%   ORIGINAL: h4/binary__ieee/float__round__roundTowardPositive__plus__infinity
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% 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/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/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/binary__ieee/float__round__def: !toneg r mode. h4/binary__ieee/float__round mode toneg r = h4/bool/LET (\x. h4/bool/COND (h4/binary__ieee/float__is__zero x) (h4/bool/COND toneg (h4/binary__ieee/float__minus__zero h4/bool/the__value) (h4/binary__ieee/float__plus__zero h4/bool/the__value)) x) (h4/binary__ieee/round mode r)
% Assm: h4/binary__ieee/infinity__properties_c0: ~h4/binary__ieee/float__is__zero (h4/binary__ieee/float__plus__infinity h4/bool/the__value)
% Assm: h4/binary__ieee/round__roundTowardPositive__plus__infinity: !x. h4/realax/real__lt (h4/binary__ieee/largest h4/bool/the__value) x ==> h4/binary__ieee/round h4/binary__ieee/roundTowardPositive x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
% Goal: !x b. h4/realax/real__lt (h4/binary__ieee/largest h4/bool/the__value) x ==> h4/binary__ieee/float__round h4/binary__ieee/roundTowardPositive b x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% 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_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_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_binaryu_u_ieees_floatu_u_roundu_u_def]: !_0. (!toneg x. happ (happ _0 toneg) x = h4/bool/COND (h4/binary__ieee/float__is__zero x) (h4/bool/COND toneg (h4/binary__ieee/float__minus__zero h4/bool/the__value) (h4/binary__ieee/float__plus__zero h4/bool/the__value)) x) ==> (!toneg r mode. h4/binary__ieee/float__round mode toneg r = h4/bool/LET (happ _0 toneg) (h4/binary__ieee/round mode r))
% Assm [h4s_binaryu_u_ieees_infinityu_u_propertiesu_c0]: ~h4/binary__ieee/float__is__zero (h4/binary__ieee/float__plus__infinity h4/bool/the__value)
% Assm [h4s_binaryu_u_ieees_roundu_u_roundTowardPositiveu_u_plusu_u_infinity]: !x. h4/realax/real__lt (h4/binary__ieee/largest h4/bool/the__value) x ==> h4/binary__ieee/round h4/binary__ieee/roundTowardPositive x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
% Goal: !x b. h4/realax/real__lt (h4/binary__ieee/largest h4/bool/the__value) x ==> h4/binary__ieee/float__round h4/binary__ieee/roundTowardPositive b x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
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_Q108176,TV_Q108172]: ![V_f, V_g]: (![V_x]: s(TV_Q108172,happ(s(t_fun(TV_Q108176,TV_Q108172),V_f),s(TV_Q108176,V_x))) = s(TV_Q108172,happ(s(t_fun(TV_Q108176,TV_Q108172),V_g),s(TV_Q108176,V_x))) => s(t_fun(TV_Q108176,TV_Q108172),V_f) = s(t_fun(TV_Q108176,TV_Q108172),V_g))).
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_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(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_f),s(TV_u_27a,V_x)))).
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_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_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_binaryu_u_ieees_floatu_u_roundu_u_def, axiom, ![TV_u_27t,TV_u_27w]: ![V_uu_0]: (![V_toneg, V_x]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_bool,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_0),s(t_bool,V_toneg))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,h4s_binaryu_u_ieees_floatu_u_isu_u_zero(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_toneg),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_zero(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_zero(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) => ![V_toneg, V_r, V_mode]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_bool,V_toneg),s(t_h4s_realaxs_real,V_r))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_let(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_bool,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_0),s(t_bool,V_toneg))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_realaxs_real,V_r))))))).
fof(ah4s_binaryu_u_ieees_infinityu_u_propertiesu_c0, axiom, ![TV_u_27t,TV_u_27w]: ~ (p(s(t_bool,h4s_binaryu_u_ieees_floatu_u_isu_u_zero(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value)))))))).
fof(ah4s_binaryu_u_ieees_roundu_u_roundTowardPositiveu_u_plusu_u_infinity, axiom, ![TV_u_27t,TV_u_27w]: ![V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_largest(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_realaxs_real,V_x)))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_round(s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardpositive),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
fof(ch4s_binaryu_u_ieees_floatu_u_roundu_u_roundTowardPositiveu_u_plusu_u_infinity, conjecture, ![TV_u_27t,TV_u_27w]: ![V_x, V_b]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_largest(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_realaxs_real,V_x)))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardpositive),s(t_bool,V_b),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
