%   ORIGINAL: h4/float/CLOSEST__IS__EVERYTHING
% 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/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/ieee/closest0: !x v s p. h4/ieee/closest v p s x = h4/min/_40 (\a. h4/ieee/is__closest v s x a /\ ((?b. h4/ieee/is__closest v s x b /\ p b) ==> p a))
% Assm: h4/float/IS__CLOSEST__EXISTS: !x v s. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> (?a. h4/ieee/is__closest v s x a)
% Goal: !x v s p. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> h4/ieee/is__closest v s x (h4/ieee/closest v p s x) /\ ((?b. h4/ieee/is__closest v s x b /\ p b) ==> p (h4/ieee/closest v p s 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_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% 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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_ieees_closest0]: !_0. (!v s x p a. happ (happ (happ (happ (happ _0 v) s) x) p) a <=> h4/ieee/is__closest v s x a /\ ((?b. h4/ieee/is__closest v s x b /\ happ p b) ==> happ p a)) ==> (!x v s p. h4/ieee/closest v p s x = h4/min/_40 (happ (happ (happ (happ _0 v) s) x) p))
% Assm [h4s_floats_ISu_u_CLOSESTu_u_EXISTS]: !x v s. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> (?a. h4/ieee/is__closest v s x a)
% Goal: !x v s p. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> h4/ieee/is__closest v s x (h4/ieee/closest v p s x) /\ ((?b. h4/ieee/is__closest v s x b /\ happ p b) ==> happ p (h4/ieee/closest v p s 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_Q98134,TV_Q98130]: ![V_f, V_g]: (![V_x]: s(TV_Q98130,happ(s(t_fun(TV_Q98134,TV_Q98130),V_f),s(TV_Q98134,V_x))) = s(TV_Q98130,happ(s(t_fun(TV_Q98134,TV_Q98130),V_g),s(TV_Q98134,V_x))) => s(t_fun(TV_Q98134,TV_Q98130),V_f) = s(t_fun(TV_Q98134,TV_Q98130),V_g))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
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_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_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_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_ieees_closest0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_v, V_s, V_x, V_p, V_a]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_v))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,V_x))),s(t_fun(TV_u_27a,t_bool),V_p))),s(TV_u_27a,V_a)))) <=> (p(s(t_bool,h4s_ieees_isu_u_closest(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_v),s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_realaxs_real,V_x),s(TV_u_27a,V_a)))) & (?[V_b]: (p(s(t_bool,h4s_ieees_isu_u_closest(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_v),s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_realaxs_real,V_x),s(TV_u_27a,V_b)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_b))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_a))))))) => ![V_x, V_v, V_s, V_p]: s(TV_u_27a,h4s_ieees_closest(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_v),s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_realaxs_real,V_x))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_v))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,V_x))),s(t_fun(TV_u_27a,t_bool),V_p))))))).
fof(ah4s_floats_ISu_u_CLOSESTu_u_EXISTS, axiom, ![V_x, V_v, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s)))) => (~ (s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),h4s_predu_u_sets_empty)) => ?[V_a]: p(s(t_bool,h4s_ieees_isu_u_closest(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_realaxs_real),V_v),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s),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)),V_a))))))).
fof(ch4s_floats_CLOSESTu_u_ISu_u_EVERYTHING, conjecture, ![V_x, V_v, V_s, V_p]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s)))) => (~ (s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),h4s_predu_u_sets_empty)) => (p(s(t_bool,h4s_ieees_isu_u_closest(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_realaxs_real),V_v),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s),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_closest(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_realaxs_real),V_v),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_p),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s),s(t_h4s_realaxs_real,V_x)))))) & (?[V_b]: (p(s(t_bool,h4s_ieees_isu_u_closest(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_realaxs_real),V_v),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s),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)),V_b)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_p),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_b))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_p),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_ieees_closest(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_realaxs_real),V_v),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_p),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),V_s),s(t_h4s_realaxs_real,V_x))))))))))).
