0.02/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.02/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.03/0.24 % Computer : n025.star.cs.uiowa.edu 0.03/0.24 % Model : x86_64 x86_64 0.03/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.24 % Memory : 32218.625MB 0.03/0.24 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.24 % CPULimit : 300 0.03/0.24 % DateTime : Sat Jul 14 05:20:25 CDT 2018 0.03/0.24 % CPUTime : 291.65/292.03 % SZS status Theorem 291.65/292.03 291.65/292.03 % SZS output start Proof 291.65/292.03 Take the following subset of the input axioms: 291.75/292.08 fof(conj_0, conjecture, 291.75/292.08 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, 291.75/292.08 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, one_one_int), 291.75/292.08 hAPP_nat_int(semiri1621563631at_int, n))), 291.75/292.08 hAPP_int_nat(number_number_of_nat, 291.75/292.08 hAPP_int_int(bit0, hAPP_int_int(bit1, pls))))!=zero_zero_int). 291.75/292.08 fof(fact_1005_nat__power__eq__0__iff, axiom, 291.75/292.08 ![Ma, Na]: 291.75/292.08 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat, Ma), 291.75/292.08 Na)=zero_zero_nat 291.75/292.08 <=> (Na!=zero_zero_nat & Ma=zero_zero_nat))). 291.75/292.08 fof(fact_1025_real__less__def, axiom, 291.75/292.08 ![Xa, Ya]: 291.75/292.08 ((Ya!=Xa 291.75/292.08 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.08 Xa), 291.75/292.08 Ya))) 291.75/292.08 <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Xa), 291.75/292.08 Ya)))). 291.75/292.08 fof(fact_120_int__less__0__conv, axiom, 291.75/292.08 ![K_2]: 291.75/292.08 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.08 hAPP_nat_int(semiri1621563631at_int, K_2)), 291.75/292.08 zero_zero_int))). 291.75/292.08 fof(fact_1256_nat__less__le, axiom, 291.75/292.08 ![Ma, Na]: 291.75/292.08 ((Ma!=Na 291.75/292.08 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Ma), 291.75/292.08 Na))) 291.75/292.08 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ma), 291.75/292.08 Na)))). 291.75/292.08 fof(fact_131_rel__simps_I46_J, axiom, 291.75/292.08 ![K_2]: pls!=hAPP_int_int(bit1, K_2)). 291.75/292.08 fof(fact_132_rel__simps_I39_J, axiom, 291.75/292.08 ![L_2]: pls!=hAPP_int_int(bit1, L_2)). 291.75/292.08 fof(fact_133_rel__simps_I50_J, axiom, 291.75/292.08 ![K_2, L_2]: hAPP_int_int(bit1, K_2)!=hAPP_int_int(bit0, L_2)). 291.75/292.08 fof(fact_134_rel__simps_I49_J, axiom, 291.75/292.08 ![K_2, L_2]: hAPP_int_int(bit1, L_2)!=hAPP_int_int(bit0, K_2)). 291.75/292.08 fof(fact_1364_not__square__less__zero, axiom, 291.75/292.08 ![A_105]: 291.75/292.08 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.08 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat, 291.75/292.08 A_105), 291.75/292.08 A_105)), 291.75/292.08 zero_zero_rat))). 291.75/292.08 fof(fact_1365_not__square__less__zero, axiom, 291.75/292.08 ![A_105]: 291.75/292.08 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.08 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.75/292.08 A_105), 291.75/292.08 A_105)), 291.75/292.08 zero_zero_real))). 291.75/292.08 fof(fact_1366_not__square__less__zero, axiom, 291.75/292.08 ![A_105]: 291.75/292.08 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.08 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.75/292.08 A_105), 291.75/292.08 A_105)), 291.75/292.08 zero_zero_int))). 291.75/292.08 fof(fact_151_Pls__def, axiom, zero_zero_int=pls). 291.75/292.08 fof(fact_1587_not__sum__squares__lt__zero, axiom, 291.75/292.08 ![X_50, Y_44]: 291.75/292.08 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.08 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat, 291.75/292.08 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat, 291.75/292.08 X_50), 291.75/292.08 X_50)), 291.75/292.08 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat, 291.75/292.08 Y_44), 291.75/292.08 Y_44))), 291.75/292.08 zero_zero_rat))). 291.75/292.08 fof(fact_1588_not__sum__squares__lt__zero, axiom, 291.75/292.08 ![X_50, Y_44]: 291.75/292.08 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.08 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.75/292.08 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.75/292.08 X_50), 291.75/292.08 X_50)), 291.75/292.08 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.75/292.08 Y_44), 291.75/292.08 Y_44))), 291.75/292.08 zero_zero_real))). 291.75/292.08 fof(fact_1589_not__sum__squares__lt__zero, axiom, 291.75/292.08 ![X_50, Y_44]: 291.75/292.08 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.08 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.75/292.08 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.75/292.08 X_50), 291.75/292.08 X_50)), 291.75/292.08 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.75/292.08 Y_44), 291.75/292.08 Y_44))), 291.75/292.08 zero_zero_int))). 291.75/292.08 fof(fact_1590_sum__squares__gt__zero__iff, axiom, 291.75/292.08 ![Xa, Ya]: 291.75/292.08 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.08 zero_zero_rat), 291.75/292.08 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat, 291.75/292.08 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat, 291.75/292.08 Xa), 291.75/292.09 Xa)), 291.75/292.09 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat, Ya), 291.75/292.09 Ya)))) 291.75/292.09 <=> (Xa!=zero_zero_rat | zero_zero_rat!=Ya))). 291.75/292.09 fof(fact_1591_sum__squares__gt__zero__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 zero_zero_real), 291.75/292.09 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.75/292.09 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.75/292.09 Xa), 291.75/292.09 Xa)), 291.75/292.09 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.75/292.09 Ya), 291.75/292.09 Ya)))) 291.75/292.09 <=> (zero_zero_real!=Ya | zero_zero_real!=Xa))). 291.75/292.09 fof(fact_1592_sum__squares__gt__zero__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((is_int(Xa) & is_int(Ya)) 291.75/292.09 => ((zero_zero_int!=Xa | zero_zero_int!=Ya) 291.75/292.09 <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.09 zero_zero_int), 291.75/292.09 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.75/292.09 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.75/292.09 Xa), 291.75/292.09 Xa)), 291.75/292.09 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.75/292.09 Ya), 291.75/292.09 Ya))))))). 291.75/292.09 fof(fact_1700_order__eq__refl, axiom, 291.75/292.09 ![Y_36, X_42]: 291.75/292.09 (X_42=Y_36 291.75/292.09 => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.75/292.09 X_42), 291.75/292.09 Y_36)))). 291.75/292.09 fof(fact_1720_order__less__asym, axiom, 291.75/292.09 ![X_39, Y_33]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_39), 291.75/292.09 Y_33)) 291.75/292.09 => ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Y_33), 291.75/292.09 X_39)))). 291.75/292.09 fof(fact_1721_order__less__asym, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 <= hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1722_order__less__asym, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => ~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)))). 291.75/292.09 fof(fact_1723_order__less__asym, axiom, 291.75/292.09 ![X_39, Y_33]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_39), 291.75/292.09 Y_33)) 291.75/292.09 => ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Y_33), 291.75/292.09 X_39)))). 291.75/292.09 fof(fact_1724_order__less__asym, axiom, 291.75/292.09 ![X_39, Y_33]: 291.75/292.09 (~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Y_33), 291.75/292.09 X_39)) 291.75/292.09 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 X_39), 291.75/292.09 Y_33)))). 291.75/292.09 fof(fact_1725_order__less__asym, axiom, 291.75/292.09 ![X_39, Y_33]: 291.75/292.09 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_39), 291.75/292.09 Y_33)) 291.75/292.09 => ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Y_33), 291.75/292.09 X_39)))). 291.75/292.09 fof(fact_1726_order__less__asym, axiom, 291.75/292.09 ![X_39, Y_33]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_39), 291.75/292.09 Y_33)) 291.75/292.09 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Y_33), 291.75/292.09 X_39)))). 291.75/292.09 fof(fact_1769_xt1_I9_J, axiom, 291.75/292.09 ![B_42, A_65]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, B_42), 291.75/292.09 A_65)) 291.75/292.09 => ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, A_65), 291.75/292.09 B_42)))). 291.75/292.09 fof(fact_1770_xt1_I9_J, axiom, 291.75/292.09 ![A_1, B_1]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 B_1), 291.75/292.09 A_1)) 291.75/292.09 => ~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 A_1), 291.75/292.09 B_1)))). 291.75/292.09 fof(fact_1771_xt1_I9_J, axiom, 291.75/292.09 ![A_1, B_1]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 B_1), 291.75/292.09 A_1)) 291.75/292.09 => ~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 A_1), 291.75/292.09 B_1)))). 291.75/292.09 fof(fact_1772_xt1_I9_J, axiom, 291.75/292.09 ![B_42, A_65]: 291.75/292.09 (~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 A_65), 291.75/292.09 B_42)) 291.75/292.09 <= hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 B_42), 291.75/292.09 A_65)))). 291.75/292.09 fof(fact_1773_xt1_I9_J, axiom, 291.75/292.09 ![B_42, A_65]: 291.75/292.09 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, B_42), 291.75/292.09 A_65)) 291.75/292.09 => ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 A_65), 291.75/292.09 B_42)))). 291.75/292.09 fof(fact_1774_xt1_I9_J, axiom, 291.75/292.09 ![B_42, A_65]: 291.75/292.09 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, A_65), 291.75/292.09 B_42)) 291.75/292.09 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, B_42), 291.75/292.09 A_65)))). 291.75/292.09 fof(fact_1775_xt1_I9_J, axiom, 291.75/292.09 ![B_42, A_65]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, B_42), 291.75/292.09 A_65)) 291.75/292.09 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_65), 291.75/292.09 B_42)))). 291.75/292.09 fof(fact_1776_order__less__asym_H, axiom, 291.75/292.09 ![A_64, B_41]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, A_64), 291.75/292.09 B_41)) 291.75/292.09 => ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, B_41), 291.75/292.09 A_64)))). 291.75/292.09 fof(fact_1777_order__less__asym_H, axiom, 291.75/292.09 ![A_1, B_1]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 A_1), 291.75/292.09 B_1)) 291.75/292.09 => ~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 B_1), 291.75/292.09 A_1)))). 291.75/292.09 fof(fact_1778_order__less__asym_H, axiom, 291.75/292.09 ![A_1, B_1]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 A_1), 291.75/292.09 B_1)) 291.75/292.09 => ~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 B_1), 291.75/292.09 A_1)))). 291.75/292.09 fof(fact_1779_order__less__asym_H, axiom, 291.75/292.09 ![A_64, B_41]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 A_64), 291.75/292.09 B_41)) 291.75/292.09 => ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 B_41), 291.75/292.09 A_64)))). 291.75/292.09 fof(fact_1780_order__less__asym_H, axiom, 291.75/292.09 ![A_64, B_41]: 291.75/292.09 (~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, B_41), 291.75/292.09 A_64)) 291.75/292.09 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 A_64), 291.75/292.09 B_41)))). 291.75/292.09 fof(fact_1781_order__less__asym_H, axiom, 291.75/292.09 ![A_64, B_41]: 291.75/292.09 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, B_41), 291.75/292.09 A_64)) 291.75/292.09 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, A_64), 291.75/292.09 B_41)))). 291.75/292.09 fof(fact_1782_order__less__asym_H, axiom, 291.75/292.09 ![A_64, B_41]: 291.75/292.09 (~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, B_41), 291.75/292.09 A_64)) 291.75/292.09 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_64), 291.75/292.09 B_41)))). 291.75/292.09 fof(fact_1790_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Y_30, X_36]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_36), 291.75/292.09 Y_30)) 291.75/292.09 => Y_30!=X_36)). 291.75/292.09 fof(fact_1791_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => Xa!=Ya)). 291.75/292.09 fof(fact_1792_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => Xa!=Ya)). 291.75/292.09 fof(fact_1793_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Y_30, X_36]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_36), 291.75/292.09 Y_30)) 291.75/292.09 => X_36!=Y_30)). 291.75/292.09 fof(fact_1794_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Y_30, X_36]: 291.75/292.09 (X_36!=Y_30 291.75/292.09 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 X_36), 291.75/292.09 Y_30)))). 291.75/292.09 fof(fact_1795_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Y_30, X_36]: 291.75/292.09 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_36), 291.75/292.09 Y_30)) 291.75/292.09 => Y_30!=X_36)). 291.75/292.09 fof(fact_1796_order__less__imp__not__eq2, axiom, 291.75/292.09 ![Y_30, X_36]: 291.75/292.09 (Y_30!=X_36 291.75/292.09 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_36), 291.75/292.09 Y_30)))). 291.75/292.09 fof(fact_1797_order__less__imp__not__eq, axiom, 291.75/292.09 ![Y_29, X_35]: 291.75/292.09 (Y_29!=X_35 291.75/292.09 <= hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_35), 291.75/292.09 Y_29)))). 291.75/292.09 fof(fact_1798_order__less__imp__not__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (Ya!=Xa 291.75/292.09 <= hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1799_order__less__imp__not__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => Xa!=Ya)). 291.75/292.09 fof(fact_1800_order__less__imp__not__eq, axiom, 291.75/292.09 ![Y_29, X_35]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_35), 291.75/292.09 Y_29)) 291.75/292.09 => X_35!=Y_29)). 291.75/292.09 fof(fact_1801_order__less__imp__not__eq, axiom, 291.75/292.09 ![Y_29, X_35]: 291.75/292.09 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_35), 291.75/292.09 Y_29)) 291.75/292.09 => Y_29!=X_35)). 291.75/292.09 fof(fact_1802_order__less__imp__not__eq, axiom, 291.75/292.09 ![Y_29, X_35]: 291.75/292.09 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_35), 291.75/292.09 Y_29)) 291.75/292.09 => X_35!=Y_29)). 291.75/292.09 fof(fact_1803_order__less__imp__not__eq, axiom, 291.75/292.09 ![Y_29, X_35]: 291.75/292.09 (Y_29!=X_35 291.75/292.09 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_35), 291.75/292.09 Y_29)))). 291.75/292.09 fof(fact_1804_order__less__imp__not__less, axiom, 291.75/292.09 ![X_34, Y_28]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_34), 291.75/292.09 Y_28)) 291.75/292.09 => ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Y_28), 291.75/292.09 X_34)))). 291.75/292.09 fof(fact_1805_order__less__imp__not__less, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 <= hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1806_order__less__imp__not__less, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => ~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)))). 291.75/292.09 fof(fact_1807_order__less__imp__not__less, axiom, 291.75/292.09 ![X_34, Y_28]: 291.75/292.09 (~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Y_28), 291.75/292.09 X_34)) 291.75/292.09 <= hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_34), 291.75/292.09 Y_28)))). 291.75/292.09 fof(fact_1808_order__less__imp__not__less, axiom, 291.75/292.09 ![X_34, Y_28]: 291.75/292.09 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_34), 291.75/292.09 Y_28)) 291.75/292.09 => ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 Y_28), 291.75/292.09 X_34)))). 291.75/292.09 fof(fact_1809_order__less__imp__not__less, axiom, 291.75/292.09 ![X_34, Y_28]: 291.75/292.09 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_34), 291.75/292.09 Y_28)) 291.75/292.09 => ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Y_28), 291.75/292.09 X_34)))). 291.75/292.09 fof(fact_180_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 (zero_zero_rat=hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, 291.75/292.09 A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa)) 291.75/292.09 <=> (zero_zero_rat=A_1 291.75/292.09 & hAPP_int_nat(number_number_of_nat, Wa)!=zero_zero_nat))). 291.75/292.09 fof(fact_1810_order__less__imp__not__less, axiom, 291.75/292.09 ![X_34, Y_28]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_34), 291.75/292.09 Y_28)) 291.75/292.09 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Y_28), 291.75/292.09 X_34)))). 291.75/292.09 fof(fact_1811_order__less__not__sym, axiom, 291.75/292.09 ![X_33, Y_27]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_33), 291.75/292.09 Y_27)) 291.75/292.09 => ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Y_27), 291.75/292.09 X_33)))). 291.75/292.09 fof(fact_1812_order__less__not__sym, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => ~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)))). 291.75/292.09 fof(fact_1813_order__less__not__sym, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 <= hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1814_order__less__not__sym, axiom, 291.75/292.09 ![X_33, Y_27]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_33), 291.75/292.09 Y_27)) 291.75/292.09 => ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Y_27), 291.75/292.09 X_33)))). 291.75/292.09 fof(fact_1815_order__less__not__sym, axiom, 291.75/292.09 ![X_33, Y_27]: 291.75/292.09 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_33), 291.75/292.09 Y_27)) 291.75/292.09 => ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 Y_27), 291.75/292.09 X_33)))). 291.75/292.09 fof(fact_1816_order__less__not__sym, axiom, 291.75/292.09 ![X_33, Y_27]: 291.75/292.09 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_33), 291.75/292.09 Y_27)) 291.75/292.09 => ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Y_27), 291.75/292.09 X_33)))). 291.75/292.09 fof(fact_1817_order__less__not__sym, axiom, 291.75/292.09 ![X_33, Y_27]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_33), 291.75/292.09 Y_27)) 291.75/292.09 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Y_27), 291.75/292.09 X_33)))). 291.75/292.09 fof(fact_1818_less__imp__neq, axiom, 291.75/292.09 ![Y_26, X_32]: 291.75/292.09 (X_32!=Y_26 291.75/292.09 <= hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_32), 291.75/292.09 Y_26)))). 291.75/292.09 fof(fact_1819_less__imp__neq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => Xa!=Ya)). 291.75/292.09 fof(fact_181_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 ((zero_zero_nat!=hAPP_int_nat(number_number_of_nat, Wa) 291.75/292.09 & zero_z891286103de_int=A_1) 291.75/292.09 <=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int, 291.75/292.09 A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa))=zero_z891286103de_int)). 291.75/292.09 fof(fact_1820_less__imp__neq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (Ya!=Xa 291.75/292.09 <= hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1821_less__imp__neq, axiom, 291.75/292.09 ![Y_26, X_32]: 291.75/292.09 (X_32!=Y_26 291.75/292.09 <= hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_32), 291.75/292.09 Y_26)))). 291.75/292.09 fof(fact_1822_less__imp__neq, axiom, 291.75/292.09 ![Y_26, X_32]: 291.75/292.09 (X_32!=Y_26 291.75/292.09 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 X_32), 291.75/292.09 Y_26)))). 291.75/292.09 fof(fact_1823_less__imp__neq, axiom, 291.75/292.09 ![Y_26, X_32]: 291.75/292.09 (Y_26!=X_32 291.75/292.09 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_32), 291.75/292.09 Y_26)))). 291.75/292.09 fof(fact_1824_less__imp__neq, axiom, 291.75/292.09 ![Y_26, X_32]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_32), 291.75/292.09 Y_26)) 291.75/292.09 => X_32!=Y_26)). 291.75/292.09 fof(fact_182_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 ((A_1=zero_zero_complex 291.75/292.09 & hAPP_int_nat(number_number_of_nat, Wa)!=zero_zero_nat) 291.75/292.09 <=> zero_zero_complex=hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex, 291.75/292.09 A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa)))). 291.75/292.09 fof(fact_183_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral, 291.75/292.09 A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa))=zero_z126310315umeral 291.75/292.09 <=> (A_1=zero_z126310315umeral 291.75/292.09 & zero_zero_nat!=hAPP_int_nat(number_number_of_nat, Wa)))). 291.75/292.09 fof(fact_1840_not__less__iff__gr__or__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Xa), 291.75/292.09 Ya)) 291.75/292.09 <=> (Xa=Ya 291.75/292.09 | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Ya), 291.75/292.09 Xa))))). 291.75/292.09 fof(fact_1841_not__less__iff__gr__or__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((Xa=Ya 291.75/292.09 | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Ya), 291.75/292.09 Xa))) 291.75/292.09 <=> ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1842_not__less__iff__gr__or__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Xa), 291.75/292.09 Ya)) 291.75/292.09 <=> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 | Ya=Xa))). 291.75/292.09 fof(fact_1843_not__less__iff__gr__or__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((Xa=Ya 291.75/292.09 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ya), 291.75/292.09 Xa))) 291.75/292.09 <=> ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1844_not__less__iff__gr__or__eq, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Ya), 291.75/292.09 Xa)) 291.75/292.09 | Ya=Xa) 291.75/292.09 <=> ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Xa), 291.75/292.09 Ya))) 291.75/292.09 <= (is_int(Ya) & is_int(Xa)))). 291.75/292.09 fof(fact_1845_linorder__neq__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (Xa!=Ya 291.75/292.09 <=> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Ya), 291.75/292.09 Xa)) 291.75/292.09 | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Xa), 291.75/292.09 Ya))))). 291.75/292.09 fof(fact_1846_linorder__neq__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Xa), 291.75/292.09 Ya))) 291.75/292.09 <=> Xa!=Ya)). 291.75/292.09 fof(fact_1847_linorder__neq__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (Xa!=Ya 291.75/292.09 <=> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Ya), 291.75/292.09 Xa))))). 291.75/292.09 fof(fact_1848_linorder__neq__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ya), 291.75/292.09 Xa)) 291.75/292.09 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Xa), 291.75/292.09 Ya))) 291.75/292.09 <=> Ya!=Xa)). 291.75/292.09 fof(fact_1849_linorder__neq__iff, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((is_int(Xa) & is_int(Ya)) 291.75/292.09 => ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Ya), 291.75/292.09 Xa)) 291.75/292.09 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Xa), 291.75/292.09 Ya))) 291.75/292.09 <=> Ya!=Xa))). 291.75/292.09 fof(fact_184_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 ((A_1=zero_zero_real 291.75/292.09 & hAPP_int_nat(number_number_of_nat, Wa)!=zero_zero_nat) 291.75/292.09 <=> zero_zero_real=hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.75/292.09 A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa)))). 291.75/292.09 fof(fact_1850_order__less__irrefl, axiom, 291.75/292.09 ![X_29]: 291.75/292.09 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_29), 291.75/292.09 X_29))). 291.75/292.09 fof(fact_1851_order__less__irrefl, axiom, 291.75/292.09 ![Xa]: 291.75/292.09 ~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Xa))). 291.75/292.09 fof(fact_1852_order__less__irrefl, axiom, 291.75/292.09 ![Xa]: 291.75/292.09 ~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Xa))). 291.75/292.09 fof(fact_1853_order__less__irrefl, axiom, 291.75/292.09 ![X_29]: 291.75/292.09 ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_29), 291.75/292.09 X_29))). 291.75/292.09 fof(fact_1854_order__less__irrefl, axiom, 291.75/292.09 ![X_29]: 291.75/292.09 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_29), 291.75/292.09 X_29))). 291.75/292.09 fof(fact_1855_order__less__irrefl, axiom, 291.75/292.09 ![X_29]: 291.75/292.09 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_29), 291.75/292.09 X_29))). 291.75/292.09 fof(fact_1856_order__less__irrefl, axiom, 291.75/292.09 ![X_29]: 291.75/292.09 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_29), 291.75/292.09 X_29))). 291.75/292.09 fof(fact_185_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 (zero_zero_nat=hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat, 291.75/292.09 A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa)) 291.75/292.09 <=> (zero_zero_nat!=hAPP_int_nat(number_number_of_nat, Wa) 291.75/292.09 & zero_zero_nat=A_1))). 291.75/292.09 fof(fact_186_power__eq__0__iff__number__of, axiom, 291.75/292.09 ![A_1, Wa]: 291.75/292.09 (((hAPP_int_nat(number_number_of_nat, Wa)!=zero_zero_nat 291.75/292.09 & A_1=zero_zero_int) 291.75/292.09 <=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, A_1), 291.75/292.09 hAPP_int_nat(number_number_of_nat, Wa))=zero_zero_int) 291.75/292.09 <= is_int(A_1))). 291.75/292.09 fof(fact_1968_linorder__antisym__conv2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Xa), 291.75/292.09 Ya)) 291.75/292.09 => (Ya=Xa 291.75/292.09 <=> ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Xa), 291.75/292.09 Ya))))). 291.75/292.09 fof(fact_1969_linorder__antisym__conv2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 => (~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 <=> Ya=Xa))). 291.75/292.09 fof(fact_1970_linorder__antisym__conv2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((Xa=Ya 291.75/292.09 <=> ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 Xa), 291.75/292.09 Ya))) 291.75/292.09 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1971_linorder__antisym__conv2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((Ya=Xa 291.75/292.09 <=> ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Xa), 291.75/292.09 Ya))) 291.75/292.09 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_1972_linorder__antisym__conv2, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Xa), 291.75/292.09 Ya)) 291.75/292.09 => (Xa=Ya 291.75/292.09 <=> ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Xa), 291.75/292.09 Ya)))) 291.75/292.09 <= (is_int(Ya) & is_int(Xa)))). 291.75/292.09 fof(fact_1980_leD, axiom, 291.75/292.09 ![Y_15, X_19]: 291.75/292.09 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Y_15), 291.75/292.09 X_19)) 291.75/292.09 => ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, X_19), 291.75/292.09 Y_15)))). 291.75/292.09 fof(fact_1981_leD, axiom, 291.75/292.09 ![Y_15, X_19]: 291.75/292.09 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.09 Y_15), 291.75/292.09 X_19)) 291.75/292.09 => ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 X_19), 291.75/292.09 Y_15)))). 291.75/292.09 fof(fact_1982_leD, axiom, 291.75/292.09 ![Y_15, X_19]: 291.75/292.09 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.09 Y_15), 291.75/292.09 X_19)) 291.75/292.09 => ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.09 X_19), 291.75/292.09 Y_15)))). 291.75/292.09 fof(fact_1983_leD, axiom, 291.75/292.09 ![Y_15, X_19]: 291.75/292.09 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Y_15), 291.75/292.09 X_19)) 291.75/292.09 => ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, X_19), 291.75/292.09 Y_15)))). 291.75/292.09 fof(fact_1984_leD, axiom, 291.75/292.09 ![Y_15, X_19]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Y_15), 291.75/292.09 X_19)) 291.75/292.09 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X_19), 291.75/292.09 Y_15)))). 291.75/292.09 fof(fact_2021_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Xa), 291.75/292.09 Ya)) 291.75/292.09 & ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Ya), 291.75/292.09 Xa))) 291.75/292.09 <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_2022_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 <=> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 & ~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa))))). 291.75/292.09 fof(fact_2023_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 <=> (~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya))))). 291.75/292.09 fof(fact_2024_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.09 Xa), 291.75/292.09 Ya))) 291.75/292.09 <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.09 Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_2025_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 & ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.09 Ya), 291.75/292.09 Xa))) 291.75/292.09 <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_2026_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Xa), 291.75/292.09 Ya)) 291.75/292.09 & ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Ya), 291.75/292.09 Xa))) 291.75/292.09 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_2027_less__le__not__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Xa), Ya)) 291.75/292.09 <=> (~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.75/292.09 Ya), 291.75/292.09 Xa)) 291.75/292.09 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Xa), 291.75/292.09 Ya))))). 291.75/292.09 fof(fact_2028_order__less__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 ((hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Xa), 291.75/292.09 Ya)) 291.75/292.09 & Xa!=Ya) 291.75/292.09 <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Xa), 291.75/292.09 Ya)))). 291.75/292.09 fof(fact_2029_order__less__le, axiom, 291.75/292.09 ![Xa, Ya]: 291.75/292.09 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya)) 291.75/292.09 <=> (Ya!=Xa 291.75/292.09 & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool, 291.75/292.09 Xa), 291.75/292.09 Ya))))). 291.75/292.09 fof(fact_202_odd__nonzero, axiom, 291.75/292.09 ![Z_1]: 291.75/292.10 zero_zero_int!=hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.75/292.10 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.75/292.10 one_one_int), 291.75/292.10 Z_1)), 291.75/292.10 Z_1)). 291.75/292.10 fof(fact_2030_order__less__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 ((Xa!=Ya 291.75/292.10 & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool, 291.75/292.10 Xa), 291.75/292.10 Ya))) 291.75/292.10 <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2031_order__less__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 ((Ya!=Xa 291.75/292.10 & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.10 Xa), 291.75/292.10 Ya))) 291.75/292.10 <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2032_order__less__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Xa), 291.75/292.10 Ya)) 291.75/292.10 <=> (Ya!=Xa 291.75/292.10 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.10 Xa), 291.75/292.10 Ya))))). 291.75/292.10 fof(fact_2033_order__less__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Xa), 291.75/292.10 Ya)) 291.75/292.10 & Xa!=Ya) 291.75/292.10 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2034_order__less__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Xa), 291.75/292.10 Ya)) 291.75/292.10 <=> (Ya!=Xa 291.75/292.10 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Xa), 291.75/292.10 Ya)))) 291.75/292.10 <= (is_int(Ya) & is_int(Xa)))). 291.75/292.10 fof(fact_2040_linorder__not__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Xa), 291.75/292.10 Ya)) 291.75/292.10 <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Ya), 291.75/292.10 Xa)))). 291.75/292.10 fof(fact_2041_linorder__not__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.10 Ya), 291.75/292.10 Xa)) 291.75/292.10 <=> ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2042_linorder__not__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Ya), 291.75/292.10 Xa)) 291.75/292.10 <=> ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2043_linorder__not__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Xa), 291.75/292.10 Ya)) 291.75/292.10 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ya), 291.75/292.10 Xa)))). 291.75/292.10 fof(fact_2044_linorder__not__le, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Ya), Xa)) 291.75/292.10 <=> ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2045_linorder__not__less, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Xa), 291.75/292.10 Ya)) 291.75/292.10 <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, 291.75/292.10 Ya), 291.75/292.10 Xa)))). 291.75/292.10 fof(fact_2046_linorder__not__less, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.75/292.10 Ya), 291.75/292.10 Xa)) 291.75/292.10 <=> ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2047_linorder__not__less, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, Ya), 291.75/292.10 Xa)) 291.75/292.10 <=> ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.10 Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2048_linorder__not__less, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Xa), 291.75/292.10 Ya)) 291.75/292.10 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, 291.75/292.10 Ya), 291.75/292.10 Xa)))). 291.75/292.10 fof(fact_2049_linorder__not__less, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Ya), 291.75/292.10 Xa)) 291.75/292.10 <=> ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Xa), 291.75/292.10 Ya)))). 291.75/292.10 fof(fact_2076_less__fun__def, axiom, 291.75/292.10 ![F, G]: 291.75/292.10 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool, 291.75/292.10 F), 291.75/292.10 G)) 291.75/292.10 <=> (~hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool, 291.75/292.10 G), 291.75/292.10 F)) 291.75/292.10 & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool, 291.75/292.10 F), 291.75/292.10 G))))). 291.75/292.10 fof(fact_2077_less__fun__def, axiom, 291.75/292.10 ![F, G]: 291.75/292.10 ((hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool, 291.75/292.10 F), 291.75/292.10 G)) 291.75/292.10 & ~hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool, 291.75/292.10 G), 291.75/292.10 F))) 291.75/292.10 <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool, 291.75/292.10 F), 291.75/292.10 G)))). 291.75/292.10 fof(fact_208_zero__less__power2, axiom, 291.75/292.10 ![A_1]: 291.75/292.10 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.10 zero_zero_rat), 291.75/292.10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, A_1), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))))) 291.75/292.10 <=> A_1!=zero_zero_rat)). 291.75/292.10 fof(fact_209_zero__less__power2, axiom, 291.75/292.10 ![A_1]: 291.75/292.10 (A_1!=zero_zero_real 291.75/292.10 <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.10 zero_zero_real), 291.75/292.10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, A_1), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls)))))))). 291.75/292.10 fof(fact_210_zero__less__power2, axiom, 291.75/292.10 ![A_1]: 291.75/292.10 ((A_1!=zero_zero_int 291.75/292.10 <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 zero_zero_int), 291.75/292.10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, A_1), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))))) 291.75/292.10 <= is_int(A_1))). 291.75/292.10 fof(fact_2119_not__real__square__gt__zero, axiom, 291.75/292.10 ![Xa]: 291.75/292.10 (~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.10 zero_zero_real), 291.75/292.10 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, Xa), Xa))) 291.75/292.10 <=> Xa=zero_zero_real)). 291.75/292.10 fof(fact_211_power2__less__0, axiom, 291.75/292.10 ![A_201]: 291.75/292.10 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, 291.75/292.10 A_201), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 zero_zero_rat))). 291.75/292.10 fof(fact_212_power2__less__0, axiom, 291.75/292.10 ![A_201]: 291.75/292.10 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.75/292.10 A_201), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 zero_zero_real))). 291.75/292.10 fof(fact_213_power2__less__0, axiom, 291.75/292.10 ![A_201]: 291.75/292.10 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, 291.75/292.10 A_201), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 zero_zero_int))). 291.75/292.10 fof(fact_214_sum__power2__gt__zero__iff, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.10 zero_zero_rat), 291.75/292.10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat, 291.75/292.10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, 291.75/292.10 Xa), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, Ya), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))))) 291.75/292.10 <=> (zero_zero_rat!=Ya | zero_zero_rat!=Xa))). 291.75/292.10 fof(fact_215_sum__power2__gt__zero__iff, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.10 zero_zero_real), 291.75/292.10 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.75/292.10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.75/292.10 Xa), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.75/292.10 Ya), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))))) 291.75/292.10 <=> (zero_zero_real!=Ya | zero_zero_real!=Xa))). 291.75/292.10 fof(fact_2165_rel__simps_I42_J, axiom, 291.75/292.10 ![L_2]: min!=hAPP_int_int(bit0, L_2)). 291.75/292.10 fof(fact_2166_rel__simps_I45_J, axiom, 291.75/292.10 ![K_2]: min!=hAPP_int_int(bit0, K_2)). 291.75/292.10 fof(fact_216_sum__power2__gt__zero__iff, axiom, 291.75/292.10 ![Xa, Ya]: 291.75/292.10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 zero_zero_int), 291.75/292.10 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.75/292.10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, 291.75/292.10 Xa), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, Ya), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))))) 291.75/292.10 <=> (Ya!=zero_zero_int | zero_zero_int!=Xa)) 291.75/292.10 <= (is_int(Ya) & is_int(Xa)))). 291.75/292.10 fof(fact_217_not__sum__power2__lt__zero, axiom, 291.75/292.10 ![X_64, Y_53]: 291.75/292.10 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat, 291.75/292.10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, 291.75/292.10 X_64), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, 291.75/292.10 Y_53), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls)))))), 291.75/292.10 zero_zero_rat))). 291.75/292.10 fof(fact_218_not__sum__power2__lt__zero, axiom, 291.75/292.10 ![X_64, Y_53]: 291.75/292.10 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.75/292.10 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.75/292.10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.75/292.10 X_64), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.75/292.10 Y_53), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls)))))), 291.75/292.10 zero_zero_real))). 291.75/292.10 fof(fact_219_not__sum__power2__lt__zero, axiom, 291.75/292.10 ![X_64, Y_53]: 291.75/292.10 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.75/292.10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, 291.75/292.10 X_64), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls))))), 291.75/292.10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, 291.75/292.10 Y_53), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls)))))), 291.75/292.10 zero_zero_int))). 291.75/292.10 fof(fact_22_zero__eq__power2, axiom, 291.75/292.10 ![A_1]: 291.75/292.10 ((A_1=zero_zero_int 291.75/292.10 <=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, A_1), 291.75/292.10 hAPP_int_nat(number_number_of_nat, 291.75/292.10 hAPP_int_int(bit0, hAPP_int_int(bit1, pls))))=zero_zero_int) 291.75/292.10 <= is_int(A_1))). 291.75/292.10 fof(fact_2401_zdvd__not__zless, axiom, 291.75/292.10 ![N, M]: 291.75/292.10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 zero_zero_int), 291.75/292.10 M)) 291.75/292.10 => (~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int, N), 291.75/292.10 M)) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, M), 291.75/292.10 N))))). 291.75/292.10 fof(fact_2464_zcong__neg__1__impl__ne__1, axiom, 291.75/292.10 ![X, P_3]: 291.75/292.10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X), 291.75/292.10 hAPP_int_int(number_number_of_int, min)), 291.75/292.10 P_3)) 291.75/292.10 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X), 291.75/292.10 one_one_int), 291.75/292.10 P_3))) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 hAPP_int_int(number_number_of_int, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls)))), 291.75/292.10 P_3)))). 291.75/292.10 fof(fact_2465_one__not__neg__one__mod__m, axiom, 291.75/292.10 ![M]: 291.75/292.10 (~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(one_one_int), 291.75/292.10 hAPP_int_int(number_number_of_int, min)), 291.75/292.10 M)) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 hAPP_int_int(number_number_of_int, 291.75/292.10 hAPP_int_int(bit0, 291.75/292.10 hAPP_int_int(bit1, 291.75/292.10 pls)))), 291.75/292.10 M)))). 291.75/292.10 fof(fact_2506_nat__dvd__not__less, axiom, 291.75/292.10 ![N, M]: 291.75/292.10 ((~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, N), M)) 291.75/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M), 291.75/292.10 N))) 291.75/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.75/292.10 zero_zero_nat), 291.75/292.10 M)))). 291.75/292.10 fof(fact_2547_divides__div__not, axiom, 291.75/292.10 ![N, X, R_2, Q_1]: 291.75/292.10 (((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, R_2), 291.75/292.10 N)) 291.75/292.10 => ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, N), 291.75/292.10 X))) 291.75/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.75/292.10 zero_zero_nat), 291.75/292.10 R_2))) 291.75/292.10 <= hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat, 291.75/292.10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat, 291.75/292.10 Q_1), 291.75/292.10 N)), 291.75/292.10 R_2)=X)). 291.75/292.10 fof(fact_2587_zcong__not__zero, axiom, 291.75/292.10 ![X, M]: 291.75/292.10 ((~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X), 291.75/292.10 zero_zero_int), 291.75/292.10 M)) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, X), 291.75/292.10 M))) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 zero_zero_int), 291.75/292.10 X)))). 291.75/292.10 fof(fact_2588_zcong__not, axiom, 291.75/292.10 ![M, A_3, B_2]: 291.75/292.10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_3), 291.75/292.10 M)) 291.75/292.10 => ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, B_2), 291.75/292.10 A_3)) 291.75/292.10 => ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3), B_2), 291.75/292.10 M))) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 zero_zero_int), 291.75/292.10 B_2)))) 291.75/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.75/292.10 zero_zero_int), 291.75/292.10 A_3)))). 291.75/292.10 fof(fact_2717_zero__less__abs__iff, axiom, 291.75/292.10 ![A_1]: 291.75/292.10 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.75/292.10 zero_zero_rat), 291.75/292.10 hAPP_rat_rat(abs_abs_rat, A_1))) 291.75/292.10 <=> zero_zero_rat!=A_1)). 291.75/292.10 fof(fact_2718_zero__less__abs__iff, axiom, 291.75/292.10 ![A_1]: 291.77/292.10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.10 zero_zero_real), 291.77/292.10 hAPP_real_real(abs_abs_real, A_1))) 291.77/292.10 <=> A_1!=zero_zero_real)). 291.77/292.10 fof(fact_2719_zero__less__abs__iff, axiom, 291.77/292.10 ![A_1]: 291.77/292.10 ((A_1!=zero_zero_int 291.77/292.10 <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 zero_zero_int), 291.77/292.10 hAPP_int_int(abs_abs_int, A_1)))) 291.77/292.10 <= is_int(A_1))). 291.77/292.10 fof(fact_2720_abs__not__less__zero, axiom, 291.77/292.10 ![A_16]: 291.77/292.10 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.77/292.10 hAPP_rat_rat(abs_abs_rat, A_16)), 291.77/292.10 zero_zero_rat))). 291.77/292.10 fof(fact_2721_abs__not__less__zero, axiom, 291.77/292.10 ![A_16]: 291.77/292.10 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.10 hAPP_real_real(abs_abs_real, A_16)), 291.77/292.10 zero_zero_real))). 291.77/292.10 fof(fact_2722_abs__not__less__zero, axiom, 291.77/292.10 ![A_16]: 291.77/292.10 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 hAPP_int_int(abs_abs_int, A_16)), 291.77/292.10 zero_zero_int))). 291.77/292.10 fof(fact_274_less__zeroE, axiom, 291.77/292.10 ![N]: 291.77/292.10 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, N), 291.77/292.10 zero_zero_nat))). 291.77/292.10 fof(fact_276_less__not__refl, axiom, 291.77/292.10 ![N]: 291.77/292.10 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, N), N))). 291.77/292.10 fof(fact_277_not__add__less1, axiom, 291.77/292.10 ![J_2, I]: 291.77/292.10 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.77/292.10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat, 291.77/292.10 I), 291.77/292.10 J_2)), 291.77/292.10 I))). 291.77/292.10 fof(fact_278_not__add__less2, axiom, 291.77/292.10 ![J_2, I]: 291.77/292.10 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.77/292.10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat, 291.77/292.10 J_2), 291.77/292.10 I)), 291.77/292.10 I))). 291.77/292.10 fof(fact_280_nat__neq__iff, axiom, 291.77/292.10 ![Ma, Na]: 291.77/292.10 (Ma!=Na 291.77/292.10 <=> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Na), 291.77/292.10 Ma)) 291.77/292.10 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ma), 291.77/292.10 Na))))). 291.77/292.10 fof(fact_2846_abs__add__one__not__less__self, axiom, 291.77/292.10 ![X]: 291.77/292.10 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.10 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.77/292.10 hAPP_real_real(abs_abs_real, 291.77/292.10 X)), 291.77/292.10 one_one_real)), 291.77/292.10 X))). 291.77/292.10 fof(fact_2852_split__zdiv, axiom, 291.77/292.10 ![Na, P_1, K]: 291.77/292.10 ((hBOOL(hAPP_int_bool(P_1, 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(div_div_int, Na), K))) 291.77/292.10 <=> ((hBOOL(hAPP_int_bool(P_1, zero_zero_int)) <= K=zero_zero_int) 291.77/292.10 & ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, K), 291.77/292.10 zero_zero_int)) 291.77/292.10 => ![I_2, J_1]: 291.77/292.10 ((is_int(I_2) & is_int(J_1)) 291.77/292.10 => ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, K), 291.77/292.10 J_1)) 291.77/292.10 & (Na=hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.77/292.10 K), 291.77/292.10 I_2)), 291.77/292.10 J_1) 291.77/292.10 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.77/292.10 J_1), 291.77/292.10 zero_zero_int)))) 291.77/292.10 => hBOOL(hAPP_int_bool(P_1, I_2))))) 291.77/292.10 & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 zero_zero_int), 291.77/292.10 K)) 291.77/292.10 => ![I_2, J_1]: 291.77/292.10 ((is_int(J_1) & is_int(I_2)) 291.77/292.10 => ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.77/292.10 zero_zero_int), 291.77/292.10 J_1)) 291.77/292.10 & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 J_1), 291.77/292.10 K)) 291.77/292.10 & Na=hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.77/292.10 K), 291.77/292.10 I_2)), 291.77/292.10 J_1))) 291.77/292.10 => hBOOL(hAPP_int_bool(P_1, I_2)))))))) 291.77/292.10 <= (is_int(K) & is_int(Na)))). 291.77/292.10 fof(fact_288_less__irrefl__nat, axiom, 291.77/292.10 ![N]: 291.77/292.10 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, N), N))). 291.77/292.10 fof(fact_289_less__not__refl2, axiom, 291.77/292.10 ![N, M]: 291.77/292.10 (M!=N 291.77/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, N), 291.77/292.10 M)))). 291.77/292.10 fof(fact_2902_inv__not__p__minus__1, axiom, 291.77/292.10 ![P_3, A_3]: 291.77/292.10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 one_one_int), 291.77/292.10 A_3)) 291.77/292.10 => (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_3), 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int, P_3), 291.77/292.10 one_one_int))) 291.77/292.10 => hAPP_int_int(inv(P_3), 291.77/292.10 A_3)!=hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int, P_3), 291.77/292.10 one_one_int))) 291.77/292.10 <= hBOOL(hAPP_int_bool(zprime, P_3)))). 291.77/292.10 fof(fact_2903_inv__not__1, axiom, 291.77/292.10 ![P_3, A_3]: 291.77/292.10 (hBOOL(hAPP_int_bool(zprime, P_3)) 291.77/292.10 => ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_3), 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int, P_3), 291.77/292.10 one_one_int))) 291.77/292.10 => one_one_int!=hAPP_int_int(inv(P_3), A_3)) 291.77/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 one_one_int), 291.77/292.10 A_3))))). 291.77/292.10 fof(fact_2904_inv__distinct, axiom, 291.77/292.10 ![P_3, A_3]: 291.77/292.10 (hBOOL(hAPP_int_bool(zprime, P_3)) 291.77/292.10 => ((hAPP_int_int(inv(P_3), A_3)!=A_3 291.77/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_3), 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int, P_3), 291.77/292.10 one_one_int)))) 291.77/292.10 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 one_one_int), 291.77/292.10 A_3))))). 291.77/292.10 fof(fact_2907_inv__not__0, axiom, 291.77/292.10 ![P_3, A_3]: 291.77/292.10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.10 one_one_int), 291.77/292.10 A_3)) 291.77/292.10 => (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_3), 291.77/292.10 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int, P_3), 291.77/292.10 one_one_int))) 291.77/292.10 => hAPP_int_int(inv(P_3), A_3)!=zero_zero_int)) 291.77/292.10 <= hBOOL(hAPP_int_bool(zprime, P_3)))). 291.77/292.10 fof(fact_290_less__not__refl3, axiom, 291.77/292.10 ![S_1, T_1]: 291.77/292.10 (T_1!=S_1 291.77/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, S_1), 291.77/292.10 T_1)))). 291.77/292.10 fof(fact_2942_d22set__le__swap, axiom, 291.77/292.10 ![A_1, B_1]: 291.77/292.10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_1), 291.77/292.10 B_1)) 291.77/292.10 => ~hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.10 B_1), 291.77/292.10 hAPP_i1948725293t_bool(d22set, A_1))))). 291.77/292.10 fof(fact_297_nat__less__cases, axiom, 291.77/292.10 ![Ma, Na, P_1]: 291.77/292.10 (((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1, Na), Ma)) 291.77/292.10 <= (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1, Na), Ma)) 291.77/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Na), 291.77/292.10 Ma)))) 291.77/292.10 <= (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1, Na), Ma)) 291.77/292.10 <= Na=Ma)) 291.77/292.10 <= (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1, Na), Ma)) 291.77/292.10 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ma), 291.77/292.10 Na))))). 291.77/292.10 fof(fact_2984_Suc__n__not__n, axiom, 291.77/292.10 ![N]: hAPP_nat_nat(suc, N)!=N). 291.77/292.10 fof(fact_2985_n__not__Suc__n, axiom, 291.77/292.10 ![N]: hAPP_nat_nat(suc, N)!=N). 291.77/292.10 fof(fact_2992_Zero__not__Suc, axiom, 291.77/292.10 ![M]: hAPP_nat_nat(suc, M)!=zero_zero_nat). 291.77/292.10 fof(fact_2993_nat_Osimps_I2_J, axiom, 291.77/292.10 ![Nat_3]: zero_zero_nat!=hAPP_nat_nat(suc, Nat_3)). 291.77/292.10 fof(fact_2994_Suc__not__Zero, axiom, 291.77/292.10 ![M]: hAPP_nat_nat(suc, M)!=zero_zero_nat). 291.77/292.10 fof(fact_2995_nat_Osimps_I3_J, axiom, 291.77/292.10 ![Nat_2]: zero_zero_nat!=hAPP_nat_nat(suc, Nat_2)). 291.77/292.10 fof(fact_2996_Zero__neq__Suc, axiom, 291.77/292.10 ![M]: zero_zero_nat!=hAPP_nat_nat(suc, M)). 291.77/292.10 fof(fact_2997_Suc__neq__Zero, axiom, 291.77/292.10 ![M]: zero_zero_nat!=hAPP_nat_nat(suc, M)). 291.77/292.10 fof(fact_299_gr__implies__not0, axiom, 291.77/292.10 ![N, M]: 291.77/292.10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M), N)) 291.77/292.10 => zero_zero_nat!=N)). 291.77/292.10 fof(fact_3014_not__less__eq, axiom, 291.77/292.10 ![Ma, Na]: 291.77/292.10 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Ma), 291.77/292.10 Na)) 291.77/292.10 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Na), 291.77/292.11 hAPP_nat_nat(suc, Ma))))). 291.77/292.11 fof(fact_3018_Suc__n__not__le__n, axiom, 291.77/292.11 ![N]: 291.77/292.11 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, 291.77/292.11 hAPP_nat_nat(suc, N)), 291.77/292.11 N))). 291.77/292.11 fof(fact_3019_not__less__eq__eq, axiom, 291.77/292.11 ![Ma, Na]: 291.77/292.11 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Ma), 291.77/292.11 Na)) 291.77/292.11 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, 291.77/292.11 hAPP_nat_nat(suc, Na)), 291.77/292.11 Ma)))). 291.77/292.11 fof(fact_301_less__nat__zero__code, axiom, 291.77/292.11 ![N]: 291.77/292.11 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, N), 291.77/292.11 zero_zero_nat))). 291.77/292.11 fof(fact_304_neq0__conv, axiom, 291.77/292.11 ![Na]: 291.77/292.11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.77/292.11 zero_zero_nat), 291.77/292.11 Na)) 291.77/292.11 <=> Na!=zero_zero_nat)). 291.77/292.11 fof(fact_305_not__less0, axiom, 291.77/292.11 ![N]: 291.77/292.11 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, N), 291.77/292.11 zero_zero_nat))). 291.77/292.11 fof(fact_3105_int__Suc, axiom, 291.77/292.11 ![M]: 291.77/292.11 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, one_one_int), 291.77/292.11 hAPP_nat_int(semiri1621563631at_int, 291.77/292.11 M))=hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, M))). 291.77/292.11 fof(fact_3125_split__zmod, axiom, 291.77/292.11 ![Na, P_1, K]: 291.77/292.11 ((is_int(Na) & is_int(K)) 291.77/292.11 => (hBOOL(hAPP_int_bool(P_1, hAPP_int_int(div_mod_int(Na), K))) 291.77/292.11 <=> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, K), 291.77/292.11 zero_zero_int)) 291.77/292.11 => ![I_2, J_1]: 291.77/292.11 ((hBOOL(hAPP_int_bool(P_1, J_1)) 291.77/292.11 <= (Na=hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.77/292.11 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.77/292.11 K), 291.77/292.11 I_2)), 291.77/292.11 J_1) 291.77/292.11 & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.77/292.11 J_1), 291.77/292.11 zero_zero_int)) 291.77/292.11 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.11 K), 291.77/292.11 J_1))))) 291.77/292.11 <= (is_int(I_2) & is_int(J_1)))) 291.77/292.11 & ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.11 zero_zero_int), 291.77/292.11 K)) 291.77/292.11 => ![I_2, J_1]: 291.77/292.11 ((is_int(I_2) & is_int(J_1)) 291.77/292.11 => (hBOOL(hAPP_int_bool(P_1, J_1)) 291.77/292.11 <= (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.11 J_1), 291.77/292.11 K)) 291.77/292.11 & (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.77/292.11 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.77/292.11 K), 291.77/292.11 I_2)), 291.77/292.11 J_1)=Na 291.77/292.11 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.77/292.11 zero_zero_int), 291.77/292.11 J_1))))))) 291.77/292.11 & (hBOOL(hAPP_int_bool(P_1, Na)) <= K=zero_zero_int)))))). 291.77/292.11 fof(fact_3149_neq__one__mod__two, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (hAPP_int_int(div_mod_int(Xa), 291.77/292.11 hAPP_int_int(number_number_of_int, 291.77/292.11 hAPP_int_int(bit0, hAPP_int_int(bit1, pls))))!=zero_zero_int 291.77/292.11 <=> one_one_int=hAPP_int_int(div_mod_int(Xa), 291.77/292.11 hAPP_int_int(number_number_of_int, 291.77/292.11 hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))))). 291.77/292.11 fof(fact_3196_StandardRes__SRStar__prop1a, axiom, 291.77/292.11 ![Xa, P_5]: 291.77/292.11 (~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa), 291.77/292.11 zero_zero_int), 291.77/292.11 P_5)) 291.77/292.11 <= hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.11 Xa), 291.77/292.11 hAPP_i1948725293t_bool(sRStar, P_5))))). 291.77/292.11 fof(fact_3210_StandardRes__SRStar__prop1, axiom, 291.77/292.11 ![Xa, P_5]: 291.77/292.11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.11 hAPP_int_int(number_number_of_int, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))), 291.77/292.11 P_5)) 291.77/292.11 => (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.11 hAPP_int_int(standardRes(P_5), Xa)), 291.77/292.11 hAPP_i1948725293t_bool(sRStar, P_5))) 291.77/292.11 <=> ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa), 291.77/292.11 zero_zero_int), 291.77/292.11 P_5))))). 291.77/292.11 fof(fact_3242_not__neg__int, axiom, 291.77/292.11 ![N]: 291.77/292.11 ~hBOOL(hAPP_int_bool(nat_neg, 291.77/292.11 hAPP_nat_int(semiri1621563631at_int, N)))). 291.77/292.11 fof(fact_3247_not__neg__eq__ge__0, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.77/292.11 zero_zero_int), 291.77/292.11 Xa)) 291.77/292.11 <=> ~hBOOL(hAPP_int_bool(nat_neg, Xa)))). 291.77/292.11 fof(fact_3251_even__power, axiom, 291.77/292.11 ![Xa, Na]: 291.77/292.11 (hBOOL(hAPP_int_bool(even_odd_even_int, 291.77/292.11 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, Xa), Na))) 291.77/292.11 <=> (zero_zero_nat!=Na 291.77/292.11 & hBOOL(hAPP_int_bool(even_odd_even_int, Xa))))). 291.77/292.11 fof(fact_3280_odd__equiv__def, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 ((~hBOOL(hAPP_int_bool(even_odd_even_int, Xa)) 291.77/292.11 <=> ?[Y_1]: 291.77/292.11 (is_int(Y_1) 291.77/292.11 & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.77/292.11 hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.77/292.11 hAPP_int_int(number_number_of_int, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))), 291.77/292.11 Y_1)), 291.77/292.11 one_one_int)=Xa)) 291.77/292.11 <= is_int(Xa))). 291.77/292.11 fof(fact_3344_odd__iff__not__even, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.11 Xa), 291.77/292.11 zOdd)) 291.77/292.11 <=> ~hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.11 Xa), 291.77/292.11 zEven)))). 291.77/292.11 fof(fact_3345_even__odd__conj, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 ~(hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.11 Xa), 291.77/292.11 zEven)) 291.77/292.11 & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.77/292.11 Xa), 291.77/292.11 zOdd)))). 291.77/292.11 fof(fact_3398_not__exp__le__zero, axiom, 291.77/292.11 ![X]: 291.77/292.11 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.77/292.11 hAPP_real_real(exp_real, X)), 291.77/292.11 zero_zero_real))). 291.77/292.11 fof(fact_3400_not__exp__less__zero, axiom, 291.77/292.11 ![X]: 291.77/292.11 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.11 hAPP_real_real(exp_real, X)), 291.77/292.11 zero_zero_real))). 291.77/292.11 fof(fact_3460_minus__Pls, axiom, 291.77/292.11 pls=hAPP_int_int(uminus_uminus_int, pls)). 291.77/292.11 fof(fact_3477_not__int__zless__negative, axiom, 291.77/292.11 ![N, M]: 291.77/292.11 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.77/292.11 hAPP_nat_int(semiri1621563631at_int, N)), 291.77/292.11 hAPP_int_int(uminus_uminus_int, 291.77/292.11 hAPP_nat_int(semiri1621563631at_int, M))))). 291.77/292.11 fof(fact_3496_not__zle__0__negative, axiom, 291.77/292.11 ![N]: 291.77/292.11 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.77/292.11 zero_zero_int), 291.77/292.11 hAPP_int_int(uminus_uminus_int, 291.77/292.11 hAPP_nat_int(semiri1621563631at_int, 291.77/292.11 hAPP_nat_nat(suc, N)))))). 291.77/292.11 fof(fact_3530_real__sqrt__not__eq__zero, axiom, 291.77/292.11 ![X]: 291.77/292.11 (hAPP_real_real(sqrt, X)!=zero_zero_real 291.77/292.11 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.11 zero_zero_real), 291.77/292.11 X)))). 291.77/292.11 fof(fact_3606_cos__arctan__not__zero, axiom, 291.77/292.11 ![X]: 291.77/292.11 zero_zero_real!=hAPP_real_real(cos, hAPP_real_real(arctan, X))). 291.77/292.11 fof(fact_3701_not__real__of__nat__less__zero, axiom, 291.77/292.11 ![N]: 291.77/292.11 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.11 hAPP_nat_real(real_nat, N)), 291.77/292.11 zero_zero_real))). 291.77/292.11 fof(fact_3784_cos__zero__iff, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (zero_zero_real=hAPP_real_real(cos, Xa) 291.77/292.11 <=> (?[N_1]: 291.77/292.11 (~hBOOL(hAPP_nat_bool(even_odd_even_nat, N_1)) 291.77/292.11 & Xa=hAPP_real_real(uminus_uminus_real, 291.77/292.11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.77/292.11 hAPP_nat_real(real_nat, 291.77/292.11 N_1)), 291.77/292.11 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.77/292.11 pi), 291.77/292.11 hAPP_int_real(number267125858f_real, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls))))))) 291.77/292.11 | ?[N_1]: 291.77/292.11 (~hBOOL(hAPP_nat_bool(even_odd_even_nat, N_1)) 291.77/292.11 & hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.77/292.11 hAPP_nat_real(real_nat, N_1)), 291.77/292.11 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.77/292.11 pi), 291.77/292.11 hAPP_int_real(number267125858f_real, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))))=Xa)))). 291.77/292.11 fof(fact_3789_even__Suc, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (hBOOL(hAPP_nat_bool(even_odd_even_nat, hAPP_nat_nat(suc, Xa))) 291.77/292.11 <=> ~hBOOL(hAPP_nat_bool(even_odd_even_nat, Xa)))). 291.77/292.11 fof(fact_3792_real__of__nat__fact__not__zero, axiom, 291.77/292.11 ![N]: 291.77/292.11 zero_zero_real!=hAPP_nat_real(real_nat, 291.77/292.11 hAPP_nat_nat(fact_fact_nat, N))). 291.77/292.11 fof(fact_3800_odd__nat__equiv__def2, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (~hBOOL(hAPP_nat_bool(even_odd_even_nat, Xa)) 291.77/292.11 <=> ?[Y_1]: 291.77/292.11 hAPP_nat_nat(suc, 291.77/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat, 291.77/292.11 hAPP_nat_nat(suc, 291.77/292.11 hAPP_nat_nat(suc, 291.77/292.11 zero_zero_nat))), 291.77/292.11 Y_1))=Xa)). 291.77/292.11 fof(fact_3805_odd__nat__equiv__def, axiom, 291.77/292.11 ![Xa]: 291.77/292.11 (~hBOOL(hAPP_nat_bool(even_odd_even_nat, Xa)) 291.77/292.11 <=> hAPP_nat_nat(suc, zero_zero_nat)=hAPP_nat_nat(div_mod_nat(Xa), 291.77/292.11 hAPP_nat_nat(suc, 291.77/292.11 hAPP_nat_nat(suc, 291.77/292.11 zero_zero_nat))))). 291.77/292.11 fof(fact_3809_even__num__iff, axiom, 291.77/292.11 ![Na]: 291.77/292.11 ((hBOOL(hAPP_nat_bool(even_odd_even_nat, Na)) 291.77/292.11 <=> ~hBOOL(hAPP_nat_bool(even_odd_even_nat, 291.77/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat, Na), 291.77/292.11 one_one_nat)))) 291.77/292.11 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.77/292.11 zero_zero_nat), 291.77/292.11 Na)))). 291.77/292.11 fof(fact_3817_odd__Suc__mult__two__ex, axiom, 291.77/292.11 ![Na]: 291.77/292.11 (~hBOOL(hAPP_nat_bool(even_odd_even_nat, Na)) 291.77/292.11 <=> ?[M_1]: 291.77/292.11 Na=hAPP_nat_nat(suc, 291.77/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat, 291.77/292.11 hAPP_int_nat(number_number_of_nat, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))), 291.77/292.11 M_1)))). 291.77/292.11 fof(fact_3822_lemma__odd__mod__4__div__2, axiom, 291.77/292.11 ![N]: 291.77/292.11 (hAPP_nat_nat(div_mod_nat(N), 291.77/292.11 hAPP_int_nat(number_number_of_nat, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))))=hAPP_int_nat(number_number_of_nat, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls))) 291.77/292.11 => ~hBOOL(hAPP_nat_bool(even_odd_even_nat, 291.77/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat, 291.77/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat, 291.77/292.11 N), 291.77/292.11 one_one_nat)), 291.77/292.11 hAPP_int_nat(number_number_of_nat, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))))))). 291.77/292.11 fof(fact_3824_cos__zero__lemma, axiom, 291.77/292.11 ![X]: 291.77/292.11 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.77/292.11 zero_zero_real), 291.77/292.11 X)) 291.77/292.11 => (zero_zero_real=hAPP_real_real(cos, X) 291.77/292.11 => ?[N_1]: 291.77/292.11 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.77/292.11 hAPP_nat_real(real_nat, N_1)), 291.77/292.11 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, pi), 291.77/292.11 hAPP_int_real(number267125858f_real, 291.77/292.11 hAPP_int_int(bit0, 291.77/292.11 hAPP_int_int(bit1, 291.77/292.11 pls)))))=X 291.77/292.11 & ~hBOOL(hAPP_nat_bool(even_odd_even_nat, N_1)))))). 291.77/292.11 fof(fact_3831_fact__nonzero__nat, axiom, 291.77/292.11 ![N]: zero_zero_nat!=hAPP_nat_nat(fact_fact_nat, N)). 291.77/292.11 fof(fact_3885_DERIV__real__root__generic, axiom, 291.77/292.11 ![Xa, Na, D]: 291.77/292.11 ((((hBOOL(hAPP_nat_bool(even_odd_even_nat, Na)) 291.77/292.11 => (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.77/292.11 zero_zero_real), 291.77/292.11 Xa)) 291.77/292.11 => D=hAPP_real_real(inverse_inverse_real, 291.77/292.11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.77/292.11 hAPP_nat_real(real_nat, 291.77/292.11 Na)), 291.77/292.11 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.11 hAPP_real_real(hAPP_n546711566l_real(root, 291.78/292.11 Na), 291.78/292.11 Xa)), 291.78/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat, 291.78/292.11 Na), 291.78/292.11 hAPP_nat_nat(suc, 291.78/292.11 zero_zero_nat))))))) 291.78/292.11 => ((hBOOL(hAPP_nat_bool(even_odd_even_nat, Na)) 291.78/292.11 => (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Xa), 291.78/292.11 zero_zero_real)) 291.78/292.11 => hAPP_real_real(uminus_uminus_real, 291.78/292.11 hAPP_real_real(inverse_inverse_real, 291.78/292.11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.11 hAPP_nat_real(real_nat, 291.78/292.11 Na)), 291.78/292.11 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.11 hAPP_real_real(hAPP_n546711566l_real(root, 291.78/292.11 Na), 291.78/292.11 Xa)), 291.78/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat, 291.78/292.11 Na), 291.78/292.11 hAPP_nat_nat(suc, 291.78/292.11 zero_zero_nat))))))=D)) 291.78/292.11 => ((D=hAPP_real_real(inverse_inverse_real, 291.78/292.11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.11 hAPP_nat_real(real_nat, 291.78/292.11 Na)), 291.78/292.11 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.11 hAPP_real_real(hAPP_n546711566l_real(root, 291.78/292.11 Na), 291.78/292.11 Xa)), 291.78/292.11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat, 291.78/292.11 Na), 291.78/292.11 hAPP_nat_nat(suc, 291.78/292.11 zero_zero_nat))))) 291.78/292.11 <= ~hBOOL(hAPP_nat_bool(even_odd_even_nat, Na))) 291.78/292.11 => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root, Na), 291.78/292.11 Xa), 291.78/292.11 D))))) 291.78/292.11 <= Xa!=zero_zero_real) 291.78/292.11 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.11 zero_zero_nat), 291.78/292.11 Na)))). 291.78/292.11 fof(fact_3888_MVT2, axiom, 291.78/292.11 ![F, A_1, B_1, F_1]: 291.78/292.11 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.11 B_1)) 291.78/292.11 => (?[Z]: 291.78/292.11 (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.11 hAPP_real_real(F, B_1)), 291.78/292.11 hAPP_real_real(F, 291.78/292.11 A_1))=hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.11 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.11 B_1), 291.78/292.11 A_1)), 291.78/292.11 hAPP_real_real(F_1, Z)) 291.78/292.11 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, Z), 291.78/292.11 B_1)) 291.78/292.11 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.11 Z)))) 291.78/292.11 <= ![X_1]: 291.78/292.11 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.11 A_1), 291.78/292.11 X_1)) 291.78/292.11 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.11 X_1), 291.78/292.11 B_1))) 291.78/292.11 => hBOOL(hAPP_real_bool(deriv_real(F, X_1), 291.78/292.11 hAPP_real_real(F_1, X_1))))))). 291.78/292.11 fof(fact_3972_complex__i__not__number__of, axiom, 291.78/292.11 ![W]: hAPP_int_complex(number528085621omplex, W)!=ii). 291.78/292.11 fof(fact_4004_int__induct, axiom, 291.78/292.11 ![P_1, K, I_1]: 291.78/292.11 (hBOOL(hAPP_int_bool(P_1, K)) 291.78/292.11 => ((![I_2]: 291.78/292.11 (((hBOOL(hAPP_int_bool(P_1, 291.78/292.11 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int, I_2), 291.78/292.11 one_one_int))) 291.78/292.11 <= hBOOL(hAPP_int_bool(P_1, I_2))) 291.78/292.11 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.11 I_2), 291.78/292.11 K))) 291.78/292.11 <= is_int(I_2)) 291.78/292.11 => hBOOL(hAPP_int_bool(P_1, I_1))) 291.78/292.11 <= ![I_2]: 291.78/292.11 (((hBOOL(hAPP_int_bool(P_1, I_2)) 291.78/292.11 => hBOOL(hAPP_int_bool(P_1, 291.78/292.11 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, 291.78/292.11 I_2), 291.78/292.11 one_one_int)))) 291.78/292.11 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, K), 291.78/292.11 I_2))) 291.78/292.11 <= is_int(I_2))))). 291.78/292.11 fof(fact_4080_code__numeral_Osimps_I3_J, axiom, 291.78/292.11 ![Code_numeral_3]: 291.78/292.11 zero_z126310315umeral!=code_S1047413653umeral(Code_numeral_3)). 291.78/292.11 fof(fact_4081_code__numeral_Osimps_I2_J, axiom, 291.78/292.11 ![Code_numeral_2]: 291.78/292.11 zero_z126310315umeral!=code_S1047413653umeral(Code_numeral_2)). 291.78/292.11 fof(fact_4177_coprime__prime, axiom, 291.78/292.11 ![P_3, A_3, B_2]: 291.78/292.11 (~(hBOOL(hAPP_nat_bool(prime, P_3)) 291.78/292.11 & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, P_3), 291.78/292.11 A_3)) 291.78/292.11 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, P_3), 291.78/292.11 B_2)))) 291.78/292.11 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime, A_3), 291.78/292.11 B_2)))). 291.78/292.11 fof(fact_4179_coprime__prime__eq, axiom, 291.78/292.11 ![A_1, B_1]: 291.78/292.11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime, A_1), B_1)) 291.78/292.11 <=> ![P_4]: 291.78/292.11 ~(hBOOL(hAPP_nat_bool(prime, P_4)) 291.78/292.11 & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, P_4), 291.78/292.11 B_1)) 291.78/292.11 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, P_4), 291.78/292.12 A_1)))))). 291.78/292.12 fof(fact_4181_prime__nd__one, axiom, 291.78/292.12 ![P_3]: 291.78/292.12 (hBOOL(hAPP_nat_bool(prime, P_3)) 291.78/292.12 => ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat, P_3), 291.78/292.12 hAPP_nat_nat(suc, zero_zero_nat))))). 291.78/292.12 fof(fact_4277_fact__nonzero__int, axiom, 291.78/292.12 ![N]: 291.78/292.12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.12 zero_zero_int), 291.78/292.12 N)) 291.78/292.12 => zero_zero_int!=hAPP_int_int(fact_fact_int, N))). 291.78/292.12 fof(fact_4290_lemma__BOLZANO2, axiom, 291.78/292.12 ![P_1]: 291.78/292.12 (![A_2, B_4]: 291.78/292.12 (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 B_4))) 291.78/292.12 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 A_2), 291.78/292.12 B_4))) 291.78/292.12 <= (![A_2, B_4, C_3]: 291.78/292.12 (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 C_3))) 291.78/292.12 <= (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 B_4), 291.78/292.12 C_3))) 291.78/292.12 & (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 B_4))) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 B_4), 291.78/292.12 C_3)) 291.78/292.12 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 A_2), 291.78/292.12 B_4)))))) 291.78/292.12 & ![X_1]: 291.78/292.12 ?[D_2]: 291.78/292.12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 zero_zero_real), 291.78/292.12 D_2)) 291.78/292.12 & ![A_2, B_4]: 291.78/292.12 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 X_1), 291.78/292.12 B_4)) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.12 B_4), 291.78/292.12 A_2)), 291.78/292.12 D_2)) 291.78/292.12 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 A_2), 291.78/292.12 X_1)))) 291.78/292.12 => hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 B_4)))))))). 291.78/292.12 fof(fact_4292_Bnor__mem__zle__swap, axiom, 291.78/292.12 ![Ma, A_1, B_1]: 291.78/292.12 (~hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int, 291.78/292.12 B_1), 291.78/292.12 hAPP_i1948725293t_bool(bnorRset(A_1), Ma))) 291.78/292.12 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, A_1), 291.78/292.12 B_1)))). 291.78/292.12 fof(fact_4295_lemma__BOLZANO, axiom, 291.78/292.12 ![A_1, B_1, P_1]: 291.78/292.12 ((![X_1]: 291.78/292.12 ?[D_2]: 291.78/292.12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 zero_zero_real), 291.78/292.12 D_2)) 291.78/292.12 & ![A_2, B_4]: 291.78/292.12 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.12 B_4), 291.78/292.12 A_2)), 291.78/292.12 D_2)) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 X_1), 291.78/292.12 B_4)) 291.78/292.12 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 A_2), 291.78/292.12 X_1)))) 291.78/292.12 => hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 B_4))))) 291.78/292.12 => (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_1), 291.78/292.12 B_1))) 291.78/292.12 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 A_1), 291.78/292.12 B_1)))) 291.78/292.12 <= ![A_2, B_4, C_3]: 291.78/292.12 (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 C_3))) 291.78/292.12 <= (hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 A_2), 291.78/292.12 B_4))) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 B_4), 291.78/292.12 C_3)) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 A_2), 291.78/292.12 B_4)) 291.78/292.12 & hBOOL(hAPP_P1333854989l_bool(P_1, 291.78/292.12 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real, 291.78/292.12 B_4), 291.78/292.12 C_3))))))))). 291.78/292.12 fof(fact_4343_gcd__pos__nat, axiom, 291.78/292.12 ![Ma, Na]: 291.78/292.12 ((Na!=zero_zero_nat | Ma!=zero_zero_nat) 291.78/292.12 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.12 zero_zero_nat), 291.78/292.12 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat, Ma), Na))))). 291.78/292.12 fof(fact_4380_powr__not__zero, axiom, 291.78/292.12 ![X, A_3]: 291.78/292.12 zero_zero_real!=hAPP_real_real(hAPP_r1250527377l_real(powr, X), 291.78/292.12 A_3)). 291.78/292.12 fof(fact_445_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 ((zero_zero_nat!=Na & A_1=zero_zero_rat) 291.78/292.12 <=> zero_zero_rat=hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat, 291.78/292.12 A_1), 291.78/292.12 Na))). 291.78/292.12 fof(fact_446_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int, 291.78/292.12 A_1), 291.78/292.12 Na)=zero_z891286103de_int 291.78/292.12 <=> (A_1=zero_z891286103de_int & Na!=zero_zero_nat))). 291.78/292.12 fof(fact_447_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 (hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex, A_1), 291.78/292.12 Na)=zero_zero_complex 291.78/292.12 <=> (zero_zero_complex=A_1 & zero_zero_nat!=Na))). 291.78/292.12 fof(fact_448_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 ((Na!=zero_zero_nat & A_1=zero_z126310315umeral) 291.78/292.12 <=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral, 291.78/292.12 A_1), 291.78/292.12 Na)=zero_z126310315umeral)). 291.78/292.12 fof(fact_449_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 ((Na!=zero_zero_nat & zero_zero_real=A_1) 291.78/292.12 <=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real, A_1), 291.78/292.12 Na)=zero_zero_real)). 291.78/292.12 fof(fact_450_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 (zero_zero_nat=hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat, 291.78/292.12 A_1), 291.78/292.12 Na) 291.78/292.12 <=> (Na!=zero_zero_nat & A_1=zero_zero_nat))). 291.78/292.12 fof(fact_451_power__eq__0__iff, axiom, 291.78/292.12 ![Na, A_1]: 291.78/292.12 (is_int(A_1) 291.78/292.12 => ((Na!=zero_zero_nat & zero_zero_int=A_1) 291.78/292.12 <=> zero_zero_int=hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, 291.78/292.12 A_1), 291.78/292.12 Na)))). 291.78/292.12 fof(fact_4521_zgcd1__iff__no__common__primedivisor, axiom, 291.78/292.12 ![A_1, B_1]: 291.78/292.12 (~?[P_4]: 291.78/292.12 (is_int(P_4) 291.78/292.12 & (hBOOL(hAPP_int_bool(zprime, P_4)) 291.78/292.12 & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int, P_4), 291.78/292.12 A_1)) 291.78/292.12 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int, P_4), 291.78/292.12 B_1))))) 291.78/292.12 <=> hAPP_int_int(legacy_zgcd(A_1), B_1)=one_one_int)). 291.78/292.12 fof(fact_452_of__nat__less__0__iff, axiom, 291.78/292.12 ![M_21]: 291.78/292.12 ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.78/292.12 hAPP_nat_rat(semiri151668891at_rat, M_21)), 291.78/292.12 zero_zero_rat))). 291.78/292.12 fof(fact_453_of__nat__less__0__iff, axiom, 291.78/292.12 ![M_21]: 291.78/292.12 ~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.78/292.12 hAPP_n522471361de_int(semiri1424489471de_int, 291.78/292.12 M_21)), 291.78/292.12 zero_z891286103de_int))). 291.78/292.12 fof(fact_454_of__nat__less__0__iff, axiom, 291.78/292.12 ![M_21]: 291.78/292.12 ~hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral, 291.78/292.12 hAPP_n1108039445umeral(semiri1619134803umeral, 291.78/292.12 M_21)), 291.78/292.12 zero_z126310315umeral))). 291.78/292.12 fof(fact_455_of__nat__less__0__iff, axiom, 291.78/292.12 ![M_21]: 291.78/292.12 ~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 hAPP_nat_real(semiri132038758t_real, M_21)), 291.78/292.12 zero_zero_real))). 291.78/292.12 fof(fact_456_of__nat__less__0__iff, axiom, 291.78/292.12 ![M_21]: 291.78/292.12 ~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.12 hAPP_nat_nat(semiri984289939at_nat, M_21)), 291.78/292.12 zero_zero_nat))). 291.78/292.12 fof(fact_457_of__nat__less__0__iff, axiom, 291.78/292.12 ![M_21]: 291.78/292.12 ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.78/292.12 hAPP_nat_int(semiri1621563631at_int, M_21)), 291.78/292.12 zero_zero_int))). 291.78/292.12 fof(fact_4606_infinite__nat__iff__unbounded__le, axiom, 291.78/292.12 ![S]: 291.78/292.12 (~hBOOL(hAPP_f54304608l_bool(finite_finite_nat, S)) 291.78/292.12 <=> ![M_1]: 291.78/292.12 ?[N_1]: 291.78/292.12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, M_1), 291.78/292.12 N_1)) 291.78/292.12 & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat, 291.78/292.12 N_1), 291.78/292.12 S))))). 291.78/292.12 fof(fact_4607_infinite__nat__iff__unbounded, axiom, 291.78/292.12 ![S]: 291.78/292.12 (~hBOOL(hAPP_f54304608l_bool(finite_finite_nat, S)) 291.78/292.12 <=> ![M_1]: 291.78/292.12 ?[N_1]: 291.78/292.12 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat, N_1), 291.78/292.12 S)) 291.78/292.12 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M_1), 291.78/292.12 N_1))))). 291.78/292.12 fof(fact_4622_Rats__abs__nat__div__natE, axiom, 291.78/292.12 ![Xa]: 291.78/292.12 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real, 291.78/292.12 Xa), 291.78/292.12 field_1210416355s_real)) 291.78/292.12 => ~![M_1, N_1]: 291.78/292.12 ((hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat, M_1), 291.78/292.12 N_1)!=one_one_nat 291.78/292.12 <= hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.12 hAPP_nat_real(real_nat, M_1)), 291.78/292.12 hAPP_nat_real(real_nat, N_1))=hAPP_real_real(abs_abs_real, Xa)) 291.78/292.12 <= N_1!=zero_zero_nat))). 291.78/292.12 fof(fact_4646_less__rat__def, axiom, 291.78/292.12 ![Wa, Z_2]: 291.78/292.12 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, Z_2), 291.78/292.12 Wa)) 291.78/292.12 <=> (Wa!=Z_2 291.78/292.12 & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, Z_2), 291.78/292.12 Wa))))). 291.78/292.12 fof(fact_4665_ratrel__iff, axiom, 291.78/292.12 ![Xa, Ya]: 291.78/292.12 ((hAPP_P1175774780nt_int(product_snd_int_int, Xa)!=zero_zero_int 291.78/292.12 & (hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.78/292.12 hAPP_P1175774780nt_int(product_fst_int_int, Xa)), 291.78/292.12 hAPP_P1175774780nt_int(product_snd_int_int, 291.78/292.12 Ya))=hAPP_int_int(hAPP_int_fun_int_int(times_times_int, 291.78/292.12 hAPP_P1175774780nt_int(product_fst_int_int, 291.78/292.12 Ya)), 291.78/292.12 hAPP_P1175774780nt_int(product_snd_int_int, 291.78/292.12 Xa)) 291.78/292.12 & hAPP_P1175774780nt_int(product_snd_int_int, Ya)!=zero_zero_int)) 291.78/292.12 <=> hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int, 291.78/292.12 hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int, 291.78/292.12 Xa), 291.78/292.12 Ya)), 291.78/292.12 ratrel)))). 291.78/292.12 fof(fact_4767_card__less, axiom, 291.78/292.12 ![I_1, M_2]: 291.78/292.12 (hAPP_f22106695ol_nat(finite_card_nat, 291.78/292.12 collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool, 291.78/292.12 hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat, 291.78/292.12 fconj), 291.78/292.12 hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool, 291.78/292.12 member_nat), 291.78/292.12 M_2))), 291.78/292.12 hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool, 291.78/292.12 ord_less_nat), 291.78/292.12 hAPP_nat_nat(suc, 291.78/292.12 I_1)))))!=zero_zero_nat 291.78/292.12 <= hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat, 291.78/292.12 zero_zero_nat), 291.78/292.12 M_2)))). 291.78/292.12 fof(fact_4778_eq__number__of__Suc, axiom, 291.78/292.12 ![Na, Va]: 291.78/292.12 (hAPP_int_nat(number_number_of_nat, Va)=hAPP_nat_nat(suc, Na) 291.78/292.12 <=> (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 & (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 => hAPP_int_nat(nat, 291.78/292.12 hAPP_int_int(number_number_of_int, 291.78/292.12 hAPP_int_int(pred, Va)))=Na)))). 291.78/292.12 fof(fact_4779_Suc__eq__number__of, axiom, 291.78/292.12 ![Na, Va]: 291.78/292.12 (hAPP_int_nat(number_number_of_nat, Va)=hAPP_nat_nat(suc, Na) 291.78/292.12 <=> (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 & (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 => hAPP_int_nat(nat, 291.78/292.12 hAPP_int_int(number_number_of_int, 291.78/292.12 hAPP_int_int(pred, Va)))=Na)))). 291.78/292.12 fof(fact_4783_less__Suc__number__of, axiom, 291.78/292.12 ![Na, Va]: 291.78/292.12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.12 hAPP_nat_nat(suc, Na)), 291.78/292.12 hAPP_int_nat(number_number_of_nat, Va))) 291.78/292.12 <=> (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 & (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, Na), 291.78/292.12 hAPP_int_nat(nat, 291.78/292.12 hAPP_int_int(number_number_of_int, 291.78/292.12 hAPP_int_int(pred, 291.78/292.12 Va))))))))). 291.78/292.12 fof(fact_4785_le__Suc__number__of, axiom, 291.78/292.12 ![Na, Va]: 291.78/292.12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, 291.78/292.12 hAPP_nat_nat(suc, Na)), 291.78/292.12 hAPP_int_nat(number_number_of_nat, Va))) 291.78/292.12 <=> (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 & (~hBOOL(hAPP_int_bool(nat_neg, 291.78/292.12 hAPP_int_int(number_number_of_int, hAPP_int_int(pred, Va)))) 291.78/292.12 => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, Na), 291.78/292.12 hAPP_int_nat(nat, 291.78/292.12 hAPP_int_int(number_number_of_int, 291.78/292.12 hAPP_int_int(pred, 291.78/292.12 Va))))))))). 291.78/292.12 fof(fact_4933_Maclaurin__objl, axiom, 291.78/292.12 ![F, Na, Diff, H]: 291.78/292.12 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 zero_zero_real), 291.78/292.12 H)) 291.78/292.12 & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.12 zero_zero_nat), 291.78/292.12 Na)) 291.78/292.12 & (F=hAPP_n546711566l_real(Diff, zero_zero_nat) 291.78/292.12 & ![M_1, T]: 291.78/292.12 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 zero_zero_real), 291.78/292.12 T)) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.12 T), 291.78/292.12 H)) 291.78/292.12 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M_1), 291.78/292.12 Na)))) 291.78/292.12 => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff, 291.78/292.12 M_1), 291.78/292.12 T), 291.78/292.12 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.12 hAPP_nat_nat(suc, 291.78/292.12 M_1)), 291.78/292.12 T))))))) 291.78/292.12 => ?[T]: 291.78/292.12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.12 zero_zero_real), 291.78/292.12 T)) 291.78/292.12 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, T), 291.78/292.12 H)) 291.78/292.12 & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.78/292.12 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real, 291.78/292.12 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.12 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.12 times_times_real), 291.78/292.12 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.12 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.12 inverse_divide_real), 291.78/292.12 hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real, 291.78/292.12 Diff), 291.78/292.12 zero_zero_real))), 291.78/292.12 hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat, 291.78/292.12 real_nat), 291.78/292.12 fact_fact_nat)))), 291.78/292.12 hAPP_r474017924t_real(power_power_real, 291.78/292.12 H))), 291.78/292.12 hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat), 291.78/292.12 Na))), 291.78/292.12 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.12 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.12 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.12 Na), 291.78/292.12 T)), 291.78/292.12 hAPP_nat_real(real_nat, 291.78/292.12 hAPP_nat_nat(fact_fact_nat, 291.78/292.12 Na)))), 291.78/292.12 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.12 H), 291.78/292.12 Na)))=hAPP_real_real(F, H))))). 291.78/292.12 fof(fact_4934_Maclaurin2__objl, axiom, 291.78/292.12 ![F, Na, Diff, H]: 291.78/292.12 (?[T]: 291.78/292.12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, T), 291.78/292.12 H)) 291.78/292.12 & (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.78/292.12 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real, 291.78/292.12 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.12 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.12 times_times_real), 291.78/292.12 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.12 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 inverse_divide_real), 291.78/292.13 hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real, 291.78/292.13 Diff), 291.78/292.13 zero_zero_real))), 291.78/292.13 hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat, 291.78/292.13 real_nat), 291.78/292.13 fact_fact_nat)))), 291.78/292.13 hAPP_r474017924t_real(power_power_real, 291.78/292.13 H))), 291.78/292.13 hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat), 291.78/292.13 Na))), 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 Na), 291.78/292.13 T)), 291.78/292.13 hAPP_nat_real(real_nat, 291.78/292.13 hAPP_nat_nat(fact_fact_nat, 291.78/292.13 Na)))), 291.78/292.13 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.13 H), 291.78/292.13 Na)))=hAPP_real_real(F, H) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.13 zero_zero_real), 291.78/292.13 T)))) 291.78/292.13 <= (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.13 zero_zero_real), 291.78/292.13 H)) 291.78/292.13 & (hAPP_n546711566l_real(Diff, zero_zero_nat)=F 291.78/292.13 & ![M_1, T]: 291.78/292.13 (hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff, M_1), 291.78/292.13 T), 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 hAPP_nat_nat(suc, 291.78/292.13 M_1)), 291.78/292.13 T))) 291.78/292.13 <= (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 T), 291.78/292.13 H)) 291.78/292.13 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 zero_zero_real), 291.78/292.13 T)) 291.78/292.13 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.13 M_1), 291.78/292.13 Na))))))))). 291.78/292.13 fof(fact_4935_Maclaurin__minus, axiom, 291.78/292.13 ![F, Na, Diff, H]: 291.78/292.13 (((hAPP_n546711566l_real(Diff, zero_zero_nat)=F 291.78/292.13 => (![M_1, T]: 291.78/292.13 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M_1), 291.78/292.13 Na)) 291.78/292.13 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 H), 291.78/292.13 T)) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, T), 291.78/292.13 zero_zero_real)))) 291.78/292.13 => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff, 291.78/292.13 M_1), 291.78/292.13 T), 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 hAPP_nat_nat(suc, 291.78/292.13 M_1)), 291.78/292.13 T)))) 291.78/292.13 => ?[T]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, T), 291.78/292.13 zero_zero_real)) 291.78/292.13 & (hAPP_real_real(F, 291.78/292.13 H)=hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.78/292.13 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real, 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 times_times_real), 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 inverse_divide_real), 291.78/292.13 hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real, 291.78/292.13 Diff), 291.78/292.13 zero_zero_real))), 291.78/292.13 hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat, 291.78/292.13 real_nat), 291.78/292.13 fact_fact_nat)))), 291.78/292.13 hAPP_r474017924t_real(power_power_real, 291.78/292.13 H))), 291.78/292.13 hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat), 291.78/292.13 Na))), 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 Na), 291.78/292.13 T)), 291.78/292.13 hAPP_nat_real(real_nat, 291.78/292.13 hAPP_nat_nat(fact_fact_nat, 291.78/292.13 Na)))), 291.78/292.13 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.13 H), 291.78/292.13 Na))) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, H), 291.78/292.13 T)))))) 291.78/292.13 <= hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.13 zero_zero_nat), 291.78/292.13 Na))) 291.78/292.13 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, H), 291.78/292.13 zero_zero_real)))). 291.78/292.13 fof(fact_4936_Maclaurin2, axiom, 291.78/292.13 ![F, Na, Diff, H]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.13 zero_zero_real), 291.78/292.13 H)) 291.78/292.13 => ((?[T]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, T), 291.78/292.13 H)) 291.78/292.13 & (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.78/292.13 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real, 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 times_times_real), 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 inverse_divide_real), 291.78/292.13 hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real, 291.78/292.13 Diff), 291.78/292.13 zero_zero_real))), 291.78/292.13 hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat, 291.78/292.13 real_nat), 291.78/292.13 fact_fact_nat)))), 291.78/292.13 hAPP_r474017924t_real(power_power_real, 291.78/292.13 H))), 291.78/292.13 hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat), 291.78/292.13 Na))), 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 Na), 291.78/292.13 T)), 291.78/292.13 hAPP_nat_real(real_nat, 291.78/292.13 hAPP_nat_nat(fact_fact_nat, 291.78/292.13 Na)))), 291.78/292.13 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.13 H), 291.78/292.13 Na)))=hAPP_real_real(F, H) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.13 zero_zero_real), 291.78/292.13 T)))) 291.78/292.13 <= ![M_1, T]: 291.78/292.13 (hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff, M_1), 291.78/292.13 T), 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 hAPP_nat_nat(suc, 291.78/292.13 M_1)), 291.78/292.13 T))) 291.78/292.13 <= (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M_1), 291.78/292.13 Na)) 291.78/292.13 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 T), 291.78/292.13 H)) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 zero_zero_real), 291.78/292.13 T)))))) 291.78/292.13 <= hAPP_n546711566l_real(Diff, zero_zero_nat)=F))). 291.78/292.13 fof(fact_4937_Maclaurin__minus__objl, axiom, 291.78/292.13 ![F, Na, Diff, H]: 291.78/292.13 (?[T]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, H), T)) 291.78/292.13 & (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.78/292.13 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real, 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 times_times_real), 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 inverse_divide_real), 291.78/292.13 hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real, 291.78/292.13 Diff), 291.78/292.13 zero_zero_real))), 291.78/292.13 hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat, 291.78/292.13 real_nat), 291.78/292.13 fact_fact_nat)))), 291.78/292.13 hAPP_r474017924t_real(power_power_real, 291.78/292.13 H))), 291.78/292.13 hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat), 291.78/292.13 Na))), 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 Na), 291.78/292.13 T)), 291.78/292.13 hAPP_nat_real(real_nat, 291.78/292.13 hAPP_nat_nat(fact_fact_nat, 291.78/292.13 Na)))), 291.78/292.13 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.13 H), 291.78/292.13 Na)))=hAPP_real_real(F, H) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, T), 291.78/292.13 zero_zero_real)))) 291.78/292.13 <= (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, H), 291.78/292.13 zero_zero_real)) 291.78/292.13 & (F=hAPP_n546711566l_real(Diff, zero_zero_nat) 291.78/292.13 & (![M_1, T]: 291.78/292.13 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, H), 291.78/292.13 T)) 291.78/292.13 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 T), 291.78/292.13 zero_zero_real)) 291.78/292.13 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, M_1), 291.78/292.13 Na)))) 291.78/292.13 => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff, 291.78/292.13 M_1), 291.78/292.13 T), 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 hAPP_nat_nat(suc, 291.78/292.13 M_1)), 291.78/292.13 T)))) 291.78/292.13 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.13 zero_zero_nat), 291.78/292.13 Na))))))). 291.78/292.13 fof(fact_4938_Maclaurin, axiom, 291.78/292.13 ![F, Na, Diff, H]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.13 zero_zero_real), 291.78/292.13 H)) 291.78/292.13 => (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.13 zero_zero_nat), 291.78/292.13 Na)) 291.78/292.13 => (F=hAPP_n546711566l_real(Diff, zero_zero_nat) 291.78/292.13 => (?[T]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.13 zero_zero_real), 291.78/292.13 T)) 291.78/292.13 & (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real, 291.78/292.13 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real, 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 times_times_real), 291.78/292.13 hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real, 291.78/292.13 hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat, 291.78/292.13 inverse_divide_real), 291.78/292.13 hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real, 291.78/292.13 Diff), 291.78/292.13 zero_zero_real))), 291.78/292.13 hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat, 291.78/292.13 real_nat), 291.78/292.13 fact_fact_nat)))), 291.78/292.13 hAPP_r474017924t_real(power_power_real, 291.78/292.13 H))), 291.78/292.13 hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat), 291.78/292.13 Na))), 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real, 291.78/292.13 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real, 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 Na), 291.78/292.13 T)), 291.78/292.13 hAPP_nat_real(real_nat, 291.78/292.13 hAPP_nat_nat(fact_fact_nat, 291.78/292.13 Na)))), 291.78/292.13 hAPP_nat_real(hAPP_r474017924t_real(power_power_real, 291.78/292.13 H), 291.78/292.13 Na)))=hAPP_real_real(F, 291.78/292.13 H) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, T), 291.78/292.13 H)))) 291.78/292.13 <= ![M_1, T]: 291.78/292.13 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 zero_zero_real), 291.78/292.13 T)) 291.78/292.13 & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 T), 291.78/292.13 H)) 291.78/292.13 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.13 M_1), 291.78/292.13 Na)))) 291.78/292.13 => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff, 291.78/292.13 M_1), 291.78/292.13 T), 291.78/292.13 hAPP_real_real(hAPP_n546711566l_real(Diff, 291.78/292.13 hAPP_nat_nat(suc, 291.78/292.13 M_1)), 291.78/292.13 T))))))))). 291.78/292.13 fof(fact_4997_gcd__pos__int, axiom, 291.78/292.13 ![Ma, Na]: 291.78/292.13 ((is_int(Na) & is_int(Ma)) 291.78/292.13 => (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.78/292.13 zero_zero_int), 291.78/292.13 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int, Ma), Na))) 291.78/292.13 <=> (Na!=zero_zero_int | zero_zero_int!=Ma)))). 291.78/292.13 fof(fact_5013_gcd__cases__int, axiom, 291.78/292.13 ![Xa, Ya, P_1]: 291.78/292.13 ((((((hBOOL(hAPP_int_bool(P_1, 291.78/292.13 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int, 291.78/292.13 hAPP_int_int(uminus_uminus_int, 291.78/292.13 Xa)), 291.78/292.13 hAPP_int_int(uminus_uminus_int, Ya)))) 291.78/292.13 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Ya), 291.78/292.13 zero_zero_int))) 291.78/292.13 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Xa), 291.78/292.13 zero_zero_int))) 291.78/292.13 => hBOOL(hAPP_int_bool(P_1, 291.78/292.13 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int, Xa), Ya)))) 291.78/292.13 <= ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.13 zero_zero_int), 291.78/292.13 Ya)) 291.78/292.13 => hBOOL(hAPP_int_bool(P_1, 291.78/292.13 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int, 291.78/292.13 hAPP_int_int(uminus_uminus_int, 291.78/292.13 Xa)), 291.78/292.13 Ya)))) 291.78/292.13 <= hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Xa), 291.78/292.13 zero_zero_int)))) 291.78/292.13 <= (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.13 zero_zero_int), 291.78/292.13 Xa)) 291.78/292.13 => (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.13 Ya), 291.78/292.13 zero_zero_int)) 291.78/292.13 => hBOOL(hAPP_int_bool(P_1, 291.78/292.13 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int, Xa), 291.78/292.13 hAPP_int_int(uminus_uminus_int, Ya))))))) 291.78/292.13 <= (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.13 zero_zero_int), 291.78/292.13 Xa)) 291.78/292.13 => (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.13 zero_zero_int), 291.78/292.13 Ya)) 291.78/292.13 => hBOOL(hAPP_int_bool(P_1, 291.78/292.13 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int, Xa), 291.78/292.13 Ya))))))). 291.78/292.13 fof(fact_5174_DERIV__isconst1, axiom, 291.78/292.13 ![F, A_1, B_1]: 291.78/292.13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.13 B_1)) 291.78/292.13 => ((![X_1]: 291.78/292.13 (hAPP_real_real(F, A_1)=hAPP_real_real(F, X_1) 291.78/292.13 <= (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 A_1), 291.78/292.13 X_1)) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 X_1), 291.78/292.13 B_1)))) 291.78/292.13 <= ![X_1]: 291.78/292.13 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_1), 291.78/292.13 B_1)) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.13 X_1))) 291.78/292.13 => hBOOL(hAPP_real_bool(deriv_real(F, X_1), zero_zero_real)))) 291.78/292.13 <= ![X_1]: 291.78/292.13 (hBOOL(hAPP_real_bool(isCont_real_real(F), X_1)) 291.78/292.13 <= (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 A_1), 291.78/292.13 X_1)) 291.78/292.13 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 X_1), 291.78/292.13 B_1))))))). 291.78/292.13 fof(fact_5175_DERIV__isconst2, axiom, 291.78/292.13 ![Xa, F, A_1, B_1]: 291.78/292.13 ((((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 A_1), 291.78/292.13 Xa)) 291.78/292.13 => (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.13 Xa), 291.78/292.13 B_1)) 291.78/292.13 => hAPP_real_real(F, Xa)=hAPP_real_real(F, A_1))) 291.78/292.13 <= ![X_1]: 291.78/292.14 (hBOOL(hAPP_real_bool(deriv_real(F, X_1), zero_zero_real)) 291.78/292.14 <= (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.14 A_1), 291.78/292.14 X_1)) 291.78/292.14 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_1), 291.78/292.14 B_1))))) 291.78/292.14 <= ![X_1]: 291.78/292.14 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 X_1), 291.78/292.14 B_1)) 291.78/292.14 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 A_1), 291.78/292.14 X_1))) 291.78/292.14 => hBOOL(hAPP_real_bool(isCont_real_real(F), X_1)))) 291.78/292.14 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.14 B_1)))). 291.78/292.14 fof(fact_5176_DERIV__isconst__end, axiom, 291.78/292.14 ![F, A_1, B_1]: 291.78/292.14 ((![X_1]: 291.78/292.14 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 A_1), 291.78/292.14 X_1)) 291.78/292.14 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 X_1), 291.78/292.14 B_1))) 291.78/292.14 => hBOOL(hAPP_real_bool(isCont_real_real(F), X_1))) 291.78/292.14 => (![X_1]: 291.78/292.14 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, X_1), 291.78/292.14 B_1)) 291.78/292.14 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.14 X_1))) 291.78/292.14 => hBOOL(hAPP_real_bool(deriv_real(F, X_1), zero_zero_real))) 291.78/292.14 => hAPP_real_real(F, A_1)=hAPP_real_real(F, B_1))) 291.78/292.14 <= hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, A_1), 291.78/292.14 B_1)))). 291.78/292.14 fof(fact_5178_lemma__isCont__inj, axiom, 291.78/292.14 ![Xa, F, G, D_1]: 291.78/292.14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.14 zero_zero_real), 291.78/292.14 D_1)) 291.78/292.14 => (![Z]: 291.78/292.14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 hAPP_real_real(abs_abs_real, 291.78/292.14 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.14 Z), 291.78/292.14 Xa))), 291.78/292.14 D_1)) 291.78/292.14 => hAPP_real_real(G, hAPP_real_real(F, Z))=Z) 291.78/292.14 => (?[Z]: 291.78/292.14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.14 hAPP_real_real(F, Xa)), 291.78/292.14 hAPP_real_real(F, Z))) 291.78/292.14 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 hAPP_real_real(abs_abs_real, 291.78/292.14 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.14 Z), 291.78/292.14 Xa))), 291.78/292.14 D_1))) 291.78/292.14 <= ![Z]: 291.78/292.14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 hAPP_real_real(abs_abs_real, 291.78/292.14 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real, 291.78/292.14 Z), 291.78/292.14 Xa))), 291.78/292.14 D_1)) 291.78/292.14 => hBOOL(hAPP_real_bool(isCont_real_real(F), Z))))))). 291.78/292.14 fof(fact_562_zless__le, axiom, 291.78/292.14 ![Wa, Z_2]: 291.78/292.14 (((Wa!=Z_2 291.78/292.14 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Z_2), 291.78/292.14 Wa))) 291.78/292.14 <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, Z_2), 291.78/292.14 Wa))) 291.78/292.14 <= (is_int(Z_2) & is_int(Wa)))). 291.78/292.14 fof(fact_689_le__number__of__eq__not__less, axiom, 291.78/292.14 ![Wa, Va]: 291.78/292.14 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat, 291.78/292.14 hAPP_int_rat(number_number_of_rat, Va)), 291.78/292.14 hAPP_int_rat(number_number_of_rat, Wa))) 291.78/292.14 <=> ~hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat, 291.78/292.14 hAPP_int_rat(number_number_of_rat, Wa)), 291.78/292.14 hAPP_int_rat(number_number_of_rat, Va))))). 291.78/292.14 fof(fact_690_le__number__of__eq__not__less, axiom, 291.78/292.14 ![Wa, Va]: 291.78/292.14 (~hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int, 291.78/292.14 hAPP_i1732201573de_int(number1226105091de_int, 291.78/292.14 Wa)), 291.78/292.14 hAPP_i1732201573de_int(number1226105091de_int, Va))) 291.78/292.14 <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int, 291.78/292.14 hAPP_i1732201573de_int(number1226105091de_int, 291.78/292.14 Va)), 291.78/292.14 hAPP_i1732201573de_int(number1226105091de_int, Wa))))). 291.78/292.14 fof(fact_691_le__number__of__eq__not__less, axiom, 291.78/292.14 ![Wa, Va]: 291.78/292.14 (~hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real, 291.78/292.14 hAPP_int_real(number267125858f_real, Wa)), 291.78/292.14 hAPP_int_real(number267125858f_real, Va))) 291.78/292.14 <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real, 291.78/292.14 hAPP_int_real(number267125858f_real, Va)), 291.78/292.14 hAPP_int_real(number267125858f_real, Wa))))). 291.78/292.14 fof(fact_692_le__number__of__eq__not__less, axiom, 291.78/292.14 ![Wa, Va]: 291.78/292.14 (~hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat, 291.78/292.14 hAPP_int_nat(number_number_of_nat, Wa)), 291.78/292.14 hAPP_int_nat(number_number_of_nat, Va))) 291.78/292.14 <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat, 291.78/292.14 hAPP_int_nat(number_number_of_nat, Va)), 291.78/292.14 hAPP_int_nat(number_number_of_nat, Wa))))). 291.78/292.14 fof(fact_693_le__number__of__eq__not__less, axiom, 291.78/292.14 ![Wa, Va]: 291.78/292.14 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, 291.78/292.14 hAPP_int_int(number_number_of_int, Va)), 291.78/292.14 hAPP_int_int(number_number_of_int, Wa))) 291.78/292.14 <=> ~hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int, 291.78/292.14 hAPP_int_int(number_number_of_int, Wa)), 291.78/292.14 hAPP_int_int(number_number_of_int, Va))))). 291.78/292.14 fof(fact_694_le__number__of__eq__not__less, axiom, 291.78/292.14 ![Wa, Va]: 291.78/292.14 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral, 291.78/292.14 hAPP_i769753017umeral(number1443263063umeral, 291.78/292.14 Va)), 291.78/292.14 hAPP_i769753017umeral(number1443263063umeral, Wa))) 291.78/292.14 <=> ~hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral, 291.78/292.14 hAPP_i769753017umeral(number1443263063umeral, 291.78/292.14 Wa)), 291.78/292.14 hAPP_i769753017umeral(number1443263063umeral, Va))))). 291.78/292.14 fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint, hypothesis, 291.78/292.14 ![B_1_1, B_2_1]: is_int(hAPP_nat_int(B_1_1, B_2_1))). 291.78/292.14 fof(help_fNot_1_1_U, axiom, 291.78/292.14 ![P]: (~hBOOL(P) | ~hBOOL(hAPP_bool_bool(fNot, P)))). 291.78/292.14 291.78/292.14 Now clausify the problem and encode Horn clauses using encoding 3 of 291.78/292.14 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 291.78/292.14 We repeatedly replace C & s=t => u=v by the two clauses: 291.78/292.14 $$fresh(y, y, x1...xn) = u 291.78/292.14 C => $$fresh(s, t, x1...xn) = v 291.78/292.14 where $$fresh is a fresh function symbol and x1..xn are the free 291.78/292.14 variables of u and v. 291.78/292.14 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 291.78/292.14 input problem has no model of domain size 1). 291.78/292.14 291.78/292.14 The encoding turns the above axioms into the following unit equations and goals: 291.78/292.14 291.78/292.14 Axiom 1118 (fact_2020_order__le__less): $$fresh6200(X, X, Y, Z) = hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, Y), Z)). 291.78/292.14 Axiom 1449 (fact_22_zero__eq__power2_1): $$fresh138(X, X, Y) = Y. 291.78/292.14 Axiom 1450 (fact_22_zero__eq__power2_1): $$fresh3419(X, X, Y) = zero_zero_int. 291.78/292.14 Axiom 2241 (fact_2803_power2__eq__iff__abs__eq): $$fresh2915(X, X, Y, Z) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, Y), hAPP_int_nat(number_number_of_nat, hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))). 291.78/292.14 Axiom 6460 (fact_1700_order__eq__refl): hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, X), X)) = $$true2. 291.78/292.14 Axiom 8439 (fact_3105_int__Suc): hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, one_one_int), hAPP_nat_int(semiri1621563631at_int, X)) = hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, X)). 291.78/292.14 Axiom 9368 (fact_3460_minus__Pls): pls = hAPP_int_int(uminus_uminus_int, pls). 291.78/292.14 Axiom 11295 (fact_22_zero__eq__power2_1): $$fresh138(is_int(X), $$true2, X) = $$fresh3419(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, X), hAPP_int_nat(number_number_of_nat, hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))), zero_zero_int, X). 291.78/292.14 Axiom 12186 (fact_151_Pls__def): zero_zero_int = pls. 291.78/292.14 Axiom 12203 (gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint): is_int(hAPP_nat_int(X, Y)) = $$true2. 291.78/292.14 Axiom 12205 (conj_0): hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, one_one_int), hAPP_nat_int(semiri1621563631at_int, n))), hAPP_int_nat(number_number_of_nat, hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))) = zero_zero_int. 291.78/292.14 291.78/292.14 Goal 1 (fact_3496_not__zle__0__negative): hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, zero_zero_int), hAPP_int_int(uminus_uminus_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, X))))) = $$true2. 291.78/292.14 The goal is true when: 291.78/292.14 X = n 291.78/292.14 291.78/292.14 Proof: 291.78/292.14 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, zero_zero_int), hAPP_int_int(uminus_uminus_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 1118 (fact_2020_order__le__less) } 291.78/292.14 $$fresh6200(?, ?, zero_zero_int, hAPP_int_int(uminus_uminus_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n)))) 291.78/292.14 = { by axiom 12186 (fact_151_Pls__def) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n)))) 291.78/292.14 = { by axiom 1449 (fact_22_zero__eq__power2_1) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh138($$true2, $$true2, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 12203 (gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh138(is_int(hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))), $$true2, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 11295 (fact_22_zero__eq__power2_1) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))), hAPP_int_nat(number_number_of_nat, hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))), zero_zero_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 2241 (fact_2803_power2__eq__iff__abs__eq) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419($$fresh2915(?, ?, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n)), ?), zero_zero_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 8439 (fact_3105_int__Suc) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419($$fresh2915(?, ?, hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, one_one_int), hAPP_nat_int(semiri1621563631at_int, n)), ?), zero_zero_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 2241 (fact_2803_power2__eq__iff__abs__eq) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int, hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int, one_one_int), hAPP_nat_int(semiri1621563631at_int, n))), hAPP_int_nat(number_number_of_nat, hAPP_int_int(bit0, hAPP_int_int(bit1, pls)))), zero_zero_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 12205 (conj_0) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419(zero_zero_int, zero_zero_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 12186 (fact_151_Pls__def) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419(pls, zero_zero_int, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 12186 (fact_151_Pls__def) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, $$fresh3419(pls, pls, hAPP_nat_int(semiri1621563631at_int, hAPP_nat_nat(suc, n))))) 291.78/292.14 = { by axiom 1450 (fact_22_zero__eq__power2_1) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, zero_zero_int)) 291.78/292.14 = { by axiom 12186 (fact_151_Pls__def) } 291.78/292.14 $$fresh6200(?, ?, pls, hAPP_int_int(uminus_uminus_int, pls)) 291.78/292.14 = { by axiom 9368 (fact_3460_minus__Pls) } 291.78/292.14 $$fresh6200(?, ?, pls, pls) 291.78/292.14 = { by axiom 1118 (fact_2020_order__le__less) } 291.78/292.14 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int, pls), pls)) 291.78/292.14 = { by axiom 6460 (fact_1700_order__eq__refl) } 291.78/292.14 $$true2 291.78/292.14 % SZS output end Proof 291.78/292.14 291.78/292.14 RESULT: Theorem (the conjecture is true). 291.96/292.34 EOF