%   ORIGINAL: h4/transc/GAUGE__MIN
% 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/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/transc/gauge0: !g E. h4/transc/gauge E g <=> (!x. E x ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (g x))
% Goal: !g2 g1 E. h4/transc/gauge E g1 /\ h4/transc/gauge E g2 ==> h4/transc/gauge E (\x. h4/bool/COND (h4/realax/real__lt (g1 x) (g2 x)) (g1 x) (g2 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_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_transcs_gauge0]: !g E. h4/transc/gauge E g <=> (!x. happ E x ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (happ g x))
% Goal: !_0. (!g1 g2 x. happ (happ (happ _0 g1) g2) x = h4/bool/COND (h4/realax/real__lt (happ g1 x) (happ g2 x)) (happ g1 x) (happ g2 x)) ==> (!g2 g1 E. h4/transc/gauge E g1 /\ h4/transc/gauge E g2 ==> h4/transc/gauge E (happ (happ _0 g1) g2))
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_Q196392,TV_Q196388]: ![V_f, V_g]: (![V_x]: s(TV_Q196388,happ(s(t_fun(TV_Q196392,TV_Q196388),V_f),s(TV_Q196392,V_x))) = s(TV_Q196388,happ(s(t_fun(TV_Q196392,TV_Q196388),V_g),s(TV_Q196392,V_x))) => s(t_fun(TV_Q196392,TV_Q196388),V_f) = s(t_fun(TV_Q196392,TV_Q196388),V_g))).
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_transcs_gauge0, axiom, ![V_g, V_E]: (p(s(t_bool,h4s_transcs_gauge(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g)))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x))))))))).
fof(ch4s_transcs_GAUGEu_u_MIN, conjecture, ![V_uu_0]: (![V_g1, V_g2, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g1))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g2))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g1),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g2),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g1),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g2),s(t_h4s_realaxs_real,V_x))))) => ![V_g2, V_g1, V_E]: ((p(s(t_bool,h4s_transcs_gauge(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g1)))) & p(s(t_bool,h4s_transcs_gauge(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g2))))) => p(s(t_bool,h4s_transcs_gauge(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g1))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g2))))))))).
