TPTP Problem File: NUM924^2.p
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%------------------------------------------------------------------------------
% File : NUM924^2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_500_thf_l102 [Bla11]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 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 : 745 ( 358 unt; 46 typ; 0 def)
% Number of atoms : 1328 ( 588 equ; 0 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 6318 ( 122 ~; 33 |; 62 &;5567 @)
% ( 157 <=>; 377 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 56 ( 56 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 43 usr; 12 con; 0-3 aty)
% Number of variables : 1346 ( 0 ^;1342 !; 4 ?;1346 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:32:41
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__RealDef__Oreal,type,
real: $tType ).
%----Explicit typings (43)
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: real > real > real ).
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_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
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_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: real > real > real ).
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_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
zero_zero_real: real ).
thf(sy_c_IntPrimes_Ozcong,type,
zcong: int > int > int > $o ).
thf(sy_c_IntPrimes_Ozprime,type,
zprime: int > $o ).
thf(sy_c_Int_OBit0,type,
bit0: int > int ).
thf(sy_c_Int_OBit1,type,
bit1: int > int ).
thf(sy_c_Int_OMin,type,
min: 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_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
number267125858f_real: int > real ).
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_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
ord_less_eq_real: real > real > $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_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Residues_OLegendre,type,
legendre: int > int > int ).
thf(sy_c_Residues_OQuadRes,type,
quadRes: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__RealDef__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_TwoSquares__Mirabelle__hgbmwyaznu_Ois__sum2sq,type,
twoSqu362149276sum2sq: int > $o ).
thf(sy_v_m,type,
m: int ).
thf(sy_v_s1____,type,
s1: int ).
thf(sy_v_s____,type,
s: int ).
thf(sy_v_t____,type,
t: int ).
%----Relevant facts (698)
thf(fact_0__096t_A_060_A0_096,axiom,
ord_less_int @ t @ zero_zero_int ).
thf(fact_1_calculation_I1_J,axiom,
ord_less_int @ t @ one_one_int ).
thf(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ zero_zero_int ) ).
thf(fact_3_t,axiom,
( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ) ).
thf(fact_4_calculation_I2_J,axiom,
( ( t = zero_zero_int )
=> ( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
= zero_zero_int ) ) ).
thf(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ( ord_less_eq_int @ one_one_int @ t ) ).
thf(fact_6_p0,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_7_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ) ).
thf(fact_8_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_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 ) ) ) ) ) @ zero_zero_int ) ).
thf(fact_9_sum__power2__gt__zero__iff,axiom,
! [X_39: real,Y_30: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_30 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_39 != zero_zero_real )
| ( Y_30 != zero_zero_real ) ) ) ).
thf(fact_10_sum__power2__gt__zero__iff,axiom,
! [X_39: int,Y_30: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_30 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_39 != zero_zero_int )
| ( Y_30 != zero_zero_int ) ) ) ).
thf(fact_11_sum__power2__eq__zero__iff,axiom,
! [X_38: real,Y_29: real] :
( ( ( plus_plus_real @ ( power_power_real @ X_38 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_29 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X_38 = zero_zero_real )
& ( Y_29 = zero_zero_real ) ) ) ).
thf(fact_12_sum__power2__eq__zero__iff,axiom,
! [X_38: int,Y_29: int] :
( ( ( plus_plus_int @ ( power_power_int @ X_38 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_29 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_int )
<=> ( ( X_38 = zero_zero_int )
& ( Y_29 = zero_zero_int ) ) ) ).
thf(fact_13_power2__less__0,axiom,
! [A_64: real] :
~ ( ord_less_real @ ( power_power_real @ A_64 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ) ).
thf(fact_14_power2__less__0,axiom,
! [A_64: int] :
~ ( ord_less_int @ ( power_power_int @ A_64 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) ).
thf(fact_15_zero__less__power2,axiom,
! [A_63: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_63 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_63 != zero_zero_real ) ) ).
thf(fact_16_zero__less__power2,axiom,
! [A_63: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_63 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_63 != zero_zero_int ) ) ).
thf(fact_17_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_real ) ).
thf(fact_18_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_19_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_20_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real ) ).
thf(fact_21_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_22_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(fact_23_zero__eq__power2,axiom,
! [A_62: real] :
( ( ( power_power_real @ A_62 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real )
<=> ( A_62 = zero_zero_real ) ) ).
thf(fact_24_zero__eq__power2,axiom,
! [A_62: int] :
( ( ( power_power_int @ A_62 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int )
<=> ( A_62 = zero_zero_int ) ) ).
thf(fact_25_add__special_I2_J,axiom,
! [W_19: int] :
( ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_19 ) )
= ( number267125858f_real @ ( plus_plus_int @ ( bit1 @ pls ) @ W_19 ) ) ) ).
thf(fact_26_add__special_I2_J,axiom,
! [W_19: int] :
( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_19 ) )
= ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_19 ) ) ) ).
thf(fact_27_add__special_I3_J,axiom,
! [V_19: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_19 ) @ one_one_real )
= ( number267125858f_real @ ( plus_plus_int @ V_19 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_28_add__special_I3_J,axiom,
! [V_19: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_19 ) @ one_one_int )
= ( number_number_of_int @ ( plus_plus_int @ V_19 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_29_t__l__p,axiom,
ord_less_int @ t @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_30__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T_1: int] :
( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
!= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ T_1 ) ) ).
thf(fact_31_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_32_qf1pt,axiom,
twoSqu362149276sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ).
thf(fact_33_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_34_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_35_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_36_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_37_times__numeral__code_I5_J,axiom,
! [V_4: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V_4 ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V_4 @ W ) ) ) ).
thf(fact_38_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_39_le__number__of,axiom,
! [X_37: int,Y_28: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_37 ) @ ( number267125858f_real @ Y_28 ) )
<=> ( ord_less_eq_int @ X_37 @ Y_28 ) ) ).
thf(fact_40_le__number__of,axiom,
! [X_37: int,Y_28: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_37 ) @ ( number_number_of_int @ Y_28 ) )
<=> ( ord_less_eq_int @ X_37 @ Y_28 ) ) ).
thf(fact_41_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
= ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
thf(fact_42_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I @ K ) ) ) ).
thf(fact_43_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_44_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z: nat] :
( ( power_power_int @ X @ ( plus_plus_nat @ Y @ Z ) )
= ( times_times_int @ ( power_power_int @ X @ Y ) @ ( power_power_int @ X @ Z ) ) ) ).
thf(fact_45_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_46_rel__simps_I34_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_47_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_48_less__eq__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_49_rel__simps_I31_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_50_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_51_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_52_eq__number__of__0,axiom,
! [V_4: int] :
( ( ( number_number_of_nat @ V_4 )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V_4 @ pls ) ) ).
thf(fact_53_eq__0__number__of,axiom,
! [V_4: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V_4 ) )
<=> ( ord_less_eq_int @ V_4 @ pls ) ) ).
thf(fact_54_semiring__mult__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( times_times_real @ ( number267125858f_real @ V_17 ) @ ( number267125858f_real @ V_18 ) )
= ( number267125858f_real @ ( times_times_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_55_semiring__mult__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_17 ) @ ( number_number_of_nat @ V_18 ) )
= ( number_number_of_nat @ ( times_times_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_56_semiring__mult__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( times_times_int @ ( number_number_of_int @ V_17 ) @ ( number_number_of_int @ V_18 ) )
= ( number_number_of_int @ ( times_times_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_57_mult__number__of__left,axiom,
! [V_16: int,W_18: int,Z_9: real] :
( ( times_times_real @ ( number267125858f_real @ V_16 ) @ ( times_times_real @ ( number267125858f_real @ W_18 ) @ Z_9 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_16 @ W_18 ) ) @ Z_9 ) ) ).
thf(fact_58_mult__number__of__left,axiom,
! [V_16: int,W_18: int,Z_9: int] :
( ( times_times_int @ ( number_number_of_int @ V_16 ) @ ( times_times_int @ ( number_number_of_int @ W_18 ) @ Z_9 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_16 @ W_18 ) ) @ Z_9 ) ) ).
thf(fact_59_arith__simps_I32_J,axiom,
! [V_15: int,W_17: int] :
( ( times_times_real @ ( number267125858f_real @ V_15 ) @ ( number267125858f_real @ W_17 ) )
= ( number267125858f_real @ ( times_times_int @ V_15 @ W_17 ) ) ) ).
thf(fact_60_arith__simps_I32_J,axiom,
! [V_15: int,W_17: int] :
( ( times_times_int @ ( number_number_of_int @ V_15 ) @ ( number_number_of_int @ W_17 ) )
= ( number_number_of_int @ ( times_times_int @ V_15 @ W_17 ) ) ) ).
thf(fact_61_number__of__mult,axiom,
! [V_14: int,W_16: int] :
( ( number267125858f_real @ ( times_times_int @ V_14 @ W_16 ) )
= ( times_times_real @ ( number267125858f_real @ V_14 ) @ ( number267125858f_real @ W_16 ) ) ) ).
thf(fact_62_number__of__mult,axiom,
! [V_14: int,W_16: int] :
( ( number_number_of_int @ ( times_times_int @ V_14 @ W_16 ) )
= ( times_times_int @ ( number_number_of_int @ V_14 ) @ ( number_number_of_int @ W_16 ) ) ) ).
thf(fact_63_sum__squares__le__zero__iff,axiom,
! [X_36: real,Y_27: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_36 @ X_36 ) @ ( times_times_real @ Y_27 @ Y_27 ) ) @ zero_zero_real )
<=> ( ( X_36 = zero_zero_real )
& ( Y_27 = zero_zero_real ) ) ) ).
thf(fact_64_sum__squares__le__zero__iff,axiom,
! [X_36: int,Y_27: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_36 @ X_36 ) @ ( times_times_int @ Y_27 @ Y_27 ) ) @ zero_zero_int )
<=> ( ( X_36 = zero_zero_int )
& ( Y_27 = zero_zero_int ) ) ) ).
thf(fact_65_sum__squares__ge__zero,axiom,
! [X_35: real,Y_26: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_35 @ X_35 ) @ ( times_times_real @ Y_26 @ Y_26 ) ) ) ).
thf(fact_66_sum__squares__ge__zero,axiom,
! [X_35: int,Y_26: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_35 @ X_35 ) @ ( times_times_int @ Y_26 @ Y_26 ) ) ) ).
thf(fact_67_le__special_I3_J,axiom,
! [X_34: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_34 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ X_34 @ pls ) ) ).
thf(fact_68_le__special_I3_J,axiom,
! [X_34: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_34 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_34 @ pls ) ) ).
thf(fact_69_le__special_I1_J,axiom,
! [Y_25: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( number267125858f_real @ Y_25 ) )
<=> ( ord_less_eq_int @ pls @ Y_25 ) ) ).
thf(fact_70_le__special_I1_J,axiom,
! [Y_25: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_25 ) )
<=> ( ord_less_eq_int @ pls @ Y_25 ) ) ).
thf(fact_71_less__0__number__of,axiom,
! [V_4: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V_4 ) )
<=> ( ord_less_int @ pls @ V_4 ) ) ).
thf(fact_72_le__number__of__eq__not__less,axiom,
! [V_13: int,W_15: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_13 ) @ ( number267125858f_real @ W_15 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_15 ) @ ( number267125858f_real @ V_13 ) ) ) ).
thf(fact_73_le__number__of__eq__not__less,axiom,
! [V_13: int,W_15: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_13 ) @ ( number_number_of_nat @ W_15 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_15 ) @ ( number_number_of_nat @ V_13 ) ) ) ).
thf(fact_74_le__number__of__eq__not__less,axiom,
! [V_13: int,W_15: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_13 ) @ ( number_number_of_int @ W_15 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_15 ) @ ( number_number_of_int @ V_13 ) ) ) ).
thf(fact_75_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_76_less__eq__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit0 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_77_rel__simps_I32_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_78_rel__simps_I27_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_eq_int @ K @ pls ) ) ).
thf(fact_79_rel__simps_I21_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_80_zadd__zless__mono,axiom,
! [Z_8: int,Z: int,W_14: int,W: int] :
( ( ord_less_int @ W_14 @ W )
=> ( ( ord_less_eq_int @ Z_8 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_14 @ Z_8 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_81_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_82_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_83_le__special_I4_J,axiom,
! [X_33: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_33 ) @ one_one_real )
<=> ( ord_less_eq_int @ X_33 @ ( bit1 @ pls ) ) ) ).
thf(fact_84_le__special_I4_J,axiom,
! [X_33: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_33 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_33 @ ( bit1 @ pls ) ) ) ).
thf(fact_85_le__special_I2_J,axiom,
! [Y_24: int] :
( ( ord_less_eq_real @ one_one_real @ ( number267125858f_real @ Y_24 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_24 ) ) ).
thf(fact_86_le__special_I2_J,axiom,
! [Y_24: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_24 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_24 ) ) ).
thf(fact_87_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_88_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_89_mult__Bit0,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit0 @ K ) @ L )
= ( bit0 @ ( times_times_int @ K @ L ) ) ) ).
thf(fact_90_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_91_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_92_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_93_plus__numeral__code_I9_J,axiom,
! [V_4: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_4 ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V_4 @ W ) ) ) ).
thf(fact_94_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_95_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_96_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_97_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_98_less__eq__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_99_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_100_less__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_101_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_102_less__nat__number__of,axiom,
! [V_4: int,V_5: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
<=> ( ( ( ord_less_int @ V_4 @ V_5 )
=> ( ord_less_int @ pls @ V_5 ) )
& ( ord_less_int @ V_4 @ V_5 ) ) ) ).
thf(fact_103_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
<=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
thf(fact_104_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_105_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_106_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
thf(fact_107_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_108_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_109_semiring__add__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( plus_plus_real @ ( number267125858f_real @ V_11 ) @ ( number267125858f_real @ V_12 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_110_semiring__add__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_11 ) @ ( number_number_of_nat @ V_12 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_111_semiring__add__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( plus_plus_int @ ( number_number_of_int @ V_11 ) @ ( number_number_of_int @ V_12 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_112_add__nat__number__of,axiom,
! [V_5: int,V_4: int] :
( ( ( ord_less_int @ V_4 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
= ( number_number_of_nat @ V_5 ) ) )
& ( ~ ( ord_less_int @ V_4 @ pls )
=> ( ( ( ord_less_int @ V_5 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
= ( number_number_of_nat @ V_4 ) ) )
& ( ~ ( ord_less_int @ V_5 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_4 @ V_5 ) ) ) ) ) ) ) ).
thf(fact_113_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
thf(fact_114_eq__number__of,axiom,
! [X_32: int,Y_23: int] :
( ( ( number267125858f_real @ X_32 )
= ( number267125858f_real @ Y_23 ) )
<=> ( X_32 = Y_23 ) ) ).
thf(fact_115_eq__number__of,axiom,
! [X_32: int,Y_23: int] :
( ( ( number_number_of_int @ X_32 )
= ( number_number_of_int @ Y_23 ) )
<=> ( X_32 = Y_23 ) ) ).
thf(fact_116_number__of__reorient,axiom,
! [W_13: int,X_31: real] :
( ( ( number267125858f_real @ W_13 )
= X_31 )
<=> ( X_31
= ( number267125858f_real @ W_13 ) ) ) ).
thf(fact_117_number__of__reorient,axiom,
! [W_13: int,X_31: int] :
( ( ( number_number_of_int @ W_13 )
= X_31 )
<=> ( X_31
= ( number_number_of_int @ W_13 ) ) ) ).
thf(fact_118_number__of__reorient,axiom,
! [W_13: int,X_31: nat] :
( ( ( number_number_of_nat @ W_13 )
= X_31 )
<=> ( X_31
= ( number_number_of_nat @ W_13 ) ) ) ).
thf(fact_119_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_120_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( ( bit0 @ K )
= ( bit0 @ L ) )
<=> ( K = L ) ) ).
thf(fact_121_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_122_sum__squares__eq__zero__iff,axiom,
! [X_30: real,Y_22: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_30 @ X_30 ) @ ( times_times_real @ Y_22 @ Y_22 ) )
= zero_zero_real )
<=> ( ( X_30 = zero_zero_real )
& ( Y_22 = zero_zero_real ) ) ) ).
thf(fact_123_sum__squares__eq__zero__iff,axiom,
! [X_30: int,Y_22: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_30 @ X_30 ) @ ( times_times_int @ Y_22 @ Y_22 ) )
= zero_zero_int )
<=> ( ( X_30 = zero_zero_int )
& ( Y_22 = zero_zero_int ) ) ) ).
thf(fact_124_left__distrib__number__of,axiom,
! [A_61: real,B_22: real,V_10: int] :
( ( times_times_real @ ( plus_plus_real @ A_61 @ B_22 ) @ ( number267125858f_real @ V_10 ) )
= ( plus_plus_real @ ( times_times_real @ A_61 @ ( number267125858f_real @ V_10 ) ) @ ( times_times_real @ B_22 @ ( number267125858f_real @ V_10 ) ) ) ) ).
thf(fact_125_left__distrib__number__of,axiom,
! [A_61: nat,B_22: nat,V_10: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_61 @ B_22 ) @ ( number_number_of_nat @ V_10 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_61 @ ( number_number_of_nat @ V_10 ) ) @ ( times_times_nat @ B_22 @ ( number_number_of_nat @ V_10 ) ) ) ) ).
thf(fact_126_left__distrib__number__of,axiom,
! [A_61: int,B_22: int,V_10: int] :
( ( times_times_int @ ( plus_plus_int @ A_61 @ B_22 ) @ ( number_number_of_int @ V_10 ) )
= ( plus_plus_int @ ( times_times_int @ A_61 @ ( number_number_of_int @ V_10 ) ) @ ( times_times_int @ B_22 @ ( number_number_of_int @ V_10 ) ) ) ) ).
thf(fact_127_right__distrib__number__of,axiom,
! [V_9: int,B_21: real,C_13: real] :
( ( times_times_real @ ( number267125858f_real @ V_9 ) @ ( plus_plus_real @ B_21 @ C_13 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_9 ) @ B_21 ) @ ( times_times_real @ ( number267125858f_real @ V_9 ) @ C_13 ) ) ) ).
thf(fact_128_right__distrib__number__of,axiom,
! [V_9: int,B_21: nat,C_13: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_9 ) @ ( plus_plus_nat @ B_21 @ C_13 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_9 ) @ B_21 ) @ ( times_times_nat @ ( number_number_of_nat @ V_9 ) @ C_13 ) ) ) ).
thf(fact_129_right__distrib__number__of,axiom,
! [V_9: int,B_21: int,C_13: int] :
( ( times_times_int @ ( number_number_of_int @ V_9 ) @ ( plus_plus_int @ B_21 @ C_13 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_9 ) @ B_21 ) @ ( times_times_int @ ( number_number_of_int @ V_9 ) @ C_13 ) ) ) ).
thf(fact_130_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_131_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ).
thf(fact_132_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_133_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_134_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_135_power2__eq__imp__eq,axiom,
! [X_29: real,Y_21: real] :
( ( ( power_power_real @ X_29 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_real @ Y_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X_29 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_21 )
=> ( X_29 = Y_21 ) ) ) ) ).
thf(fact_136_power2__eq__imp__eq,axiom,
! [X_29: nat,Y_21: nat] :
( ( ( power_power_nat @ X_29 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_nat @ Y_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X_29 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_21 )
=> ( X_29 = Y_21 ) ) ) ) ).
thf(fact_137_power2__eq__imp__eq,axiom,
! [X_29: int,Y_21: int] :
( ( ( power_power_int @ X_29 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ Y_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X_29 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_21 )
=> ( X_29 = Y_21 ) ) ) ) ).
thf(fact_138_power2__le__imp__le,axiom,
! [X_28: real,Y_20: real] :
( ( ord_less_eq_real @ ( power_power_real @ X_28 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_20 )
=> ( ord_less_eq_real @ X_28 @ Y_20 ) ) ) ).
thf(fact_139_power2__le__imp__le,axiom,
! [X_28: nat,Y_20: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X_28 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_20 )
=> ( ord_less_eq_nat @ X_28 @ Y_20 ) ) ) ).
thf(fact_140_power2__le__imp__le,axiom,
! [X_28: int,Y_20: int] :
( ( ord_less_eq_int @ ( power_power_int @ X_28 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_20 )
=> ( ord_less_eq_int @ X_28 @ Y_20 ) ) ) ).
thf(fact_141_zero__le__power2,axiom,
! [A_60: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_60 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_142_zero__le__power2,axiom,
! [A_60: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_60 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_143_power2__less__imp__less,axiom,
! [X_27: real,Y_19: real] :
( ( ord_less_real @ ( power_power_real @ X_27 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_19 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_19 )
=> ( ord_less_real @ X_27 @ Y_19 ) ) ) ).
thf(fact_144_power2__less__imp__less,axiom,
! [X_27: nat,Y_19: nat] :
( ( ord_less_nat @ ( power_power_nat @ X_27 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_19 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_19 )
=> ( ord_less_nat @ X_27 @ Y_19 ) ) ) ).
thf(fact_145_power2__less__imp__less,axiom,
! [X_27: int,Y_19: int] :
( ( ord_less_int @ ( power_power_int @ X_27 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_19 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_19 )
=> ( ord_less_int @ X_27 @ Y_19 ) ) ) ).
thf(fact_146_sum__power2__le__zero__iff,axiom,
! [X_26: real,Y_18: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real )
<=> ( ( X_26 = zero_zero_real )
& ( Y_18 = zero_zero_real ) ) ) ).
thf(fact_147_sum__power2__le__zero__iff,axiom,
! [X_26: int,Y_18: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int )
<=> ( ( X_26 = zero_zero_int )
& ( Y_18 = zero_zero_int ) ) ) ).
thf(fact_148_sum__power2__ge__zero,axiom,
! [X_25: real,Y_17: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_25 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_17 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_149_sum__power2__ge__zero,axiom,
! [X_25: int,Y_17: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_25 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_17 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_150_sum__squares__gt__zero__iff,axiom,
! [X_24: real,Y_16: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_24 @ X_24 ) @ ( times_times_real @ Y_16 @ Y_16 ) ) )
<=> ( ( X_24 != zero_zero_real )
| ( Y_16 != zero_zero_real ) ) ) ).
thf(fact_151_sum__squares__gt__zero__iff,axiom,
! [X_24: int,Y_16: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_24 @ X_24 ) @ ( times_times_int @ Y_16 @ Y_16 ) ) )
<=> ( ( X_24 != zero_zero_int )
| ( Y_16 != zero_zero_int ) ) ) ).
thf(fact_152_not__sum__squares__lt__zero,axiom,
! [X_23: real,Y_15: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X_23 @ X_23 ) @ ( times_times_real @ Y_15 @ Y_15 ) ) @ zero_zero_real ) ).
thf(fact_153_not__sum__squares__lt__zero,axiom,
! [X_23: int,Y_15: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_23 @ X_23 ) @ ( times_times_int @ Y_15 @ Y_15 ) ) @ zero_zero_int ) ).
thf(fact_154_mult__numeral__1,axiom,
! [A_59: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit1 @ pls ) ) @ A_59 )
= A_59 ) ).
thf(fact_155_mult__numeral__1,axiom,
! [A_59: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit1 @ pls ) ) @ A_59 )
= A_59 ) ).
thf(fact_156_mult__numeral__1__right,axiom,
! [A_58: real] :
( ( times_times_real @ A_58 @ ( number267125858f_real @ ( bit1 @ pls ) ) )
= A_58 ) ).
thf(fact_157_mult__numeral__1__right,axiom,
! [A_58: int] :
( ( times_times_int @ A_58 @ ( number_number_of_int @ ( bit1 @ pls ) ) )
= A_58 ) ).
thf(fact_158_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_159_mult__Bit1,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit1 @ K ) @ L )
= ( plus_plus_int @ ( bit0 @ ( times_times_int @ K @ L ) ) @ L ) ) ).
thf(fact_160_pos__zmult__eq__1__iff,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
thf(fact_161_double__eq__0__iff,axiom,
! [A_57: real] :
( ( ( plus_plus_real @ A_57 @ A_57 )
= zero_zero_real )
<=> ( A_57 = zero_zero_real ) ) ).
thf(fact_162_double__eq__0__iff,axiom,
! [A_57: int] :
( ( ( plus_plus_int @ A_57 @ A_57 )
= zero_zero_int )
<=> ( A_57 = zero_zero_int ) ) ).
thf(fact_163_rel__simps_I46_J,axiom,
! [K: int] :
( ( bit1 @ K )
!= pls ) ).
thf(fact_164_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_165_rel__simps_I50_J,axiom,
! [K: int,L: int] :
( ( bit1 @ K )
!= ( bit0 @ L ) ) ).
thf(fact_166_rel__simps_I49_J,axiom,
! [K: int,L: int] :
( ( bit0 @ K )
!= ( bit1 @ L ) ) ).
thf(fact_167_rel__simps_I44_J,axiom,
! [K: int] :
( ( ( bit0 @ K )
= pls )
<=> ( K = pls ) ) ).
thf(fact_168_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_169_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_170_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_171_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_172_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_173_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_174_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_175_rel__simps_I14_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_176_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_177_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_178_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_179_add__Bit0__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_180_Bit0__def,axiom,
! [K: int] :
( ( bit0 @ K )
= ( plus_plus_int @ K @ K ) ) ).
thf(fact_181_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_182_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_183_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_184_double__number__of__Bit0,axiom,
! [W_12: int] :
( ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ ( number267125858f_real @ W_12 ) )
= ( number267125858f_real @ ( bit0 @ W_12 ) ) ) ).
thf(fact_185_double__number__of__Bit0,axiom,
! [W_12: int] :
( ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ ( number_number_of_int @ W_12 ) )
= ( number_number_of_int @ ( bit0 @ W_12 ) ) ) ).
thf(fact_186_power3__eq__cube,axiom,
! [A_56: real] :
( ( power_power_real @ A_56 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( times_times_real @ A_56 @ A_56 ) @ A_56 ) ) ).
thf(fact_187_power3__eq__cube,axiom,
! [A_56: nat] :
( ( power_power_nat @ A_56 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( times_times_nat @ A_56 @ A_56 ) @ A_56 ) ) ).
thf(fact_188_power3__eq__cube,axiom,
! [A_56: int] :
( ( power_power_int @ A_56 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( times_times_int @ A_56 @ A_56 ) @ A_56 ) ) ).
thf(fact_189_semiring__mult__2,axiom,
! [Z_7: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_7 )
= ( plus_plus_real @ Z_7 @ Z_7 ) ) ).
thf(fact_190_semiring__mult__2,axiom,
! [Z_7: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_7 )
= ( plus_plus_nat @ Z_7 @ Z_7 ) ) ).
thf(fact_191_semiring__mult__2,axiom,
! [Z_7: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_7 )
= ( plus_plus_int @ Z_7 @ Z_7 ) ) ).
thf(fact_192_mult__2,axiom,
! [Z_6: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_6 )
= ( plus_plus_real @ Z_6 @ Z_6 ) ) ).
thf(fact_193_mult__2,axiom,
! [Z_6: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_6 )
= ( plus_plus_int @ Z_6 @ Z_6 ) ) ).
thf(fact_194_semiring__mult__2__right,axiom,
! [Z_5: real] :
( ( times_times_real @ Z_5 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_5 @ Z_5 ) ) ).
thf(fact_195_semiring__mult__2__right,axiom,
! [Z_5: nat] :
( ( times_times_nat @ Z_5 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ Z_5 @ Z_5 ) ) ).
thf(fact_196_semiring__mult__2__right,axiom,
! [Z_5: int] :
( ( times_times_int @ Z_5 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_5 @ Z_5 ) ) ).
thf(fact_197_mult__2__right,axiom,
! [Z_4: real] :
( ( times_times_real @ Z_4 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_4 @ Z_4 ) ) ).
thf(fact_198_mult__2__right,axiom,
! [Z_4: int] :
( ( times_times_int @ Z_4 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_4 @ Z_4 ) ) ).
thf(fact_199_power2__eq__square,axiom,
! [A_55: real] :
( ( power_power_real @ A_55 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ A_55 @ A_55 ) ) ).
thf(fact_200_power2__eq__square,axiom,
! [A_55: nat] :
( ( power_power_nat @ A_55 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ A_55 @ A_55 ) ) ).
thf(fact_201_power2__eq__square,axiom,
! [A_55: int] :
( ( power_power_int @ A_55 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ A_55 @ A_55 ) ) ).
thf(fact_202_even__less__0__iff,axiom,
! [A_54: real] :
( ( ord_less_real @ ( plus_plus_real @ A_54 @ A_54 ) @ zero_zero_real )
<=> ( ord_less_real @ A_54 @ zero_zero_real ) ) ).
thf(fact_203_even__less__0__iff,axiom,
! [A_54: int] :
( ( ord_less_int @ ( plus_plus_int @ A_54 @ A_54 ) @ zero_zero_int )
<=> ( ord_less_int @ A_54 @ zero_zero_int ) ) ).
thf(fact_204_semiring__numeral__0__eq__0,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_205_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_206_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_207_number__of__Pls,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_208_number__of__Pls,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_209_semiring__norm_I112_J,axiom,
( zero_zero_real
= ( number267125858f_real @ pls ) ) ).
thf(fact_210_semiring__norm_I112_J,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_211_add__numeral__0,axiom,
! [A_53: real] :
( ( plus_plus_real @ ( number267125858f_real @ pls ) @ A_53 )
= A_53 ) ).
thf(fact_212_add__numeral__0,axiom,
! [A_53: int] :
( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_53 )
= A_53 ) ).
thf(fact_213_add__numeral__0__right,axiom,
! [A_52: real] :
( ( plus_plus_real @ A_52 @ ( number267125858f_real @ pls ) )
= A_52 ) ).
thf(fact_214_add__numeral__0__right,axiom,
! [A_52: int] :
( ( plus_plus_int @ A_52 @ ( number_number_of_int @ pls ) )
= A_52 ) ).
thf(fact_215_less__number__of,axiom,
! [X_22: int,Y_14: int] :
( ( ord_less_real @ ( number267125858f_real @ X_22 ) @ ( number267125858f_real @ Y_14 ) )
<=> ( ord_less_int @ X_22 @ Y_14 ) ) ).
thf(fact_216_less__number__of,axiom,
! [X_22: int,Y_14: int] :
( ( ord_less_int @ ( number_number_of_int @ X_22 ) @ ( number_number_of_int @ Y_14 ) )
<=> ( ord_less_int @ X_22 @ Y_14 ) ) ).
thf(fact_217_add__number__of__left,axiom,
! [V_8: int,W_11: int,Z_3: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_8 ) @ ( plus_plus_real @ ( number267125858f_real @ W_11 ) @ Z_3 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_8 @ W_11 ) ) @ Z_3 ) ) ).
thf(fact_218_add__number__of__left,axiom,
! [V_8: int,W_11: int,Z_3: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_8 ) @ ( plus_plus_int @ ( number_number_of_int @ W_11 ) @ Z_3 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_8 @ W_11 ) ) @ Z_3 ) ) ).
thf(fact_219_add__number__of__eq,axiom,
! [V_7: int,W_10: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_7 ) @ ( number267125858f_real @ W_10 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_7 @ W_10 ) ) ) ).
thf(fact_220_add__number__of__eq,axiom,
! [V_7: int,W_10: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_7 ) @ ( number_number_of_int @ W_10 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_7 @ W_10 ) ) ) ).
thf(fact_221_number__of__add,axiom,
! [V_6: int,W_9: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_6 @ W_9 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_6 ) @ ( number267125858f_real @ W_9 ) ) ) ).
thf(fact_222_number__of__add,axiom,
! [V_6: int,W_9: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_6 @ W_9 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_6 ) @ ( number_number_of_int @ W_9 ) ) ) ).
thf(fact_223_rel__simps_I12_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_224_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_225_rel__simps_I16_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_226_bin__less__0__simps_I4_J,axiom,
! [W: int] :
( ( ord_less_int @ ( bit1 @ W ) @ zero_zero_int )
<=> ( ord_less_int @ W @ zero_zero_int ) ) ).
thf(fact_227_rel__simps_I10_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_228_rel__simps_I4_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_int @ pls @ K ) ) ).
thf(fact_229_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_230_bin__less__0__simps_I3_J,axiom,
! [W: int] :
( ( ord_less_int @ ( bit0 @ W ) @ zero_zero_int )
<=> ( ord_less_int @ W @ zero_zero_int ) ) ).
thf(fact_231_add__Bit1__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_232_add__Bit0__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_233_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_234_Bit1__def,axiom,
! [K: int] :
( ( bit1 @ K )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
thf(fact_235_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_236_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
thf(fact_237_power2__eq__square__number__of,axiom,
! [W_8: int] :
( ( power_power_real @ ( number267125858f_real @ W_8 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( number267125858f_real @ W_8 ) @ ( number267125858f_real @ W_8 ) ) ) ).
thf(fact_238_power2__eq__square__number__of,axiom,
! [W_8: int] :
( ( power_power_nat @ ( number_number_of_nat @ W_8 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ W_8 ) @ ( number_number_of_nat @ W_8 ) ) ) ).
thf(fact_239_power2__eq__square__number__of,axiom,
! [W_8: int] :
( ( power_power_int @ ( number_number_of_int @ W_8 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( number_number_of_int @ W_8 ) @ ( number_number_of_int @ W_8 ) ) ) ).
thf(fact_240_power2__sum,axiom,
! [X_21: real,Y_13: real] :
( ( power_power_real @ ( plus_plus_real @ X_21 @ Y_13 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_21 ) @ Y_13 ) ) ) ).
thf(fact_241_power2__sum,axiom,
! [X_21: nat,Y_13: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X_21 @ Y_13 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_21 ) @ Y_13 ) ) ) ).
thf(fact_242_power2__sum,axiom,
! [X_21: int,Y_13: int] :
( ( power_power_int @ ( plus_plus_int @ X_21 @ Y_13 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_21 ) @ Y_13 ) ) ) ).
thf(fact_243_number__of__Bit0,axiom,
! [W_7: int] :
( ( number267125858f_real @ ( bit0 @ W_7 ) )
= ( plus_plus_real @ ( plus_plus_real @ zero_zero_real @ ( number267125858f_real @ W_7 ) ) @ ( number267125858f_real @ W_7 ) ) ) ).
thf(fact_244_number__of__Bit0,axiom,
! [W_7: int] :
( ( number_number_of_int @ ( bit0 @ W_7 ) )
= ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ ( number_number_of_int @ W_7 ) ) @ ( number_number_of_int @ W_7 ) ) ) ).
thf(fact_245_number__of__Bit1,axiom,
! [W_6: int] :
( ( number267125858f_real @ ( bit1 @ W_6 ) )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_6 ) ) @ ( number267125858f_real @ W_6 ) ) ) ).
thf(fact_246_number__of__Bit1,axiom,
! [W_6: int] :
( ( number_number_of_int @ ( bit1 @ W_6 ) )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_6 ) ) @ ( number_number_of_int @ W_6 ) ) ) ).
thf(fact_247_semiring__numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_248_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_249_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_250_numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_251_numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_252_semiring__norm_I110_J,axiom,
( one_one_real
= ( number267125858f_real @ ( bit1 @ pls ) ) ) ).
thf(fact_253_semiring__norm_I110_J,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_254_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_255_less__special_I3_J,axiom,
! [X_20: int] :
( ( ord_less_real @ ( number267125858f_real @ X_20 ) @ zero_zero_real )
<=> ( ord_less_int @ X_20 @ pls ) ) ).
thf(fact_256_less__special_I3_J,axiom,
! [X_20: int] :
( ( ord_less_int @ ( number_number_of_int @ X_20 ) @ zero_zero_int )
<=> ( ord_less_int @ X_20 @ pls ) ) ).
thf(fact_257_less__special_I1_J,axiom,
! [Y_12: int] :
( ( ord_less_real @ zero_zero_real @ ( number267125858f_real @ Y_12 ) )
<=> ( ord_less_int @ pls @ Y_12 ) ) ).
thf(fact_258_less__special_I1_J,axiom,
! [Y_12: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_12 ) )
<=> ( ord_less_int @ pls @ Y_12 ) ) ).
thf(fact_259_semiring__one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_260_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_261_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_262_one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_263_one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_264_less__special_I4_J,axiom,
! [X_19: int] :
( ( ord_less_real @ ( number267125858f_real @ X_19 ) @ one_one_real )
<=> ( ord_less_int @ X_19 @ ( bit1 @ pls ) ) ) ).
thf(fact_265_less__special_I4_J,axiom,
! [X_19: int] :
( ( ord_less_int @ ( number_number_of_int @ X_19 ) @ one_one_int )
<=> ( ord_less_int @ X_19 @ ( bit1 @ pls ) ) ) ).
thf(fact_266_less__special_I2_J,axiom,
! [Y_11: int] :
( ( ord_less_real @ one_one_real @ ( number267125858f_real @ Y_11 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_11 ) ) ).
thf(fact_267_less__special_I2_J,axiom,
! [Y_11: int] :
( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_11 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_11 ) ) ).
thf(fact_268__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
dvd_dvd_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) ).
thf(fact_269_zadd__power3,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_270_zadd__power2,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) @ B ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_271_int__pos__lt__two__imp__zero__or__one,axiom,
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( X = zero_zero_int )
| ( X = one_one_int ) ) ) ) ).
thf(fact_272_s0p,axiom,
( ( ord_less_eq_int @ zero_zero_int @ s )
& ( ord_less_int @ s @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ s @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_273_cube__square,axiom,
! [A: int] :
( ( times_times_int @ A @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_274_power2__ge__self,axiom,
! [X: int] : ( ord_less_eq_int @ X @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_275_self__quotient__aux1,axiom,
! [R: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_int @ R @ A )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_276_self__quotient__aux2,axiom,
! [R: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_277_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_278_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_18: real,N_35: nat] :
( ( power_power_real @ X_18 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_35 ) )
= ( times_times_real @ ( power_power_real @ X_18 @ N_35 ) @ ( power_power_real @ X_18 @ N_35 ) ) ) ).
thf(fact_279_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_18: nat,N_35: nat] :
( ( power_power_nat @ X_18 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_35 ) )
= ( times_times_nat @ ( power_power_nat @ X_18 @ N_35 ) @ ( power_power_nat @ X_18 @ N_35 ) ) ) ).
thf(fact_280_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_18: int,N_35: nat] :
( ( power_power_int @ X_18 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_35 ) )
= ( times_times_int @ ( power_power_int @ X_18 @ N_35 ) @ ( power_power_int @ X_18 @ N_35 ) ) ) ).
thf(fact_281__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ s1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_282__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ X_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ Y_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_283__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
~ ! [S: int] :
~ ( ( ord_less_eq_int @ zero_zero_int @ S )
& ( ord_less_int @ S @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ S @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_284_s1,axiom,
zcong @ ( power_power_int @ s1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_285_zero__less__power__nat__eq,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
<=> ( ( N = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ X ) ) ) ).
thf(fact_286_zprime__zdvd__power,axiom,
! [A: int,N: nat,P: int] :
( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ A @ N ) )
=> ( dvd_dvd_int @ P @ A ) ) ) ).
thf(fact_287_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_17: real,P_2: nat,Q_4: nat] :
( ( power_power_real @ ( power_power_real @ X_17 @ P_2 ) @ Q_4 )
= ( power_power_real @ X_17 @ ( times_times_nat @ P_2 @ Q_4 ) ) ) ).
thf(fact_288_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_17: nat,P_2: nat,Q_4: nat] :
( ( power_power_nat @ ( power_power_nat @ X_17 @ P_2 ) @ Q_4 )
= ( power_power_nat @ X_17 @ ( times_times_nat @ P_2 @ Q_4 ) ) ) ).
thf(fact_289_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_17: int,P_2: nat,Q_4: nat] :
( ( power_power_int @ ( power_power_int @ X_17 @ P_2 ) @ Q_4 )
= ( power_power_int @ X_17 @ ( times_times_nat @ P_2 @ Q_4 ) ) ) ).
thf(fact_290_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B: int,P: int] :
( ( zprime @ P )
=> ( ~ ( dvd_dvd_int @ P @ B )
=> ( ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ A ) ) ) ) ).
thf(fact_291_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ~ ( dvd_dvd_int @ P @ A )
=> ( ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ B ) ) ) ) ).
thf(fact_292_zpower__zpower,axiom,
! [X: int,Y: nat,Z: nat] :
( ( power_power_int @ ( power_power_int @ X @ Y ) @ Z )
= ( power_power_int @ X @ ( times_times_nat @ Y @ Z ) ) ) ).
thf(fact_293_zdvd__not__zless,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
thf(fact_294_zdvd__antisym__nonneg,axiom,
! [N: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
thf(fact_295_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
thf(fact_296_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
<=> ( dvd_dvd_int @ K @ N ) ) ).
thf(fact_297_zdvd__period,axiom,
! [C: int,X: int,T: int,A: int,D: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
<=> ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
thf(fact_298_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_299_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
thf(fact_300_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( ( twoSqu362149276sum2sq @ X )
=> ( ( twoSqu362149276sum2sq @ Y )
=> ( twoSqu362149276sum2sq @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_301_le__nat__number__of,axiom,
! [V_4: int,V_5: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
<=> ( ~ ( ord_less_eq_int @ V_4 @ V_5 )
=> ( ord_less_eq_int @ V_4 @ pls ) ) ) ).
thf(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: real,Ly_4: real,Rx_6: real,Ry_4: real] :
( ( times_times_real @ ( times_times_real @ Lx_6 @ Ly_4 ) @ ( times_times_real @ Rx_6 @ Ry_4 ) )
= ( times_times_real @ ( times_times_real @ Lx_6 @ Rx_6 ) @ ( times_times_real @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_6 @ Ly_4 ) @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_6 @ Rx_6 ) @ ( times_times_nat @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_6 @ Ly_4 ) @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ ( times_times_int @ Lx_6 @ Rx_6 ) @ ( times_times_int @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: real,Ly_3: real,Rx_5: real,Ry_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_5 @ Ly_3 ) @ ( times_times_real @ Rx_5 @ Ry_3 ) )
= ( times_times_real @ Rx_5 @ ( times_times_real @ ( times_times_real @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_306_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ Rx_5 @ ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_307_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ Rx_5 @ ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: real,Ly_2: real,Rx_4: real,Ry_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_4 @ Ly_2 ) @ ( times_times_real @ Rx_4 @ Ry_2 ) )
= ( times_times_real @ Lx_4 @ ( times_times_real @ Ly_2 @ ( times_times_real @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_309_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_2 ) @ ( times_times_nat @ Rx_4 @ Ry_2 ) )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_310_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_2 ) @ ( times_times_int @ Rx_4 @ Ry_2 ) )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_311_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_1: real,Rx_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_real @ ( times_times_real @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_312_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_1: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_314_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: real,Ly: real,Rx_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_real @ Lx_2 @ ( times_times_real @ Ly @ Rx_2 ) ) ) ).
thf(fact_315_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly @ Rx_2 ) ) ) ).
thf(fact_316_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly @ Rx_2 ) ) ) ).
thf(fact_317_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: real,Rx_1: real,Ry_1: real] :
( ( times_times_real @ Lx_1 @ ( times_times_real @ Rx_1 @ Ry_1 ) )
= ( times_times_real @ ( times_times_real @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_318_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ Lx_1 @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ Lx_1 @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ ( times_times_int @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: real,Rx: real,Ry: real] :
( ( times_times_real @ Lx @ ( times_times_real @ Rx @ Ry ) )
= ( times_times_real @ Rx @ ( times_times_real @ Lx @ Ry ) ) ) ).
thf(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ Lx @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ Rx @ ( times_times_nat @ Lx @ Ry ) ) ) ).
thf(fact_322_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] :
( ( times_times_int @ Lx @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ Rx @ ( times_times_int @ Lx @ Ry ) ) ) ).
thf(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_51: real,B_20: real] :
( ( times_times_real @ A_51 @ B_20 )
= ( times_times_real @ B_20 @ A_51 ) ) ).
thf(fact_324_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_51: nat,B_20: nat] :
( ( times_times_nat @ A_51 @ B_20 )
= ( times_times_nat @ B_20 @ A_51 ) ) ).
thf(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_51: int,B_20: int] :
( ( times_times_int @ A_51 @ B_20 )
= ( times_times_int @ B_20 @ A_51 ) ) ).
thf(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_50: real,C_12: real] :
( ( plus_plus_real @ A_50 @ C_12 )
= ( plus_plus_real @ C_12 @ A_50 ) ) ).
thf(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_50: nat,C_12: nat] :
( ( plus_plus_nat @ A_50 @ C_12 )
= ( plus_plus_nat @ C_12 @ A_50 ) ) ).
thf(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_50: int,C_12: int] :
( ( plus_plus_int @ A_50 @ C_12 )
= ( plus_plus_int @ C_12 @ A_50 ) ) ).
thf(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_49: real,C_11: real,D_6: real] :
( ( plus_plus_real @ A_49 @ ( plus_plus_real @ C_11 @ D_6 ) )
= ( plus_plus_real @ C_11 @ ( plus_plus_real @ A_49 @ D_6 ) ) ) ).
thf(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_49: nat,C_11: nat,D_6: nat] :
( ( plus_plus_nat @ A_49 @ ( plus_plus_nat @ C_11 @ D_6 ) )
= ( plus_plus_nat @ C_11 @ ( plus_plus_nat @ A_49 @ D_6 ) ) ) ).
thf(fact_331_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_49: int,C_11: int,D_6: int] :
( ( plus_plus_int @ A_49 @ ( plus_plus_int @ C_11 @ D_6 ) )
= ( plus_plus_int @ C_11 @ ( plus_plus_int @ A_49 @ D_6 ) ) ) ).
thf(fact_332_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_48: real,C_10: real,D_5: real] :
( ( plus_plus_real @ A_48 @ ( plus_plus_real @ C_10 @ D_5 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_48 @ C_10 ) @ D_5 ) ) ).
thf(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_48: nat,C_10: nat,D_5: nat] :
( ( plus_plus_nat @ A_48 @ ( plus_plus_nat @ C_10 @ D_5 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_48 @ C_10 ) @ D_5 ) ) ).
thf(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_48: int,C_10: int,D_5: int] :
( ( plus_plus_int @ A_48 @ ( plus_plus_int @ C_10 @ D_5 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_48 @ C_10 ) @ D_5 ) ) ).
thf(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_47: real,B_19: real,C_9: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_47 @ B_19 ) @ C_9 )
= ( plus_plus_real @ A_47 @ ( plus_plus_real @ B_19 @ C_9 ) ) ) ).
thf(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_47: nat,B_19: nat,C_9: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_47 @ B_19 ) @ C_9 )
= ( plus_plus_nat @ A_47 @ ( plus_plus_nat @ B_19 @ C_9 ) ) ) ).
thf(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_47: int,B_19: int,C_9: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_47 @ B_19 ) @ C_9 )
= ( plus_plus_int @ A_47 @ ( plus_plus_int @ B_19 @ C_9 ) ) ) ).
thf(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_46: real,B_18: real,C_8: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_46 @ B_18 ) @ C_8 )
= ( plus_plus_real @ ( plus_plus_real @ A_46 @ C_8 ) @ B_18 ) ) ).
thf(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_46: nat,B_18: nat,C_8: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_46 @ B_18 ) @ C_8 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_46 @ C_8 ) @ B_18 ) ) ).
thf(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_46: int,B_18: int,C_8: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_46 @ B_18 ) @ C_8 )
= ( plus_plus_int @ ( plus_plus_int @ A_46 @ C_8 ) @ B_18 ) ) ).
thf(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_45: real,B_17: real,C_7: real,D_4: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_45 @ B_17 ) @ ( plus_plus_real @ C_7 @ D_4 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_45 @ C_7 ) @ ( plus_plus_real @ B_17 @ D_4 ) ) ) ).
thf(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_45: nat,B_17: nat,C_7: nat,D_4: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_45 @ B_17 ) @ ( plus_plus_nat @ C_7 @ D_4 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_45 @ C_7 ) @ ( plus_plus_nat @ B_17 @ D_4 ) ) ) ).
thf(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_45: int,B_17: int,C_7: int,D_4: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_45 @ B_17 ) @ ( plus_plus_int @ C_7 @ D_4 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_45 @ C_7 ) @ ( plus_plus_int @ B_17 @ D_4 ) ) ) ).
thf(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_16: real] :
( ( power_power_real @ X_16 @ one_one_nat )
= X_16 ) ).
thf(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_16: nat] :
( ( power_power_nat @ X_16 @ one_one_nat )
= X_16 ) ).
thf(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_16: int] :
( ( power_power_int @ X_16 @ one_one_nat )
= X_16 ) ).
thf(fact_347_zero__less__power__nat__eq__number__of,axiom,
! [X: nat,W: int] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ ( number_number_of_nat @ W ) ) )
<=> ( ( ( number_number_of_nat @ W )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ X ) ) ) ).
thf(fact_348_nat__mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
thf(fact_349_nat__mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
thf(fact_350_mult__nat__number__of,axiom,
! [V_5: int,V_4: int] :
( ( ( ord_less_int @ V_4 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V_4 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_4 ) @ ( number_number_of_nat @ V_5 ) )
= ( number_number_of_nat @ ( times_times_int @ V_4 @ V_5 ) ) ) ) ) ).
thf(fact_351_nat__number__of__mult__left,axiom,
! [V_5: int,K: nat,V_4: int] :
( ( ( ord_less_int @ V_4 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_4 ) @ ( times_times_nat @ ( number_number_of_nat @ V_5 ) @ K ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V_4 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_4 ) @ ( times_times_nat @ ( number_number_of_nat @ V_5 ) @ K ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V_4 @ V_5 ) ) @ K ) ) ) ) ).
thf(fact_352_power__even__eq,axiom,
! [A_44: real,N_34: nat] :
( ( power_power_real @ A_44 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_34 ) )
= ( power_power_real @ ( power_power_real @ A_44 @ N_34 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_353_power__even__eq,axiom,
! [A_44: nat,N_34: nat] :
( ( power_power_nat @ A_44 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_34 ) )
= ( power_power_nat @ ( power_power_nat @ A_44 @ N_34 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_354_power__even__eq,axiom,
! [A_44: int,N_34: nat] :
( ( power_power_int @ A_44 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_34 ) )
= ( power_power_int @ ( power_power_int @ A_44 @ N_34 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_355_even__power__le__0__imp__0,axiom,
! [A_43: real,K_2: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A_43 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_2 ) ) @ zero_zero_real )
=> ( A_43 = zero_zero_real ) ) ).
thf(fact_356_even__power__le__0__imp__0,axiom,
! [A_43: int,K_2: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A_43 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_2 ) ) @ zero_zero_int )
=> ( A_43 = zero_zero_int ) ) ).
thf(fact_357_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_42: real] :
( ( times_times_real @ zero_zero_real @ A_42 )
= zero_zero_real ) ).
thf(fact_358_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_42: nat] :
( ( times_times_nat @ zero_zero_nat @ A_42 )
= zero_zero_nat ) ).
thf(fact_359_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_42: int] :
( ( times_times_int @ zero_zero_int @ A_42 )
= zero_zero_int ) ).
thf(fact_360_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_41: real] :
( ( times_times_real @ A_41 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_361_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_41: nat] :
( ( times_times_nat @ A_41 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_41: int] :
( ( times_times_int @ A_41 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40: real] :
( ( plus_plus_real @ zero_zero_real @ A_40 )
= A_40 ) ).
thf(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_40 )
= A_40 ) ).
thf(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40: int] :
( ( plus_plus_int @ zero_zero_int @ A_40 )
= A_40 ) ).
thf(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39: real] :
( ( plus_plus_real @ A_39 @ zero_zero_real )
= A_39 ) ).
thf(fact_367_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39: nat] :
( ( plus_plus_nat @ A_39 @ zero_zero_nat )
= A_39 ) ).
thf(fact_368_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39: int] :
( ( plus_plus_int @ A_39 @ zero_zero_int )
= A_39 ) ).
thf(fact_369_add__0__iff,axiom,
! [B_16: real,A_38: real] :
( ( B_16
= ( plus_plus_real @ B_16 @ A_38 ) )
<=> ( A_38 = zero_zero_real ) ) ).
thf(fact_370_add__0__iff,axiom,
! [B_16: nat,A_38: nat] :
( ( B_16
= ( plus_plus_nat @ B_16 @ A_38 ) )
<=> ( A_38 = zero_zero_nat ) ) ).
thf(fact_371_add__0__iff,axiom,
! [B_16: int,A_38: int] :
( ( B_16
= ( plus_plus_int @ B_16 @ A_38 ) )
<=> ( A_38 = zero_zero_int ) ) ).
thf(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_15: real,Y_10: real,Z_2: real] :
( ( times_times_real @ X_15 @ ( plus_plus_real @ Y_10 @ Z_2 ) )
= ( plus_plus_real @ ( times_times_real @ X_15 @ Y_10 ) @ ( times_times_real @ X_15 @ Z_2 ) ) ) ).
thf(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_15: nat,Y_10: nat,Z_2: nat] :
( ( times_times_nat @ X_15 @ ( plus_plus_nat @ Y_10 @ Z_2 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_15 @ Y_10 ) @ ( times_times_nat @ X_15 @ Z_2 ) ) ) ).
thf(fact_374_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_15: int,Y_10: int,Z_2: int] :
( ( times_times_int @ X_15 @ ( plus_plus_int @ Y_10 @ Z_2 ) )
= ( plus_plus_int @ ( times_times_int @ X_15 @ Y_10 ) @ ( times_times_int @ X_15 @ Z_2 ) ) ) ).
thf(fact_375_crossproduct__noteq,axiom,
! [C_6: real,D_3: real,A_37: real,B_15: real] :
( ( ( A_37 != B_15 )
& ( C_6 != D_3 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_37 @ C_6 ) @ ( times_times_real @ B_15 @ D_3 ) )
!= ( plus_plus_real @ ( times_times_real @ A_37 @ D_3 ) @ ( times_times_real @ B_15 @ C_6 ) ) ) ) ).
thf(fact_376_crossproduct__noteq,axiom,
! [C_6: nat,D_3: nat,A_37: nat,B_15: nat] :
( ( ( A_37 != B_15 )
& ( C_6 != D_3 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_37 @ C_6 ) @ ( times_times_nat @ B_15 @ D_3 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_37 @ D_3 ) @ ( times_times_nat @ B_15 @ C_6 ) ) ) ) ).
thf(fact_377_crossproduct__noteq,axiom,
! [C_6: int,D_3: int,A_37: int,B_15: int] :
( ( ( A_37 != B_15 )
& ( C_6 != D_3 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_37 @ C_6 ) @ ( times_times_int @ B_15 @ D_3 ) )
!= ( plus_plus_int @ ( times_times_int @ A_37 @ D_3 ) @ ( times_times_int @ B_15 @ C_6 ) ) ) ) ).
thf(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_36: real,B_14: real,C_5: real] :
( ( times_times_real @ ( plus_plus_real @ A_36 @ B_14 ) @ C_5 )
= ( plus_plus_real @ ( times_times_real @ A_36 @ C_5 ) @ ( times_times_real @ B_14 @ C_5 ) ) ) ).
thf(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_36: nat,B_14: nat,C_5: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_36 @ B_14 ) @ C_5 )
= ( plus_plus_nat @ ( times_times_nat @ A_36 @ C_5 ) @ ( times_times_nat @ B_14 @ C_5 ) ) ) ).
thf(fact_380_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_36: int,B_14: int,C_5: int] :
( ( times_times_int @ ( plus_plus_int @ A_36 @ B_14 ) @ C_5 )
= ( plus_plus_int @ ( times_times_int @ A_36 @ C_5 ) @ ( times_times_int @ B_14 @ C_5 ) ) ) ).
thf(fact_381_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_35: real,M_13: real,B_13: real] :
( ( plus_plus_real @ ( times_times_real @ A_35 @ M_13 ) @ ( times_times_real @ B_13 @ M_13 ) )
= ( times_times_real @ ( plus_plus_real @ A_35 @ B_13 ) @ M_13 ) ) ).
thf(fact_382_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_35: nat,M_13: nat,B_13: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_35 @ M_13 ) @ ( times_times_nat @ B_13 @ M_13 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_35 @ B_13 ) @ M_13 ) ) ).
thf(fact_383_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_35: int,M_13: int,B_13: int] :
( ( plus_plus_int @ ( times_times_int @ A_35 @ M_13 ) @ ( times_times_int @ B_13 @ M_13 ) )
= ( times_times_int @ ( plus_plus_int @ A_35 @ B_13 ) @ M_13 ) ) ).
thf(fact_384_crossproduct__eq,axiom,
! [W_5: real,Y_9: real,X_14: real,Z_1: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_5 @ Y_9 ) @ ( times_times_real @ X_14 @ Z_1 ) )
= ( plus_plus_real @ ( times_times_real @ W_5 @ Z_1 ) @ ( times_times_real @ X_14 @ Y_9 ) ) )
<=> ( ( W_5 = X_14 )
| ( Y_9 = Z_1 ) ) ) ).
thf(fact_385_crossproduct__eq,axiom,
! [W_5: nat,Y_9: nat,X_14: nat,Z_1: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_5 @ Y_9 ) @ ( times_times_nat @ X_14 @ Z_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_5 @ Z_1 ) @ ( times_times_nat @ X_14 @ Y_9 ) ) )
<=> ( ( W_5 = X_14 )
| ( Y_9 = Z_1 ) ) ) ).
thf(fact_386_crossproduct__eq,axiom,
! [W_5: int,Y_9: int,X_14: int,Z_1: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_5 @ Y_9 ) @ ( times_times_int @ X_14 @ Z_1 ) )
= ( plus_plus_int @ ( times_times_int @ W_5 @ Z_1 ) @ ( times_times_int @ X_14 @ Y_9 ) ) )
<=> ( ( W_5 = X_14 )
| ( Y_9 = Z_1 ) ) ) ).
thf(fact_387_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_34: real] :
( ( times_times_real @ one_one_real @ A_34 )
= A_34 ) ).
thf(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_34: nat] :
( ( times_times_nat @ one_one_nat @ A_34 )
= A_34 ) ).
thf(fact_389_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_34: int] :
( ( times_times_int @ one_one_int @ A_34 )
= A_34 ) ).
thf(fact_390_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_33: real] :
( ( times_times_real @ A_33 @ one_one_real )
= A_33 ) ).
thf(fact_391_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_33: nat] :
( ( times_times_nat @ A_33 @ one_one_nat )
= A_33 ) ).
thf(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_33: int] :
( ( times_times_int @ A_33 @ one_one_int )
= A_33 ) ).
thf(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_13: real,Y_8: real,Q_3: nat] :
( ( power_power_real @ ( times_times_real @ X_13 @ Y_8 ) @ Q_3 )
= ( times_times_real @ ( power_power_real @ X_13 @ Q_3 ) @ ( power_power_real @ Y_8 @ Q_3 ) ) ) ).
thf(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_13: nat,Y_8: nat,Q_3: nat] :
( ( power_power_nat @ ( times_times_nat @ X_13 @ Y_8 ) @ Q_3 )
= ( times_times_nat @ ( power_power_nat @ X_13 @ Q_3 ) @ ( power_power_nat @ Y_8 @ Q_3 ) ) ) ).
thf(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_13: int,Y_8: int,Q_3: nat] :
( ( power_power_int @ ( times_times_int @ X_13 @ Y_8 ) @ Q_3 )
= ( times_times_int @ ( power_power_int @ X_13 @ Q_3 ) @ ( power_power_int @ Y_8 @ Q_3 ) ) ) ).
thf(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_12: real,P_1: nat,Q_2: nat] :
( ( times_times_real @ ( power_power_real @ X_12 @ P_1 ) @ ( power_power_real @ X_12 @ Q_2 ) )
= ( power_power_real @ X_12 @ ( plus_plus_nat @ P_1 @ Q_2 ) ) ) ).
thf(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_12: nat,P_1: nat,Q_2: nat] :
( ( times_times_nat @ ( power_power_nat @ X_12 @ P_1 ) @ ( power_power_nat @ X_12 @ Q_2 ) )
= ( power_power_nat @ X_12 @ ( plus_plus_nat @ P_1 @ Q_2 ) ) ) ).
thf(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_12: int,P_1: nat,Q_2: nat] :
( ( times_times_int @ ( power_power_int @ X_12 @ P_1 ) @ ( power_power_int @ X_12 @ Q_2 ) )
= ( power_power_int @ X_12 @ ( plus_plus_nat @ P_1 @ Q_2 ) ) ) ).
thf(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_11: real] :
( ( power_power_real @ X_11 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_11: nat] :
( ( power_power_nat @ X_11 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_11: int] :
( ( power_power_int @ X_11 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_402_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_403_zero__le__even__power_H,axiom,
! [A_32: real,N_33: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_32 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_33 ) ) ) ).
thf(fact_404_zero__le__even__power_H,axiom,
! [A_32: int,N_33: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_32 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_33 ) ) ) ).
thf(fact_405_add__scale__eq__noteq,axiom,
! [C_4: real,D_2: real,A_31: real,B_12: real,R_2: real] :
( ( R_2 != zero_zero_real )
=> ( ( ( A_31 = B_12 )
& ( C_4 != D_2 ) )
=> ( ( plus_plus_real @ A_31 @ ( times_times_real @ R_2 @ C_4 ) )
!= ( plus_plus_real @ B_12 @ ( times_times_real @ R_2 @ D_2 ) ) ) ) ) ).
thf(fact_406_add__scale__eq__noteq,axiom,
! [C_4: nat,D_2: nat,A_31: nat,B_12: nat,R_2: nat] :
( ( R_2 != zero_zero_nat )
=> ( ( ( A_31 = B_12 )
& ( C_4 != D_2 ) )
=> ( ( plus_plus_nat @ A_31 @ ( times_times_nat @ R_2 @ C_4 ) )
!= ( plus_plus_nat @ B_12 @ ( times_times_nat @ R_2 @ D_2 ) ) ) ) ) ).
thf(fact_407_add__scale__eq__noteq,axiom,
! [C_4: int,D_2: int,A_31: int,B_12: int,R_2: int] :
( ( R_2 != zero_zero_int )
=> ( ( ( A_31 = B_12 )
& ( C_4 != D_2 ) )
=> ( ( plus_plus_int @ A_31 @ ( times_times_int @ R_2 @ C_4 ) )
!= ( plus_plus_int @ B_12 @ ( times_times_int @ R_2 @ D_2 ) ) ) ) ) ).
thf(fact_408_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_12: real] :
( ( plus_plus_real @ M_12 @ M_12 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_12 ) ) ).
thf(fact_409_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_12: nat] :
( ( plus_plus_nat @ M_12 @ M_12 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_12 ) ) ).
thf(fact_410_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_12: int] :
( ( plus_plus_int @ M_12 @ M_12 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_12 ) ) ).
thf(fact_411_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: real,A_30: real] :
( ( plus_plus_real @ M_11 @ ( times_times_real @ A_30 @ M_11 ) )
= ( times_times_real @ ( plus_plus_real @ A_30 @ one_one_real ) @ M_11 ) ) ).
thf(fact_412_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: nat,A_30: nat] :
( ( plus_plus_nat @ M_11 @ ( times_times_nat @ A_30 @ M_11 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_30 @ one_one_nat ) @ M_11 ) ) ).
thf(fact_413_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: int,A_30: int] :
( ( plus_plus_int @ M_11 @ ( times_times_int @ A_30 @ M_11 ) )
= ( times_times_int @ ( plus_plus_int @ A_30 @ one_one_int ) @ M_11 ) ) ).
thf(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_29: real,M_10: real] :
( ( plus_plus_real @ ( times_times_real @ A_29 @ M_10 ) @ M_10 )
= ( times_times_real @ ( plus_plus_real @ A_29 @ one_one_real ) @ M_10 ) ) ).
thf(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_29: nat,M_10: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_29 @ M_10 ) @ M_10 )
= ( times_times_nat @ ( plus_plus_nat @ A_29 @ one_one_nat ) @ M_10 ) ) ).
thf(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_29: int,M_10: int] :
( ( plus_plus_int @ ( times_times_int @ A_29 @ M_10 ) @ M_10 )
= ( times_times_int @ ( plus_plus_int @ A_29 @ one_one_int ) @ M_10 ) ) ).
thf(fact_417_power__eq__0__iff__number__of,axiom,
! [A_28: real,W_4: int] :
( ( ( power_power_real @ A_28 @ ( number_number_of_nat @ W_4 ) )
= zero_zero_real )
<=> ( ( A_28 = zero_zero_real )
& ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat ) ) ) ).
thf(fact_418_power__eq__0__iff__number__of,axiom,
! [A_28: nat,W_4: int] :
( ( ( power_power_nat @ A_28 @ ( number_number_of_nat @ W_4 ) )
= zero_zero_nat )
<=> ( ( A_28 = zero_zero_nat )
& ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat ) ) ) ).
thf(fact_419_power__eq__0__iff__number__of,axiom,
! [A_28: int,W_4: int] :
( ( ( power_power_int @ A_28 @ ( number_number_of_nat @ W_4 ) )
= zero_zero_int )
<=> ( ( A_28 = zero_zero_int )
& ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat ) ) ) ).
thf(fact_420_pos__zmult__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
thf(fact_421_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_422_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_423_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_424_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_425_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( ( number_number_of_nat @ W_3 )
= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_3 ) )
= one_one_real ) )
& ( ( ( number_number_of_nat @ W_3 )
!= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_3 ) )
= zero_zero_real ) ) ) ).
thf(fact_426_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( ( number_number_of_nat @ W_3 )
= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_3 ) )
= one_one_nat ) )
& ( ( ( number_number_of_nat @ W_3 )
!= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_3 ) )
= zero_zero_nat ) ) ) ).
thf(fact_427_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( ( number_number_of_nat @ W_3 )
= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_3 ) )
= one_one_int ) )
& ( ( ( number_number_of_nat @ W_3 )
!= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_3 ) )
= zero_zero_int ) ) ) ).
thf(fact_428_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
thf(fact_429_q__pos__lemma,axiom,
! [B_11: int,Q_1: int,R_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_11 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_int @ R_1 @ B_11 )
=> ( ( ord_less_int @ zero_zero_int @ B_11 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_1 ) ) ) ) ).
thf(fact_430_q__neg__lemma,axiom,
! [B_11: int,Q_1: int,R_1: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_11 @ Q_1 ) @ R_1 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_11 )
=> ( ord_less_eq_int @ Q_1 @ zero_zero_int ) ) ) ) ).
thf(fact_431_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_1 ) @ R_1 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ( ord_less_int @ R @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
thf(fact_432_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R: int,B_11: int,Q_1: int,R_1: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_11 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_11 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_int @ R_1 @ B_11 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B_11 )
=> ( ( ord_less_eq_int @ B_11 @ B )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
thf(fact_433_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_1 ) @ R_1 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
thf(fact_434_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R: int,B_11: int,Q_1: int,R_1: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_11 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_11 @ Q_1 ) @ R_1 ) @ zero_zero_int )
=> ( ( ord_less_int @ R @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_11 )
=> ( ( ord_less_eq_int @ B_11 @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
thf(fact_435_quartic__square__square,axiom,
! [X: int] :
( ( power_power_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_10: real] :
( ( times_times_real @ X_10 @ X_10 )
= ( power_power_real @ X_10 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_10: nat] :
( ( times_times_nat @ X_10 @ X_10 )
= ( power_power_nat @ X_10 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_438_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_10: int] :
( ( times_times_int @ X_10 @ X_10 )
= ( power_power_int @ X_10 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_439_s,axiom,
zcong @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_440_Euler_Oaux____1,axiom,
! [Y: int,X: int,P: int] :
( ~ ( zcong @ X @ zero_zero_int @ P )
=> ( ( zcong @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X @ P )
=> ~ ( dvd_dvd_int @ P @ Y ) ) ) ).
thf(fact_441_zprime__def,axiom,
! [P: int] :
( ( zprime @ P )
<=> ( ( ord_less_int @ one_one_int @ P )
& ! [M_9: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M_9 )
& ( dvd_dvd_int @ M_9 @ P ) )
=> ( ( M_9 = one_one_int )
| ( M_9 = P ) ) ) ) ) ).
thf(fact_442_prime__g__5,axiom,
! [P: int] :
( ( zprime @ P )
=> ( ( P
!= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( P
!= ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P ) ) ) ) ).
thf(fact_443_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_444__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
~ ! [S1: int] :
~ ( zcong @ ( power_power_int @ S1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ).
thf(fact_445__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
= one_one_int ) ).
thf(fact_446_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_447_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_448_rel__simps_I47_J,axiom,
! [K: int] :
( ( ( bit1 @ K )
= min )
<=> ( K = min ) ) ).
thf(fact_449_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_450_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_451_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_452_rel__simps_I45_J,axiom,
! [K: int] :
( ( bit0 @ K )
!= min ) ).
thf(fact_453_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_454_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_455_rel__simps_I13_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_int @ K @ min ) ) ).
thf(fact_456_rel__simps_I9_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_457_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_458_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_459_rel__simps_I8_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_460_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_461_rel__simps_I26_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ min @ K ) ) ).
thf(fact_462_rel__simps_I30_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_463_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_464_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_465_rel__simps_I28_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_466_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_467_rel__simps_I11_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_468_rel__simps_I25_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_469_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( number_number_of_int @ min ) )
& ( N
= ( number_number_of_int @ min ) ) ) ) ) ).
thf(fact_470_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_471_zcong__sym,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ B @ A @ M ) ) ).
thf(fact_472_zcong__refl,axiom,
! [K: int,M: int] : ( zcong @ K @ K @ M ) ).
thf(fact_473_zcong__trans,axiom,
! [C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ B @ C @ M )
=> ( zcong @ A @ C @ M ) ) ) ).
thf(fact_474_order__le__neq__implies__less,axiom,
! [X_9: real,Y_7: real] :
( ( ord_less_eq_real @ X_9 @ Y_7 )
=> ( ( X_9 != Y_7 )
=> ( ord_less_real @ X_9 @ Y_7 ) ) ) ).
thf(fact_475_order__le__neq__implies__less,axiom,
! [X_9: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_9 @ Y_7 )
=> ( ( X_9 != Y_7 )
=> ( ord_less_nat @ X_9 @ Y_7 ) ) ) ).
thf(fact_476_order__le__neq__implies__less,axiom,
! [X_9: int,Y_7: int] :
( ( ord_less_eq_int @ X_9 @ Y_7 )
=> ( ( X_9 != Y_7 )
=> ( ord_less_int @ X_9 @ Y_7 ) ) ) ).
thf(fact_477_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ A @ C )
=> ( ( ord_less_int @ B @ C )
=> ( ( ord_less_eq_int @ A @ B )
| ( ord_less_eq_int @ B @ A ) ) ) ) ).
thf(fact_478_IntPrimes_Ozcong__zero,axiom,
! [A: int,B: int] :
( ( zcong @ A @ B @ zero_zero_int )
<=> ( A = B ) ) ).
thf(fact_479_zcong__1,axiom,
! [A: int,B: int] : ( zcong @ A @ B @ one_one_int ) ).
thf(fact_480_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : ( zcong @ ( times_times_int @ A @ M ) @ ( times_times_int @ B @ M ) @ M ) ).
thf(fact_481_zcong__scalar,axiom,
! [K: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( times_times_int @ A @ K ) @ ( times_times_int @ B @ K ) @ M ) ) ).
thf(fact_482_zcong__scalar2,axiom,
! [K: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( times_times_int @ K @ A ) @ ( times_times_int @ K @ B ) @ M ) ) ).
thf(fact_483_zcong__zmult,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) @ M ) ) ) ).
thf(fact_484_zcong__zadd,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) @ M ) ) ) ).
thf(fact_485_power__m1__even,axiom,
! [N_32: nat] :
( ( power_power_real @ ( number267125858f_real @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_32 ) )
= one_one_real ) ).
thf(fact_486_power__m1__even,axiom,
! [N_32: nat] :
( ( power_power_int @ ( number_number_of_int @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_32 ) )
= one_one_int ) ).
thf(fact_487_zcong__not,axiom,
! [B: int,M: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ A )
=> ~ ( zcong @ A @ B @ M ) ) ) ) ) ).
thf(fact_488_zcong__iff__lin,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ? [K_1: int] :
( B
= ( plus_plus_int @ A @ ( times_times_int @ M @ K_1 ) ) ) ) ).
thf(fact_489_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_490_zcong__zless__0,axiom,
! [M: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( zcong @ A @ zero_zero_int @ M )
=> ( A = zero_zero_int ) ) ) ) ).
thf(fact_491_zcong__zless__imp__eq,axiom,
! [B: int,M: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ M )
=> ( ( zcong @ A @ B @ M )
=> ( A = B ) ) ) ) ) ) ).
thf(fact_492_zcong__zpower__zmult,axiom,
! [Z: nat,X: int,Y: nat,P: int] :
( ( zcong @ ( power_power_int @ X @ Y ) @ one_one_int @ P )
=> ( zcong @ ( power_power_int @ X @ ( times_times_nat @ Y @ Z ) ) @ one_one_int @ P ) ) ).
thf(fact_493_zprime__zdvd__zmult,axiom,
! [N: int,P: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P @ M )
| ( dvd_dvd_int @ P @ N ) ) ) ) ) ).
thf(fact_494__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
quadRes @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( number_number_of_int @ min ) ).
thf(fact_495__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
dvd_dvd_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) ) ).
thf(fact_496_neg__one__power__eq__mod__m,axiom,
! [J: nat,K: nat,M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ( ( zcong @ ( power_power_int @ ( number_number_of_int @ min ) @ J ) @ ( power_power_int @ ( number_number_of_int @ min ) @ K ) @ M )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ J )
= ( power_power_int @ ( number_number_of_int @ min ) @ K ) ) ) ) ).
thf(fact_497_zcong__neg__1__impl__ne__1,axiom,
! [X: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zcong @ X @ ( number_number_of_int @ min ) @ P )
=> ~ ( zcong @ X @ one_one_int @ P ) ) ) ).
thf(fact_498__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
( ( minus_minus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) )
= ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) ) ).
thf(fact_499__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
( ~ ( quadRes @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( number_number_of_int @ min ) )
=> ( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
!= one_one_int ) ) ).
thf(fact_500_not__real__square__gt__zero,axiom,
! [X: real] :
( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) )
<=> ( X = zero_zero_real ) ) ).
thf(fact_501_zcong__zdiff,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) @ M ) ) ) ).
thf(fact_502_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_503_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_504_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_505_diff__bin__simps_I7_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ K @ L ) ) ) ).
thf(fact_506_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_507_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_508_number__of__diff,axiom,
! [V_3: int,W_2: int] :
( ( number267125858f_real @ ( minus_minus_int @ V_3 @ W_2 ) )
= ( minus_minus_real @ ( number267125858f_real @ V_3 ) @ ( number267125858f_real @ W_2 ) ) ) ).
thf(fact_509_number__of__diff,axiom,
! [V_3: int,W_2: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_3 @ W_2 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_3 ) @ ( number_number_of_int @ W_2 ) ) ) ).
thf(fact_510_right__diff__distrib__number__of,axiom,
! [V_2: int,B_10: real,C_3: real] :
( ( times_times_real @ ( number267125858f_real @ V_2 ) @ ( minus_minus_real @ B_10 @ C_3 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_2 ) @ B_10 ) @ ( times_times_real @ ( number267125858f_real @ V_2 ) @ C_3 ) ) ) ).
thf(fact_511_right__diff__distrib__number__of,axiom,
! [V_2: int,B_10: int,C_3: int] :
( ( times_times_int @ ( number_number_of_int @ V_2 ) @ ( minus_minus_int @ B_10 @ C_3 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_2 ) @ B_10 ) @ ( times_times_int @ ( number_number_of_int @ V_2 ) @ C_3 ) ) ) ).
thf(fact_512_left__diff__distrib__number__of,axiom,
! [A_27: real,B_9: real,V_1: int] :
( ( times_times_real @ ( minus_minus_real @ A_27 @ B_9 ) @ ( number267125858f_real @ V_1 ) )
= ( minus_minus_real @ ( times_times_real @ A_27 @ ( number267125858f_real @ V_1 ) ) @ ( times_times_real @ B_9 @ ( number267125858f_real @ V_1 ) ) ) ) ).
thf(fact_513_left__diff__distrib__number__of,axiom,
! [A_27: int,B_9: int,V_1: int] :
( ( times_times_int @ ( minus_minus_int @ A_27 @ B_9 ) @ ( number_number_of_int @ V_1 ) )
= ( minus_minus_int @ ( times_times_int @ A_27 @ ( number_number_of_int @ V_1 ) ) @ ( times_times_int @ B_9 @ ( number_number_of_int @ V_1 ) ) ) ) ).
thf(fact_514_diff__bin__simps_I9_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ K @ L ) ) ) ).
thf(fact_515_diff__bin__simps_I10_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ K @ L ) ) ) ).
thf(fact_516_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_517_less__bin__lemma,axiom,
! [K: int,L: int] :
( ( ord_less_int @ K @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
thf(fact_518_xzgcda__linear__aux1,axiom,
! [A: int,R: int,B: int,M: int,C: int,D: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ ( times_times_int @ R @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R @ D ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A @ M ) @ ( times_times_int @ C @ N ) ) @ ( times_times_int @ R @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D @ N ) ) ) ) ) ).
thf(fact_519_zcong__def,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A @ B ) ) ) ).
thf(fact_520_add__number__of__diff1,axiom,
! [V: int,W_1: int,C_2: real] :
( ( plus_plus_real @ ( number267125858f_real @ V ) @ ( minus_minus_real @ ( number267125858f_real @ W_1 ) @ C_2 ) )
= ( minus_minus_real @ ( number267125858f_real @ ( plus_plus_int @ V @ W_1 ) ) @ C_2 ) ) ).
thf(fact_521_add__number__of__diff1,axiom,
! [V: int,W_1: int,C_2: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( minus_minus_int @ ( number_number_of_int @ W_1 ) @ C_2 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V @ W_1 ) ) @ C_2 ) ) ).
thf(fact_522_Euler_Oaux1,axiom,
! [A: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ A )
=> ( ( X
!= ( minus_minus_int @ A @ one_one_int ) )
=> ( ord_less_int @ X @ ( minus_minus_int @ A @ one_one_int ) ) ) ) ) ).
thf(fact_523_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_524_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_525_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_526_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_527_inv__not__p__minus__1__aux,axiom,
! [A: int,P: int] :
( ( zcong @ ( times_times_int @ A @ ( minus_minus_int @ P @ one_one_int ) ) @ one_one_int @ P )
<=> ( zcong @ A @ ( minus_minus_int @ P @ one_one_int ) @ P ) ) ).
thf(fact_528_zcong__eq__trans,axiom,
! [D: int,C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( B = C )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ A @ D @ M ) ) ) ) ).
thf(fact_529_zcong__square__zless,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P )
=> ( ( A = one_one_int )
| ( A
= ( minus_minus_int @ P @ one_one_int ) ) ) ) ) ) ) ).
thf(fact_530_zcong__square,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P )
=> ( ( zcong @ A @ one_one_int @ P )
| ( zcong @ A @ ( minus_minus_int @ P @ one_one_int ) @ P ) ) ) ) ) ).
thf(fact_531_zspecial__product,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_532_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_533_zcong__zmult__prop2,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ ( times_times_int @ D @ A ) @ M )
<=> ( zcong @ C @ ( times_times_int @ D @ B ) @ M ) ) ) ).
thf(fact_534_zcong__zmult__prop1,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ ( times_times_int @ A @ D ) @ M )
<=> ( zcong @ C @ ( times_times_int @ B @ D ) @ M ) ) ) ).
thf(fact_535_zcong__shift,axiom,
! [C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) @ M ) ) ).
thf(fact_536_zcong__zpower,axiom,
! [Z: nat,X: int,Y: int,M: int] :
( ( zcong @ X @ Y @ M )
=> ( zcong @ ( power_power_int @ X @ Z ) @ ( power_power_int @ Y @ Z ) @ M ) ) ).
thf(fact_537_power2__diff,axiom,
! [X_8: real,Y_6: real] :
( ( power_power_real @ ( minus_minus_real @ X_8 @ Y_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_8 ) @ Y_6 ) ) ) ).
thf(fact_538_power2__diff,axiom,
! [X_8: int,Y_6: int] :
( ( power_power_int @ ( minus_minus_int @ X_8 @ Y_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_8 ) @ Y_6 ) ) ) ).
thf(fact_539_zdiff__power2,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( minus_minus_int @ A @ B ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) @ B ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_540_zdiff__power3,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( minus_minus_int @ A @ B ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_541_zcong__less__eq,axiom,
! [M: int,Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ X @ Y @ M )
=> ( ( ord_less_int @ X @ M )
=> ( ( ord_less_int @ Y @ M )
=> ( X = Y ) ) ) ) ) ) ) ).
thf(fact_542_zcong__not__zero,axiom,
! [M: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ M )
=> ~ ( zcong @ X @ zero_zero_int @ M ) ) ) ).
thf(fact_543_zdvd__bounds,axiom,
! [N: int,M: int] :
( ( dvd_dvd_int @ N @ M )
=> ( ( ord_less_eq_int @ M @ zero_zero_int )
| ( ord_less_eq_int @ N @ M ) ) ) ).
thf(fact_544_zcong__zero__equiv__div,axiom,
! [A: int,M: int] :
( ( zcong @ A @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A ) ) ).
thf(fact_545_zcong__eq__zdvd__prop,axiom,
! [X: int,P: int] :
( ( zcong @ X @ zero_zero_int @ P )
<=> ( dvd_dvd_int @ P @ X ) ) ).
thf(fact_546_zprime__zdvd__zmult__better,axiom,
! [M: int,N: int,P: int] :
( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P @ M )
| ( dvd_dvd_int @ P @ N ) ) ) ) ).
thf(fact_547_Int2_Ozcong__zero,axiom,
! [M: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ M )
=> ( ( zcong @ X @ zero_zero_int @ M )
=> ( X = zero_zero_int ) ) ) ) ).
thf(fact_548_zpower__zdvd__prop1,axiom,
! [P: int,Y: int,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P @ Y )
=> ( dvd_dvd_int @ P @ ( power_power_int @ Y @ N ) ) ) ) ).
thf(fact_549_neg__one__power,axiom,
! [N: nat] :
( ( ( power_power_int @ ( number_number_of_int @ min ) @ N )
= one_one_int )
| ( ( power_power_int @ ( number_number_of_int @ min ) @ N )
= ( number_number_of_int @ min ) ) ) ).
thf(fact_550_zcong__zmult__prop3,axiom,
! [Y: int,X: int,P: int] :
( ( zprime @ P )
=> ( ~ ( zcong @ X @ zero_zero_int @ P )
=> ( ~ ( zcong @ Y @ zero_zero_int @ P )
=> ~ ( zcong @ ( times_times_int @ X @ Y ) @ zero_zero_int @ P ) ) ) ) ).
thf(fact_551_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ~ ( zcong @ A @ zero_zero_int @ P )
& ~ ( zcong @ B @ zero_zero_int @ P ) )
=> ~ ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P ) ) ) ) ).
thf(fact_552_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P )
=> ( ( zcong @ A @ zero_zero_int @ P )
| ( zcong @ B @ zero_zero_int @ P ) ) ) ) ) ).
thf(fact_553_zpower__zdvd__prop2,axiom,
! [Y: int,N: nat,P: int] :
( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ Y @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( dvd_dvd_int @ P @ Y ) ) ) ) ).
thf(fact_554_Legendre__1mod4,axiom,
! [M: int] :
( ( zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ M ) @ one_one_int ) )
=> ( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ M ) @ one_one_int ) )
= one_one_int ) ) ).
thf(fact_555_one__not__neg__one__mod__m,axiom,
! [M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ~ ( zcong @ one_one_int @ ( number_number_of_int @ min ) @ M ) ) ).
thf(fact_556_Legendre__def,axiom,
! [A: int,P: int] :
( ( ( zcong @ A @ zero_zero_int @ P )
=> ( ( legendre @ A @ P )
= zero_zero_int ) )
& ( ~ ( zcong @ A @ zero_zero_int @ P )
=> ( ( ( quadRes @ P @ A )
=> ( ( legendre @ A @ P )
= one_one_int ) )
& ( ~ ( quadRes @ P @ A )
=> ( ( legendre @ A @ P )
= ( number_number_of_int @ min ) ) ) ) ) ) ).
thf(fact_557_QuadRes__def,axiom,
! [M: int,X: int] :
( ( quadRes @ M @ X )
<=> ? [Y_1: int] : ( zcong @ ( power_power_int @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X @ M ) ) ).
thf(fact_558_real__sum__squared__expand,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) @ Y ) ) ) ).
thf(fact_559_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_560_power__eq__if,axiom,
! [P: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( power_power_nat @ P @ M )
= one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( power_power_nat @ P @ M )
= ( times_times_nat @ P @ ( power_power_nat @ P @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).
thf(fact_561_diff__square,axiom,
! [X: nat,Y: nat] :
( ( minus_minus_nat @ ( power_power_nat @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( times_times_nat @ ( plus_plus_nat @ X @ Y ) @ ( minus_minus_nat @ X @ Y ) ) ) ).
thf(fact_562_realpow__two__sum__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
thf(fact_563_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_564_real__le__eq__diff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ord_less_eq_real @ ( minus_minus_real @ X @ Y ) @ zero_zero_real ) ) ).
thf(fact_565_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_566_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_567_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] :
( ( times_times_real @ ( times_times_real @ Z1 @ Z2 ) @ Z3 )
= ( times_times_real @ Z1 @ ( times_times_real @ Z2 @ Z3 ) ) ) ).
thf(fact_568_real__mult__commute,axiom,
! [Z: real,W: real] :
( ( times_times_real @ Z @ W )
= ( times_times_real @ W @ Z ) ) ).
thf(fact_569_real__mult__1,axiom,
! [Z: real] :
( ( times_times_real @ one_one_real @ Z )
= Z ) ).
thf(fact_570_real__add__left__mono,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( plus_plus_real @ Z @ X ) @ ( plus_plus_real @ Z @ Y ) ) ) ).
thf(fact_571_add__diff__add,axiom,
! [A_26: real,C_1: real,B_8: real,D_1: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_26 @ C_1 ) @ ( plus_plus_real @ B_8 @ D_1 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_26 @ B_8 ) @ ( minus_minus_real @ C_1 @ D_1 ) ) ) ).
thf(fact_572_add__diff__add,axiom,
! [A_26: int,C_1: int,B_8: int,D_1: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_26 @ C_1 ) @ ( plus_plus_int @ B_8 @ D_1 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_26 @ B_8 ) @ ( minus_minus_int @ C_1 @ D_1 ) ) ) ).
thf(fact_573_real__mult__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
<=> ( A = B ) ) ) ).
thf(fact_574_real__mult__right__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
<=> ( A = B ) ) ) ).
thf(fact_575_nat__mult__eq__one,axiom,
! [N: nat,M: nat] :
( ( ( times_times_nat @ N @ M )
= one_one_nat )
<=> ( ( N = one_one_nat )
& ( M = one_one_nat ) ) ) ).
thf(fact_576_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W: real] :
( ( times_times_real @ ( plus_plus_real @ Z1 @ Z2 ) @ W )
= ( plus_plus_real @ ( times_times_real @ Z1 @ W ) @ ( times_times_real @ Z2 @ W ) ) ) ).
thf(fact_577_nat__power__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ M @ N )
= zero_zero_nat )
<=> ( ( N != zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
thf(fact_578_mult__diff__mult,axiom,
! [X_7: real,Y_5: real,A_25: real,B_7: real] :
( ( minus_minus_real @ ( times_times_real @ X_7 @ Y_5 ) @ ( times_times_real @ A_25 @ B_7 ) )
= ( plus_plus_real @ ( times_times_real @ X_7 @ ( minus_minus_real @ Y_5 @ B_7 ) ) @ ( times_times_real @ ( minus_minus_real @ X_7 @ A_25 ) @ B_7 ) ) ) ).
thf(fact_579_mult__diff__mult,axiom,
! [X_7: int,Y_5: int,A_25: int,B_7: int] :
( ( minus_minus_int @ ( times_times_int @ X_7 @ Y_5 ) @ ( times_times_int @ A_25 @ B_7 ) )
= ( plus_plus_int @ ( times_times_int @ X_7 @ ( minus_minus_int @ Y_5 @ B_7 ) ) @ ( times_times_int @ ( minus_minus_int @ X_7 @ A_25 ) @ B_7 ) ) ) ).
thf(fact_580_real__mult__le__cancel__iff1,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_581_real__mult__le__cancel__iff2,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_582_real__mult__less__mono2,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) ) ) ) ).
thf(fact_583_real__mult__order,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ Y ) ) ) ) ).
thf(fact_584_real__mult__less__iff1,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
<=> ( ord_less_real @ X @ Y ) ) ) ).
thf(fact_585_real__two__squares__add__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
thf(fact_586_exp__eq__1,axiom,
! [X: nat,N: nat] :
( ( ( power_power_nat @ X @ N )
= one_one_nat )
<=> ( ( X = one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_587_real__squared__diff__one__factored,axiom,
! [X_6: real] :
( ( minus_minus_real @ ( times_times_real @ X_6 @ X_6 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X_6 @ one_one_real ) @ ( minus_minus_real @ X_6 @ one_one_real ) ) ) ).
thf(fact_588_real__squared__diff__one__factored,axiom,
! [X_6: int] :
( ( minus_minus_int @ ( times_times_int @ X_6 @ X_6 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_6 @ one_one_int ) @ ( minus_minus_int @ X_6 @ one_one_int ) ) ) ).
thf(fact_589_realpow__minus__mult,axiom,
! [X_5: real,N_31: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( ( times_times_real @ ( power_power_real @ X_5 @ ( minus_minus_nat @ N_31 @ one_one_nat ) ) @ X_5 )
= ( power_power_real @ X_5 @ N_31 ) ) ) ).
thf(fact_590_realpow__minus__mult,axiom,
! [X_5: nat,N_31: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( ( times_times_nat @ ( power_power_nat @ X_5 @ ( minus_minus_nat @ N_31 @ one_one_nat ) ) @ X_5 )
= ( power_power_nat @ X_5 @ N_31 ) ) ) ).
thf(fact_591_realpow__minus__mult,axiom,
! [X_5: int,N_31: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( ( times_times_int @ ( power_power_int @ X_5 @ ( minus_minus_nat @ N_31 @ one_one_nat ) ) @ X_5 )
= ( power_power_int @ X_5 @ N_31 ) ) ) ).
thf(fact_592_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_593_realpow__num__eq__if,axiom,
! [M_8: real,N_30: nat] :
( ( ( N_30 = zero_zero_nat )
=> ( ( power_power_real @ M_8 @ N_30 )
= one_one_real ) )
& ( ( N_30 != zero_zero_nat )
=> ( ( power_power_real @ M_8 @ N_30 )
= ( times_times_real @ M_8 @ ( power_power_real @ M_8 @ ( minus_minus_nat @ N_30 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_594_realpow__num__eq__if,axiom,
! [M_8: nat,N_30: nat] :
( ( ( N_30 = zero_zero_nat )
=> ( ( power_power_nat @ M_8 @ N_30 )
= one_one_nat ) )
& ( ( N_30 != zero_zero_nat )
=> ( ( power_power_nat @ M_8 @ N_30 )
= ( times_times_nat @ M_8 @ ( power_power_nat @ M_8 @ ( minus_minus_nat @ N_30 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_595_realpow__num__eq__if,axiom,
! [M_8: int,N_30: nat] :
( ( ( N_30 = zero_zero_nat )
=> ( ( power_power_int @ M_8 @ N_30 )
= one_one_int ) )
& ( ( N_30 != zero_zero_nat )
=> ( ( power_power_int @ M_8 @ N_30 )
= ( times_times_int @ M_8 @ ( power_power_int @ M_8 @ ( minus_minus_nat @ N_30 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_596_power__strict__mono,axiom,
! [N_29: nat,A_24: real,B_6: real] :
( ( ord_less_real @ A_24 @ B_6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_24 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_29 )
=> ( ord_less_real @ ( power_power_real @ A_24 @ N_29 ) @ ( power_power_real @ B_6 @ N_29 ) ) ) ) ) ).
thf(fact_597_power__strict__mono,axiom,
! [N_29: nat,A_24: nat,B_6: nat] :
( ( ord_less_nat @ A_24 @ B_6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_24 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_29 )
=> ( ord_less_nat @ ( power_power_nat @ A_24 @ N_29 ) @ ( power_power_nat @ B_6 @ N_29 ) ) ) ) ) ).
thf(fact_598_power__strict__mono,axiom,
! [N_29: nat,A_24: int,B_6: int] :
( ( ord_less_int @ A_24 @ B_6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_24 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_29 )
=> ( ord_less_int @ ( power_power_int @ A_24 @ N_29 ) @ ( power_power_int @ B_6 @ N_29 ) ) ) ) ) ).
thf(fact_599_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_600_real__le__linear,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
| ( ord_less_eq_real @ W @ Z ) ) ).
thf(fact_601_real__le__trans,axiom,
! [K: real,I: real,J: real] :
( ( ord_less_eq_real @ I @ J )
=> ( ( ord_less_eq_real @ J @ K )
=> ( ord_less_eq_real @ I @ K ) ) ) ).
thf(fact_602_real__le__antisym,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_603_field__power__not__zero,axiom,
! [N_28: nat,A_23: real] :
( ( A_23 != zero_zero_real )
=> ( ( power_power_real @ A_23 @ N_28 )
!= zero_zero_real ) ) ).
thf(fact_604_field__power__not__zero,axiom,
! [N_28: nat,A_23: int] :
( ( A_23 != zero_zero_int )
=> ( ( power_power_int @ A_23 @ N_28 )
!= zero_zero_int ) ) ).
thf(fact_605_power__mult__distrib,axiom,
! [A_22: real,B_5: real,N_27: nat] :
( ( power_power_real @ ( times_times_real @ A_22 @ B_5 ) @ N_27 )
= ( times_times_real @ ( power_power_real @ A_22 @ N_27 ) @ ( power_power_real @ B_5 @ N_27 ) ) ) ).
thf(fact_606_power__mult__distrib,axiom,
! [A_22: nat,B_5: nat,N_27: nat] :
( ( power_power_nat @ ( times_times_nat @ A_22 @ B_5 ) @ N_27 )
= ( times_times_nat @ ( power_power_nat @ A_22 @ N_27 ) @ ( power_power_nat @ B_5 @ N_27 ) ) ) ).
thf(fact_607_power__mult__distrib,axiom,
! [A_22: int,B_5: int,N_27: nat] :
( ( power_power_int @ ( times_times_int @ A_22 @ B_5 ) @ N_27 )
= ( times_times_int @ ( power_power_int @ A_22 @ N_27 ) @ ( power_power_int @ B_5 @ N_27 ) ) ) ).
thf(fact_608_power__commutes,axiom,
! [A_21: real,N_26: nat] :
( ( times_times_real @ ( power_power_real @ A_21 @ N_26 ) @ A_21 )
= ( times_times_real @ A_21 @ ( power_power_real @ A_21 @ N_26 ) ) ) ).
thf(fact_609_power__commutes,axiom,
! [A_21: nat,N_26: nat] :
( ( times_times_nat @ ( power_power_nat @ A_21 @ N_26 ) @ A_21 )
= ( times_times_nat @ A_21 @ ( power_power_nat @ A_21 @ N_26 ) ) ) ).
thf(fact_610_power__commutes,axiom,
! [A_21: int,N_26: nat] :
( ( times_times_int @ ( power_power_int @ A_21 @ N_26 ) @ A_21 )
= ( times_times_int @ A_21 @ ( power_power_int @ A_21 @ N_26 ) ) ) ).
thf(fact_611_power__one,axiom,
! [N_25: nat] :
( ( power_power_real @ one_one_real @ N_25 )
= one_one_real ) ).
thf(fact_612_power__one,axiom,
! [N_25: nat] :
( ( power_power_nat @ one_one_nat @ N_25 )
= one_one_nat ) ).
thf(fact_613_power__one,axiom,
! [N_25: nat] :
( ( power_power_int @ one_one_int @ N_25 )
= one_one_int ) ).
thf(fact_614_dvd__power__same,axiom,
! [N_24: nat,X_4: nat,Y_4: nat] :
( ( dvd_dvd_nat @ X_4 @ Y_4 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_4 @ N_24 ) @ ( power_power_nat @ Y_4 @ N_24 ) ) ) ).
thf(fact_615_dvd__power__same,axiom,
! [N_24: nat,X_4: int,Y_4: int] :
( ( dvd_dvd_int @ X_4 @ Y_4 )
=> ( dvd_dvd_int @ ( power_power_int @ X_4 @ N_24 ) @ ( power_power_int @ Y_4 @ N_24 ) ) ) ).
thf(fact_616_dvd__power__same,axiom,
! [N_24: nat,X_4: real,Y_4: real] :
( ( dvd_dvd_real @ X_4 @ Y_4 )
=> ( dvd_dvd_real @ ( power_power_real @ X_4 @ N_24 ) @ ( power_power_real @ Y_4 @ N_24 ) ) ) ).
thf(fact_617_power__mult,axiom,
! [A_20: real,M_7: nat,N_23: nat] :
( ( power_power_real @ A_20 @ ( times_times_nat @ M_7 @ N_23 ) )
= ( power_power_real @ ( power_power_real @ A_20 @ M_7 ) @ N_23 ) ) ).
thf(fact_618_power__mult,axiom,
! [A_20: nat,M_7: nat,N_23: nat] :
( ( power_power_nat @ A_20 @ ( times_times_nat @ M_7 @ N_23 ) )
= ( power_power_nat @ ( power_power_nat @ A_20 @ M_7 ) @ N_23 ) ) ).
thf(fact_619_power__mult,axiom,
! [A_20: int,M_7: nat,N_23: nat] :
( ( power_power_int @ A_20 @ ( times_times_nat @ M_7 @ N_23 ) )
= ( power_power_int @ ( power_power_int @ A_20 @ M_7 ) @ N_23 ) ) ).
thf(fact_620_power__one__right,axiom,
! [A_19: real] :
( ( power_power_real @ A_19 @ one_one_nat )
= A_19 ) ).
thf(fact_621_power__one__right,axiom,
! [A_19: nat] :
( ( power_power_nat @ A_19 @ one_one_nat )
= A_19 ) ).
thf(fact_622_power__one__right,axiom,
! [A_19: int] :
( ( power_power_int @ A_19 @ one_one_nat )
= A_19 ) ).
thf(fact_623_power__mono,axiom,
! [N_22: nat,A_18: real,B_4: real] :
( ( ord_less_eq_real @ A_18 @ B_4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_18 )
=> ( ord_less_eq_real @ ( power_power_real @ A_18 @ N_22 ) @ ( power_power_real @ B_4 @ N_22 ) ) ) ) ).
thf(fact_624_power__mono,axiom,
! [N_22: nat,A_18: nat,B_4: nat] :
( ( ord_less_eq_nat @ A_18 @ B_4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_18 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_18 @ N_22 ) @ ( power_power_nat @ B_4 @ N_22 ) ) ) ) ).
thf(fact_625_power__mono,axiom,
! [N_22: nat,A_18: int,B_4: int] :
( ( ord_less_eq_int @ A_18 @ B_4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_18 )
=> ( ord_less_eq_int @ ( power_power_int @ A_18 @ N_22 ) @ ( power_power_int @ B_4 @ N_22 ) ) ) ) ).
thf(fact_626_zero__le__power,axiom,
! [N_21: nat,A_17: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_17 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_17 @ N_21 ) ) ) ).
thf(fact_627_zero__le__power,axiom,
! [N_21: nat,A_17: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_17 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A_17 @ N_21 ) ) ) ).
thf(fact_628_zero__le__power,axiom,
! [N_21: nat,A_17: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_17 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_17 @ N_21 ) ) ) ).
thf(fact_629_zero__less__power,axiom,
! [N_20: nat,A_16: real] :
( ( ord_less_real @ zero_zero_real @ A_16 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_16 @ N_20 ) ) ) ).
thf(fact_630_zero__less__power,axiom,
! [N_20: nat,A_16: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_16 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A_16 @ N_20 ) ) ) ).
thf(fact_631_zero__less__power,axiom,
! [N_20: nat,A_16: int] :
( ( ord_less_int @ zero_zero_int @ A_16 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_16 @ N_20 ) ) ) ).
thf(fact_632_one__le__power,axiom,
! [N_19: nat,A_15: real] :
( ( ord_less_eq_real @ one_one_real @ A_15 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A_15 @ N_19 ) ) ) ).
thf(fact_633_one__le__power,axiom,
! [N_19: nat,A_15: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A_15 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A_15 @ N_19 ) ) ) ).
thf(fact_634_one__le__power,axiom,
! [N_19: nat,A_15: int] :
( ( ord_less_eq_int @ one_one_int @ A_15 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A_15 @ N_19 ) ) ) ).
thf(fact_635_power__inject__exp,axiom,
! [M_6: nat,N_18: nat,A_14: real] :
( ( ord_less_real @ one_one_real @ A_14 )
=> ( ( ( power_power_real @ A_14 @ M_6 )
= ( power_power_real @ A_14 @ N_18 ) )
<=> ( M_6 = N_18 ) ) ) ).
thf(fact_636_power__inject__exp,axiom,
! [M_6: nat,N_18: nat,A_14: nat] :
( ( ord_less_nat @ one_one_nat @ A_14 )
=> ( ( ( power_power_nat @ A_14 @ M_6 )
= ( power_power_nat @ A_14 @ N_18 ) )
<=> ( M_6 = N_18 ) ) ) ).
thf(fact_637_power__inject__exp,axiom,
! [M_6: nat,N_18: nat,A_14: int] :
( ( ord_less_int @ one_one_int @ A_14 )
=> ( ( ( power_power_int @ A_14 @ M_6 )
= ( power_power_int @ A_14 @ N_18 ) )
<=> ( M_6 = N_18 ) ) ) ).
thf(fact_638_power__eq__0__iff,axiom,
! [A_13: real,N_17: nat] :
( ( ( power_power_real @ A_13 @ N_17 )
= zero_zero_real )
<=> ( ( A_13 = zero_zero_real )
& ( N_17 != zero_zero_nat ) ) ) ).
thf(fact_639_power__eq__0__iff,axiom,
! [A_13: nat,N_17: nat] :
( ( ( power_power_nat @ A_13 @ N_17 )
= zero_zero_nat )
<=> ( ( A_13 = zero_zero_nat )
& ( N_17 != zero_zero_nat ) ) ) ).
thf(fact_640_power__eq__0__iff,axiom,
! [A_13: int,N_17: nat] :
( ( ( power_power_int @ A_13 @ N_17 )
= zero_zero_int )
<=> ( ( A_13 = zero_zero_int )
& ( N_17 != zero_zero_nat ) ) ) ).
thf(fact_641_power__0,axiom,
! [A_12: real] :
( ( power_power_real @ A_12 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_642_power__0,axiom,
! [A_12: nat] :
( ( power_power_nat @ A_12 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_643_power__0,axiom,
! [A_12: int] :
( ( power_power_int @ A_12 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_644_power__add,axiom,
! [A_11: real,M_5: nat,N_16: nat] :
( ( power_power_real @ A_11 @ ( plus_plus_nat @ M_5 @ N_16 ) )
= ( times_times_real @ ( power_power_real @ A_11 @ M_5 ) @ ( power_power_real @ A_11 @ N_16 ) ) ) ).
thf(fact_645_power__add,axiom,
! [A_11: nat,M_5: nat,N_16: nat] :
( ( power_power_nat @ A_11 @ ( plus_plus_nat @ M_5 @ N_16 ) )
= ( times_times_nat @ ( power_power_nat @ A_11 @ M_5 ) @ ( power_power_nat @ A_11 @ N_16 ) ) ) ).
thf(fact_646_power__add,axiom,
! [A_11: int,M_5: nat,N_16: nat] :
( ( power_power_int @ A_11 @ ( plus_plus_nat @ M_5 @ N_16 ) )
= ( times_times_int @ ( power_power_int @ A_11 @ M_5 ) @ ( power_power_int @ A_11 @ N_16 ) ) ) ).
thf(fact_647_power__le__dvd,axiom,
! [M_4: nat,A_10: nat,N_15: nat,B_3: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A_10 @ N_15 ) @ B_3 )
=> ( ( ord_less_eq_nat @ M_4 @ N_15 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_10 @ M_4 ) @ B_3 ) ) ) ).
thf(fact_648_power__le__dvd,axiom,
! [M_4: nat,A_10: int,N_15: nat,B_3: int] :
( ( dvd_dvd_int @ ( power_power_int @ A_10 @ N_15 ) @ B_3 )
=> ( ( ord_less_eq_nat @ M_4 @ N_15 )
=> ( dvd_dvd_int @ ( power_power_int @ A_10 @ M_4 ) @ B_3 ) ) ) ).
thf(fact_649_power__le__dvd,axiom,
! [M_4: nat,A_10: real,N_15: nat,B_3: real] :
( ( dvd_dvd_real @ ( power_power_real @ A_10 @ N_15 ) @ B_3 )
=> ( ( ord_less_eq_nat @ M_4 @ N_15 )
=> ( dvd_dvd_real @ ( power_power_real @ A_10 @ M_4 ) @ B_3 ) ) ) ).
thf(fact_650_dvd__power__le,axiom,
! [N_14: nat,M_3: nat,X_3: nat,Y_3: nat] :
( ( dvd_dvd_nat @ X_3 @ Y_3 )
=> ( ( ord_less_eq_nat @ N_14 @ M_3 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_3 @ N_14 ) @ ( power_power_nat @ Y_3 @ M_3 ) ) ) ) ).
thf(fact_651_dvd__power__le,axiom,
! [N_14: nat,M_3: nat,X_3: int,Y_3: int] :
( ( dvd_dvd_int @ X_3 @ Y_3 )
=> ( ( ord_less_eq_nat @ N_14 @ M_3 )
=> ( dvd_dvd_int @ ( power_power_int @ X_3 @ N_14 ) @ ( power_power_int @ Y_3 @ M_3 ) ) ) ) ).
thf(fact_652_dvd__power__le,axiom,
! [N_14: nat,M_3: nat,X_3: real,Y_3: real] :
( ( dvd_dvd_real @ X_3 @ Y_3 )
=> ( ( ord_less_eq_nat @ N_14 @ M_3 )
=> ( dvd_dvd_real @ ( power_power_real @ X_3 @ N_14 ) @ ( power_power_real @ Y_3 @ M_3 ) ) ) ) ).
thf(fact_653_le__imp__power__dvd,axiom,
! [A_9: nat,M_2: nat,N_13: nat] :
( ( ord_less_eq_nat @ M_2 @ N_13 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_9 @ M_2 ) @ ( power_power_nat @ A_9 @ N_13 ) ) ) ).
thf(fact_654_le__imp__power__dvd,axiom,
! [A_9: int,M_2: nat,N_13: nat] :
( ( ord_less_eq_nat @ M_2 @ N_13 )
=> ( dvd_dvd_int @ ( power_power_int @ A_9 @ M_2 ) @ ( power_power_int @ A_9 @ N_13 ) ) ) ).
thf(fact_655_le__imp__power__dvd,axiom,
! [A_9: real,M_2: nat,N_13: nat] :
( ( ord_less_eq_nat @ M_2 @ N_13 )
=> ( dvd_dvd_real @ ( power_power_real @ A_9 @ M_2 ) @ ( power_power_real @ A_9 @ N_13 ) ) ) ).
thf(fact_656_nat__power__less__imp__less,axiom,
! [M: nat,N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_657_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_658_power__less__imp__less__base,axiom,
! [A_8: real,N_12: nat,B_2: real] :
( ( ord_less_real @ ( power_power_real @ A_8 @ N_12 ) @ ( power_power_real @ B_2 @ N_12 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_2 )
=> ( ord_less_real @ A_8 @ B_2 ) ) ) ).
thf(fact_659_power__less__imp__less__base,axiom,
! [A_8: nat,N_12: nat,B_2: nat] :
( ( ord_less_nat @ ( power_power_nat @ A_8 @ N_12 ) @ ( power_power_nat @ B_2 @ N_12 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_2 )
=> ( ord_less_nat @ A_8 @ B_2 ) ) ) ).
thf(fact_660_power__less__imp__less__base,axiom,
! [A_8: int,N_12: nat,B_2: int] :
( ( ord_less_int @ ( power_power_int @ A_8 @ N_12 ) @ ( power_power_int @ B_2 @ N_12 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_2 )
=> ( ord_less_int @ A_8 @ B_2 ) ) ) ).
thf(fact_661_power__less__power__Suc,axiom,
! [N_11: nat,A_7: real] :
( ( ord_less_real @ one_one_real @ A_7 )
=> ( ord_less_real @ ( power_power_real @ A_7 @ N_11 ) @ ( times_times_real @ A_7 @ ( power_power_real @ A_7 @ N_11 ) ) ) ) ).
thf(fact_662_power__less__power__Suc,axiom,
! [N_11: nat,A_7: nat] :
( ( ord_less_nat @ one_one_nat @ A_7 )
=> ( ord_less_nat @ ( power_power_nat @ A_7 @ N_11 ) @ ( times_times_nat @ A_7 @ ( power_power_nat @ A_7 @ N_11 ) ) ) ) ).
thf(fact_663_power__less__power__Suc,axiom,
! [N_11: nat,A_7: int] :
( ( ord_less_int @ one_one_int @ A_7 )
=> ( ord_less_int @ ( power_power_int @ A_7 @ N_11 ) @ ( times_times_int @ A_7 @ ( power_power_int @ A_7 @ N_11 ) ) ) ) ).
thf(fact_664_power__gt1__lemma,axiom,
! [N_10: nat,A_6: real] :
( ( ord_less_real @ one_one_real @ A_6 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A_6 @ ( power_power_real @ A_6 @ N_10 ) ) ) ) ).
thf(fact_665_power__gt1__lemma,axiom,
! [N_10: nat,A_6: nat] :
( ( ord_less_nat @ one_one_nat @ A_6 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A_6 @ ( power_power_nat @ A_6 @ N_10 ) ) ) ) ).
thf(fact_666_power__gt1__lemma,axiom,
! [N_10: nat,A_6: int] :
( ( ord_less_int @ one_one_int @ A_6 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A_6 @ ( power_power_int @ A_6 @ N_10 ) ) ) ) ).
thf(fact_667_power__0__left,axiom,
! [N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_9 )
= one_one_real ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_9 )
= zero_zero_real ) ) ) ).
thf(fact_668_power__0__left,axiom,
! [N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_9 )
= one_one_nat ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_9 )
= zero_zero_nat ) ) ) ).
thf(fact_669_power__0__left,axiom,
! [N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_9 )
= one_one_int ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_9 )
= zero_zero_int ) ) ) ).
thf(fact_670_power__strict__increasing,axiom,
! [A_5: real,N_8: nat,N_7: nat] :
( ( ord_less_nat @ N_8 @ N_7 )
=> ( ( ord_less_real @ one_one_real @ A_5 )
=> ( ord_less_real @ ( power_power_real @ A_5 @ N_8 ) @ ( power_power_real @ A_5 @ N_7 ) ) ) ) ).
thf(fact_671_power__strict__increasing,axiom,
! [A_5: nat,N_8: nat,N_7: nat] :
( ( ord_less_nat @ N_8 @ N_7 )
=> ( ( ord_less_nat @ one_one_nat @ A_5 )
=> ( ord_less_nat @ ( power_power_nat @ A_5 @ N_8 ) @ ( power_power_nat @ A_5 @ N_7 ) ) ) ) ).
thf(fact_672_power__strict__increasing,axiom,
! [A_5: int,N_8: nat,N_7: nat] :
( ( ord_less_nat @ N_8 @ N_7 )
=> ( ( ord_less_int @ one_one_int @ A_5 )
=> ( ord_less_int @ ( power_power_int @ A_5 @ N_8 ) @ ( power_power_int @ A_5 @ N_7 ) ) ) ) ).
thf(fact_673_power__less__imp__less__exp,axiom,
! [M_1: nat,N_6: nat,A_4: real] :
( ( ord_less_real @ one_one_real @ A_4 )
=> ( ( ord_less_real @ ( power_power_real @ A_4 @ M_1 ) @ ( power_power_real @ A_4 @ N_6 ) )
=> ( ord_less_nat @ M_1 @ N_6 ) ) ) ).
thf(fact_674_power__less__imp__less__exp,axiom,
! [M_1: nat,N_6: nat,A_4: nat] :
( ( ord_less_nat @ one_one_nat @ A_4 )
=> ( ( ord_less_nat @ ( power_power_nat @ A_4 @ M_1 ) @ ( power_power_nat @ A_4 @ N_6 ) )
=> ( ord_less_nat @ M_1 @ N_6 ) ) ) ).
thf(fact_675_power__less__imp__less__exp,axiom,
! [M_1: nat,N_6: nat,A_4: int] :
( ( ord_less_int @ one_one_int @ A_4 )
=> ( ( ord_less_int @ ( power_power_int @ A_4 @ M_1 ) @ ( power_power_int @ A_4 @ N_6 ) )
=> ( ord_less_nat @ M_1 @ N_6 ) ) ) ).
thf(fact_676_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_1: real] :
( ( ord_less_real @ one_one_real @ B_1 )
=> ( ( ord_less_real @ ( power_power_real @ B_1 @ X_2 ) @ ( power_power_real @ B_1 @ Y_2 ) )
<=> ( ord_less_nat @ X_2 @ Y_2 ) ) ) ).
thf(fact_677_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_1: nat] :
( ( ord_less_nat @ one_one_nat @ B_1 )
=> ( ( ord_less_nat @ ( power_power_nat @ B_1 @ X_2 ) @ ( power_power_nat @ B_1 @ Y_2 ) )
<=> ( ord_less_nat @ X_2 @ Y_2 ) ) ) ).
thf(fact_678_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_1: int] :
( ( ord_less_int @ one_one_int @ B_1 )
=> ( ( ord_less_int @ ( power_power_int @ B_1 @ X_2 ) @ ( power_power_int @ B_1 @ Y_2 ) )
<=> ( ord_less_nat @ X_2 @ Y_2 ) ) ) ).
thf(fact_679_power__increasing,axiom,
! [A_3: real,N_5: nat,N_4: nat] :
( ( ord_less_eq_nat @ N_5 @ N_4 )
=> ( ( ord_less_eq_real @ one_one_real @ A_3 )
=> ( ord_less_eq_real @ ( power_power_real @ A_3 @ N_5 ) @ ( power_power_real @ A_3 @ N_4 ) ) ) ) ).
thf(fact_680_power__increasing,axiom,
! [A_3: nat,N_5: nat,N_4: nat] :
( ( ord_less_eq_nat @ N_5 @ N_4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A_3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_3 @ N_5 ) @ ( power_power_nat @ A_3 @ N_4 ) ) ) ) ).
thf(fact_681_power__increasing,axiom,
! [A_3: int,N_5: nat,N_4: nat] :
( ( ord_less_eq_nat @ N_5 @ N_4 )
=> ( ( ord_less_eq_int @ one_one_int @ A_3 )
=> ( ord_less_eq_int @ ( power_power_int @ A_3 @ N_5 ) @ ( power_power_int @ A_3 @ N_4 ) ) ) ) ).
thf(fact_682_power__Suc__less,axiom,
! [N_3: nat,A_2: real] :
( ( ord_less_real @ zero_zero_real @ A_2 )
=> ( ( ord_less_real @ A_2 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A_2 @ ( power_power_real @ A_2 @ N_3 ) ) @ ( power_power_real @ A_2 @ N_3 ) ) ) ) ).
thf(fact_683_power__Suc__less,axiom,
! [N_3: nat,A_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_2 )
=> ( ( ord_less_nat @ A_2 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_2 @ ( power_power_nat @ A_2 @ N_3 ) ) @ ( power_power_nat @ A_2 @ N_3 ) ) ) ) ).
thf(fact_684_power__Suc__less,axiom,
! [N_3: nat,A_2: int] :
( ( ord_less_int @ zero_zero_int @ A_2 )
=> ( ( ord_less_int @ A_2 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A_2 @ ( power_power_int @ A_2 @ N_3 ) ) @ ( power_power_int @ A_2 @ N_3 ) ) ) ) ).
thf(fact_685_power__strict__decreasing,axiom,
! [A_1: int,N_2: nat,N_1: nat] :
( ( ord_less_nat @ N_2 @ N_1 )
=> ( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A_1 @ N_1 ) @ ( power_power_int @ A_1 @ N_2 ) ) ) ) ) ).
thf(fact_686_realpow__pos__nth__unique,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X_1: real] :
( ( ord_less_real @ zero_zero_real @ X_1 )
& ( ( power_power_real @ X_1 @ N )
= A )
& ! [Y_1: real] :
( ( ( ord_less_real @ zero_zero_real @ Y_1 )
& ( ( power_power_real @ Y_1 @ N )
= A ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_687_divides__ge,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = zero_zero_nat )
| ( ord_less_eq_nat @ A @ B ) ) ) ).
thf(fact_688_divides__exp,axiom,
! [N: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
thf(fact_689_divides__mul__l,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
thf(fact_690_divides__mul__r,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
thf(fact_691_divides__add__revr,axiom,
! [B: nat,D: nat,A: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
=> ( dvd_dvd_nat @ D @ B ) ) ) ).
thf(fact_692_nat__mult__dvd__cancel__disj_H,axiom,
! [M: nat,K: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_693_divides__rev,axiom,
! [A: nat,N: nat,B: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_694_divides__exp2,axiom,
! [X: nat,Y: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_695_divides__div__not,axiom,
! [X: nat,Q: nat,N: nat,R: nat] :
( ( X
= ( plus_plus_nat @ ( times_times_nat @ Q @ N ) @ R ) )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( ord_less_nat @ R @ N )
=> ~ ( dvd_dvd_nat @ N @ X ) ) ) ) ).
thf(fact_696_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_697_divides__cases,axiom,
! [N: nat,M: nat] :
( ( dvd_dvd_nat @ N @ M )
=> ( ( M = zero_zero_nat )
| ( M = N )
| ( ord_less_eq_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) @ M ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
ord_less_int @ ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) @ zero_zero_int ).
%------------------------------------------------------------------------------