%   ORIGINAL: h4/float/FLOAT__MUL__FINITE
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/float/FLOAT__MUL: !b a. h4/ieee/Finite a /\ h4/ieee/Finite b /\ h4/realax/real__lt (h4/real/abs (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b))) (h4/ieee/threshold h4/ieee/float__format) ==> h4/ieee/Finite (h4/ieee/float__mul a b) /\ h4/ieee/Val (h4/ieee/float__mul a b) = h4/realax/real__add (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b)) (h4/float/error (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b)))
% Goal: !b a. h4/ieee/Finite a /\ h4/ieee/Finite b /\ h4/realax/real__lt (h4/real/abs (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b))) (h4/ieee/threshold h4/ieee/float__format) ==> h4/ieee/Finite (h4/ieee/float__mul a b)
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_floats_FLOATu_u_MUL]: !b a. h4/ieee/Finite a /\ h4/ieee/Finite b /\ h4/realax/real__lt (h4/real/abs (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b))) (h4/ieee/threshold h4/ieee/float__format) ==> h4/ieee/Finite (h4/ieee/float__mul a b) /\ h4/ieee/Val (h4/ieee/float__mul a b) = h4/realax/real__add (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b)) (h4/float/error (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b)))
% Goal: !b a. h4/ieee/Finite a /\ h4/ieee/Finite b /\ h4/realax/real__lt (h4/real/abs (h4/realax/real__mul (h4/ieee/Val a) (h4/ieee/Val b))) (h4/ieee/threshold h4/ieee/float__format) ==> h4/ieee/Finite (h4/ieee/float__mul a b)
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_Q100456,TV_Q100452]: ![V_f, V_g]: (![V_x]: s(TV_Q100452,happ(s(t_fun(TV_Q100456,TV_Q100452),V_f),s(TV_Q100456,V_x))) = s(TV_Q100452,happ(s(t_fun(TV_Q100456,TV_Q100452),V_g),s(TV_Q100456,V_x))) => s(t_fun(TV_Q100456,TV_Q100452),V_f) = s(t_fun(TV_Q100456,TV_Q100452),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
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_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_floats_FLOATu_u_MUL, axiom, ![V_b, V_a]: ((p(s(t_bool,h4s_ieees_finite(s(t_h4s_ieees_float,V_a)))) & (p(s(t_bool,h4s_ieees_finite(s(t_h4s_ieees_float,V_b)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_a))),s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_b))))))),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,h4s_ieees_floatu_u_mul(s(t_h4s_ieees_float,V_a),s(t_h4s_ieees_float,V_b)))))) & s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,h4s_ieees_floatu_u_mul(s(t_h4s_ieees_float,V_a),s(t_h4s_ieees_float,V_b))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_a))),s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_b))))),s(t_h4s_realaxs_real,h4s_floats_error(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_a))),s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_b)))))))))))).
fof(ch4s_floats_FLOATu_u_MULu_u_FINITE, conjecture, ![V_b, V_a]: ((p(s(t_bool,h4s_ieees_finite(s(t_h4s_ieees_float,V_a)))) & (p(s(t_bool,h4s_ieees_finite(s(t_h4s_ieees_float,V_b)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_a))),s(t_h4s_realaxs_real,h4s_ieees_val(s(t_h4s_ieees_float,V_b))))))),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,h4s_ieees_floatu_u_mul(s(t_h4s_ieees_float,V_a),s(t_h4s_ieees_float,V_b)))))))).
