%   ORIGINAL: h4/util__prob/INF__DEF__ALT
% 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/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% 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/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/real/inf__def: !p. h4/real/inf p = h4/realax/real__neg (h4/real/sup (\r. p (h4/realax/real__neg r)))
% Goal: !p. h4/real/inf p = h4/realax/real__neg (h4/real/sup (\r. h4/bool/IN (h4/realax/real__neg r) p))
%   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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% 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_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_reals_infu_u_def]: !_0. (!p r. happ (happ _0 p) r <=> happ p (h4/realax/real__neg r)) ==> (!p. h4/real/inf p = h4/realax/real__neg (h4/real/sup (happ _0 p)))
% Goal: !_0. (!p r. happ (happ _0 p) r <=> h4/bool/IN (h4/realax/real__neg r) p) ==> (!p. h4/real/inf p = h4/realax/real__neg (h4/real/sup (happ _0 p)))
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_Q186766,TV_Q186762]: ![V_f, V_g]: (![V_x]: s(TV_Q186762,happ(s(t_fun(TV_Q186766,TV_Q186762),V_f),s(TV_Q186766,V_x))) = s(TV_Q186762,happ(s(t_fun(TV_Q186766,TV_Q186762),V_g),s(TV_Q186766,V_x))) => s(t_fun(TV_Q186766,TV_Q186762),V_f) = s(t_fun(TV_Q186766,TV_Q186762),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_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_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> 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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![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),V_g),s(TV_u_27a,V_x))))).
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_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_reals_infu_u_def, axiom, ![V_uu_0]: (![V_p, V_r]: s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_bool),t_fun(t_h4s_realaxs_real,t_bool)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_bool),V_p))),s(t_h4s_realaxs_real,V_r))) = s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_p),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_r))))) => ![V_p]: s(t_h4s_realaxs_real,h4s_reals_inf(s(t_fun(t_h4s_realaxs_real,t_bool),V_p))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_reals_sup(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_bool),t_fun(t_h4s_realaxs_real,t_bool)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_bool),V_p))))))))).
fof(ch4s_utilu_u_probs_INFu_u_DEFu_u_ALT, conjecture, ![V_uu_0]: (![V_p, V_r]: s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_bool),t_fun(t_h4s_realaxs_real,t_bool)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_bool),V_p))),s(t_h4s_realaxs_real,V_r))) = s(t_bool,h4s_bools_in(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_r))),s(t_fun(t_h4s_realaxs_real,t_bool),V_p))) => ![V_p]: s(t_h4s_realaxs_real,h4s_reals_inf(s(t_fun(t_h4s_realaxs_real,t_bool),V_p))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_reals_sup(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_bool),t_fun(t_h4s_realaxs_real,t_bool)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_bool),V_p))))))))).
