%   ORIGINAL: h4/binary__ieee/round__roundTowardPositive__bottom
% 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/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/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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/pred__set/GSPEC__F: h4/pred__set/GSPEC (\x. h4/pair/_2C x F) = h4/pred__set/EMPTY
% Assm: h4/binary__ieee/rounding__case__def_c1: !v3 v2 v1 v0. h4/binary__ieee/rounding__CASE h4/binary__ieee/roundTowardPositive v0 v1 v2 v3 = v1
% Assm: h4/binary__ieee/round__def: !x mode. h4/binary__ieee/round mode x = h4/binary__ieee/rounding__CASE mode (h4/bool/LET (\t. h4/bool/COND (h4/real/real__lte x (h4/realax/real__neg t)) (h4/binary__ieee/float__minus__infinity h4/bool/the__value) (h4/bool/COND (h4/real/real__ge x t) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/closest__such (\a. ~h4/words/word__lsb (h4/binary__ieee/float__Significand a)) h4/binary__ieee/float__is__finite x))) (h4/binary__ieee/threshold h4/bool/the__value)) (h4/bool/LET (\t. h4/bool/COND (h4/realax/real__lt x (h4/realax/real__neg t)) (h4/binary__ieee/float__bottom h4/bool/the__value) (h4/bool/COND (h4/real/real__gt x t) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/closest (h4/pred__set/GSPEC (\a. h4/pair/_2C a (h4/binary__ieee/float__is__finite a /\ h4/real/real__ge (h4/binary__ieee/float__to__real a) x))) x))) (h4/binary__ieee/largest h4/bool/the__value)) (h4/bool/LET (\t. h4/bool/COND (h4/realax/real__lt x (h4/realax/real__neg t)) (h4/binary__ieee/float__minus__infinity h4/bool/the__value) (h4/bool/COND (h4/real/real__gt x t) (h4/binary__ieee/float__top h4/bool/the__value) (h4/binary__ieee/closest (h4/pred__set/GSPEC (\a. h4/pair/_2C a (h4/binary__ieee/float__is__finite a /\ h4/real/real__lte (h4/binary__ieee/float__to__real a) x))) x))) (h4/binary__ieee/largest h4/bool/the__value)) (h4/bool/LET (\t. h4/bool/COND (h4/realax/real__lt x (h4/realax/real__neg t)) (h4/binary__ieee/float__bottom h4/bool/the__value) (h4/bool/COND (h4/real/real__gt x t) (h4/binary__ieee/float__top h4/bool/the__value) (h4/binary__ieee/closest (h4/pred__set/GSPEC (\a. h4/pair/_2C a (h4/binary__ieee/float__is__finite a /\ h4/real/real__lte (h4/real/abs (h4/binary__ieee/float__to__real a)) (h4/real/abs x)))) x))) (h4/binary__ieee/largest h4/bool/the__value))
% Goal: !x. h4/realax/real__lt x (h4/realax/real__neg (h4/binary__ieee/largest h4/bool/the__value)) ==> h4/binary__ieee/round h4/binary__ieee/roundTowardPositive x = h4/binary__ieee/float__bottom 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_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_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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_predu_u_sets_GSPECu_u_F]: !_0. (!x. happ _0 x = h4/pair/_2C x F) ==> h4/pred__set/GSPEC _0 = h4/pred__set/EMPTY
% Assm [h4s_binaryu_u_ieees_roundingu_u_caseu_u_defu_c1]: !v3 v2 v1 v0. h4/binary__ieee/rounding__CASE h4/binary__ieee/roundTowardPositive v0 v1 v2 v3 = v1
% Assm [h4s_binaryu_u_ieees_roundu_u_def]: !_7. (!x a. ?v. (v <=> happ h4/binary__ieee/float__is__finite a /\ h4/real/real__lte (h4/real/abs (h4/binary__ieee/float__to__real a)) (h4/real/abs x)) /\ happ (happ _7 x) a = h4/pair/_2C a v) ==> (!_6. (!x t. happ (happ _6 x) t = h4/bool/COND (h4/realax/real__lt x (h4/realax/real__neg t)) (h4/binary__ieee/float__bottom h4/bool/the__value) (h4/bool/COND (h4/real/real__gt x t) (h4/binary__ieee/float__top h4/bool/the__value) (h4/binary__ieee/closest (h4/pred__set/GSPEC (happ _7 x)) x))) ==> (!_5. (!x a. ?v. (v <=> happ h4/binary__ieee/float__is__finite a /\ h4/real/real__lte (h4/binary__ieee/float__to__real a) x) /\ happ (happ _5 x) a = h4/pair/_2C a v) ==> (!_4. (!x t. happ (happ _4 x) t = h4/bool/COND (h4/realax/real__lt x (h4/realax/real__neg t)) (h4/binary__ieee/float__minus__infinity h4/bool/the__value) (h4/bool/COND (h4/real/real__gt x t) (h4/binary__ieee/float__top h4/bool/the__value) (h4/binary__ieee/closest (h4/pred__set/GSPEC (happ _5 x)) x))) ==> (!_3. (!x a. ?v. (v <=> happ h4/binary__ieee/float__is__finite a /\ h4/real/real__ge (h4/binary__ieee/float__to__real a) x) /\ happ (happ _3 x) a = h4/pair/_2C a v) ==> (!_2. (!x t. happ (happ _2 x) t = h4/bool/COND (h4/realax/real__lt x (h4/realax/real__neg t)) (h4/binary__ieee/float__bottom h4/bool/the__value) (h4/bool/COND (h4/real/real__gt x t) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/closest (h4/pred__set/GSPEC (happ _3 x)) x))) ==> (!_1. (!a. happ _1 a <=> ~h4/words/word__lsb (h4/binary__ieee/float__Significand a)) ==> (!_0. (!x t. happ (happ _0 x) t = h4/bool/COND (h4/real/real__lte x (h4/realax/real__neg t)) (h4/binary__ieee/float__minus__infinity h4/bool/the__value) (h4/bool/COND (h4/real/real__ge x t) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/closest__such _1 h4/binary__ieee/float__is__finite x))) ==> (!x mode. h4/binary__ieee/round mode x = h4/binary__ieee/rounding__CASE mode (h4/bool/LET (happ _0 x) (h4/binary__ieee/threshold h4/bool/the__value)) (h4/bool/LET (happ _2 x) (h4/binary__ieee/largest h4/bool/the__value)) (h4/bool/LET (happ _4 x) (h4/binary__ieee/largest h4/bool/the__value)) (h4/bool/LET (happ _6 x) (h4/binary__ieee/largest h4/bool/the__value))))))))))
% Goal: !x. h4/realax/real__lt x (h4/realax/real__neg (h4/binary__ieee/largest h4/bool/the__value)) ==> h4/binary__ieee/round h4/binary__ieee/roundTowardPositive x = h4/binary__ieee/float__bottom 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_Q108042,TV_Q108038]: ![V_f, V_g]: (![V_x]: s(TV_Q108038,happ(s(t_fun(TV_Q108042,TV_Q108038),V_f),s(TV_Q108042,V_x))) = s(TV_Q108038,happ(s(t_fun(TV_Q108042,TV_Q108038),V_g),s(TV_Q108042,V_x))) => s(t_fun(TV_Q108042,TV_Q108038),V_f) = s(t_fun(TV_Q108042,TV_Q108038),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_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_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_predu_u_sets_GSPECu_u_F, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,f))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_binaryu_u_ieees_roundingu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v3, V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_binaryu_u_ieees_roundingu_u_case(s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardpositive),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2),s(TV_u_27a,V_v3))) = s(TV_u_27a,V_v1)).
fof(ah4s_binaryu_u_ieees_roundu_u_def, axiom, ![TV_u_27t,TV_u_27w]: ![V_uu_7]: (![V_x, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_binaryu_u_ieees_floatu_u_isu_u_finite),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_floatu_u_tou_u_real(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a))))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x)))))))) & s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool))),V_uu_7),s(t_h4s_realaxs_real,V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a))) = s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_pairs_u_2c(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a),s(t_bool,V_v)))) => ![V_uu_6]: (![V_x, V_t]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_6),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_t))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_t))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_bottom(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_bools_cond(s(t_bool,h4s_reals_realu_u_gt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_t))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_top(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_closest(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool))),V_uu_7),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_x))))))) => ![V_uu_5]: (![V_x, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_binaryu_u_ieees_floatu_u_isu_u_finite),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_floatu_u_tou_u_real(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a))),s(t_h4s_realaxs_real,V_x)))))) & s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool))),V_uu_5),s(t_h4s_realaxs_real,V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a))) = s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_pairs_u_2c(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a),s(t_bool,V_v)))) => ![V_uu_4]: (![V_x, V_t]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_4),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_t))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_t))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(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_bools_cond(s(t_bool,h4s_reals_realu_u_gt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_t))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_top(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_closest(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool))),V_uu_5),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_x))))))) => ![V_uu_3]: (![V_x, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_binaryu_u_ieees_floatu_u_isu_u_finite),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a)))) & p(s(t_bool,h4s_reals_realu_u_ge(s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_floatu_u_tou_u_real(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a))),s(t_h4s_realaxs_real,V_x)))))) & s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool))),V_uu_3),s(t_h4s_realaxs_real,V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a))) = s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_pairs_u_2c(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a),s(t_bool,V_v)))) => ![V_uu_2]: (![V_x, V_t]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_2),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_t))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_t))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_bottom(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_bools_cond(s(t_bool,h4s_reals_realu_u_gt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_t))),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))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_closest(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool))),V_uu_3),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_x))))))) => ![V_uu_1]: (![V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),V_uu_1),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a)))) <=> ~ (p(s(t_bool,h4s_wordss_wordu_u_lsb(s(t_h4s_fcps_cart(t_bool,TV_u_27t),h4s_binaryu_u_ieees_floatu_u_significand(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_a)))))))) => ![V_uu_0]: (![V_x, V_t]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_0),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_t))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_t))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(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_bools_cond(s(t_bool,h4s_reals_realu_u_ge(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_t))),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))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_closestu_u_such(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),V_uu_1),s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_bool),h4s_binaryu_u_ieees_floatu_u_isu_u_finite),s(t_h4s_realaxs_real,V_x))))))) => ![V_x, V_mode]: 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_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_roundingu_u_case(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_let(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_0),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_threshold(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_bools_let(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_2),s(t_h4s_realaxs_real,V_x))),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_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_let(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_4),s(t_h4s_realaxs_real,V_x))),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_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_let(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_6),s(t_h4s_realaxs_real,V_x))),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)))))))))))))))).
fof(ch4s_binaryu_u_ieees_roundu_u_roundTowardPositiveu_u_bottom, conjecture, ![TV_u_27t,TV_u_27w]: ![V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(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_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_bottom(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
