4/float/error x = h4/real/real__sub (h4/ieee/Val (h4/ieee/float0 (h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x))) x
% Assm: h4/float/BOUND__AT__WORST__LEMMA: !x a. h4/realax/real__lt (h4/real/abs x) (h4/ieee/threshold h4/ieee/float__format) /\ h4/ieee/is__finite h4/ieee/float__format a ==> h4/real/real__lte (h4/real/abs (h4/real/real__sub (h4/ieee/valof h4/ieee/float__format (h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x)) x)) (h4/real/abs (h4/real/real__sub (h4/ieee/valof h4/ieee/float__format a) x))
% Goal: !x a. h4/realax/real__lt (h4/real/abs x) (h4/ieee/threshold h4/ieee/float__format) /\ h4/ieee/Finite a ==> h4/real/real__lte (h4/real/abs (h4/float/error x)) (h4/real/abs (h4/real/real__sub (h4/ieee/Val a) 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_ieees_Val0]: !a. h4/ieee/Val a = h4/ieee/valof h4/ieee/float__format (h4/ieee/defloat a)
% Assm [h4s_floats_DEFLOATu_u_FLOATu_u_ROUND]: !x. h4/ieee/defloat (h4/ieee/float0 (h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x)) = h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x
% Assm [h4s_floats_ISFINITE]: !a. h4/ieee/Finite a <=> h4/ieee/is__finite h4/ieee/float__format (h4/ieee/defloat a)
% Assm [h4s_floats_erroru_u_def]: !x. h4/float/error x = h4/real/real__sub (h4/ieee/Val (h4/ieee/float0 (h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x))) x
% Assm [h4s_floats_BOUNDu_u_ATu_u_WORSTu_u_LEMMA]: !x a. h4/realax/real__lt (h4/real/abs x) (h4/ieee/threshold h4/ieee/float__format) /\ h4/ieee/is__finite h4/ieee/float__format a ==> h4/real/real__lte (h4/real/abs (h4/real/real__sub (h4/ieee/valof h4/ieee/float__format (h4/ieee/round h4/ieee/float__format h4/ieee/To__nearest x)) x)) (h4/real/abs (h4/real/real__sub (h4/ieee/valof h4/ieee/float__format a) x))
% Goal: !x a. h4/realax/real__lt (h4/real/abs x) (h4/ieee/threshold h4/ieee/float__format) /\ h4/ieee/Finite a ==> h4/real/real__lte (h4/real/abs (h4/float/error x)) (h4/real/abs (h4/real/real__sub (h4/ieee/Val a) 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_Q99068,TV_Q99064]: ![V_f, V_g]: (![V_x]: s(TV_Q99064,happ(s(t_fun(TV_Q99068,TV_Q99064),V_f),s(TV_Q99068,V_x))) = s(TV_Q99064,happ(s(t_fun(TV_Q99068,TV_Q99064),V_g),s(TV_Q99068,V_x))) => s(t_fun(TV_Q99068,TV_Q99064),V_f) = s(t_fun(TV_Q99068,TV_Q99064),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: 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_ieees_Val0, axiom, ![V_a]: s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_a))) = s(t_h4s_realaxs_real,h4s_ieees_valof(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_defloat(s(t_h4s_ieees_float,V_a)))))).
fof(ah4s_floats_DEFLOATu_u_FLOATu_u_ROUND, axiom, ![V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_defloat(s(t_h4s_ieees_float,h4s_ieees_float0(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_round(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_ieees_roundmode,h4s_ieees_tou_u_nearest),s(t_h4s_realaxs_real,V_x))))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_round(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_ieees_roundmode,h4s_ieees_tou_u_nearest),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_floats_ISFINITE, axiom, ![V_a]: s(t_bool,h4s_ieees_finite(s(t_h4s_ieees_float,V_a))) = s(t_bool,h4s_ieees_isu_u_finite(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_defloat(s(t_h4s_ieees_float,V_a)))))).
fof(ah4s_floats_erroru_u_def, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_floats_error(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,h4s_ieees_float0(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_round(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_ieees_roundmode,h4s_ieees_tou_u_nearest),s(t_h4s_realaxs_real,V_x))))))),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_floats_BOUNDu_u_ATu_u_WORSTu_u_LEMMA, axiom, ![V_x, V_a]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_ieees_threshold(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format)))))) & p(s(t_bool,h4s_ieees_isu_u_finite(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_ieees_valof(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_round(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_ieees_roundmode,h4s_ieees_tou_u_nearest),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_ieees_valof(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))),s(t_h4s_realaxs_real,V_x)))))))))).
fof(ch4s_floats_ERRORu_u_ATu_u_WORSTu_u_LEMMA, conjecture, ![V_x, V_a]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_ieees_threshold(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_ieees_floatu_u_format)))))) & p(s(t_bool,h4s_ieees_finite(s(t_h4s_ieees_float,V_a))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_floats_error(s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_a))),s(t_h4s_realaxs_real,V_x)))))))))).
