%   ORIGINAL: h4/transc/PI2__BOUNDS_c0
% 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__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~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/real/REAL__10: ~(h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/real/real__of__num h4/num/0)
% Assm: h4/real/REAL__LT__LE: !y x. h4/realax/real__lt x y <=> h4/real/real__lte x y /\ ~(x = y)
% Assm: h4/real/REAL__LT__IMP__NE: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm: h4/transc/COS__0: h4/transc/cos (h4/real/real__of__num h4/num/0) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/transc/COS__2: h4/realax/real__lt (h4/transc/cos (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) (h4/real/real__of__num h4/num/0)
% Assm: h4/transc/COS__ISZERO: h4/bool/_3F_21 (\x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) /\ h4/transc/cos x = h4/real/real__of__num h4/num/0)
% Assm: h4/transc/PI2: h4/real/_2F h4/transc/pi (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) = h4/min/_40 (\x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) /\ h4/transc/cos x = h4/real/real__of__num h4/num/0)
% Assm: h4/transc/COS__PI2: h4/transc/cos (h4/real/_2F h4/transc/pi (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) = h4/real/real__of__num h4/num/0
% Goal: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F h4/transc/pi (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))))
%   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_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~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_reals_REALu_u_10]: ~(h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/real/real__of__num h4/num/0)
% Assm [h4s_reals_REALu_u_LTu_u_LE]: !y x. h4/realax/real__lt x y <=> h4/real/real__lte x y /\ ~(x = y)
% Assm [h4s_reals_REALu_u_LTu_u_IMPu_u_NE]: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm [h4s_transcs_COSu_u_0]: h4/transc/cos (h4/real/real__of__num h4/num/0) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_transcs_COSu_u_2]: h4/realax/real__lt (h4/transc/cos (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) (h4/real/real__of__num h4/num/0)
% Assm [h4s_transcs_COSu_u_ISZERO]: !_0. (!x. happ _0 x <=> h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) /\ h4/transc/cos x = h4/real/real__of__num h4/num/0) ==> h4/bool/_3F_21 _0
% Assm [h4s_transcs_PI2]: !_0. (!x. happ _0 x <=> h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) /\ h4/transc/cos x = h4/real/real__of__num h4/num/0) ==> h4/real/_2F h4/transc/pi (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) = h4/min/_40 _0
% Assm [h4s_transcs_COSu_u_PI2]: h4/transc/cos (h4/real/_2F h4/transc/pi (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) = h4/real/real__of__num h4/num/0
% Goal: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F h4/transc/pi (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))))
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_Q194186,TV_Q194182]: ![V_f, V_g]: (![V_x]: s(TV_Q194182,happ(s(t_fun(TV_Q194186,TV_Q194182),V_f),s(TV_Q194186,V_x))) = s(TV_Q194182,happ(s(t_fun(TV_Q194186,TV_Q194182),V_g),s(TV_Q194186,V_x))) => s(t_fun(TV_Q194186,TV_Q194182),V_f) = s(t_fun(TV_Q194186,TV_Q194182),V_g))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
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_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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_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_reals_REALu_u_10, axiom, ~ (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_reals_REALu_u_LTu_u_LE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) <=> (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & ~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))))).
fof(ah4s_reals_REALu_u_LTu_u_IMPu_u_NE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) => ~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_transcs_COSu_u_0, axiom, s(t_h4s_realaxs_real,h4s_transcs_cos(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,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_transcs_COSu_u_2, axiom, p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_transcs_cos(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_transcs_COSu_u_ISZERO, axiom, ![V_uu_0]: (![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_realaxs_real,V_x)))) <=> (p(s(t_bool,h4s_reals_realu_u_lte(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,V_x)))) & (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) & s(t_h4s_realaxs_real,h4s_transcs_cos(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0)))))).
fof(ah4s_transcs_PI2, axiom, ![V_uu_0]: (![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_realaxs_real,V_x)))) <=> (p(s(t_bool,h4s_reals_realu_u_lte(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,V_x)))) & (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) & s(t_h4s_realaxs_real,h4s_transcs_cos(s(t_h4s_realaxs_real,V_x))) = 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,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_transcs_pi),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_realaxs_real,h4s_mins_u_40(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0))))).
fof(ah4s_transcs_COSu_u_PI2, axiom, s(t_h4s_realaxs_real,h4s_transcs_cos(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_transcs_pi),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ch4s_transcs_PI2u_u_BOUNDSu_c0, conjecture, 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,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_transcs_pi),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))))).
