% COMMAND-LINE: --uf-ho % EXPECT: % SZS status Theorem for NUM925^1 %------------------------------------------------------------------------------ % File : NUM925^1 : TPTP v7.2.0. Released v5.3.0. % Domain : Number Theory % Problem : Sum of two squares line 192, 100 axioms selected % Version : Especial. % English : % Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day % : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe % Source : [Bla11] % Names : s2s_100_thf_l192 [Bla11] % Status : Theorem % Rating : 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0 % Syntax : Number of formulae : 128 ( 0 unit; 21 type; 0 defn) % Number of atoms : 868 ( 105 equality; 251 variable) % Maximal formula depth : 11 ( 5 average) % Number of connectives : 571 ( 20 ~; 4 |; 7 &; 495 @) % ( 36 <=>; 9 =>; 0 <=; 0 <~>) % ( 0 ~|; 0 ~&) % Number of type conns : 18 ( 18 >; 0 *; 0 +; 0 <<) % Number of symbols : 23 ( 21 :; 0 =) % Number of variables : 118 ( 0 sgn; 118 !; 0 ?; 0 ^) % ( 118 :; 0 !>; 0 ?*) % ( 0 @-; 0 @+) % SPC : TH0_THM_EQU_NAR % Comments : This file was generated by Isabelle (most likely Sledgehammer) % 2011-08-09 19:10:38 %------------------------------------------------------------------------------ %----Should-be-implicit typings (2) thf(ty_ty_tc__Int__Oint,type,( int: $tType )). thf(ty_ty_tc__Nat__Onat,type,( nat: $tType )). %----Explicit typings (19) thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,( one_one_int: int )). thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,( one_one_nat: nat )). thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,( plus_plus_int: int > int > int )). thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,( plus_plus_nat: nat > nat > nat )). thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,( zero_zero_int: int )). thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,( zero_zero_nat: nat )). thf(sy_c_Int_OBit0,type,( bit0: int > int )). thf(sy_c_Int_OBit1,type,( bit1: int > int )). thf(sy_c_Int_OPls,type,( pls: int )). thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,( number_number_of_int: int > int )). thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,( number_number_of_nat: int > nat )). thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,( semiri1621563631at_int: nat > int )). thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Nat__Onat,type,( semiri984289939at_nat: nat > nat )). thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,( ord_less_int: int > int > $o )). thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,( ord_less_nat: nat > nat > $o )). thf(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,( power_power_int: int > nat > int )). thf(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,( power_power_nat: nat > nat > nat )). thf(sy_v_n____,type,( n: nat )). thf(sy_v_t____,type,( t: int )). %----Relevant facts (106) thf(fact_0_n1pos,axiom, ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) )). thf(fact_1_t1,axiom, ( ord_less_int @ one_one_int @ t )). thf(fact_2_sum__power2__eq__zero__iff,axiom,( ! [X_3: int,Y_3: int] : ( ( ( plus_plus_int @ ( power_power_int @ X_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) = zero_zero_int ) <=> ( ( X_3 = zero_zero_int ) & ( Y_3 = zero_zero_int ) ) ) )). thf(fact_3_one__power2,axiom, ( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) = one_one_int )). thf(fact_4_one__power2,axiom, ( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) = one_one_nat )). thf(fact_5_zero__power2,axiom, ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) = zero_zero_int )). thf(fact_6_zero__power2,axiom, ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) = zero_zero_nat )). thf(fact_7_zero__eq__power2,axiom,( ! [A_7: int] : ( ( ( power_power_int @ A_7 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) = zero_zero_int ) <=> ( A_7 = zero_zero_int ) ) )). thf(fact_8_add__special_I2_J,axiom,( ! [W_7: int] : ( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_7 ) ) = ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_7 ) ) ) )). thf(fact_9_add__special_I3_J,axiom,( ! [V_5: int] : ( ( plus_plus_int @ ( number_number_of_int @ V_5 ) @ one_one_int ) = ( number_number_of_int @ ( plus_plus_int @ V_5 @ ( bit1 @ pls ) ) ) ) )). thf(fact_10_one__add__one__is__two,axiom, ( ( plus_plus_int @ one_one_int @ one_one_int ) = ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )). thf(fact_11_semiring__one__add__one__is__two,axiom, ( ( plus_plus_int @ one_one_int @ one_one_int ) = ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )). thf(fact_12_semiring__one__add__one__is__two,axiom, ( ( plus_plus_nat @ one_one_nat @ one_one_nat ) = ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )). thf(fact_13_quartic__square__square,axiom,( ! [X_3: int] : ( ( power_power_int @ ( power_power_int @ X_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) = ( power_power_int @ X_3 @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) )). thf(fact_14_power__0__left__number__of,axiom,( ! [W_6: int] : ( ( ( ( number_number_of_nat @ W_6 ) = zero_zero_nat ) => ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_6 ) ) = one_one_int ) ) & ( ( ( number_number_of_nat @ W_6 ) != zero_zero_nat ) => ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_6 ) ) = zero_zero_int ) ) ) )). thf(fact_15_power__0__left__number__of,axiom,( ! [W_6: int] : ( ( ( ( number_number_of_nat @ W_6 ) = zero_zero_nat ) => ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_6 ) ) = one_one_nat ) ) & ( ( ( number_number_of_nat @ W_6 ) != zero_zero_nat ) => ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_6 ) ) = zero_zero_nat ) ) ) )). thf(fact_16_semiring__norm_I110_J,axiom, ( one_one_int = ( number_number_of_int @ ( bit1 @ pls ) ) )). thf(fact_17_numeral__1__eq__1,axiom, ( ( number_number_of_int @ ( bit1 @ pls ) ) = one_one_int )). thf(fact_18_n0,axiom, ( ord_less_nat @ zero_zero_nat @ n )). thf(fact_19_zless__linear,axiom,( ! [X_3: int,Y_3: int] : ( ( ord_less_int @ X_3 @ Y_3 ) | ( X_3 = Y_3 ) | ( ord_less_int @ Y_3 @ X_3 ) ) )). thf(fact_20_less__number__of__int__code,axiom,( ! [K: int,L: int] : ( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) ) <=> ( ord_less_int @ K @ L ) ) )). thf(fact_21_plus__numeral__code_I9_J,axiom,( ! [V_3: int,W_4: int] : ( ( plus_plus_int @ ( number_number_of_int @ V_3 ) @ ( number_number_of_int @ W_4 ) ) = ( number_number_of_int @ ( plus_plus_int @ V_3 @ W_4 ) ) ) )). thf(fact_22_less__number__of,axiom,( ! [X_6: int,Y_5: int] : ( ( ord_less_int @ ( number_number_of_int @ X_6 ) @ ( number_number_of_int @ Y_5 ) ) <=> ( ord_less_int @ X_6 @ Y_5 ) ) )). thf(fact_23_zero__is__num__zero,axiom, ( zero_zero_int = ( number_number_of_int @ pls ) )). thf(fact_24_zpower__int,axiom,( ! [M: nat,N_1: nat] : ( ( power_power_int @ ( semiri1621563631at_int @ M ) @ N_1 ) = ( semiri1621563631at_int @ ( power_power_nat @ M @ N_1 ) ) ) )). thf(fact_25_int__power,axiom,( ! [M: nat,N_1: nat] : ( ( semiri1621563631at_int @ ( power_power_nat @ M @ N_1 ) ) = ( power_power_int @ ( semiri1621563631at_int @ M ) @ N_1 ) ) )). thf(fact_26_zadd__int__left,axiom,( ! [M: nat,N_1: nat,Z: int] : ( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N_1 ) @ Z ) ) = ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N_1 ) ) @ Z ) ) )). thf(fact_27_zadd__int,axiom,( ! [M: nat,N_1: nat] : ( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) ) = ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N_1 ) ) ) )). thf(fact_28_int__1,axiom, ( ( semiri1621563631at_int @ one_one_nat ) = one_one_int )). thf(fact_29_nat__number__of__Pls,axiom, ( ( number_number_of_nat @ pls ) = zero_zero_nat )). thf(fact_30_semiring__norm_I113_J,axiom, ( zero_zero_nat = ( number_number_of_nat @ pls ) )). thf(fact_31_int__eq__0__conv,axiom,( ! [N_1: nat] : ( ( ( semiri1621563631at_int @ N_1 ) = zero_zero_int ) <=> ( N_1 = zero_zero_nat ) ) )). thf(fact_32_int__0,axiom, ( ( semiri1621563631at_int @ zero_zero_nat ) = zero_zero_int )). thf(fact_33_nat__1__add__1,axiom, ( ( plus_plus_nat @ one_one_nat @ one_one_nat ) = ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )). thf(fact_34_less__int__code_I16_J,axiom,( ! [K1: int,K2: int] : ( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) ) <=> ( ord_less_int @ K1 @ K2 ) ) )). thf(fact_35_rel__simps_I17_J,axiom,( ! [K: int,L: int] : ( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) ) <=> ( ord_less_int @ K @ L ) ) )). thf(fact_36_rel__simps_I2_J,axiom,( ~ ( ord_less_int @ pls @ pls ) )). thf(fact_37_less__int__code_I13_J,axiom,( ! [K1: int,K2: int] : ( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) ) <=> ( ord_less_int @ K1 @ K2 ) ) )). thf(fact_38_rel__simps_I14_J,axiom,( ! [K: int,L: int] : ( ( ord_less_int @ ( bit0 @ K ) @ ( bit0 @ L ) ) <=> ( ord_less_int @ K @ L ) ) )). thf(fact_39_zadd__strict__right__mono,axiom,( ! [K: int,I: int,J: int] : ( ( ord_less_int @ I @ J ) => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) )). thf(fact_40_add__nat__number__of,axiom,( ! [V_4: int,V_3: int] : ( ( ( ord_less_int @ V_3 @ pls ) => ( ( plus_plus_nat @ ( number_number_of_nat @ V_3 ) @ ( number_number_of_nat @ V_4 ) ) = ( number_number_of_nat @ V_4 ) ) ) & ( ~ ( ord_less_int @ V_3 @ pls ) => ( ( ( ord_less_int @ V_4 @ pls ) => ( ( plus_plus_nat @ ( number_number_of_nat @ V_3 ) @ ( number_number_of_nat @ V_4 ) ) = ( number_number_of_nat @ V_3 ) ) ) & ( ~ ( ord_less_int @ V_4 @ pls ) => ( ( plus_plus_nat @ ( number_number_of_nat @ V_3 ) @ ( number_number_of_nat @ V_4 ) ) = ( number_number_of_nat @ ( plus_plus_int @ V_3 @ V_4 ) ) ) ) ) ) ) )). thf(fact_41_one__is__num__one,axiom, ( one_one_int = ( number_number_of_int @ ( bit1 @ pls ) ) )). thf(fact_42_nat__numeral__1__eq__1,axiom, ( ( number_number_of_nat @ ( bit1 @ pls ) ) = one_one_nat )). thf(fact_43_Numeral1__eq1__nat,axiom, ( one_one_nat = ( number_number_of_nat @ ( bit1 @ pls ) ) )). thf(fact_44_eq__number__of,axiom,( ! [X_5: int,Y_4: int] : ( ( ( number_number_of_int @ X_5 ) = ( number_number_of_int @ Y_4 ) ) <=> ( X_5 = Y_4 ) ) )). thf(fact_45_number__of__reorient,axiom,( ! [W_5: int,X_4: nat] : ( ( ( number_number_of_nat @ W_5 ) = X_4 ) <=> ( X_4 = ( number_number_of_nat @ W_5 ) ) ) )). thf(fact_46_number__of__reorient,axiom,( ! [W_5: int,X_4: int] : ( ( ( number_number_of_int @ W_5 ) = X_4 ) <=> ( X_4 = ( number_number_of_int @ W_5 ) ) ) )). thf(fact_47_rel__simps_I51_J,axiom,( ! [K: int,L: int] : ( ( ( bit1 @ K ) = ( bit1 @ L ) ) <=> ( K = L ) ) )). thf(fact_48_rel__simps_I48_J,axiom,( ! [K: int,L: int] : ( ( ( bit0 @ K ) = ( bit0 @ L ) ) <=> ( K = L ) ) )). thf(fact_49_even__less__0__iff,axiom,( ! [A_6: int] : ( ( ord_less_int @ ( plus_plus_int @ A_6 @ A_6 ) @ zero_zero_int ) <=> ( ord_less_int @ A_6 @ zero_zero_int ) ) )). thf(fact_50_zadd__assoc,axiom,( ! [Z1: int,Z2: int,Z3: int] : ( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 ) = ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) )). thf(fact_51_zadd__left__commute,axiom,( ! [X_3: int,Y_3: int,Z: int] : ( ( plus_plus_int @ X_3 @ ( plus_plus_int @ Y_3 @ Z ) ) = ( plus_plus_int @ Y_3 @ ( plus_plus_int @ X_3 @ Z ) ) ) )). thf(fact_52_zadd__commute,axiom,( ! [Z: int,W_4: int] : ( ( plus_plus_int @ Z @ W_4 ) = ( plus_plus_int @ W_4 @ Z ) ) )). thf(fact_53_int__int__eq,axiom,( ! [M: nat,N_1: nat] : ( ( ( semiri1621563631at_int @ M ) = ( semiri1621563631at_int @ N_1 ) ) <=> ( M = N_1 ) ) )). thf(fact_54_less__special_I3_J,axiom,( ! [X_2: int] : ( ( ord_less_int @ ( number_number_of_int @ X_2 ) @ zero_zero_int ) <=> ( ord_less_int @ X_2 @ pls ) ) )). thf(fact_55_less__special_I1_J,axiom,( ! [Y_2: int] : ( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_2 ) ) <=> ( ord_less_int @ pls @ Y_2 ) ) )). thf(fact_56_rel__simps_I12_J,axiom,( ! [K: int] : ( ( ord_less_int @ ( bit1 @ K ) @ pls ) <=> ( ord_less_int @ K @ pls ) ) )). thf(fact_57_less__int__code_I15_J,axiom,( ! [K1: int,K2: int] : ( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) ) <=> ( ord_less_int @ K1 @ K2 ) ) )). thf(fact_58_rel__simps_I16_J,axiom,( ! [K: int,L: int] : ( ( ord_less_int @ ( bit1 @ K ) @ ( bit0 @ L ) ) <=> ( ord_less_int @ K @ L ) ) )). thf(fact_59_rel__simps_I10_J,axiom,( ! [K: int] : ( ( ord_less_int @ ( bit0 @ K ) @ pls ) <=> ( ord_less_int @ K @ pls ) ) )). thf(fact_60_rel__simps_I4_J,axiom,( ! [K: int] : ( ( ord_less_int @ pls @ ( bit0 @ K ) ) <=> ( ord_less_int @ pls @ K ) ) )). thf(fact_61_bin__less__0__simps_I4_J,axiom,( ! [W_4: int] : ( ( ord_less_int @ ( bit1 @ W_4 ) @ zero_zero_int ) <=> ( ord_less_int @ W_4 @ zero_zero_int ) ) )). thf(fact_62_bin__less__0__simps_I1_J,axiom,( ~ ( ord_less_int @ pls @ zero_zero_int ) )). thf(fact_63_bin__less__0__simps_I3_J,axiom,( ! [W_4: int] : ( ( ord_less_int @ ( bit0 @ W_4 ) @ zero_zero_int ) <=> ( ord_less_int @ W_4 @ zero_zero_int ) ) )). thf(fact_64_int__0__less__1,axiom, ( ord_less_int @ zero_zero_int @ one_one_int )). thf(fact_65_zless__add1__eq,axiom,( ! [W_4: int,Z: int] : ( ( ord_less_int @ W_4 @ ( plus_plus_int @ Z @ one_one_int ) ) <=> ( ( ord_less_int @ W_4 @ Z ) | ( W_4 = Z ) ) ) )). thf(fact_66_int__less__0__conv,axiom,( ! [K: nat] : ~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) )). thf(fact_67_less__special_I4_J,axiom,( ! [X_1: int] : ( ( ord_less_int @ ( number_number_of_int @ X_1 ) @ one_one_int ) <=> ( ord_less_int @ X_1 @ ( bit1 @ pls ) ) ) )). thf(fact_68_less__special_I2_J,axiom,( ! [Y_1: int] : ( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_1 ) ) <=> ( ord_less_int @ ( bit1 @ pls ) @ Y_1 ) ) )). thf(fact_69_odd__less__0,axiom,( ! [Z: int] : ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int ) <=> ( ord_less_int @ Z @ zero_zero_int ) ) )). thf(fact_70_double__eq__0__iff,axiom,( ! [A_5: int] : ( ( ( plus_plus_int @ A_5 @ A_5 ) = zero_zero_int ) <=> ( A_5 = zero_zero_int ) ) )). thf(fact_71_rel__simps_I46_J,axiom,( ! [K: int] : ( ( bit1 @ K ) != pls ) )). thf(fact_72_rel__simps_I39_J,axiom,( ! [L: int] : ( pls != ( bit1 @ L ) ) )). thf(fact_73_rel__simps_I50_J,axiom,( ! [K: int,L: int] : ( ( bit1 @ K ) != ( bit0 @ L ) ) )). thf(fact_74_rel__simps_I49_J,axiom,( ! [K: int,L: int] : ( ( bit0 @ K ) != ( bit1 @ L ) ) )). thf(fact_75_rel__simps_I44_J,axiom,( ! [K: int] : ( ( ( bit0 @ K ) = pls ) <=> ( K = pls ) ) )). thf(fact_76_rel__simps_I38_J,axiom,( ! [L: int] : ( ( pls = ( bit0 @ L ) ) <=> ( pls = L ) ) )). thf(fact_77_Bit0__Pls,axiom, ( ( bit0 @ pls ) = pls )). thf(fact_78_Pls__def,axiom,( pls = zero_zero_int )). thf(fact_79_int__0__neq__1,axiom,( zero_zero_int != one_one_int )). thf(fact_80_add__Pls__right,axiom,( ! [K: int] : ( ( plus_plus_int @ K @ pls ) = K ) )). thf(fact_81_add__Pls,axiom,( ! [K: int] : ( ( plus_plus_int @ pls @ K ) = K ) )). thf(fact_82_add__Bit0__Bit0,axiom,( ! [K: int,L: int] : ( ( plus_plus_int @ ( bit0 @ K ) @ ( bit0 @ L ) ) = ( bit0 @ ( plus_plus_int @ K @ L ) ) ) )). thf(fact_83_Bit0__def,axiom,( ! [K: int] : ( ( bit0 @ K ) = ( plus_plus_int @ K @ K ) ) )). thf(fact_84_zadd__0__right,axiom,( ! [Z: int] : ( ( plus_plus_int @ Z @ zero_zero_int ) = Z ) )). thf(fact_85_zadd__0,axiom,( ! [Z: int] : ( ( plus_plus_int @ zero_zero_int @ Z ) = Z ) )). thf(fact_86_semiring__numeral__0__eq__0,axiom, ( ( number_number_of_int @ pls ) = zero_zero_int )). thf(fact_87_semiring__numeral__0__eq__0,axiom, ( ( number_number_of_nat @ pls ) = zero_zero_nat )). thf(fact_88_number__of__Pls,axiom, ( ( number_number_of_int @ pls ) = zero_zero_int )). thf(fact_89_semiring__norm_I112_J,axiom, ( zero_zero_int = ( number_number_of_int @ pls ) )). thf(fact_90_add__numeral__0,axiom,( ! [A_4: int] : ( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_4 ) = A_4 ) )). thf(fact_91_add__numeral__0__right,axiom,( ! [A_3: int] : ( ( plus_plus_int @ A_3 @ ( number_number_of_int @ pls ) ) = A_3 ) )). thf(fact_92_power__eq__0__iff__number__of,axiom,( ! [A_2: int,W_3: int] : ( ( ( power_power_int @ A_2 @ ( number_number_of_nat @ W_3 ) ) = zero_zero_int ) <=> ( ( A_2 = zero_zero_int ) & ( ( number_number_of_nat @ W_3 ) != zero_zero_nat ) ) ) )). thf(fact_93_power__eq__0__iff__number__of,axiom,( ! [A_2: nat,W_3: int] : ( ( ( power_power_nat @ A_2 @ ( number_number_of_nat @ W_3 ) ) = zero_zero_nat ) <=> ( ( A_2 = zero_zero_nat ) & ( ( number_number_of_nat @ W_3 ) != zero_zero_nat ) ) ) )). thf(fact_94_add__number__of__left,axiom,( ! [V_2: int,W_2: int,Z_1: int] : ( ( plus_plus_int @ ( number_number_of_int @ V_2 ) @ ( plus_plus_int @ ( number_number_of_int @ W_2 ) @ Z_1 ) ) = ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_2 @ W_2 ) ) @ Z_1 ) ) )). thf(fact_95_add__number__of__eq,axiom,( ! [V_1: int,W_1: int] : ( ( plus_plus_int @ ( number_number_of_int @ V_1 ) @ ( number_number_of_int @ W_1 ) ) = ( number_number_of_int @ ( plus_plus_int @ V_1 @ W_1 ) ) ) )). thf(fact_96_number__of__add,axiom,( ! [V: int,W: int] : ( ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) = ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) )). thf(fact_97_add__Bit1__Bit0,axiom,( ! [K: int,L: int] : ( ( plus_plus_int @ ( bit1 @ K ) @ ( bit0 @ L ) ) = ( bit1 @ ( plus_plus_int @ K @ L ) ) ) )). thf(fact_98_add__Bit0__Bit1,axiom,( ! [K: int,L: int] : ( ( plus_plus_int @ ( bit0 @ K ) @ ( bit1 @ L ) ) = ( bit1 @ ( plus_plus_int @ K @ L ) ) ) )). thf(fact_99_Bit1__def,axiom,( ! [K: int] : ( ( bit1 @ K ) = ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) )). thf(fact_100_odd__nonzero,axiom,( ! [Z: int] : ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) != zero_zero_int ) )). thf(fact_101_number__of__int,axiom,( ! [N: nat] : ( ( number_number_of_nat @ ( semiri1621563631at_int @ N ) ) = ( semiri984289939at_nat @ N ) ) )). thf(fact_102_number__of__int,axiom,( ! [N: nat] : ( ( number_number_of_int @ ( semiri1621563631at_int @ N ) ) = ( semiri1621563631at_int @ N ) ) )). thf(fact_103_zero__less__power2,axiom,( ! [A_1: int] : ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) <=> ( A_1 != zero_zero_int ) ) )). thf(fact_104_power2__less__0,axiom,( ! [A: int] : ~ ( ord_less_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) )). thf(fact_105_sum__power2__gt__zero__iff,axiom,( ! [X: int,Y: int] : ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) <=> ( ( X != zero_zero_int ) | ( Y != zero_zero_int ) ) ) )). %----Conjectures (1) thf(conj_0,conjecture,( ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) != zero_zero_int )). %------------------------------------------------------------------------------