TPTP Problem File: NUM925^3.p
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%------------------------------------------------------------------------------
% File : NUM925^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_1000_thf_l192 [Bla11]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.57 v6.0.0, 0.71 v5.5.0, 0.50 v5.4.0, 0.80 v5.3.0
% Syntax : Number of formulae : 1266 ( 471 unt; 61 typ; 0 def)
% Number of atoms : 2562 ( 890 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10361 ( 219 ~; 66 |; 148 &;8742 @)
% ( 335 <=>; 851 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 87 ( 87 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 58 usr; 19 con; 0-3 aty)
% Number of variables : 2501 ( 0 ^;2482 !; 19 ?;2501 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:49:12
%------------------------------------------------------------------------------
%----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 (58)
thf(sy_c_All,type,
all: ( nat > $o ) > $o ).
thf(sy_c_Ex,type,
ex: ( nat > $o ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
abs_abs_real: real > real ).
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_If_000tc__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_000tc__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
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_Onat,type,
nat_1: int > nat ).
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_Int_Osucc,type,
succ: int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Nat__Onat,type,
semiri984289939at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__RealDef__Oreal,type,
semiri132038758t_real: nat > 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_TwoSquares__Mirabelle__dzzvbppuls_Ois__sum2sq,type,
twoSqu919416604sum2sq: int > $o ).
thf(sy_v_m,type,
m: int ).
thf(sy_v_m1____,type,
m1: int ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_s1____,type,
s1: int ).
thf(sy_v_s____,type,
s: int ).
thf(sy_v_t____,type,
t: int ).
thf(sy_v_tn____,type,
tn: nat ).
thf(sy_v_x____,type,
x: int ).
thf(sy_v_y____,type,
y: int ).
%----Relevant facts (1198)
thf(fact_0_n1pos,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ).
thf(fact_1_t1,axiom,
ord_less_int @ one_one_int @ t ).
thf(fact_2_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: 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 )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
thf(fact_3_sum__power2__eq__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_4_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_5_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_6_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_real ) ).
thf(fact_7_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(fact_8_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_9_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real ) ).
thf(fact_10_zero__eq__power2,axiom,
! [A_136: int] :
( ( ( power_power_int @ A_136 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int )
<=> ( A_136 = zero_zero_int ) ) ).
thf(fact_11_zero__eq__power2,axiom,
! [A_136: real] :
( ( ( power_power_real @ A_136 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real )
<=> ( A_136 = zero_zero_real ) ) ).
thf(fact_12_add__special_I2_J,axiom,
! [W_16: int] :
( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_16 ) )
= ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_16 ) ) ) ).
thf(fact_13_add__special_I2_J,axiom,
! [W_16: int] :
( ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_16 ) )
= ( number267125858f_real @ ( plus_plus_int @ ( bit1 @ pls ) @ W_16 ) ) ) ).
thf(fact_14_add__special_I3_J,axiom,
! [V_16: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_16 ) @ one_one_int )
= ( number_number_of_int @ ( plus_plus_int @ V_16 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_15_add__special_I3_J,axiom,
! [V_16: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_16 ) @ one_one_real )
= ( number267125858f_real @ ( plus_plus_int @ V_16 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_16_one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_17_one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_18_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_19_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_20_semiring__one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_21_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_22_power__0__left__number__of,axiom,
! [W_15: int] :
( ( ( ( number_number_of_nat @ W_15 )
= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_15 ) )
= one_one_int ) )
& ( ( ( number_number_of_nat @ W_15 )
!= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_15 ) )
= zero_zero_int ) ) ) ).
thf(fact_23_power__0__left__number__of,axiom,
! [W_15: int] :
( ( ( ( number_number_of_nat @ W_15 )
= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_15 ) )
= one_one_nat ) )
& ( ( ( number_number_of_nat @ W_15 )
!= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_15 ) )
= zero_zero_nat ) ) ) ).
thf(fact_24_power__0__left__number__of,axiom,
! [W_15: int] :
( ( ( ( number_number_of_nat @ W_15 )
= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_15 ) )
= one_one_real ) )
& ( ( ( number_number_of_nat @ W_15 )
!= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_15 ) )
= zero_zero_real ) ) ) ).
thf(fact_25_semiring__norm_I110_J,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_26_semiring__norm_I110_J,axiom,
( one_one_real
= ( number267125858f_real @ ( bit1 @ pls ) ) ) ).
thf(fact_27_numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_28_numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_29_n0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
thf(fact_30_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_31_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_32_plus__numeral__code_I9_J,axiom,
! [V_1: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_1 ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V_1 @ W ) ) ) ).
thf(fact_33_less__number__of,axiom,
! [X_30: int,Y_23: int] :
( ( ord_less_int @ ( number_number_of_int @ X_30 ) @ ( number_number_of_int @ Y_23 ) )
<=> ( ord_less_int @ X_30 @ Y_23 ) ) ).
thf(fact_34_less__number__of,axiom,
! [X_30: int,Y_23: int] :
( ( ord_less_real @ ( number267125858f_real @ X_30 ) @ ( number267125858f_real @ Y_23 ) )
<=> ( ord_less_int @ X_30 @ Y_23 ) ) ).
thf(fact_35_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_36_zpower__int,axiom,
! [M: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ M ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) ) ) ).
thf(fact_37_int__power,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1621563631at_int @ M ) @ N ) ) ).
thf(fact_38_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
thf(fact_39_zadd__int,axiom,
! [M: nat,N: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_40_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_41_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_42_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_43_int__eq__0__conv,axiom,
! [N: nat] :
( ( ( semiri1621563631at_int @ N )
= zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_44_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_45_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_46_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_47_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_48_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_49_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_50_rel__simps_I14_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_51_zadd__strict__right__mono,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ord_less_int @ ( plus_plus_int @ I_1 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
thf(fact_52_add__nat__number__of,axiom,
! [V_2: int,V_1: int] :
( ( ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ V_2 ) ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( ( ord_less_int @ V_2 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ V_1 ) ) )
& ( ~ ( ord_less_int @ V_2 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_1 @ V_2 ) ) ) ) ) ) ) ).
thf(fact_53_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_54_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_55_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_56_eq__number__of,axiom,
! [X_29: int,Y_22: int] :
( ( ( number_number_of_int @ X_29 )
= ( number_number_of_int @ Y_22 ) )
<=> ( X_29 = Y_22 ) ) ).
thf(fact_57_eq__number__of,axiom,
! [X_29: int,Y_22: int] :
( ( ( number267125858f_real @ X_29 )
= ( number267125858f_real @ Y_22 ) )
<=> ( X_29 = Y_22 ) ) ).
thf(fact_58_number__of__reorient,axiom,
! [W_14: int,X_28: nat] :
( ( ( number_number_of_nat @ W_14 )
= X_28 )
<=> ( X_28
= ( number_number_of_nat @ W_14 ) ) ) ).
thf(fact_59_number__of__reorient,axiom,
! [W_14: int,X_28: int] :
( ( ( number_number_of_int @ W_14 )
= X_28 )
<=> ( X_28
= ( number_number_of_int @ W_14 ) ) ) ).
thf(fact_60_number__of__reorient,axiom,
! [W_14: int,X_28: real] :
( ( ( number267125858f_real @ W_14 )
= X_28 )
<=> ( X_28
= ( number267125858f_real @ W_14 ) ) ) ).
thf(fact_61_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_62_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( ( bit0 @ K )
= ( bit0 @ L ) )
<=> ( K = L ) ) ).
thf(fact_63_even__less__0__iff,axiom,
! [A_135: int] :
( ( ord_less_int @ ( plus_plus_int @ A_135 @ A_135 ) @ zero_zero_int )
<=> ( ord_less_int @ A_135 @ zero_zero_int ) ) ).
thf(fact_64_even__less__0__iff,axiom,
! [A_135: real] :
( ( ord_less_real @ ( plus_plus_real @ A_135 @ A_135 ) @ zero_zero_real )
<=> ( ord_less_real @ A_135 @ zero_zero_real ) ) ).
thf(fact_65_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_66_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_67_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_68_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
thf(fact_69_less__special_I3_J,axiom,
! [X_27: int] :
( ( ord_less_int @ ( number_number_of_int @ X_27 ) @ zero_zero_int )
<=> ( ord_less_int @ X_27 @ pls ) ) ).
thf(fact_70_less__special_I3_J,axiom,
! [X_27: int] :
( ( ord_less_real @ ( number267125858f_real @ X_27 ) @ zero_zero_real )
<=> ( ord_less_int @ X_27 @ pls ) ) ).
thf(fact_71_less__special_I1_J,axiom,
! [Y_21: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_21 ) )
<=> ( ord_less_int @ pls @ Y_21 ) ) ).
thf(fact_72_less__special_I1_J,axiom,
! [Y_21: int] :
( ( ord_less_real @ zero_zero_real @ ( number267125858f_real @ Y_21 ) )
<=> ( ord_less_int @ pls @ Y_21 ) ) ).
thf(fact_73_rel__simps_I12_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_74_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_75_rel__simps_I16_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_76_rel__simps_I10_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_77_rel__simps_I4_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_int @ pls @ K ) ) ).
thf(fact_78_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_79_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_80_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_81_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_82_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_83_int__less__0__conv,axiom,
! [K: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
thf(fact_84_less__special_I4_J,axiom,
! [X_26: int] :
( ( ord_less_int @ ( number_number_of_int @ X_26 ) @ one_one_int )
<=> ( ord_less_int @ X_26 @ ( bit1 @ pls ) ) ) ).
thf(fact_85_less__special_I4_J,axiom,
! [X_26: int] :
( ( ord_less_real @ ( number267125858f_real @ X_26 ) @ one_one_real )
<=> ( ord_less_int @ X_26 @ ( bit1 @ pls ) ) ) ).
thf(fact_86_less__special_I2_J,axiom,
! [Y_20: int] :
( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_20 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_20 ) ) ).
thf(fact_87_less__special_I2_J,axiom,
! [Y_20: int] :
( ( ord_less_real @ one_one_real @ ( number267125858f_real @ Y_20 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_20 ) ) ).
thf(fact_88_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_89_double__eq__0__iff,axiom,
! [A_134: int] :
( ( ( plus_plus_int @ A_134 @ A_134 )
= zero_zero_int )
<=> ( A_134 = zero_zero_int ) ) ).
thf(fact_90_double__eq__0__iff,axiom,
! [A_134: real] :
( ( ( plus_plus_real @ A_134 @ A_134 )
= zero_zero_real )
<=> ( A_134 = zero_zero_real ) ) ).
thf(fact_91_rel__simps_I46_J,axiom,
! [K: int] :
( ( bit1 @ K )
!= pls ) ).
thf(fact_92_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_93_rel__simps_I50_J,axiom,
! [K: int,L: int] :
( ( bit1 @ K )
!= ( bit0 @ L ) ) ).
thf(fact_94_rel__simps_I49_J,axiom,
! [K: int,L: int] :
( ( bit0 @ K )
!= ( bit1 @ L ) ) ).
thf(fact_95_rel__simps_I44_J,axiom,
! [K: int] :
( ( ( bit0 @ K )
= pls )
<=> ( K = pls ) ) ).
thf(fact_96_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_97_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_98_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_99_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_100_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_101_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_102_add__Bit0__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_103_Bit0__def,axiom,
! [K: int] :
( ( bit0 @ K )
= ( plus_plus_int @ K @ K ) ) ).
thf(fact_104_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_105_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_106_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_107_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_108_semiring__numeral__0__eq__0,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_109_number__of__Pls,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_110_number__of__Pls,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_111_semiring__norm_I112_J,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_112_semiring__norm_I112_J,axiom,
( zero_zero_real
= ( number267125858f_real @ pls ) ) ).
thf(fact_113_add__numeral__0,axiom,
! [A_133: int] :
( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_133 )
= A_133 ) ).
thf(fact_114_add__numeral__0,axiom,
! [A_133: real] :
( ( plus_plus_real @ ( number267125858f_real @ pls ) @ A_133 )
= A_133 ) ).
thf(fact_115_add__numeral__0__right,axiom,
! [A_132: int] :
( ( plus_plus_int @ A_132 @ ( number_number_of_int @ pls ) )
= A_132 ) ).
thf(fact_116_add__numeral__0__right,axiom,
! [A_132: real] :
( ( plus_plus_real @ A_132 @ ( number267125858f_real @ pls ) )
= A_132 ) ).
thf(fact_117_power__eq__0__iff__number__of,axiom,
! [A_131: int,W_13: int] :
( ( ( power_power_int @ A_131 @ ( number_number_of_nat @ W_13 ) )
= zero_zero_int )
<=> ( ( A_131 = zero_zero_int )
& ( ( number_number_of_nat @ W_13 )
!= zero_zero_nat ) ) ) ).
thf(fact_118_power__eq__0__iff__number__of,axiom,
! [A_131: nat,W_13: int] :
( ( ( power_power_nat @ A_131 @ ( number_number_of_nat @ W_13 ) )
= zero_zero_nat )
<=> ( ( A_131 = zero_zero_nat )
& ( ( number_number_of_nat @ W_13 )
!= zero_zero_nat ) ) ) ).
thf(fact_119_power__eq__0__iff__number__of,axiom,
! [A_131: real,W_13: int] :
( ( ( power_power_real @ A_131 @ ( number_number_of_nat @ W_13 ) )
= zero_zero_real )
<=> ( ( A_131 = zero_zero_real )
& ( ( number_number_of_nat @ W_13 )
!= zero_zero_nat ) ) ) ).
thf(fact_120_add__number__of__left,axiom,
! [V_15: int,W_12: int,Z_5: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_15 ) @ ( plus_plus_int @ ( number_number_of_int @ W_12 ) @ Z_5 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_15 @ W_12 ) ) @ Z_5 ) ) ).
thf(fact_121_add__number__of__left,axiom,
! [V_15: int,W_12: int,Z_5: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_15 ) @ ( plus_plus_real @ ( number267125858f_real @ W_12 ) @ Z_5 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_15 @ W_12 ) ) @ Z_5 ) ) ).
thf(fact_122_add__number__of__eq,axiom,
! [V_14: int,W_11: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_14 ) @ ( number_number_of_int @ W_11 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_14 @ W_11 ) ) ) ).
thf(fact_123_add__number__of__eq,axiom,
! [V_14: int,W_11: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_14 ) @ ( number267125858f_real @ W_11 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_14 @ W_11 ) ) ) ).
thf(fact_124_number__of__add,axiom,
! [V_13: int,W_10: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_13 @ W_10 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_13 ) @ ( number_number_of_int @ W_10 ) ) ) ).
thf(fact_125_number__of__add,axiom,
! [V_13: int,W_10: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_13 @ W_10 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_13 ) @ ( number267125858f_real @ W_10 ) ) ) ).
thf(fact_126_add__Bit1__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_127_add__Bit0__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_128_Bit1__def,axiom,
! [K: int] :
( ( bit1 @ K )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
thf(fact_129_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_130_number__of__int,axiom,
! [N_32: nat] :
( ( number_number_of_nat @ ( semiri1621563631at_int @ N_32 ) )
= ( semiri984289939at_nat @ N_32 ) ) ).
thf(fact_131_number__of__int,axiom,
! [N_32: nat] :
( ( number_number_of_int @ ( semiri1621563631at_int @ N_32 ) )
= ( semiri1621563631at_int @ N_32 ) ) ).
thf(fact_132_number__of__int,axiom,
! [N_32: nat] :
( ( number267125858f_real @ ( semiri1621563631at_int @ N_32 ) )
= ( semiri132038758t_real @ N_32 ) ) ).
thf(fact_133_zero__less__power2,axiom,
! [A_130: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_130 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_130 != zero_zero_int ) ) ).
thf(fact_134_zero__less__power2,axiom,
! [A_130: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_130 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_130 != zero_zero_real ) ) ).
thf(fact_135_power2__less__0,axiom,
! [A_129: int] :
~ ( ord_less_int @ ( power_power_int @ A_129 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) ).
thf(fact_136_power2__less__0,axiom,
! [A_129: real] :
~ ( ord_less_real @ ( power_power_real @ A_129 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ) ).
thf(fact_137_sum__power2__gt__zero__iff,axiom,
! [X_25: int,Y_19: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_25 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_19 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_25 != zero_zero_int )
| ( Y_19 != zero_zero_int ) ) ) ).
thf(fact_138_sum__power2__gt__zero__iff,axiom,
! [X_25: real,Y_19: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_25 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_19 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_25 != zero_zero_real )
| ( Y_19 != zero_zero_real ) ) ) ).
thf(fact_139_not__sum__power2__lt__zero,axiom,
! [X_24: int,Y_18: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X_24 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int ) ).
thf(fact_140_not__sum__power2__lt__zero,axiom,
! [X_24: real,Y_18: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X_24 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ) ).
thf(fact_141_number__of__Bit0,axiom,
! [W_9: int] :
( ( number_number_of_int @ ( bit0 @ W_9 ) )
= ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ ( number_number_of_int @ W_9 ) ) @ ( number_number_of_int @ W_9 ) ) ) ).
thf(fact_142_number__of__Bit0,axiom,
! [W_9: int] :
( ( number267125858f_real @ ( bit0 @ W_9 ) )
= ( plus_plus_real @ ( plus_plus_real @ zero_zero_real @ ( number267125858f_real @ W_9 ) ) @ ( number267125858f_real @ W_9 ) ) ) ).
thf(fact_143_number__of__Bit1,axiom,
! [W_8: int] :
( ( number_number_of_int @ ( bit1 @ W_8 ) )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_8 ) ) @ ( number_number_of_int @ W_8 ) ) ) ).
thf(fact_144_number__of__Bit1,axiom,
! [W_8: int] :
( ( number267125858f_real @ ( bit1 @ W_8 ) )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_8 ) ) @ ( number267125858f_real @ W_8 ) ) ) ).
thf(fact_145_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_146_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_147_semiring__numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_148_mn,axiom,
ord_less_int @ m1 @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ).
thf(fact_149_of__nat__less__two__power,axiom,
! [N_31: nat] : ( ord_less_int @ ( semiri1621563631at_int @ N_31 ) @ ( power_power_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_31 ) ) ).
thf(fact_150_of__nat__less__two__power,axiom,
! [N_31: nat] : ( ord_less_real @ ( semiri132038758t_real @ N_31 ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_31 ) ) ).
thf(fact_151_transfer__int__nat__numerals_I3_J,axiom,
( ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) )
= ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_152_transfer__int__nat__numerals_I4_J,axiom,
( ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) )
= ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_153_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_154_of__nat__0__less__iff,axiom,
! [N_30: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N_30 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_30 ) ) ).
thf(fact_155_of__nat__0__less__iff,axiom,
! [N_30: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri984289939at_nat @ N_30 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_30 ) ) ).
thf(fact_156_of__nat__0__less__iff,axiom,
! [N_30: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri132038758t_real @ N_30 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_30 ) ) ).
thf(fact_157_one__less__power,axiom,
! [N_29: nat,A_128: int] :
( ( ord_less_int @ one_one_int @ A_128 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_29 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A_128 @ N_29 ) ) ) ) ).
thf(fact_158_one__less__power,axiom,
! [N_29: nat,A_128: nat] :
( ( ord_less_nat @ one_one_nat @ A_128 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_29 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A_128 @ N_29 ) ) ) ) ).
thf(fact_159_one__less__power,axiom,
! [N_29: nat,A_128: real] :
( ( ord_less_real @ one_one_real @ A_128 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_29 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A_128 @ N_29 ) ) ) ) ).
thf(fact_160_power__0__left,axiom,
! [N_28: nat] :
( ( ( N_28 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_28 )
= one_one_int ) )
& ( ( N_28 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_28 )
= zero_zero_int ) ) ) ).
thf(fact_161_power__0__left,axiom,
! [N_28: nat] :
( ( ( N_28 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_28 )
= one_one_nat ) )
& ( ( N_28 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_28 )
= zero_zero_nat ) ) ) ).
thf(fact_162_power__0__left,axiom,
! [N_28: nat] :
( ( ( N_28 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_28 )
= one_one_real ) )
& ( ( N_28 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_28 )
= zero_zero_real ) ) ) ).
thf(fact_163_power__strict__decreasing,axiom,
! [A_127: int,N_27: nat,N_26: nat] :
( ( ord_less_nat @ N_27 @ N_26 )
=> ( ( ord_less_int @ zero_zero_int @ A_127 )
=> ( ( ord_less_int @ A_127 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A_127 @ N_26 ) @ ( power_power_int @ A_127 @ N_27 ) ) ) ) ) ).
thf(fact_164_power__strict__decreasing,axiom,
! [A_127: nat,N_27: nat,N_26: nat] :
( ( ord_less_nat @ N_27 @ N_26 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_127 )
=> ( ( ord_less_nat @ A_127 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A_127 @ N_26 ) @ ( power_power_nat @ A_127 @ N_27 ) ) ) ) ) ).
thf(fact_165_power__strict__decreasing,axiom,
! [A_127: real,N_27: nat,N_26: nat] :
( ( ord_less_nat @ N_27 @ N_26 )
=> ( ( ord_less_real @ zero_zero_real @ A_127 )
=> ( ( ord_less_real @ A_127 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A_127 @ N_26 ) @ ( power_power_real @ A_127 @ N_27 ) ) ) ) ) ).
thf(fact_166_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_167_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_168_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
thf(fact_169_int__gr__induct,axiom,
! [P: int > $o,K: int,I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_170_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_171_tn0,axiom,
ord_less_nat @ zero_zero_nat @ tn ).
thf(fact_172_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_173_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_174_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_175_not__add__less1,axiom,
! [I_1: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ I_1 ) ).
thf(fact_176_not__add__less2,axiom,
! [J: nat,I_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I_1 ) @ I_1 ) ).
thf(fact_177_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_178_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_179_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_180_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z ) ) ) ).
thf(fact_181_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
thf(fact_182_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N ) )
<=> ( M = N ) ) ).
thf(fact_183_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M = N ) ) ).
thf(fact_184_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_185_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_186_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_187_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_188_less__not__refl3,axiom,
! [S_1: nat,T: nat] :
( ( ord_less_nat @ S_1 @ T )
=> ( S_1 != T ) ) ).
thf(fact_189_trans__less__add1,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_190_trans__less__add2,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_191_add__less__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_192_add__less__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_193_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_194_add__lessD1,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
=> ( ord_less_nat @ I_1 @ K ) ) ).
thf(fact_195_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P @ N @ M ) )
=> ( ( ( M = N )
=> ( P @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P @ N @ M ) )
=> ( P @ N @ M ) ) ) ) ).
thf(fact_196_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_197_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_198_nat__power__less__imp__less,axiom,
! [M: nat,N: nat,I_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ I_1 )
=> ( ( ord_less_nat @ ( power_power_nat @ I_1 @ M ) @ ( power_power_nat @ I_1 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_199_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_200_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_201_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_202_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_203_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_204_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_205_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_nat @ X @ Y ) ) ).
thf(fact_206_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_207_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_208_less__nat__number__of,axiom,
! [V_1: int,V_2: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
<=> ( ( ( ord_less_int @ V_1 @ V_2 )
=> ( ord_less_int @ pls @ V_2 ) )
& ( ord_less_int @ V_1 @ V_2 ) ) ) ).
thf(fact_209_linorder__neqE__linordered__idom,axiom,
! [X_23: int,Y_17: int] :
( ( X_23 != Y_17 )
=> ( ~ ( ord_less_int @ X_23 @ Y_17 )
=> ( ord_less_int @ Y_17 @ X_23 ) ) ) ).
thf(fact_210_linorder__neqE__linordered__idom,axiom,
! [X_23: real,Y_17: real] :
( ( X_23 != Y_17 )
=> ( ~ ( ord_less_real @ X_23 @ Y_17 )
=> ( ord_less_real @ Y_17 @ X_23 ) ) ) ).
thf(fact_211_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_212_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_213_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_214_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_215_power__one__right,axiom,
! [A_126: int] :
( ( power_power_int @ A_126 @ one_one_nat )
= A_126 ) ).
thf(fact_216_power__one__right,axiom,
! [A_126: nat] :
( ( power_power_nat @ A_126 @ one_one_nat )
= A_126 ) ).
thf(fact_217_power__one__right,axiom,
! [A_126: real] :
( ( power_power_real @ A_126 @ one_one_nat )
= A_126 ) ).
thf(fact_218_of__nat__eq__iff,axiom,
! [M_14: nat,N_25: nat] :
( ( ( semiri132038758t_real @ M_14 )
= ( semiri132038758t_real @ N_25 ) )
<=> ( M_14 = N_25 ) ) ).
thf(fact_219_of__nat__eq__iff,axiom,
! [M_14: nat,N_25: nat] :
( ( ( semiri984289939at_nat @ M_14 )
= ( semiri984289939at_nat @ N_25 ) )
<=> ( M_14 = N_25 ) ) ).
thf(fact_220_of__nat__eq__iff,axiom,
! [M_14: nat,N_25: nat] :
( ( ( semiri1621563631at_int @ M_14 )
= ( semiri1621563631at_int @ N_25 ) )
<=> ( M_14 = N_25 ) ) ).
thf(fact_221_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_222_int__if__cong,axiom,
! [X: nat,Y: nat,P: $o] :
( ( P
=> ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( semiri1621563631at_int @ Y )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
thf(fact_223_less__0__number__of,axiom,
! [V_1: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V_1 ) )
<=> ( ord_less_int @ pls @ V_1 ) ) ).
thf(fact_224_zero__less__int__conv,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_225_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_226_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_227_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_228_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_229_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_230_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_231_field__power__not__zero,axiom,
! [N_24: nat,A_125: int] :
( ( A_125 != zero_zero_int )
=> ( ( power_power_int @ A_125 @ N_24 )
!= zero_zero_int ) ) ).
thf(fact_232_field__power__not__zero,axiom,
! [N_24: nat,A_125: real] :
( ( A_125 != zero_zero_real )
=> ( ( power_power_real @ A_125 @ N_24 )
!= zero_zero_real ) ) ).
thf(fact_233_power__one,axiom,
! [N_23: nat] :
( ( power_power_int @ one_one_int @ N_23 )
= one_one_int ) ).
thf(fact_234_power__one,axiom,
! [N_23: nat] :
( ( power_power_nat @ one_one_nat @ N_23 )
= one_one_nat ) ).
thf(fact_235_power__one,axiom,
! [N_23: nat] :
( ( power_power_real @ one_one_real @ N_23 )
= one_one_real ) ).
thf(fact_236_of__nat__less__iff,axiom,
! [M_13: nat,N_22: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M_13 ) @ ( semiri1621563631at_int @ N_22 ) )
<=> ( ord_less_nat @ M_13 @ N_22 ) ) ).
thf(fact_237_of__nat__less__iff,axiom,
! [M_13: nat,N_22: nat] :
( ( ord_less_nat @ ( semiri984289939at_nat @ M_13 ) @ ( semiri984289939at_nat @ N_22 ) )
<=> ( ord_less_nat @ M_13 @ N_22 ) ) ).
thf(fact_238_of__nat__less__iff,axiom,
! [M_13: nat,N_22: nat] :
( ( ord_less_real @ ( semiri132038758t_real @ M_13 ) @ ( semiri132038758t_real @ N_22 ) )
<=> ( ord_less_nat @ M_13 @ N_22 ) ) ).
thf(fact_239_less__imp__of__nat__less,axiom,
! [M_12: nat,N_21: nat] :
( ( ord_less_nat @ M_12 @ N_21 )
=> ( ord_less_int @ ( semiri1621563631at_int @ M_12 ) @ ( semiri1621563631at_int @ N_21 ) ) ) ).
thf(fact_240_less__imp__of__nat__less,axiom,
! [M_12: nat,N_21: nat] :
( ( ord_less_nat @ M_12 @ N_21 )
=> ( ord_less_nat @ ( semiri984289939at_nat @ M_12 ) @ ( semiri984289939at_nat @ N_21 ) ) ) ).
thf(fact_241_less__imp__of__nat__less,axiom,
! [M_12: nat,N_21: nat] :
( ( ord_less_nat @ M_12 @ N_21 )
=> ( ord_less_real @ ( semiri132038758t_real @ M_12 ) @ ( semiri132038758t_real @ N_21 ) ) ) ).
thf(fact_242_of__nat__less__imp__less,axiom,
! [M_11: nat,N_20: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M_11 ) @ ( semiri1621563631at_int @ N_20 ) )
=> ( ord_less_nat @ M_11 @ N_20 ) ) ).
thf(fact_243_of__nat__less__imp__less,axiom,
! [M_11: nat,N_20: nat] :
( ( ord_less_nat @ ( semiri984289939at_nat @ M_11 ) @ ( semiri984289939at_nat @ N_20 ) )
=> ( ord_less_nat @ M_11 @ N_20 ) ) ).
thf(fact_244_of__nat__less__imp__less,axiom,
! [M_11: nat,N_20: nat] :
( ( ord_less_real @ ( semiri132038758t_real @ M_11 ) @ ( semiri132038758t_real @ N_20 ) )
=> ( ord_less_nat @ M_11 @ N_20 ) ) ).
thf(fact_245_of__nat__add,axiom,
! [M_10: nat,N_19: nat] :
( ( semiri1621563631at_int @ ( plus_plus_nat @ M_10 @ N_19 ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ M_10 ) @ ( semiri1621563631at_int @ N_19 ) ) ) ).
thf(fact_246_of__nat__add,axiom,
! [M_10: nat,N_19: nat] :
( ( semiri984289939at_nat @ ( plus_plus_nat @ M_10 @ N_19 ) )
= ( plus_plus_nat @ ( semiri984289939at_nat @ M_10 ) @ ( semiri984289939at_nat @ N_19 ) ) ) ).
thf(fact_247_of__nat__add,axiom,
! [M_10: nat,N_19: nat] :
( ( semiri132038758t_real @ ( plus_plus_nat @ M_10 @ N_19 ) )
= ( plus_plus_real @ ( semiri132038758t_real @ M_10 ) @ ( semiri132038758t_real @ N_19 ) ) ) ).
thf(fact_248_of__nat__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_249_of__nat__1,axiom,
( ( semiri984289939at_nat @ one_one_nat )
= one_one_nat ) ).
thf(fact_250_of__nat__1,axiom,
( ( semiri132038758t_real @ one_one_nat )
= one_one_real ) ).
thf(fact_251_of__nat__power,axiom,
! [M_9: nat,N_18: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M_9 @ N_18 ) )
= ( power_power_int @ ( semiri1621563631at_int @ M_9 ) @ N_18 ) ) ).
thf(fact_252_of__nat__power,axiom,
! [M_9: nat,N_18: nat] :
( ( semiri984289939at_nat @ ( power_power_nat @ M_9 @ N_18 ) )
= ( power_power_nat @ ( semiri984289939at_nat @ M_9 ) @ N_18 ) ) ).
thf(fact_253_of__nat__power,axiom,
! [M_9: nat,N_18: nat] :
( ( semiri132038758t_real @ ( power_power_nat @ M_9 @ N_18 ) )
= ( power_power_real @ ( semiri132038758t_real @ M_9 ) @ N_18 ) ) ).
thf(fact_254_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_255_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
thf(fact_256_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ X ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ X @ N ) ) ) ).
thf(fact_257_pos__add__strict,axiom,
! [B_89: int,C_63: int,A_124: int] :
( ( ord_less_int @ zero_zero_int @ A_124 )
=> ( ( ord_less_int @ B_89 @ C_63 )
=> ( ord_less_int @ B_89 @ ( plus_plus_int @ A_124 @ C_63 ) ) ) ) ).
thf(fact_258_pos__add__strict,axiom,
! [B_89: nat,C_63: nat,A_124: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_124 )
=> ( ( ord_less_nat @ B_89 @ C_63 )
=> ( ord_less_nat @ B_89 @ ( plus_plus_nat @ A_124 @ C_63 ) ) ) ) ).
thf(fact_259_pos__add__strict,axiom,
! [B_89: real,C_63: real,A_124: real] :
( ( ord_less_real @ zero_zero_real @ A_124 )
=> ( ( ord_less_real @ B_89 @ C_63 )
=> ( ord_less_real @ B_89 @ ( plus_plus_real @ A_124 @ C_63 ) ) ) ) ).
thf(fact_260_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
thf(fact_261_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_262_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
thf(fact_263_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_264_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_265_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
thf(fact_266_zero__less__power,axiom,
! [N_17: nat,A_123: int] :
( ( ord_less_int @ zero_zero_int @ A_123 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_123 @ N_17 ) ) ) ).
thf(fact_267_zero__less__power,axiom,
! [N_17: nat,A_123: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_123 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A_123 @ N_17 ) ) ) ).
thf(fact_268_zero__less__power,axiom,
! [N_17: nat,A_123: real] :
( ( ord_less_real @ zero_zero_real @ A_123 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_123 @ N_17 ) ) ) ).
thf(fact_269_less__add__one,axiom,
! [A_122: int] : ( ord_less_int @ A_122 @ ( plus_plus_int @ A_122 @ one_one_int ) ) ).
thf(fact_270_less__add__one,axiom,
! [A_122: nat] : ( ord_less_nat @ A_122 @ ( plus_plus_nat @ A_122 @ one_one_nat ) ) ).
thf(fact_271_less__add__one,axiom,
! [A_122: real] : ( ord_less_real @ A_122 @ ( plus_plus_real @ A_122 @ one_one_real ) ) ).
thf(fact_272_power__inject__exp,axiom,
! [M_8: nat,N_16: nat,A_121: int] :
( ( ord_less_int @ one_one_int @ A_121 )
=> ( ( ( power_power_int @ A_121 @ M_8 )
= ( power_power_int @ A_121 @ N_16 ) )
<=> ( M_8 = N_16 ) ) ) ).
thf(fact_273_power__inject__exp,axiom,
! [M_8: nat,N_16: nat,A_121: nat] :
( ( ord_less_nat @ one_one_nat @ A_121 )
=> ( ( ( power_power_nat @ A_121 @ M_8 )
= ( power_power_nat @ A_121 @ N_16 ) )
<=> ( M_8 = N_16 ) ) ) ).
thf(fact_274_power__inject__exp,axiom,
! [M_8: nat,N_16: nat,A_121: real] :
( ( ord_less_real @ one_one_real @ A_121 )
=> ( ( ( power_power_real @ A_121 @ M_8 )
= ( power_power_real @ A_121 @ N_16 ) )
<=> ( M_8 = N_16 ) ) ) ).
thf(fact_275_power__strict__increasing__iff,axiom,
! [X_22: nat,Y_16: nat,B_88: int] :
( ( ord_less_int @ one_one_int @ B_88 )
=> ( ( ord_less_int @ ( power_power_int @ B_88 @ X_22 ) @ ( power_power_int @ B_88 @ Y_16 ) )
<=> ( ord_less_nat @ X_22 @ Y_16 ) ) ) ).
thf(fact_276_power__strict__increasing__iff,axiom,
! [X_22: nat,Y_16: nat,B_88: nat] :
( ( ord_less_nat @ one_one_nat @ B_88 )
=> ( ( ord_less_nat @ ( power_power_nat @ B_88 @ X_22 ) @ ( power_power_nat @ B_88 @ Y_16 ) )
<=> ( ord_less_nat @ X_22 @ Y_16 ) ) ) ).
thf(fact_277_power__strict__increasing__iff,axiom,
! [X_22: nat,Y_16: nat,B_88: real] :
( ( ord_less_real @ one_one_real @ B_88 )
=> ( ( ord_less_real @ ( power_power_real @ B_88 @ X_22 ) @ ( power_power_real @ B_88 @ Y_16 ) )
<=> ( ord_less_nat @ X_22 @ Y_16 ) ) ) ).
thf(fact_278_power__less__imp__less__exp,axiom,
! [M_7: nat,N_15: nat,A_120: int] :
( ( ord_less_int @ one_one_int @ A_120 )
=> ( ( ord_less_int @ ( power_power_int @ A_120 @ M_7 ) @ ( power_power_int @ A_120 @ N_15 ) )
=> ( ord_less_nat @ M_7 @ N_15 ) ) ) ).
thf(fact_279_power__less__imp__less__exp,axiom,
! [M_7: nat,N_15: nat,A_120: nat] :
( ( ord_less_nat @ one_one_nat @ A_120 )
=> ( ( ord_less_nat @ ( power_power_nat @ A_120 @ M_7 ) @ ( power_power_nat @ A_120 @ N_15 ) )
=> ( ord_less_nat @ M_7 @ N_15 ) ) ) ).
thf(fact_280_power__less__imp__less__exp,axiom,
! [M_7: nat,N_15: nat,A_120: real] :
( ( ord_less_real @ one_one_real @ A_120 )
=> ( ( ord_less_real @ ( power_power_real @ A_120 @ M_7 ) @ ( power_power_real @ A_120 @ N_15 ) )
=> ( ord_less_nat @ M_7 @ N_15 ) ) ) ).
thf(fact_281_power__strict__increasing,axiom,
! [A_119: int,N_14: nat,N_13: nat] :
( ( ord_less_nat @ N_14 @ N_13 )
=> ( ( ord_less_int @ one_one_int @ A_119 )
=> ( ord_less_int @ ( power_power_int @ A_119 @ N_14 ) @ ( power_power_int @ A_119 @ N_13 ) ) ) ) ).
thf(fact_282_power__strict__increasing,axiom,
! [A_119: nat,N_14: nat,N_13: nat] :
( ( ord_less_nat @ N_14 @ N_13 )
=> ( ( ord_less_nat @ one_one_nat @ A_119 )
=> ( ord_less_nat @ ( power_power_nat @ A_119 @ N_14 ) @ ( power_power_nat @ A_119 @ N_13 ) ) ) ) ).
thf(fact_283_power__strict__increasing,axiom,
! [A_119: real,N_14: nat,N_13: nat] :
( ( ord_less_nat @ N_14 @ N_13 )
=> ( ( ord_less_real @ one_one_real @ A_119 )
=> ( ord_less_real @ ( power_power_real @ A_119 @ N_14 ) @ ( power_power_real @ A_119 @ N_13 ) ) ) ) ).
thf(fact_284_power__eq__0__iff,axiom,
! [A_118: int,N_12: nat] :
( ( ( power_power_int @ A_118 @ N_12 )
= zero_zero_int )
<=> ( ( A_118 = zero_zero_int )
& ( N_12 != zero_zero_nat ) ) ) ).
thf(fact_285_power__eq__0__iff,axiom,
! [A_118: nat,N_12: nat] :
( ( ( power_power_nat @ A_118 @ N_12 )
= zero_zero_nat )
<=> ( ( A_118 = zero_zero_nat )
& ( N_12 != zero_zero_nat ) ) ) ).
thf(fact_286_power__eq__0__iff,axiom,
! [A_118: real,N_12: nat] :
( ( ( power_power_real @ A_118 @ N_12 )
= zero_zero_real )
<=> ( ( A_118 = zero_zero_real )
& ( N_12 != zero_zero_nat ) ) ) ).
thf(fact_287_of__nat__less__0__iff,axiom,
! [M_6: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ M_6 ) @ zero_zero_int ) ).
thf(fact_288_of__nat__less__0__iff,axiom,
! [M_6: nat] :
~ ( ord_less_nat @ ( semiri984289939at_nat @ M_6 ) @ zero_zero_nat ) ).
thf(fact_289_of__nat__less__0__iff,axiom,
! [M_6: nat] :
~ ( ord_less_real @ ( semiri132038758t_real @ M_6 ) @ zero_zero_real ) ).
thf(fact_290_power__0,axiom,
! [A_117: int] :
( ( power_power_int @ A_117 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_291_power__0,axiom,
! [A_117: nat] :
( ( power_power_nat @ A_117 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_292_power__0,axiom,
! [A_117: real] :
( ( power_power_real @ A_117 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_293_of__nat__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_294_of__nat__0,axiom,
( ( semiri984289939at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_295_of__nat__0,axiom,
( ( semiri132038758t_real @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_296_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_297_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
& ( K
= ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_298_less__imp__add__positive,axiom,
! [I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I_1 @ K_1 )
= J ) ) ) ).
thf(fact_299_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_300_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_21: int] :
( ( power_power_int @ X_21 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_21: nat] :
( ( power_power_nat @ X_21 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_21: real] :
( ( power_power_real @ X_21 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_303_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_116: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_116 @ A_116 ) )
<=> ( ord_less_int @ zero_zero_int @ A_116 ) ) ).
thf(fact_304_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_116: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_116 @ A_116 ) )
<=> ( ord_less_real @ zero_zero_real @ A_116 ) ) ).
thf(fact_305_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_115: int] :
( ( ord_less_int @ ( plus_plus_int @ A_115 @ A_115 ) @ zero_zero_int )
<=> ( ord_less_int @ A_115 @ zero_zero_int ) ) ).
thf(fact_306_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_115: real] :
( ( ord_less_real @ ( plus_plus_real @ A_115 @ A_115 ) @ zero_zero_real )
<=> ( ord_less_real @ A_115 @ zero_zero_real ) ) ).
thf(fact_307_add__pos__pos,axiom,
! [B_87: int,A_114: int] :
( ( ord_less_int @ zero_zero_int @ A_114 )
=> ( ( ord_less_int @ zero_zero_int @ B_87 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_114 @ B_87 ) ) ) ) ).
thf(fact_308_add__pos__pos,axiom,
! [B_87: nat,A_114: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_114 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_87 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_114 @ B_87 ) ) ) ) ).
thf(fact_309_add__pos__pos,axiom,
! [B_87: real,A_114: real] :
( ( ord_less_real @ zero_zero_real @ A_114 )
=> ( ( ord_less_real @ zero_zero_real @ B_87 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_114 @ B_87 ) ) ) ) ).
thf(fact_310_add__neg__neg,axiom,
! [B_86: int,A_113: int] :
( ( ord_less_int @ A_113 @ zero_zero_int )
=> ( ( ord_less_int @ B_86 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_113 @ B_86 ) @ zero_zero_int ) ) ) ).
thf(fact_311_add__neg__neg,axiom,
! [B_86: nat,A_113: nat] :
( ( ord_less_nat @ A_113 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_86 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_113 @ B_86 ) @ zero_zero_nat ) ) ) ).
thf(fact_312_add__neg__neg,axiom,
! [B_86: real,A_113: real] :
( ( ord_less_real @ A_113 @ zero_zero_real )
=> ( ( ord_less_real @ B_86 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_113 @ B_86 ) @ zero_zero_real ) ) ) ).
thf(fact_313_zero__reorient,axiom,
! [X_20: int] :
( ( zero_zero_int = X_20 )
<=> ( X_20 = zero_zero_int ) ) ).
thf(fact_314_zero__reorient,axiom,
! [X_20: nat] :
( ( zero_zero_nat = X_20 )
<=> ( X_20 = zero_zero_nat ) ) ).
thf(fact_315_zero__reorient,axiom,
! [X_20: real] :
( ( zero_zero_real = X_20 )
<=> ( X_20 = zero_zero_real ) ) ).
thf(fact_316_add__right__imp__eq,axiom,
! [B_85: int,A_112: int,C_62: int] :
( ( ( plus_plus_int @ B_85 @ A_112 )
= ( plus_plus_int @ C_62 @ A_112 ) )
=> ( B_85 = C_62 ) ) ).
thf(fact_317_add__right__imp__eq,axiom,
! [B_85: nat,A_112: nat,C_62: nat] :
( ( ( plus_plus_nat @ B_85 @ A_112 )
= ( plus_plus_nat @ C_62 @ A_112 ) )
=> ( B_85 = C_62 ) ) ).
thf(fact_318_add__right__imp__eq,axiom,
! [B_85: real,A_112: real,C_62: real] :
( ( ( plus_plus_real @ B_85 @ A_112 )
= ( plus_plus_real @ C_62 @ A_112 ) )
=> ( B_85 = C_62 ) ) ).
thf(fact_319_add__imp__eq,axiom,
! [A_111: int,B_84: int,C_61: int] :
( ( ( plus_plus_int @ A_111 @ B_84 )
= ( plus_plus_int @ A_111 @ C_61 ) )
=> ( B_84 = C_61 ) ) ).
thf(fact_320_add__imp__eq,axiom,
! [A_111: nat,B_84: nat,C_61: nat] :
( ( ( plus_plus_nat @ A_111 @ B_84 )
= ( plus_plus_nat @ A_111 @ C_61 ) )
=> ( B_84 = C_61 ) ) ).
thf(fact_321_add__imp__eq,axiom,
! [A_111: real,B_84: real,C_61: real] :
( ( ( plus_plus_real @ A_111 @ B_84 )
= ( plus_plus_real @ A_111 @ C_61 ) )
=> ( B_84 = C_61 ) ) ).
thf(fact_322_add__left__imp__eq,axiom,
! [A_110: int,B_83: int,C_60: int] :
( ( ( plus_plus_int @ A_110 @ B_83 )
= ( plus_plus_int @ A_110 @ C_60 ) )
=> ( B_83 = C_60 ) ) ).
thf(fact_323_add__left__imp__eq,axiom,
! [A_110: nat,B_83: nat,C_60: nat] :
( ( ( plus_plus_nat @ A_110 @ B_83 )
= ( plus_plus_nat @ A_110 @ C_60 ) )
=> ( B_83 = C_60 ) ) ).
thf(fact_324_add__left__imp__eq,axiom,
! [A_110: real,B_83: real,C_60: real] :
( ( ( plus_plus_real @ A_110 @ B_83 )
= ( plus_plus_real @ A_110 @ C_60 ) )
=> ( B_83 = C_60 ) ) ).
thf(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_109: int,B_82: int,C_59: int,D_23: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_109 @ B_82 ) @ ( plus_plus_int @ C_59 @ D_23 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_109 @ C_59 ) @ ( plus_plus_int @ B_82 @ D_23 ) ) ) ).
thf(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_109: nat,B_82: nat,C_59: nat,D_23: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_109 @ B_82 ) @ ( plus_plus_nat @ C_59 @ D_23 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_109 @ C_59 ) @ ( plus_plus_nat @ B_82 @ D_23 ) ) ) ).
thf(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_109: real,B_82: real,C_59: real,D_23: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_109 @ B_82 ) @ ( plus_plus_real @ C_59 @ D_23 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_109 @ C_59 ) @ ( plus_plus_real @ B_82 @ D_23 ) ) ) ).
thf(fact_328_add__right__cancel,axiom,
! [B_81: int,A_108: int,C_58: int] :
( ( ( plus_plus_int @ B_81 @ A_108 )
= ( plus_plus_int @ C_58 @ A_108 ) )
<=> ( B_81 = C_58 ) ) ).
thf(fact_329_add__right__cancel,axiom,
! [B_81: nat,A_108: nat,C_58: nat] :
( ( ( plus_plus_nat @ B_81 @ A_108 )
= ( plus_plus_nat @ C_58 @ A_108 ) )
<=> ( B_81 = C_58 ) ) ).
thf(fact_330_add__right__cancel,axiom,
! [B_81: real,A_108: real,C_58: real] :
( ( ( plus_plus_real @ B_81 @ A_108 )
= ( plus_plus_real @ C_58 @ A_108 ) )
<=> ( B_81 = C_58 ) ) ).
thf(fact_331_add__left__cancel,axiom,
! [A_107: int,B_80: int,C_57: int] :
( ( ( plus_plus_int @ A_107 @ B_80 )
= ( plus_plus_int @ A_107 @ C_57 ) )
<=> ( B_80 = C_57 ) ) ).
thf(fact_332_add__left__cancel,axiom,
! [A_107: nat,B_80: nat,C_57: nat] :
( ( ( plus_plus_nat @ A_107 @ B_80 )
= ( plus_plus_nat @ A_107 @ C_57 ) )
<=> ( B_80 = C_57 ) ) ).
thf(fact_333_add__left__cancel,axiom,
! [A_107: real,B_80: real,C_57: real] :
( ( ( plus_plus_real @ A_107 @ B_80 )
= ( plus_plus_real @ A_107 @ C_57 ) )
<=> ( B_80 = C_57 ) ) ).
thf(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_106: int,B_79: int,C_56: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_106 @ B_79 ) @ C_56 )
= ( plus_plus_int @ ( plus_plus_int @ A_106 @ C_56 ) @ B_79 ) ) ).
thf(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_106: nat,B_79: nat,C_56: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_106 @ B_79 ) @ C_56 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_106 @ C_56 ) @ B_79 ) ) ).
thf(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_106: real,B_79: real,C_56: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_106 @ B_79 ) @ C_56 )
= ( plus_plus_real @ ( plus_plus_real @ A_106 @ C_56 ) @ B_79 ) ) ).
thf(fact_337_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_105: int,B_78: int,C_55: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_105 @ B_78 ) @ C_55 )
= ( plus_plus_int @ A_105 @ ( plus_plus_int @ B_78 @ C_55 ) ) ) ).
thf(fact_338_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_105: nat,B_78: nat,C_55: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_105 @ B_78 ) @ C_55 )
= ( plus_plus_nat @ A_105 @ ( plus_plus_nat @ B_78 @ C_55 ) ) ) ).
thf(fact_339_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_105: real,B_78: real,C_55: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_105 @ B_78 ) @ C_55 )
= ( plus_plus_real @ A_105 @ ( plus_plus_real @ B_78 @ C_55 ) ) ) ).
thf(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_104: int,B_77: int,C_54: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_104 @ B_77 ) @ C_54 )
= ( plus_plus_int @ A_104 @ ( plus_plus_int @ B_77 @ C_54 ) ) ) ).
thf(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_104: nat,B_77: nat,C_54: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_104 @ B_77 ) @ C_54 )
= ( plus_plus_nat @ A_104 @ ( plus_plus_nat @ B_77 @ C_54 ) ) ) ).
thf(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_104: real,B_77: real,C_54: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_104 @ B_77 ) @ C_54 )
= ( plus_plus_real @ A_104 @ ( plus_plus_real @ B_77 @ C_54 ) ) ) ).
thf(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_103: int,C_53: int,D_22: int] :
( ( plus_plus_int @ A_103 @ ( plus_plus_int @ C_53 @ D_22 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_103 @ C_53 ) @ D_22 ) ) ).
thf(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_103: nat,C_53: nat,D_22: nat] :
( ( plus_plus_nat @ A_103 @ ( plus_plus_nat @ C_53 @ D_22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_103 @ C_53 ) @ D_22 ) ) ).
thf(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_103: real,C_53: real,D_22: real] :
( ( plus_plus_real @ A_103 @ ( plus_plus_real @ C_53 @ D_22 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_103 @ C_53 ) @ D_22 ) ) ).
thf(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_102: int,C_52: int,D_21: int] :
( ( plus_plus_int @ A_102 @ ( plus_plus_int @ C_52 @ D_21 ) )
= ( plus_plus_int @ C_52 @ ( plus_plus_int @ A_102 @ D_21 ) ) ) ).
thf(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_102: nat,C_52: nat,D_21: nat] :
( ( plus_plus_nat @ A_102 @ ( plus_plus_nat @ C_52 @ D_21 ) )
= ( plus_plus_nat @ C_52 @ ( plus_plus_nat @ A_102 @ D_21 ) ) ) ).
thf(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_102: real,C_52: real,D_21: real] :
( ( plus_plus_real @ A_102 @ ( plus_plus_real @ C_52 @ D_21 ) )
= ( plus_plus_real @ C_52 @ ( plus_plus_real @ A_102 @ D_21 ) ) ) ).
thf(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_101: int,C_51: int] :
( ( plus_plus_int @ A_101 @ C_51 )
= ( plus_plus_int @ C_51 @ A_101 ) ) ).
thf(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_101: nat,C_51: nat] :
( ( plus_plus_nat @ A_101 @ C_51 )
= ( plus_plus_nat @ C_51 @ A_101 ) ) ).
thf(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_101: real,C_51: real] :
( ( plus_plus_real @ A_101 @ C_51 )
= ( plus_plus_real @ C_51 @ A_101 ) ) ).
thf(fact_352_one__reorient,axiom,
! [X_19: int] :
( ( one_one_int = X_19 )
<=> ( X_19 = one_one_int ) ) ).
thf(fact_353_one__reorient,axiom,
! [X_19: nat] :
( ( one_one_nat = X_19 )
<=> ( X_19 = one_one_nat ) ) ).
thf(fact_354_one__reorient,axiom,
! [X_19: real] :
( ( one_one_real = X_19 )
<=> ( X_19 = one_one_real ) ) ).
thf(fact_355_add__0__iff,axiom,
! [B_76: int,A_100: int] :
( ( B_76
= ( plus_plus_int @ B_76 @ A_100 ) )
<=> ( A_100 = zero_zero_int ) ) ).
thf(fact_356_add__0__iff,axiom,
! [B_76: nat,A_100: nat] :
( ( B_76
= ( plus_plus_nat @ B_76 @ A_100 ) )
<=> ( A_100 = zero_zero_nat ) ) ).
thf(fact_357_add__0__iff,axiom,
! [B_76: real,A_100: real] :
( ( B_76
= ( plus_plus_real @ B_76 @ A_100 ) )
<=> ( A_100 = zero_zero_real ) ) ).
thf(fact_358_add_Ocomm__neutral,axiom,
! [A_99: int] :
( ( plus_plus_int @ A_99 @ zero_zero_int )
= A_99 ) ).
thf(fact_359_add_Ocomm__neutral,axiom,
! [A_99: nat] :
( ( plus_plus_nat @ A_99 @ zero_zero_nat )
= A_99 ) ).
thf(fact_360_add_Ocomm__neutral,axiom,
! [A_99: real] :
( ( plus_plus_real @ A_99 @ zero_zero_real )
= A_99 ) ).
thf(fact_361_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_98: int] :
( ( plus_plus_int @ A_98 @ zero_zero_int )
= A_98 ) ).
thf(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_98: nat] :
( ( plus_plus_nat @ A_98 @ zero_zero_nat )
= A_98 ) ).
thf(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_98: real] :
( ( plus_plus_real @ A_98 @ zero_zero_real )
= A_98 ) ).
thf(fact_364_add__0__right,axiom,
! [A_97: int] :
( ( plus_plus_int @ A_97 @ zero_zero_int )
= A_97 ) ).
thf(fact_365_add__0__right,axiom,
! [A_97: nat] :
( ( plus_plus_nat @ A_97 @ zero_zero_nat )
= A_97 ) ).
thf(fact_366_add__0__right,axiom,
! [A_97: real] :
( ( plus_plus_real @ A_97 @ zero_zero_real )
= A_97 ) ).
thf(fact_367_double__zero__sym,axiom,
! [A_96: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_96 @ A_96 ) )
<=> ( A_96 = zero_zero_int ) ) ).
thf(fact_368_double__zero__sym,axiom,
! [A_96: real] :
( ( zero_zero_real
= ( plus_plus_real @ A_96 @ A_96 ) )
<=> ( A_96 = zero_zero_real ) ) ).
thf(fact_369_add__0,axiom,
! [A_95: int] :
( ( plus_plus_int @ zero_zero_int @ A_95 )
= A_95 ) ).
thf(fact_370_add__0,axiom,
! [A_95: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_95 )
= A_95 ) ).
thf(fact_371_add__0,axiom,
! [A_95: real] :
( ( plus_plus_real @ zero_zero_real @ A_95 )
= A_95 ) ).
thf(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_94: int] :
( ( plus_plus_int @ zero_zero_int @ A_94 )
= A_94 ) ).
thf(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_94: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_94 )
= A_94 ) ).
thf(fact_374_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_94: real] :
( ( plus_plus_real @ zero_zero_real @ A_94 )
= A_94 ) ).
thf(fact_375_add__0__left,axiom,
! [A_93: int] :
( ( plus_plus_int @ zero_zero_int @ A_93 )
= A_93 ) ).
thf(fact_376_add__0__left,axiom,
! [A_93: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_93 )
= A_93 ) ).
thf(fact_377_add__0__left,axiom,
! [A_93: real] :
( ( plus_plus_real @ zero_zero_real @ A_93 )
= A_93 ) ).
thf(fact_378_add__less__imp__less__left,axiom,
! [C_50: int,A_92: int,B_75: int] :
( ( ord_less_int @ ( plus_plus_int @ C_50 @ A_92 ) @ ( plus_plus_int @ C_50 @ B_75 ) )
=> ( ord_less_int @ A_92 @ B_75 ) ) ).
thf(fact_379_add__less__imp__less__left,axiom,
! [C_50: nat,A_92: nat,B_75: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_50 @ A_92 ) @ ( plus_plus_nat @ C_50 @ B_75 ) )
=> ( ord_less_nat @ A_92 @ B_75 ) ) ).
thf(fact_380_add__less__imp__less__left,axiom,
! [C_50: real,A_92: real,B_75: real] :
( ( ord_less_real @ ( plus_plus_real @ C_50 @ A_92 ) @ ( plus_plus_real @ C_50 @ B_75 ) )
=> ( ord_less_real @ A_92 @ B_75 ) ) ).
thf(fact_381_add__less__imp__less__right,axiom,
! [A_91: int,C_49: int,B_74: int] :
( ( ord_less_int @ ( plus_plus_int @ A_91 @ C_49 ) @ ( plus_plus_int @ B_74 @ C_49 ) )
=> ( ord_less_int @ A_91 @ B_74 ) ) ).
thf(fact_382_add__less__imp__less__right,axiom,
! [A_91: nat,C_49: nat,B_74: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_91 @ C_49 ) @ ( plus_plus_nat @ B_74 @ C_49 ) )
=> ( ord_less_nat @ A_91 @ B_74 ) ) ).
thf(fact_383_add__less__imp__less__right,axiom,
! [A_91: real,C_49: real,B_74: real] :
( ( ord_less_real @ ( plus_plus_real @ A_91 @ C_49 ) @ ( plus_plus_real @ B_74 @ C_49 ) )
=> ( ord_less_real @ A_91 @ B_74 ) ) ).
thf(fact_384_add__strict__mono,axiom,
! [C_48: int,D_20: int,A_90: int,B_73: int] :
( ( ord_less_int @ A_90 @ B_73 )
=> ( ( ord_less_int @ C_48 @ D_20 )
=> ( ord_less_int @ ( plus_plus_int @ A_90 @ C_48 ) @ ( plus_plus_int @ B_73 @ D_20 ) ) ) ) ).
thf(fact_385_add__strict__mono,axiom,
! [C_48: nat,D_20: nat,A_90: nat,B_73: nat] :
( ( ord_less_nat @ A_90 @ B_73 )
=> ( ( ord_less_nat @ C_48 @ D_20 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_90 @ C_48 ) @ ( plus_plus_nat @ B_73 @ D_20 ) ) ) ) ).
thf(fact_386_add__strict__mono,axiom,
! [C_48: real,D_20: real,A_90: real,B_73: real] :
( ( ord_less_real @ A_90 @ B_73 )
=> ( ( ord_less_real @ C_48 @ D_20 )
=> ( ord_less_real @ ( plus_plus_real @ A_90 @ C_48 ) @ ( plus_plus_real @ B_73 @ D_20 ) ) ) ) ).
thf(fact_387_add__strict__left__mono,axiom,
! [C_47: int,A_89: int,B_72: int] :
( ( ord_less_int @ A_89 @ B_72 )
=> ( ord_less_int @ ( plus_plus_int @ C_47 @ A_89 ) @ ( plus_plus_int @ C_47 @ B_72 ) ) ) ).
thf(fact_388_add__strict__left__mono,axiom,
! [C_47: nat,A_89: nat,B_72: nat] :
( ( ord_less_nat @ A_89 @ B_72 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_47 @ A_89 ) @ ( plus_plus_nat @ C_47 @ B_72 ) ) ) ).
thf(fact_389_add__strict__left__mono,axiom,
! [C_47: real,A_89: real,B_72: real] :
( ( ord_less_real @ A_89 @ B_72 )
=> ( ord_less_real @ ( plus_plus_real @ C_47 @ A_89 ) @ ( plus_plus_real @ C_47 @ B_72 ) ) ) ).
thf(fact_390_add__strict__right__mono,axiom,
! [C_46: int,A_88: int,B_71: int] :
( ( ord_less_int @ A_88 @ B_71 )
=> ( ord_less_int @ ( plus_plus_int @ A_88 @ C_46 ) @ ( plus_plus_int @ B_71 @ C_46 ) ) ) ).
thf(fact_391_add__strict__right__mono,axiom,
! [C_46: nat,A_88: nat,B_71: nat] :
( ( ord_less_nat @ A_88 @ B_71 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_88 @ C_46 ) @ ( plus_plus_nat @ B_71 @ C_46 ) ) ) ).
thf(fact_392_add__strict__right__mono,axiom,
! [C_46: real,A_88: real,B_71: real] :
( ( ord_less_real @ A_88 @ B_71 )
=> ( ord_less_real @ ( plus_plus_real @ A_88 @ C_46 ) @ ( plus_plus_real @ B_71 @ C_46 ) ) ) ).
thf(fact_393_add__less__cancel__left,axiom,
! [C_45: int,A_87: int,B_70: int] :
( ( ord_less_int @ ( plus_plus_int @ C_45 @ A_87 ) @ ( plus_plus_int @ C_45 @ B_70 ) )
<=> ( ord_less_int @ A_87 @ B_70 ) ) ).
thf(fact_394_add__less__cancel__left,axiom,
! [C_45: nat,A_87: nat,B_70: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_45 @ A_87 ) @ ( plus_plus_nat @ C_45 @ B_70 ) )
<=> ( ord_less_nat @ A_87 @ B_70 ) ) ).
thf(fact_395_add__less__cancel__left,axiom,
! [C_45: real,A_87: real,B_70: real] :
( ( ord_less_real @ ( plus_plus_real @ C_45 @ A_87 ) @ ( plus_plus_real @ C_45 @ B_70 ) )
<=> ( ord_less_real @ A_87 @ B_70 ) ) ).
thf(fact_396_add__less__cancel__right,axiom,
! [A_86: int,C_44: int,B_69: int] :
( ( ord_less_int @ ( plus_plus_int @ A_86 @ C_44 ) @ ( plus_plus_int @ B_69 @ C_44 ) )
<=> ( ord_less_int @ A_86 @ B_69 ) ) ).
thf(fact_397_add__less__cancel__right,axiom,
! [A_86: nat,C_44: nat,B_69: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_86 @ C_44 ) @ ( plus_plus_nat @ B_69 @ C_44 ) )
<=> ( ord_less_nat @ A_86 @ B_69 ) ) ).
thf(fact_398_add__less__cancel__right,axiom,
! [A_86: real,C_44: real,B_69: real] :
( ( ord_less_real @ ( plus_plus_real @ A_86 @ C_44 ) @ ( plus_plus_real @ B_69 @ C_44 ) )
<=> ( ord_less_real @ A_86 @ B_69 ) ) ).
thf(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_18: int] :
( ( power_power_int @ X_18 @ one_one_nat )
= X_18 ) ).
thf(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_18: nat] :
( ( power_power_nat @ X_18 @ one_one_nat )
= X_18 ) ).
thf(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_18: real] :
( ( power_power_real @ X_18 @ one_one_nat )
= X_18 ) ).
thf(fact_402_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_403_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_404_realpow__pos__nth,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
thf(fact_405_tpos,axiom,
ord_less_eq_int @ one_one_int @ t ).
thf(fact_406_nat__number__of__add__1,axiom,
! [V_1: int] :
( ( ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ one_one_nat )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ one_one_nat )
= ( number_number_of_nat @ ( succ @ V_1 ) ) ) ) ) ).
thf(fact_407_nat__1__add__number__of,axiom,
! [V_1: int] :
( ( ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V_1 ) )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ ( succ @ V_1 ) ) ) ) ) ).
thf(fact_408_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_409_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_410_tn,axiom,
( tn
= ( minus_minus_nat @ ( nat_1 @ t ) @ one_one_nat ) ) ).
thf(fact_411__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0_,axiom,
ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( nat_1 @ t ) @ one_one_nat ) ).
thf(fact_412_le__iff__diff__le__0,axiom,
! [A_85: int,B_68: int] :
( ( ord_less_eq_int @ A_85 @ B_68 )
<=> ( ord_less_eq_int @ ( minus_minus_int @ A_85 @ B_68 ) @ zero_zero_int ) ) ).
thf(fact_413_le__iff__diff__le__0,axiom,
! [A_85: real,B_68: real] :
( ( ord_less_eq_real @ A_85 @ B_68 )
<=> ( ord_less_eq_real @ ( minus_minus_real @ A_85 @ B_68 ) @ zero_zero_real ) ) ).
thf(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_84: int,B_67: int] :
( ( times_times_int @ A_84 @ B_67 )
= ( times_times_int @ B_67 @ A_84 ) ) ).
thf(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_84: nat,B_67: nat] :
( ( times_times_nat @ A_84 @ B_67 )
= ( times_times_nat @ B_67 @ A_84 ) ) ).
thf(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_84: real,B_67: real] :
( ( times_times_real @ A_84 @ B_67 )
= ( times_times_real @ B_67 @ A_84 ) ) ).
thf(fact_417_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ Lx_6 @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ Rx_6 @ ( times_times_int @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_418_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ Lx_6 @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ Rx_6 @ ( times_times_nat @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_419_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: real,Rx_6: real,Ry_4: real] :
( ( times_times_real @ Lx_6 @ ( times_times_real @ Rx_6 @ Ry_4 ) )
= ( times_times_real @ Rx_6 @ ( times_times_real @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_420_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ Lx_5 @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ ( times_times_int @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_421_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ Lx_5 @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_422_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: real,Rx_5: real,Ry_3: real] :
( ( times_times_real @ Lx_5 @ ( times_times_real @ Rx_5 @ Ry_3 ) )
= ( times_times_real @ ( times_times_real @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_423_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_83: int,B_66: int,C_43: int] :
( ( times_times_int @ ( times_times_int @ A_83 @ B_66 ) @ C_43 )
= ( times_times_int @ A_83 @ ( times_times_int @ B_66 @ C_43 ) ) ) ).
thf(fact_424_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_83: nat,B_66: nat,C_43: nat] :
( ( times_times_nat @ ( times_times_nat @ A_83 @ B_66 ) @ C_43 )
= ( times_times_nat @ A_83 @ ( times_times_nat @ B_66 @ C_43 ) ) ) ).
thf(fact_425_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_83: real,B_66: real,C_43: real] :
( ( times_times_real @ ( times_times_real @ A_83 @ B_66 ) @ C_43 )
= ( times_times_real @ A_83 @ ( times_times_real @ B_66 @ C_43 ) ) ) ).
thf(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: real,Ly_4: real,Rx_4: real] :
( ( times_times_real @ ( times_times_real @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_real @ Lx_4 @ ( times_times_real @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_3: real,Rx_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_real @ ( times_times_real @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly_2 ) @ ( times_times_int @ Rx_2 @ Ry_2 ) )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly_2 ) @ ( times_times_nat @ Rx_2 @ Ry_2 ) )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: real,Ly_2: real,Rx_2: real,Ry_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_2 @ Ly_2 ) @ ( times_times_real @ Rx_2 @ Ry_2 ) )
= ( times_times_real @ Lx_2 @ ( times_times_real @ Ly_2 @ ( times_times_real @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: real,Ly_1: real,Rx_1: real,Ry_1: real] :
( ( times_times_real @ ( times_times_real @ Lx_1 @ Ly_1 ) @ ( times_times_real @ Rx_1 @ Ry_1 ) )
= ( times_times_real @ Rx_1 @ ( times_times_real @ ( times_times_real @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_1 @ Ly_1 ) @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ Rx_1 @ ( times_times_nat @ ( times_times_nat @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ Rx_1 @ ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_438_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: real,Ly: real,Rx: real,Ry: real] :
( ( times_times_real @ ( times_times_real @ Lx @ Ly ) @ ( times_times_real @ Rx @ Ry ) )
= ( times_times_real @ ( times_times_real @ Lx @ Rx ) @ ( times_times_real @ Ly @ Ry ) ) ) ).
thf(fact_439_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx @ Ly ) @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ ( times_times_nat @ Lx @ Rx ) @ ( times_times_nat @ Ly @ Ry ) ) ) ).
thf(fact_440_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] :
( ( times_times_int @ ( times_times_int @ Lx @ Ly ) @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ ( times_times_int @ Lx @ Rx ) @ ( times_times_int @ Ly @ Ry ) ) ) ).
thf(fact_441_diff__eq__diff__eq,axiom,
! [A_82: int,B_65: int,C_42: int,D_19: int] :
( ( ( minus_minus_int @ A_82 @ B_65 )
= ( minus_minus_int @ C_42 @ D_19 ) )
=> ( ( A_82 = B_65 )
<=> ( C_42 = D_19 ) ) ) ).
thf(fact_442_diff__eq__diff__eq,axiom,
! [A_82: real,B_65: real,C_42: real,D_19: real] :
( ( ( minus_minus_real @ A_82 @ B_65 )
= ( minus_minus_real @ C_42 @ D_19 ) )
=> ( ( A_82 = B_65 )
<=> ( C_42 = D_19 ) ) ) ).
thf(fact_443_diff__eq__diff__less__eq,axiom,
! [A_81: int,B_64: int,C_41: int,D_18: int] :
( ( ( minus_minus_int @ A_81 @ B_64 )
= ( minus_minus_int @ C_41 @ D_18 ) )
=> ( ( ord_less_eq_int @ A_81 @ B_64 )
<=> ( ord_less_eq_int @ C_41 @ D_18 ) ) ) ).
thf(fact_444_diff__eq__diff__less__eq,axiom,
! [A_81: real,B_64: real,C_41: real,D_18: real] :
( ( ( minus_minus_real @ A_81 @ B_64 )
= ( minus_minus_real @ C_41 @ D_18 ) )
=> ( ( ord_less_eq_real @ A_81 @ B_64 )
<=> ( ord_less_eq_real @ C_41 @ D_18 ) ) ) ).
thf(fact_445_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_real @ ( number267125858f_real @ V_11 ) @ ( number267125858f_real @ V_12 ) )
= ( number267125858f_real @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_446_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_11 ) @ ( number_number_of_nat @ V_12 ) )
= ( number_number_of_nat @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_447_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_int @ ( number_number_of_int @ V_11 ) @ ( number_number_of_int @ V_12 ) )
= ( number_number_of_int @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_448_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_449_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_450_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_451_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_452_zle__trans,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_eq_int @ I_1 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I_1 @ K ) ) ) ).
thf(fact_453_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_454_le__number__of,axiom,
! [X_17: int,Y_15: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_17 ) @ ( number267125858f_real @ Y_15 ) )
<=> ( ord_less_eq_int @ X_17 @ Y_15 ) ) ).
thf(fact_455_le__number__of,axiom,
! [X_17: int,Y_15: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_17 ) @ ( number_number_of_int @ Y_15 ) )
<=> ( ord_less_eq_int @ X_17 @ Y_15 ) ) ).
thf(fact_456_number__of__mult,axiom,
! [V_10: int,W_7: int] :
( ( number267125858f_real @ ( times_times_int @ V_10 @ W_7 ) )
= ( times_times_real @ ( number267125858f_real @ V_10 ) @ ( number267125858f_real @ W_7 ) ) ) ).
thf(fact_457_number__of__mult,axiom,
! [V_10: int,W_7: int] :
( ( number_number_of_int @ ( times_times_int @ V_10 @ W_7 ) )
= ( times_times_int @ ( number_number_of_int @ V_10 ) @ ( number_number_of_int @ W_7 ) ) ) ).
thf(fact_458_arith__simps_I32_J,axiom,
! [V_9: int,W_6: int] :
( ( times_times_real @ ( number267125858f_real @ V_9 ) @ ( number267125858f_real @ W_6 ) )
= ( number267125858f_real @ ( times_times_int @ V_9 @ W_6 ) ) ) ).
thf(fact_459_arith__simps_I32_J,axiom,
! [V_9: int,W_6: int] :
( ( times_times_int @ ( number_number_of_int @ V_9 ) @ ( number_number_of_int @ W_6 ) )
= ( number_number_of_int @ ( times_times_int @ V_9 @ W_6 ) ) ) ).
thf(fact_460_mult__number__of__left,axiom,
! [V_8: int,W_5: int,Z_4: real] :
( ( times_times_real @ ( number267125858f_real @ V_8 ) @ ( times_times_real @ ( number267125858f_real @ W_5 ) @ Z_4 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_8 @ W_5 ) ) @ Z_4 ) ) ).
thf(fact_461_mult__number__of__left,axiom,
! [V_8: int,W_5: int,Z_4: int] :
( ( times_times_int @ ( number_number_of_int @ V_8 ) @ ( times_times_int @ ( number_number_of_int @ W_5 ) @ Z_4 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_8 @ W_5 ) ) @ Z_4 ) ) ).
thf(fact_462_eq__nat__nat__iff,axiom,
! [Z_1: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( ( nat_1 @ Z )
= ( nat_1 @ Z_1 ) )
<=> ( Z = Z_1 ) ) ) ) ).
thf(fact_463_right__diff__distrib__number__of,axiom,
! [V_7: int,B_63: int,C_40: int] :
( ( times_times_int @ ( number_number_of_int @ V_7 ) @ ( minus_minus_int @ B_63 @ C_40 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_7 ) @ B_63 ) @ ( times_times_int @ ( number_number_of_int @ V_7 ) @ C_40 ) ) ) ).
thf(fact_464_right__diff__distrib__number__of,axiom,
! [V_7: int,B_63: real,C_40: real] :
( ( times_times_real @ ( number267125858f_real @ V_7 ) @ ( minus_minus_real @ B_63 @ C_40 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_7 ) @ B_63 ) @ ( times_times_real @ ( number267125858f_real @ V_7 ) @ C_40 ) ) ) ).
thf(fact_465_left__diff__distrib__number__of,axiom,
! [A_80: int,B_62: int,V_6: int] :
( ( times_times_int @ ( minus_minus_int @ A_80 @ B_62 ) @ ( number_number_of_int @ V_6 ) )
= ( minus_minus_int @ ( times_times_int @ A_80 @ ( number_number_of_int @ V_6 ) ) @ ( times_times_int @ B_62 @ ( number_number_of_int @ V_6 ) ) ) ) ).
thf(fact_466_left__diff__distrib__number__of,axiom,
! [A_80: real,B_62: real,V_6: int] :
( ( times_times_real @ ( minus_minus_real @ A_80 @ B_62 ) @ ( number267125858f_real @ V_6 ) )
= ( minus_minus_real @ ( times_times_real @ A_80 @ ( number267125858f_real @ V_6 ) ) @ ( times_times_real @ B_62 @ ( number267125858f_real @ V_6 ) ) ) ) ).
thf(fact_467_number__of__diff,axiom,
! [V_5: int,W_4: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_5 @ W_4 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_5 ) @ ( number_number_of_int @ W_4 ) ) ) ).
thf(fact_468_number__of__diff,axiom,
! [V_5: int,W_4: int] :
( ( number267125858f_real @ ( minus_minus_int @ V_5 @ W_4 ) )
= ( minus_minus_real @ ( number267125858f_real @ V_5 ) @ ( number267125858f_real @ W_4 ) ) ) ).
thf(fact_469_of__nat__mult,axiom,
! [M_5: nat,N_11: nat] :
( ( semiri132038758t_real @ ( times_times_nat @ M_5 @ N_11 ) )
= ( times_times_real @ ( semiri132038758t_real @ M_5 ) @ ( semiri132038758t_real @ N_11 ) ) ) ).
thf(fact_470_of__nat__mult,axiom,
! [M_5: nat,N_11: nat] :
( ( semiri984289939at_nat @ ( times_times_nat @ M_5 @ N_11 ) )
= ( times_times_nat @ ( semiri984289939at_nat @ M_5 ) @ ( semiri984289939at_nat @ N_11 ) ) ) ).
thf(fact_471_of__nat__mult,axiom,
! [M_5: nat,N_11: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M_5 @ N_11 ) )
= ( times_times_int @ ( semiri1621563631at_int @ M_5 ) @ ( semiri1621563631at_int @ N_11 ) ) ) ).
thf(fact_472_of__nat__le__iff,axiom,
! [M_4: nat,N_10: nat] :
( ( ord_less_eq_real @ ( semiri132038758t_real @ M_4 ) @ ( semiri132038758t_real @ N_10 ) )
<=> ( ord_less_eq_nat @ M_4 @ N_10 ) ) ).
thf(fact_473_of__nat__le__iff,axiom,
! [M_4: nat,N_10: nat] :
( ( ord_less_eq_nat @ ( semiri984289939at_nat @ M_4 ) @ ( semiri984289939at_nat @ N_10 ) )
<=> ( ord_less_eq_nat @ M_4 @ N_10 ) ) ).
thf(fact_474_of__nat__le__iff,axiom,
! [M_4: nat,N_10: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M_4 ) @ ( semiri1621563631at_int @ N_10 ) )
<=> ( ord_less_eq_nat @ M_4 @ N_10 ) ) ).
thf(fact_475_diff__commute,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K ) @ J ) ) ).
thf(fact_476_nat__if__cong,axiom,
! [X: int,Y: int,P: $o] :
( ( P
=> ( ( nat_1 @ X )
= ( nat_1 @ ( if_int @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( nat_1 @ Y )
= ( nat_1 @ ( if_int @ P @ X @ Y ) ) ) ) ) ).
thf(fact_477_split__mult__neg__le,axiom,
! [B_61: real,A_79: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_79 )
& ( ord_less_eq_real @ B_61 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_79 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_61 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_79 @ B_61 ) @ zero_zero_real ) ) ).
thf(fact_478_split__mult__neg__le,axiom,
! [B_61: nat,A_79: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_79 )
& ( ord_less_eq_nat @ B_61 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_79 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_61 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_79 @ B_61 ) @ zero_zero_nat ) ) ).
thf(fact_479_split__mult__neg__le,axiom,
! [B_61: int,A_79: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_79 )
& ( ord_less_eq_int @ B_61 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_79 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_61 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_79 @ B_61 ) @ zero_zero_int ) ) ).
thf(fact_480_split__mult__pos__le,axiom,
! [B_60: real,A_78: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_78 )
& ( ord_less_eq_real @ zero_zero_real @ B_60 ) )
| ( ( ord_less_eq_real @ A_78 @ zero_zero_real )
& ( ord_less_eq_real @ B_60 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_78 @ B_60 ) ) ) ).
thf(fact_481_split__mult__pos__le,axiom,
! [B_60: int,A_78: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_78 )
& ( ord_less_eq_int @ zero_zero_int @ B_60 ) )
| ( ( ord_less_eq_int @ A_78 @ zero_zero_int )
& ( ord_less_eq_int @ B_60 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_78 @ B_60 ) ) ) ).
thf(fact_482_mult__mono,axiom,
! [C_39: real,D_17: real,A_77: real,B_59: real] :
( ( ord_less_eq_real @ A_77 @ B_59 )
=> ( ( ord_less_eq_real @ C_39 @ D_17 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_59 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_39 )
=> ( ord_less_eq_real @ ( times_times_real @ A_77 @ C_39 ) @ ( times_times_real @ B_59 @ D_17 ) ) ) ) ) ) ).
thf(fact_483_mult__mono,axiom,
! [C_39: nat,D_17: nat,A_77: nat,B_59: nat] :
( ( ord_less_eq_nat @ A_77 @ B_59 )
=> ( ( ord_less_eq_nat @ C_39 @ D_17 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_59 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_39 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_77 @ C_39 ) @ ( times_times_nat @ B_59 @ D_17 ) ) ) ) ) ) ).
thf(fact_484_mult__mono,axiom,
! [C_39: int,D_17: int,A_77: int,B_59: int] :
( ( ord_less_eq_int @ A_77 @ B_59 )
=> ( ( ord_less_eq_int @ C_39 @ D_17 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_59 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_39 )
=> ( ord_less_eq_int @ ( times_times_int @ A_77 @ C_39 ) @ ( times_times_int @ B_59 @ D_17 ) ) ) ) ) ) ).
thf(fact_485_mult__mono_H,axiom,
! [C_38: real,D_16: real,A_76: real,B_58: real] :
( ( ord_less_eq_real @ A_76 @ B_58 )
=> ( ( ord_less_eq_real @ C_38 @ D_16 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_76 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_38 )
=> ( ord_less_eq_real @ ( times_times_real @ A_76 @ C_38 ) @ ( times_times_real @ B_58 @ D_16 ) ) ) ) ) ) ).
thf(fact_486_mult__mono_H,axiom,
! [C_38: nat,D_16: nat,A_76: nat,B_58: nat] :
( ( ord_less_eq_nat @ A_76 @ B_58 )
=> ( ( ord_less_eq_nat @ C_38 @ D_16 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_76 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_38 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_76 @ C_38 ) @ ( times_times_nat @ B_58 @ D_16 ) ) ) ) ) ) ).
thf(fact_487_mult__mono_H,axiom,
! [C_38: int,D_16: int,A_76: int,B_58: int] :
( ( ord_less_eq_int @ A_76 @ B_58 )
=> ( ( ord_less_eq_int @ C_38 @ D_16 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_76 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_38 )
=> ( ord_less_eq_int @ ( times_times_int @ A_76 @ C_38 ) @ ( times_times_int @ B_58 @ D_16 ) ) ) ) ) ) ).
thf(fact_488_mult__left__mono__neg,axiom,
! [C_37: real,B_57: real,A_75: real] :
( ( ord_less_eq_real @ B_57 @ A_75 )
=> ( ( ord_less_eq_real @ C_37 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_37 @ A_75 ) @ ( times_times_real @ C_37 @ B_57 ) ) ) ) ).
thf(fact_489_mult__left__mono__neg,axiom,
! [C_37: int,B_57: int,A_75: int] :
( ( ord_less_eq_int @ B_57 @ A_75 )
=> ( ( ord_less_eq_int @ C_37 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_37 @ A_75 ) @ ( times_times_int @ C_37 @ B_57 ) ) ) ) ).
thf(fact_490_mult__right__mono__neg,axiom,
! [C_36: real,B_56: real,A_74: real] :
( ( ord_less_eq_real @ B_56 @ A_74 )
=> ( ( ord_less_eq_real @ C_36 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_74 @ C_36 ) @ ( times_times_real @ B_56 @ C_36 ) ) ) ) ).
thf(fact_491_mult__right__mono__neg,axiom,
! [C_36: int,B_56: int,A_74: int] :
( ( ord_less_eq_int @ B_56 @ A_74 )
=> ( ( ord_less_eq_int @ C_36 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_74 @ C_36 ) @ ( times_times_int @ B_56 @ C_36 ) ) ) ) ).
thf(fact_492_comm__mult__left__mono,axiom,
! [C_35: real,A_73: real,B_55: real] :
( ( ord_less_eq_real @ A_73 @ B_55 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_35 )
=> ( ord_less_eq_real @ ( times_times_real @ C_35 @ A_73 ) @ ( times_times_real @ C_35 @ B_55 ) ) ) ) ).
thf(fact_493_comm__mult__left__mono,axiom,
! [C_35: nat,A_73: nat,B_55: nat] :
( ( ord_less_eq_nat @ A_73 @ B_55 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_35 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_35 @ A_73 ) @ ( times_times_nat @ C_35 @ B_55 ) ) ) ) ).
thf(fact_494_comm__mult__left__mono,axiom,
! [C_35: int,A_73: int,B_55: int] :
( ( ord_less_eq_int @ A_73 @ B_55 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_35 )
=> ( ord_less_eq_int @ ( times_times_int @ C_35 @ A_73 ) @ ( times_times_int @ C_35 @ B_55 ) ) ) ) ).
thf(fact_495_mult__left__mono,axiom,
! [C_34: real,A_72: real,B_54: real] :
( ( ord_less_eq_real @ A_72 @ B_54 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_34 )
=> ( ord_less_eq_real @ ( times_times_real @ C_34 @ A_72 ) @ ( times_times_real @ C_34 @ B_54 ) ) ) ) ).
thf(fact_496_mult__left__mono,axiom,
! [C_34: nat,A_72: nat,B_54: nat] :
( ( ord_less_eq_nat @ A_72 @ B_54 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_34 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_34 @ A_72 ) @ ( times_times_nat @ C_34 @ B_54 ) ) ) ) ).
thf(fact_497_mult__left__mono,axiom,
! [C_34: int,A_72: int,B_54: int] :
( ( ord_less_eq_int @ A_72 @ B_54 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_34 )
=> ( ord_less_eq_int @ ( times_times_int @ C_34 @ A_72 ) @ ( times_times_int @ C_34 @ B_54 ) ) ) ) ).
thf(fact_498_mult__right__mono,axiom,
! [C_33: real,A_71: real,B_53: real] :
( ( ord_less_eq_real @ A_71 @ B_53 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_33 )
=> ( ord_less_eq_real @ ( times_times_real @ A_71 @ C_33 ) @ ( times_times_real @ B_53 @ C_33 ) ) ) ) ).
thf(fact_499_mult__right__mono,axiom,
! [C_33: nat,A_71: nat,B_53: nat] :
( ( ord_less_eq_nat @ A_71 @ B_53 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_33 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_71 @ C_33 ) @ ( times_times_nat @ B_53 @ C_33 ) ) ) ) ).
thf(fact_500_mult__right__mono,axiom,
! [C_33: int,A_71: int,B_53: int] :
( ( ord_less_eq_int @ A_71 @ B_53 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_33 )
=> ( ord_less_eq_int @ ( times_times_int @ A_71 @ C_33 ) @ ( times_times_int @ B_53 @ C_33 ) ) ) ) ).
thf(fact_501_mult__nonpos__nonpos,axiom,
! [B_52: real,A_70: real] :
( ( ord_less_eq_real @ A_70 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_52 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_70 @ B_52 ) ) ) ) ).
thf(fact_502_mult__nonpos__nonpos,axiom,
! [B_52: int,A_70: int] :
( ( ord_less_eq_int @ A_70 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_52 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_70 @ B_52 ) ) ) ) ).
thf(fact_503_mult__nonpos__nonneg,axiom,
! [B_51: real,A_69: real] :
( ( ord_less_eq_real @ A_69 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_51 )
=> ( ord_less_eq_real @ ( times_times_real @ A_69 @ B_51 ) @ zero_zero_real ) ) ) ).
thf(fact_504_mult__nonpos__nonneg,axiom,
! [B_51: nat,A_69: nat] :
( ( ord_less_eq_nat @ A_69 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_51 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_69 @ B_51 ) @ zero_zero_nat ) ) ) ).
thf(fact_505_mult__nonpos__nonneg,axiom,
! [B_51: int,A_69: int] :
( ( ord_less_eq_int @ A_69 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_51 )
=> ( ord_less_eq_int @ ( times_times_int @ A_69 @ B_51 ) @ zero_zero_int ) ) ) ).
thf(fact_506_mult__nonneg__nonpos2,axiom,
! [B_50: real,A_68: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_68 )
=> ( ( ord_less_eq_real @ B_50 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_50 @ A_68 ) @ zero_zero_real ) ) ) ).
thf(fact_507_mult__nonneg__nonpos2,axiom,
! [B_50: nat,A_68: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_68 )
=> ( ( ord_less_eq_nat @ B_50 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_50 @ A_68 ) @ zero_zero_nat ) ) ) ).
thf(fact_508_mult__nonneg__nonpos2,axiom,
! [B_50: int,A_68: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_68 )
=> ( ( ord_less_eq_int @ B_50 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_50 @ A_68 ) @ zero_zero_int ) ) ) ).
thf(fact_509_mult__nonneg__nonpos,axiom,
! [B_49: real,A_67: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_67 )
=> ( ( ord_less_eq_real @ B_49 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_67 @ B_49 ) @ zero_zero_real ) ) ) ).
thf(fact_510_mult__nonneg__nonpos,axiom,
! [B_49: nat,A_67: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_67 )
=> ( ( ord_less_eq_nat @ B_49 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_67 @ B_49 ) @ zero_zero_nat ) ) ) ).
thf(fact_511_mult__nonneg__nonpos,axiom,
! [B_49: int,A_67: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_67 )
=> ( ( ord_less_eq_int @ B_49 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_67 @ B_49 ) @ zero_zero_int ) ) ) ).
thf(fact_512_transfer__nat__int__relations_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( nat_1 @ X )
= ( nat_1 @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(fact_513_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_514_mult__nonneg__nonneg,axiom,
! [B_48: real,A_66: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_66 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_48 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_66 @ B_48 ) ) ) ) ).
thf(fact_515_mult__nonneg__nonneg,axiom,
! [B_48: nat,A_66: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_66 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_48 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_66 @ B_48 ) ) ) ) ).
thf(fact_516_mult__nonneg__nonneg,axiom,
! [B_48: int,A_66: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_66 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_48 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_66 @ B_48 ) ) ) ) ).
thf(fact_517_le__add__iff1,axiom,
! [A_65: real,E_6: real,C_32: real,B_47: real,D_15: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_65 @ E_6 ) @ C_32 ) @ ( plus_plus_real @ ( times_times_real @ B_47 @ E_6 ) @ D_15 ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_65 @ B_47 ) @ E_6 ) @ C_32 ) @ D_15 ) ) ).
thf(fact_518_le__add__iff1,axiom,
! [A_65: int,E_6: int,C_32: int,B_47: int,D_15: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_65 @ E_6 ) @ C_32 ) @ ( plus_plus_int @ ( times_times_int @ B_47 @ E_6 ) @ D_15 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_65 @ B_47 ) @ E_6 ) @ C_32 ) @ D_15 ) ) ).
thf(fact_519_eq__add__iff1,axiom,
! [A_64: real,E_5: real,C_31: real,B_46: real,D_14: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_64 @ E_5 ) @ C_31 )
= ( plus_plus_real @ ( times_times_real @ B_46 @ E_5 ) @ D_14 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_64 @ B_46 ) @ E_5 ) @ C_31 )
= D_14 ) ) ).
thf(fact_520_eq__add__iff1,axiom,
! [A_64: int,E_5: int,C_31: int,B_46: int,D_14: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_64 @ E_5 ) @ C_31 )
= ( plus_plus_int @ ( times_times_int @ B_46 @ E_5 ) @ D_14 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_64 @ B_46 ) @ E_5 ) @ C_31 )
= D_14 ) ) ).
thf(fact_521_le__add__iff2,axiom,
! [A_63: real,E_4: real,C_30: real,B_45: real,D_13: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_63 @ E_4 ) @ C_30 ) @ ( plus_plus_real @ ( times_times_real @ B_45 @ E_4 ) @ D_13 ) )
<=> ( ord_less_eq_real @ C_30 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_45 @ A_63 ) @ E_4 ) @ D_13 ) ) ) ).
thf(fact_522_le__add__iff2,axiom,
! [A_63: int,E_4: int,C_30: int,B_45: int,D_13: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_63 @ E_4 ) @ C_30 ) @ ( plus_plus_int @ ( times_times_int @ B_45 @ E_4 ) @ D_13 ) )
<=> ( ord_less_eq_int @ C_30 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_45 @ A_63 ) @ E_4 ) @ D_13 ) ) ) ).
thf(fact_523_eq__add__iff2,axiom,
! [A_62: real,E_3: real,C_29: real,B_44: real,D_12: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_62 @ E_3 ) @ C_29 )
= ( plus_plus_real @ ( times_times_real @ B_44 @ E_3 ) @ D_12 ) )
<=> ( C_29
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_44 @ A_62 ) @ E_3 ) @ D_12 ) ) ) ).
thf(fact_524_eq__add__iff2,axiom,
! [A_62: int,E_3: int,C_29: int,B_44: int,D_12: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_62 @ E_3 ) @ C_29 )
= ( plus_plus_int @ ( times_times_int @ B_44 @ E_3 ) @ D_12 ) )
<=> ( C_29
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_44 @ A_62 ) @ E_3 ) @ D_12 ) ) ) ).
thf(fact_525_mult__diff__mult,axiom,
! [X_16: real,Y_14: real,A_61: real,B_43: real] :
( ( minus_minus_real @ ( times_times_real @ X_16 @ Y_14 ) @ ( times_times_real @ A_61 @ B_43 ) )
= ( plus_plus_real @ ( times_times_real @ X_16 @ ( minus_minus_real @ Y_14 @ B_43 ) ) @ ( times_times_real @ ( minus_minus_real @ X_16 @ A_61 ) @ B_43 ) ) ) ).
thf(fact_526_mult__diff__mult,axiom,
! [X_16: int,Y_14: int,A_61: int,B_43: int] :
( ( minus_minus_int @ ( times_times_int @ X_16 @ Y_14 ) @ ( times_times_int @ A_61 @ B_43 ) )
= ( plus_plus_int @ ( times_times_int @ X_16 @ ( minus_minus_int @ Y_14 @ B_43 ) ) @ ( times_times_int @ ( minus_minus_int @ X_16 @ A_61 ) @ B_43 ) ) ) ).
thf(fact_527_mult__le__0__iff,axiom,
! [A_60: real,B_42: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_60 @ B_42 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_60 )
& ( ord_less_eq_real @ B_42 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_60 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_42 ) ) ) ) ).
thf(fact_528_mult__le__0__iff,axiom,
! [A_60: int,B_42: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_60 @ B_42 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_60 )
& ( ord_less_eq_int @ B_42 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_60 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_42 ) ) ) ) ).
thf(fact_529_zero__le__mult__iff,axiom,
! [A_59: real,B_41: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_59 @ B_41 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_59 )
& ( ord_less_eq_real @ zero_zero_real @ B_41 ) )
| ( ( ord_less_eq_real @ A_59 @ zero_zero_real )
& ( ord_less_eq_real @ B_41 @ zero_zero_real ) ) ) ) ).
thf(fact_530_zero__le__mult__iff,axiom,
! [A_59: int,B_41: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_59 @ B_41 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_59 )
& ( ord_less_eq_int @ zero_zero_int @ B_41 ) )
| ( ( ord_less_eq_int @ A_59 @ zero_zero_int )
& ( ord_less_eq_int @ B_41 @ zero_zero_int ) ) ) ) ).
thf(fact_531_all__nat,axiom,
! [P: nat > $o] :
( ( all @ P )
<=> ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ ( nat_1 @ X_1 ) ) ) ) ).
thf(fact_532_ex__nat,axiom,
! [P: nat > $o] :
( ( ex @ P )
<=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ ( nat_1 @ X_1 ) ) ) ) ).
thf(fact_533_zero__le__square,axiom,
! [A_58: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_58 @ A_58 ) ) ).
thf(fact_534_zero__le__square,axiom,
! [A_58: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_58 @ A_58 ) ) ).
thf(fact_535_less__add__iff1,axiom,
! [A_57: real,E_2: real,C_28: real,B_40: real,D_11: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_57 @ E_2 ) @ C_28 ) @ ( plus_plus_real @ ( times_times_real @ B_40 @ E_2 ) @ D_11 ) )
<=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_57 @ B_40 ) @ E_2 ) @ C_28 ) @ D_11 ) ) ).
thf(fact_536_less__add__iff1,axiom,
! [A_57: int,E_2: int,C_28: int,B_40: int,D_11: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_57 @ E_2 ) @ C_28 ) @ ( plus_plus_int @ ( times_times_int @ B_40 @ E_2 ) @ D_11 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_57 @ B_40 ) @ E_2 ) @ C_28 ) @ D_11 ) ) ).
thf(fact_537_less__add__iff2,axiom,
! [A_56: real,E_1: real,C_27: real,B_39: real,D_10: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_56 @ E_1 ) @ C_27 ) @ ( plus_plus_real @ ( times_times_real @ B_39 @ E_1 ) @ D_10 ) )
<=> ( ord_less_real @ C_27 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_39 @ A_56 ) @ E_1 ) @ D_10 ) ) ) ).
thf(fact_538_less__add__iff2,axiom,
! [A_56: int,E_1: int,C_27: int,B_39: int,D_10: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_56 @ E_1 ) @ C_27 ) @ ( plus_plus_int @ ( times_times_int @ B_39 @ E_1 ) @ D_10 ) )
<=> ( ord_less_int @ C_27 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_39 @ A_56 ) @ E_1 ) @ D_10 ) ) ) ).
thf(fact_539_real__squared__diff__one__factored,axiom,
! [X_15: real] :
( ( minus_minus_real @ ( times_times_real @ X_15 @ X_15 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X_15 @ one_one_real ) @ ( minus_minus_real @ X_15 @ one_one_real ) ) ) ).
thf(fact_540_real__squared__diff__one__factored,axiom,
! [X_15: int] :
( ( minus_minus_int @ ( times_times_int @ X_15 @ X_15 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_15 @ one_one_int ) @ ( minus_minus_int @ X_15 @ one_one_int ) ) ) ).
thf(fact_541_mult__left__le__imp__le,axiom,
! [C_26: real,A_55: real,B_38: real] :
( ( ord_less_eq_real @ ( times_times_real @ C_26 @ A_55 ) @ ( times_times_real @ C_26 @ B_38 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_26 )
=> ( ord_less_eq_real @ A_55 @ B_38 ) ) ) ).
thf(fact_542_mult__left__le__imp__le,axiom,
! [C_26: nat,A_55: nat,B_38: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C_26 @ A_55 ) @ ( times_times_nat @ C_26 @ B_38 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_26 )
=> ( ord_less_eq_nat @ A_55 @ B_38 ) ) ) ).
thf(fact_543_mult__left__le__imp__le,axiom,
! [C_26: int,A_55: int,B_38: int] :
( ( ord_less_eq_int @ ( times_times_int @ C_26 @ A_55 ) @ ( times_times_int @ C_26 @ B_38 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_26 )
=> ( ord_less_eq_int @ A_55 @ B_38 ) ) ) ).
thf(fact_544_mult__right__le__imp__le,axiom,
! [A_54: real,C_25: real,B_37: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_54 @ C_25 ) @ ( times_times_real @ B_37 @ C_25 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_25 )
=> ( ord_less_eq_real @ A_54 @ B_37 ) ) ) ).
thf(fact_545_mult__right__le__imp__le,axiom,
! [A_54: nat,C_25: nat,B_37: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A_54 @ C_25 ) @ ( times_times_nat @ B_37 @ C_25 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_25 )
=> ( ord_less_eq_nat @ A_54 @ B_37 ) ) ) ).
thf(fact_546_mult__right__le__imp__le,axiom,
! [A_54: int,C_25: int,B_37: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_54 @ C_25 ) @ ( times_times_int @ B_37 @ C_25 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_25 )
=> ( ord_less_eq_int @ A_54 @ B_37 ) ) ) ).
thf(fact_547_mult__less__imp__less__left,axiom,
! [C_24: real,A_53: real,B_36: real] :
( ( ord_less_real @ ( times_times_real @ C_24 @ A_53 ) @ ( times_times_real @ C_24 @ B_36 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_24 )
=> ( ord_less_real @ A_53 @ B_36 ) ) ) ).
thf(fact_548_mult__less__imp__less__left,axiom,
! [C_24: nat,A_53: nat,B_36: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_24 @ A_53 ) @ ( times_times_nat @ C_24 @ B_36 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_24 )
=> ( ord_less_nat @ A_53 @ B_36 ) ) ) ).
thf(fact_549_mult__less__imp__less__left,axiom,
! [C_24: int,A_53: int,B_36: int] :
( ( ord_less_int @ ( times_times_int @ C_24 @ A_53 ) @ ( times_times_int @ C_24 @ B_36 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_24 )
=> ( ord_less_int @ A_53 @ B_36 ) ) ) ).
thf(fact_550_mult__left__less__imp__less,axiom,
! [C_23: real,A_52: real,B_35: real] :
( ( ord_less_real @ ( times_times_real @ C_23 @ A_52 ) @ ( times_times_real @ C_23 @ B_35 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_23 )
=> ( ord_less_real @ A_52 @ B_35 ) ) ) ).
thf(fact_551_mult__left__less__imp__less,axiom,
! [C_23: nat,A_52: nat,B_35: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_23 @ A_52 ) @ ( times_times_nat @ C_23 @ B_35 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_23 )
=> ( ord_less_nat @ A_52 @ B_35 ) ) ) ).
thf(fact_552_mult__left__less__imp__less,axiom,
! [C_23: int,A_52: int,B_35: int] :
( ( ord_less_int @ ( times_times_int @ C_23 @ A_52 ) @ ( times_times_int @ C_23 @ B_35 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_23 )
=> ( ord_less_int @ A_52 @ B_35 ) ) ) ).
thf(fact_553_mult__less__imp__less__right,axiom,
! [A_51: real,C_22: real,B_34: real] :
( ( ord_less_real @ ( times_times_real @ A_51 @ C_22 ) @ ( times_times_real @ B_34 @ C_22 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_22 )
=> ( ord_less_real @ A_51 @ B_34 ) ) ) ).
thf(fact_554_mult__less__imp__less__right,axiom,
! [A_51: nat,C_22: nat,B_34: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_51 @ C_22 ) @ ( times_times_nat @ B_34 @ C_22 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_22 )
=> ( ord_less_nat @ A_51 @ B_34 ) ) ) ).
thf(fact_555_mult__less__imp__less__right,axiom,
! [A_51: int,C_22: int,B_34: int] :
( ( ord_less_int @ ( times_times_int @ A_51 @ C_22 ) @ ( times_times_int @ B_34 @ C_22 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_22 )
=> ( ord_less_int @ A_51 @ B_34 ) ) ) ).
thf(fact_556_mult__right__less__imp__less,axiom,
! [A_50: real,C_21: real,B_33: real] :
( ( ord_less_real @ ( times_times_real @ A_50 @ C_21 ) @ ( times_times_real @ B_33 @ C_21 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_21 )
=> ( ord_less_real @ A_50 @ B_33 ) ) ) ).
thf(fact_557_mult__right__less__imp__less,axiom,
! [A_50: nat,C_21: nat,B_33: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_50 @ C_21 ) @ ( times_times_nat @ B_33 @ C_21 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_21 )
=> ( ord_less_nat @ A_50 @ B_33 ) ) ) ).
thf(fact_558_mult__right__less__imp__less,axiom,
! [A_50: int,C_21: int,B_33: int] :
( ( ord_less_int @ ( times_times_int @ A_50 @ C_21 ) @ ( times_times_int @ B_33 @ C_21 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_21 )
=> ( ord_less_int @ A_50 @ B_33 ) ) ) ).
thf(fact_559_mult__le__less__imp__less,axiom,
! [C_20: real,D_9: real,A_49: real,B_32: real] :
( ( ord_less_eq_real @ A_49 @ B_32 )
=> ( ( ord_less_real @ C_20 @ D_9 )
=> ( ( ord_less_real @ zero_zero_real @ A_49 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_20 )
=> ( ord_less_real @ ( times_times_real @ A_49 @ C_20 ) @ ( times_times_real @ B_32 @ D_9 ) ) ) ) ) ) ).
thf(fact_560_mult__le__less__imp__less,axiom,
! [C_20: nat,D_9: nat,A_49: nat,B_32: nat] :
( ( ord_less_eq_nat @ A_49 @ B_32 )
=> ( ( ord_less_nat @ C_20 @ D_9 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_49 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_20 )
=> ( ord_less_nat @ ( times_times_nat @ A_49 @ C_20 ) @ ( times_times_nat @ B_32 @ D_9 ) ) ) ) ) ) ).
thf(fact_561_mult__le__less__imp__less,axiom,
! [C_20: int,D_9: int,A_49: int,B_32: int] :
( ( ord_less_eq_int @ A_49 @ B_32 )
=> ( ( ord_less_int @ C_20 @ D_9 )
=> ( ( ord_less_int @ zero_zero_int @ A_49 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_20 )
=> ( ord_less_int @ ( times_times_int @ A_49 @ C_20 ) @ ( times_times_int @ B_32 @ D_9 ) ) ) ) ) ) ).
thf(fact_562_mult__less__le__imp__less,axiom,
! [C_19: real,D_8: real,A_48: real,B_31: real] :
( ( ord_less_real @ A_48 @ B_31 )
=> ( ( ord_less_eq_real @ C_19 @ D_8 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_48 )
=> ( ( ord_less_real @ zero_zero_real @ C_19 )
=> ( ord_less_real @ ( times_times_real @ A_48 @ C_19 ) @ ( times_times_real @ B_31 @ D_8 ) ) ) ) ) ) ).
thf(fact_563_mult__less__le__imp__less,axiom,
! [C_19: nat,D_8: nat,A_48: nat,B_31: nat] :
( ( ord_less_nat @ A_48 @ B_31 )
=> ( ( ord_less_eq_nat @ C_19 @ D_8 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_48 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_19 )
=> ( ord_less_nat @ ( times_times_nat @ A_48 @ C_19 ) @ ( times_times_nat @ B_31 @ D_8 ) ) ) ) ) ) ).
thf(fact_564_mult__less__le__imp__less,axiom,
! [C_19: int,D_8: int,A_48: int,B_31: int] :
( ( ord_less_int @ A_48 @ B_31 )
=> ( ( ord_less_eq_int @ C_19 @ D_8 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_48 )
=> ( ( ord_less_int @ zero_zero_int @ C_19 )
=> ( ord_less_int @ ( times_times_int @ A_48 @ C_19 ) @ ( times_times_int @ B_31 @ D_8 ) ) ) ) ) ) ).
thf(fact_565_mult__strict__mono_H,axiom,
! [C_18: real,D_7: real,A_47: real,B_30: real] :
( ( ord_less_real @ A_47 @ B_30 )
=> ( ( ord_less_real @ C_18 @ D_7 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_47 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_18 )
=> ( ord_less_real @ ( times_times_real @ A_47 @ C_18 ) @ ( times_times_real @ B_30 @ D_7 ) ) ) ) ) ) ).
thf(fact_566_mult__strict__mono_H,axiom,
! [C_18: nat,D_7: nat,A_47: nat,B_30: nat] :
( ( ord_less_nat @ A_47 @ B_30 )
=> ( ( ord_less_nat @ C_18 @ D_7 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_47 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_18 )
=> ( ord_less_nat @ ( times_times_nat @ A_47 @ C_18 ) @ ( times_times_nat @ B_30 @ D_7 ) ) ) ) ) ) ).
thf(fact_567_mult__strict__mono_H,axiom,
! [C_18: int,D_7: int,A_47: int,B_30: int] :
( ( ord_less_int @ A_47 @ B_30 )
=> ( ( ord_less_int @ C_18 @ D_7 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_47 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_18 )
=> ( ord_less_int @ ( times_times_int @ A_47 @ C_18 ) @ ( times_times_int @ B_30 @ D_7 ) ) ) ) ) ) ).
thf(fact_568_mult__strict__mono,axiom,
! [C_17: real,D_6: real,A_46: real,B_29: real] :
( ( ord_less_real @ A_46 @ B_29 )
=> ( ( ord_less_real @ C_17 @ D_6 )
=> ( ( ord_less_real @ zero_zero_real @ B_29 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_17 )
=> ( ord_less_real @ ( times_times_real @ A_46 @ C_17 ) @ ( times_times_real @ B_29 @ D_6 ) ) ) ) ) ) ).
thf(fact_569_mult__strict__mono,axiom,
! [C_17: nat,D_6: nat,A_46: nat,B_29: nat] :
( ( ord_less_nat @ A_46 @ B_29 )
=> ( ( ord_less_nat @ C_17 @ D_6 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_29 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_17 )
=> ( ord_less_nat @ ( times_times_nat @ A_46 @ C_17 ) @ ( times_times_nat @ B_29 @ D_6 ) ) ) ) ) ) ).
thf(fact_570_mult__strict__mono,axiom,
! [C_17: int,D_6: int,A_46: int,B_29: int] :
( ( ord_less_int @ A_46 @ B_29 )
=> ( ( ord_less_int @ C_17 @ D_6 )
=> ( ( ord_less_int @ zero_zero_int @ B_29 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_17 )
=> ( ord_less_int @ ( times_times_int @ A_46 @ C_17 ) @ ( times_times_int @ B_29 @ D_6 ) ) ) ) ) ) ).
thf(fact_571_mult__le__cancel__left__neg,axiom,
! [A_45: real,B_28: real,C_16: real] :
( ( ord_less_real @ C_16 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_16 @ A_45 ) @ ( times_times_real @ C_16 @ B_28 ) )
<=> ( ord_less_eq_real @ B_28 @ A_45 ) ) ) ).
thf(fact_572_mult__le__cancel__left__neg,axiom,
! [A_45: int,B_28: int,C_16: int] :
( ( ord_less_int @ C_16 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_16 @ A_45 ) @ ( times_times_int @ C_16 @ B_28 ) )
<=> ( ord_less_eq_int @ B_28 @ A_45 ) ) ) ).
thf(fact_573_mult__le__cancel__left__pos,axiom,
! [A_44: real,B_27: real,C_15: real] :
( ( ord_less_real @ zero_zero_real @ C_15 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_15 @ A_44 ) @ ( times_times_real @ C_15 @ B_27 ) )
<=> ( ord_less_eq_real @ A_44 @ B_27 ) ) ) ).
thf(fact_574_mult__le__cancel__left__pos,axiom,
! [A_44: int,B_27: int,C_15: int] :
( ( ord_less_int @ zero_zero_int @ C_15 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_15 @ A_44 ) @ ( times_times_int @ C_15 @ B_27 ) )
<=> ( ord_less_eq_int @ A_44 @ B_27 ) ) ) ).
thf(fact_575_sum__squares__ge__zero,axiom,
! [X_14: real,Y_13: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_14 @ X_14 ) @ ( times_times_real @ Y_13 @ Y_13 ) ) ) ).
thf(fact_576_sum__squares__ge__zero,axiom,
! [X_14: int,Y_13: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_14 @ X_14 ) @ ( times_times_int @ Y_13 @ Y_13 ) ) ) ).
thf(fact_577_sum__squares__le__zero__iff,axiom,
! [X_13: real,Y_12: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_13 @ X_13 ) @ ( times_times_real @ Y_12 @ Y_12 ) ) @ zero_zero_real )
<=> ( ( X_13 = zero_zero_real )
& ( Y_12 = zero_zero_real ) ) ) ).
thf(fact_578_sum__squares__le__zero__iff,axiom,
! [X_13: int,Y_12: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_13 @ X_13 ) @ ( times_times_int @ Y_12 @ Y_12 ) ) @ zero_zero_int )
<=> ( ( X_13 = zero_zero_int )
& ( Y_12 = zero_zero_int ) ) ) ).
thf(fact_579_mult__right__le__one__le,axiom,
! [Y_11: real,X_12: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_12 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_11 )
=> ( ( ord_less_eq_real @ Y_11 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X_12 @ Y_11 ) @ X_12 ) ) ) ) ).
thf(fact_580_mult__right__le__one__le,axiom,
! [Y_11: int,X_12: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_12 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_11 )
=> ( ( ord_less_eq_int @ Y_11 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X_12 @ Y_11 ) @ X_12 ) ) ) ) ).
thf(fact_581_mult__left__le__one__le,axiom,
! [Y_10: real,X_11: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_11 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_10 )
=> ( ( ord_less_eq_real @ Y_10 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y_10 @ X_11 ) @ X_11 ) ) ) ) ).
thf(fact_582_mult__left__le__one__le,axiom,
! [Y_10: int,X_11: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_11 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_10 )
=> ( ( ord_less_eq_int @ Y_10 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y_10 @ X_11 ) @ X_11 ) ) ) ) ).
thf(fact_583_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat_1 @ Z )
= zero_zero_nat ) ) ).
thf(fact_584_nat__0__iff,axiom,
! [I_1: int] :
( ( ( nat_1 @ I_1 )
= zero_zero_nat )
<=> ( ord_less_eq_int @ I_1 @ zero_zero_int ) ) ).
thf(fact_585_power__increasing,axiom,
! [A_43: real,N_9: nat,N_8: nat] :
( ( ord_less_eq_nat @ N_9 @ N_8 )
=> ( ( ord_less_eq_real @ one_one_real @ A_43 )
=> ( ord_less_eq_real @ ( power_power_real @ A_43 @ N_9 ) @ ( power_power_real @ A_43 @ N_8 ) ) ) ) ).
thf(fact_586_power__increasing,axiom,
! [A_43: nat,N_9: nat,N_8: nat] :
( ( ord_less_eq_nat @ N_9 @ N_8 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A_43 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_43 @ N_9 ) @ ( power_power_nat @ A_43 @ N_8 ) ) ) ) ).
thf(fact_587_power__increasing,axiom,
! [A_43: int,N_9: nat,N_8: nat] :
( ( ord_less_eq_nat @ N_9 @ N_8 )
=> ( ( ord_less_eq_int @ one_one_int @ A_43 )
=> ( ord_less_eq_int @ ( power_power_int @ A_43 @ N_9 ) @ ( power_power_int @ A_43 @ N_8 ) ) ) ) ).
thf(fact_588_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z ) )
= zero_zero_int ) ) ) ).
thf(fact_589_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1621563631at_int @ M )
= Z )
<=> ( ( M
= ( nat_1 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
thf(fact_590_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z ) )
= Z ) ) ).
thf(fact_591_right__minus__eq,axiom,
! [A_42: real,B_26: real] :
( ( ( minus_minus_real @ A_42 @ B_26 )
= zero_zero_real )
<=> ( A_42 = B_26 ) ) ).
thf(fact_592_right__minus__eq,axiom,
! [A_42: int,B_26: int] :
( ( ( minus_minus_int @ A_42 @ B_26 )
= zero_zero_int )
<=> ( A_42 = B_26 ) ) ).
thf(fact_593_eq__iff__diff__eq__0,axiom,
! [A_41: real,B_25: real] :
( ( A_41 = B_25 )
<=> ( ( minus_minus_real @ A_41 @ B_25 )
= zero_zero_real ) ) ).
thf(fact_594_eq__iff__diff__eq__0,axiom,
! [A_41: int,B_25: int] :
( ( A_41 = B_25 )
<=> ( ( minus_minus_int @ A_41 @ B_25 )
= zero_zero_int ) ) ).
thf(fact_595_diff__self,axiom,
! [A_40: real] :
( ( minus_minus_real @ A_40 @ A_40 )
= zero_zero_real ) ).
thf(fact_596_diff__self,axiom,
! [A_40: int] :
( ( minus_minus_int @ A_40 @ A_40 )
= zero_zero_int ) ).
thf(fact_597_diff__0__right,axiom,
! [A_39: real] :
( ( minus_minus_real @ A_39 @ zero_zero_real )
= A_39 ) ).
thf(fact_598_diff__0__right,axiom,
! [A_39: int] :
( ( minus_minus_int @ A_39 @ zero_zero_int )
= A_39 ) ).
thf(fact_599_diff__eq__diff__less,axiom,
! [A_38: int,B_24: int,C_14: int,D_5: int] :
( ( ( minus_minus_int @ A_38 @ B_24 )
= ( minus_minus_int @ C_14 @ D_5 ) )
=> ( ( ord_less_int @ A_38 @ B_24 )
<=> ( ord_less_int @ C_14 @ D_5 ) ) ) ).
thf(fact_600_diff__eq__diff__less,axiom,
! [A_38: real,B_24: real,C_14: real,D_5: real] :
( ( ( minus_minus_real @ A_38 @ B_24 )
= ( minus_minus_real @ C_14 @ D_5 ) )
=> ( ( ord_less_real @ A_38 @ B_24 )
<=> ( ord_less_real @ C_14 @ D_5 ) ) ) ).
thf(fact_601_add__diff__add,axiom,
! [A_37: real,C_13: real,B_23: real,D_4: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_37 @ C_13 ) @ ( plus_plus_real @ B_23 @ D_4 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_37 @ B_23 ) @ ( minus_minus_real @ C_13 @ D_4 ) ) ) ).
thf(fact_602_add__diff__add,axiom,
! [A_37: int,C_13: int,B_23: int,D_4: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_37 @ C_13 ) @ ( plus_plus_int @ B_23 @ D_4 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_37 @ B_23 ) @ ( minus_minus_int @ C_13 @ D_4 ) ) ) ).
thf(fact_603_add__diff__cancel,axiom,
! [A_36: real,B_22: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_36 @ B_22 ) @ B_22 )
= A_36 ) ).
thf(fact_604_add__diff__cancel,axiom,
! [A_36: int,B_22: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_36 @ B_22 ) @ B_22 )
= A_36 ) ).
thf(fact_605_diff__add__cancel,axiom,
! [A_35: real,B_21: real] :
( ( plus_plus_real @ ( minus_minus_real @ A_35 @ B_21 ) @ B_21 )
= A_35 ) ).
thf(fact_606_diff__add__cancel,axiom,
! [A_35: int,B_21: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_35 @ B_21 ) @ B_21 )
= A_35 ) ).
thf(fact_607_mult__zero__left,axiom,
! [A_34: real] :
( ( times_times_real @ zero_zero_real @ A_34 )
= zero_zero_real ) ).
thf(fact_608_mult__zero__left,axiom,
! [A_34: nat] :
( ( times_times_nat @ zero_zero_nat @ A_34 )
= zero_zero_nat ) ).
thf(fact_609_mult__zero__left,axiom,
! [A_34: int] :
( ( times_times_int @ zero_zero_int @ A_34 )
= zero_zero_int ) ).
thf(fact_610_mult__zero__right,axiom,
! [A_33: real] :
( ( times_times_real @ A_33 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_611_mult__zero__right,axiom,
! [A_33: nat] :
( ( times_times_nat @ A_33 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_612_mult__zero__right,axiom,
! [A_33: int] :
( ( times_times_int @ A_33 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_613_mult__eq__0__iff,axiom,
! [A_32: real,B_20: real] :
( ( ( times_times_real @ A_32 @ B_20 )
= zero_zero_real )
<=> ( ( A_32 = zero_zero_real )
| ( B_20 = zero_zero_real ) ) ) ).
thf(fact_614_mult__eq__0__iff,axiom,
! [A_32: int,B_20: int] :
( ( ( times_times_int @ A_32 @ B_20 )
= zero_zero_int )
<=> ( ( A_32 = zero_zero_int )
| ( B_20 = zero_zero_int ) ) ) ).
thf(fact_615_no__zero__divisors,axiom,
! [B_19: real,A_31: real] :
( ( A_31 != zero_zero_real )
=> ( ( B_19 != zero_zero_real )
=> ( ( times_times_real @ A_31 @ B_19 )
!= zero_zero_real ) ) ) ).
thf(fact_616_no__zero__divisors,axiom,
! [B_19: nat,A_31: nat] :
( ( A_31 != zero_zero_nat )
=> ( ( B_19 != zero_zero_nat )
=> ( ( times_times_nat @ A_31 @ B_19 )
!= zero_zero_nat ) ) ) ).
thf(fact_617_no__zero__divisors,axiom,
! [B_19: int,A_31: int] :
( ( A_31 != zero_zero_int )
=> ( ( B_19 != zero_zero_int )
=> ( ( times_times_int @ A_31 @ B_19 )
!= zero_zero_int ) ) ) ).
thf(fact_618_divisors__zero,axiom,
! [A_30: real,B_18: real] :
( ( ( times_times_real @ A_30 @ B_18 )
= zero_zero_real )
=> ( ( A_30 = zero_zero_real )
| ( B_18 = zero_zero_real ) ) ) ).
thf(fact_619_divisors__zero,axiom,
! [A_30: nat,B_18: nat] :
( ( ( times_times_nat @ A_30 @ B_18 )
= zero_zero_nat )
=> ( ( A_30 = zero_zero_nat )
| ( B_18 = zero_zero_nat ) ) ) ).
thf(fact_620_divisors__zero,axiom,
! [A_30: int,B_18: int] :
( ( ( times_times_int @ A_30 @ B_18 )
= zero_zero_int )
=> ( ( A_30 = zero_zero_int )
| ( B_18 = zero_zero_int ) ) ) ).
thf(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_29: real] :
( ( times_times_real @ A_29 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_29: nat] :
( ( times_times_nat @ A_29 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_623_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_29: int] :
( ( times_times_int @ A_29 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_624_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_28: real] :
( ( times_times_real @ zero_zero_real @ A_28 )
= zero_zero_real ) ).
thf(fact_625_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_28: nat] :
( ( times_times_nat @ zero_zero_nat @ A_28 )
= zero_zero_nat ) ).
thf(fact_626_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_28: int] :
( ( times_times_int @ zero_zero_int @ A_28 )
= zero_zero_int ) ).
thf(fact_627_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
thf(fact_628_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_629_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_630_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_631_comm__semiring__class_Odistrib,axiom,
! [A_27: real,B_17: real,C_12: real] :
( ( times_times_real @ ( plus_plus_real @ A_27 @ B_17 ) @ C_12 )
= ( plus_plus_real @ ( times_times_real @ A_27 @ C_12 ) @ ( times_times_real @ B_17 @ C_12 ) ) ) ).
thf(fact_632_comm__semiring__class_Odistrib,axiom,
! [A_27: nat,B_17: nat,C_12: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_27 @ B_17 ) @ C_12 )
= ( plus_plus_nat @ ( times_times_nat @ A_27 @ C_12 ) @ ( times_times_nat @ B_17 @ C_12 ) ) ) ).
thf(fact_633_comm__semiring__class_Odistrib,axiom,
! [A_27: int,B_17: int,C_12: int] :
( ( times_times_int @ ( plus_plus_int @ A_27 @ B_17 ) @ C_12 )
= ( plus_plus_int @ ( times_times_int @ A_27 @ C_12 ) @ ( times_times_int @ B_17 @ C_12 ) ) ) ).
thf(fact_634_combine__common__factor,axiom,
! [A_26: real,E: real,B_16: real,C_11: real] :
( ( plus_plus_real @ ( times_times_real @ A_26 @ E ) @ ( plus_plus_real @ ( times_times_real @ B_16 @ E ) @ C_11 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_26 @ B_16 ) @ E ) @ C_11 ) ) ).
thf(fact_635_combine__common__factor,axiom,
! [A_26: nat,E: nat,B_16: nat,C_11: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_26 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B_16 @ E ) @ C_11 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_26 @ B_16 ) @ E ) @ C_11 ) ) ).
thf(fact_636_combine__common__factor,axiom,
! [A_26: int,E: int,B_16: int,C_11: int] :
( ( plus_plus_int @ ( times_times_int @ A_26 @ E ) @ ( plus_plus_int @ ( times_times_int @ B_16 @ E ) @ C_11 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_26 @ B_16 ) @ E ) @ C_11 ) ) ).
thf(fact_637_crossproduct__eq,axiom,
! [W_3: real,Y_9: real,X_10: real,Z_3: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_3 @ Y_9 ) @ ( times_times_real @ X_10 @ Z_3 ) )
= ( plus_plus_real @ ( times_times_real @ W_3 @ Z_3 ) @ ( times_times_real @ X_10 @ Y_9 ) ) )
<=> ( ( W_3 = X_10 )
| ( Y_9 = Z_3 ) ) ) ).
thf(fact_638_crossproduct__eq,axiom,
! [W_3: nat,Y_9: nat,X_10: nat,Z_3: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_3 @ Y_9 ) @ ( times_times_nat @ X_10 @ Z_3 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_3 @ Z_3 ) @ ( times_times_nat @ X_10 @ Y_9 ) ) )
<=> ( ( W_3 = X_10 )
| ( Y_9 = Z_3 ) ) ) ).
thf(fact_639_crossproduct__eq,axiom,
! [W_3: int,Y_9: int,X_10: int,Z_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_3 @ Y_9 ) @ ( times_times_int @ X_10 @ Z_3 ) )
= ( plus_plus_int @ ( times_times_int @ W_3 @ Z_3 ) @ ( times_times_int @ X_10 @ Y_9 ) ) )
<=> ( ( W_3 = X_10 )
| ( Y_9 = Z_3 ) ) ) ).
thf(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_25: real,M_3: real,B_15: real] :
( ( plus_plus_real @ ( times_times_real @ A_25 @ M_3 ) @ ( times_times_real @ B_15 @ M_3 ) )
= ( times_times_real @ ( plus_plus_real @ A_25 @ B_15 ) @ M_3 ) ) ).
thf(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_25: nat,M_3: nat,B_15: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_25 @ M_3 ) @ ( times_times_nat @ B_15 @ M_3 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_25 @ B_15 ) @ M_3 ) ) ).
thf(fact_642_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_25: int,M_3: int,B_15: int] :
( ( plus_plus_int @ ( times_times_int @ A_25 @ M_3 ) @ ( times_times_int @ B_15 @ M_3 ) )
= ( times_times_int @ ( plus_plus_int @ A_25 @ B_15 ) @ M_3 ) ) ).
thf(fact_643_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_24: real,B_14: real,C_10: real] :
( ( times_times_real @ ( plus_plus_real @ A_24 @ B_14 ) @ C_10 )
= ( plus_plus_real @ ( times_times_real @ A_24 @ C_10 ) @ ( times_times_real @ B_14 @ C_10 ) ) ) ).
thf(fact_644_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_24: nat,B_14: nat,C_10: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_24 @ B_14 ) @ C_10 )
= ( plus_plus_nat @ ( times_times_nat @ A_24 @ C_10 ) @ ( times_times_nat @ B_14 @ C_10 ) ) ) ).
thf(fact_645_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_24: int,B_14: int,C_10: int] :
( ( times_times_int @ ( plus_plus_int @ A_24 @ B_14 ) @ C_10 )
= ( plus_plus_int @ ( times_times_int @ A_24 @ C_10 ) @ ( times_times_int @ B_14 @ C_10 ) ) ) ).
thf(fact_646_crossproduct__noteq,axiom,
! [C_9: real,D_3: real,A_23: real,B_13: real] :
( ( ( A_23 != B_13 )
& ( C_9 != D_3 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_23 @ C_9 ) @ ( times_times_real @ B_13 @ D_3 ) )
!= ( plus_plus_real @ ( times_times_real @ A_23 @ D_3 ) @ ( times_times_real @ B_13 @ C_9 ) ) ) ) ).
thf(fact_647_crossproduct__noteq,axiom,
! [C_9: nat,D_3: nat,A_23: nat,B_13: nat] :
( ( ( A_23 != B_13 )
& ( C_9 != D_3 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_23 @ C_9 ) @ ( times_times_nat @ B_13 @ D_3 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_23 @ D_3 ) @ ( times_times_nat @ B_13 @ C_9 ) ) ) ) ).
thf(fact_648_crossproduct__noteq,axiom,
! [C_9: int,D_3: int,A_23: int,B_13: int] :
( ( ( A_23 != B_13 )
& ( C_9 != D_3 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_23 @ C_9 ) @ ( times_times_int @ B_13 @ D_3 ) )
!= ( plus_plus_int @ ( times_times_int @ A_23 @ D_3 ) @ ( times_times_int @ B_13 @ C_9 ) ) ) ) ).
thf(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_9: real,Y_8: real,Z_2: real] :
( ( times_times_real @ X_9 @ ( plus_plus_real @ Y_8 @ Z_2 ) )
= ( plus_plus_real @ ( times_times_real @ X_9 @ Y_8 ) @ ( times_times_real @ X_9 @ Z_2 ) ) ) ).
thf(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_9: nat,Y_8: nat,Z_2: nat] :
( ( times_times_nat @ X_9 @ ( plus_plus_nat @ Y_8 @ Z_2 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_9 @ Y_8 ) @ ( times_times_nat @ X_9 @ Z_2 ) ) ) ).
thf(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_9: int,Y_8: int,Z_2: int] :
( ( times_times_int @ X_9 @ ( plus_plus_int @ Y_8 @ Z_2 ) )
= ( plus_plus_int @ ( times_times_int @ X_9 @ Y_8 ) @ ( times_times_int @ X_9 @ Z_2 ) ) ) ).
thf(fact_652_nat__int,axiom,
! [N: nat] :
( ( nat_1 @ ( semiri1621563631at_int @ N ) )
= N ) ).
thf(fact_653_mult_Ocomm__neutral,axiom,
! [A_22: real] :
( ( times_times_real @ A_22 @ one_one_real )
= A_22 ) ).
thf(fact_654_mult_Ocomm__neutral,axiom,
! [A_22: nat] :
( ( times_times_nat @ A_22 @ one_one_nat )
= A_22 ) ).
thf(fact_655_mult_Ocomm__neutral,axiom,
! [A_22: int] :
( ( times_times_int @ A_22 @ one_one_int )
= A_22 ) ).
thf(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_21: real] :
( ( times_times_real @ A_21 @ one_one_real )
= A_21 ) ).
thf(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_21: nat] :
( ( times_times_nat @ A_21 @ one_one_nat )
= A_21 ) ).
thf(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_21: int] :
( ( times_times_int @ A_21 @ one_one_int )
= A_21 ) ).
thf(fact_659_mult__1__right,axiom,
! [A_20: real] :
( ( times_times_real @ A_20 @ one_one_real )
= A_20 ) ).
thf(fact_660_mult__1__right,axiom,
! [A_20: nat] :
( ( times_times_nat @ A_20 @ one_one_nat )
= A_20 ) ).
thf(fact_661_mult__1__right,axiom,
! [A_20: int] :
( ( times_times_int @ A_20 @ one_one_int )
= A_20 ) ).
thf(fact_662_mult__1,axiom,
! [A_19: real] :
( ( times_times_real @ one_one_real @ A_19 )
= A_19 ) ).
thf(fact_663_mult__1,axiom,
! [A_19: nat] :
( ( times_times_nat @ one_one_nat @ A_19 )
= A_19 ) ).
thf(fact_664_mult__1,axiom,
! [A_19: int] :
( ( times_times_int @ one_one_int @ A_19 )
= A_19 ) ).
thf(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_18: real] :
( ( times_times_real @ one_one_real @ A_18 )
= A_18 ) ).
thf(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_18: nat] :
( ( times_times_nat @ one_one_nat @ A_18 )
= A_18 ) ).
thf(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_18: int] :
( ( times_times_int @ one_one_int @ A_18 )
= A_18 ) ).
thf(fact_668_mult__1__left,axiom,
! [A_17: real] :
( ( times_times_real @ one_one_real @ A_17 )
= A_17 ) ).
thf(fact_669_mult__1__left,axiom,
! [A_17: nat] :
( ( times_times_nat @ one_one_nat @ A_17 )
= A_17 ) ).
thf(fact_670_mult__1__left,axiom,
! [A_17: int] :
( ( times_times_int @ one_one_int @ A_17 )
= A_17 ) ).
thf(fact_671_add__le__imp__le__left,axiom,
! [C_8: real,A_16: real,B_12: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_8 @ A_16 ) @ ( plus_plus_real @ C_8 @ B_12 ) )
=> ( ord_less_eq_real @ A_16 @ B_12 ) ) ).
thf(fact_672_add__le__imp__le__left,axiom,
! [C_8: nat,A_16: nat,B_12: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_8 @ A_16 ) @ ( plus_plus_nat @ C_8 @ B_12 ) )
=> ( ord_less_eq_nat @ A_16 @ B_12 ) ) ).
thf(fact_673_add__le__imp__le__left,axiom,
! [C_8: int,A_16: int,B_12: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_8 @ A_16 ) @ ( plus_plus_int @ C_8 @ B_12 ) )
=> ( ord_less_eq_int @ A_16 @ B_12 ) ) ).
thf(fact_674_add__le__imp__le__right,axiom,
! [A_15: real,C_7: real,B_11: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_15 @ C_7 ) @ ( plus_plus_real @ B_11 @ C_7 ) )
=> ( ord_less_eq_real @ A_15 @ B_11 ) ) ).
thf(fact_675_add__le__imp__le__right,axiom,
! [A_15: nat,C_7: nat,B_11: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_15 @ C_7 ) @ ( plus_plus_nat @ B_11 @ C_7 ) )
=> ( ord_less_eq_nat @ A_15 @ B_11 ) ) ).
thf(fact_676_add__le__imp__le__right,axiom,
! [A_15: int,C_7: int,B_11: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_15 @ C_7 ) @ ( plus_plus_int @ B_11 @ C_7 ) )
=> ( ord_less_eq_int @ A_15 @ B_11 ) ) ).
thf(fact_677_add__mono,axiom,
! [C_6: real,D_2: real,A_14: real,B_10: real] :
( ( ord_less_eq_real @ A_14 @ B_10 )
=> ( ( ord_less_eq_real @ C_6 @ D_2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_14 @ C_6 ) @ ( plus_plus_real @ B_10 @ D_2 ) ) ) ) ).
thf(fact_678_add__mono,axiom,
! [C_6: nat,D_2: nat,A_14: nat,B_10: nat] :
( ( ord_less_eq_nat @ A_14 @ B_10 )
=> ( ( ord_less_eq_nat @ C_6 @ D_2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_14 @ C_6 ) @ ( plus_plus_nat @ B_10 @ D_2 ) ) ) ) ).
thf(fact_679_add__mono,axiom,
! [C_6: int,D_2: int,A_14: int,B_10: int] :
( ( ord_less_eq_int @ A_14 @ B_10 )
=> ( ( ord_less_eq_int @ C_6 @ D_2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_14 @ C_6 ) @ ( plus_plus_int @ B_10 @ D_2 ) ) ) ) ).
thf(fact_680_add__left__mono,axiom,
! [C_5: real,A_13: real,B_9: real] :
( ( ord_less_eq_real @ A_13 @ B_9 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C_5 @ A_13 ) @ ( plus_plus_real @ C_5 @ B_9 ) ) ) ).
thf(fact_681_add__left__mono,axiom,
! [C_5: nat,A_13: nat,B_9: nat] :
( ( ord_less_eq_nat @ A_13 @ B_9 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_5 @ A_13 ) @ ( plus_plus_nat @ C_5 @ B_9 ) ) ) ).
thf(fact_682_add__left__mono,axiom,
! [C_5: int,A_13: int,B_9: int] :
( ( ord_less_eq_int @ A_13 @ B_9 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_5 @ A_13 ) @ ( plus_plus_int @ C_5 @ B_9 ) ) ) ).
thf(fact_683_add__right__mono,axiom,
! [C_4: real,A_12: real,B_8: real] :
( ( ord_less_eq_real @ A_12 @ B_8 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_12 @ C_4 ) @ ( plus_plus_real @ B_8 @ C_4 ) ) ) ).
thf(fact_684_add__right__mono,axiom,
! [C_4: nat,A_12: nat,B_8: nat] :
( ( ord_less_eq_nat @ A_12 @ B_8 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_12 @ C_4 ) @ ( plus_plus_nat @ B_8 @ C_4 ) ) ) ).
thf(fact_685_add__right__mono,axiom,
! [C_4: int,A_12: int,B_8: int] :
( ( ord_less_eq_int @ A_12 @ B_8 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_12 @ C_4 ) @ ( plus_plus_int @ B_8 @ C_4 ) ) ) ).
thf(fact_686_add__le__cancel__left,axiom,
! [C_3: real,A_11: real,B_7: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_3 @ A_11 ) @ ( plus_plus_real @ C_3 @ B_7 ) )
<=> ( ord_less_eq_real @ A_11 @ B_7 ) ) ).
thf(fact_687_add__le__cancel__left,axiom,
! [C_3: nat,A_11: nat,B_7: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_3 @ A_11 ) @ ( plus_plus_nat @ C_3 @ B_7 ) )
<=> ( ord_less_eq_nat @ A_11 @ B_7 ) ) ).
thf(fact_688_add__le__cancel__left,axiom,
! [C_3: int,A_11: int,B_7: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_3 @ A_11 ) @ ( plus_plus_int @ C_3 @ B_7 ) )
<=> ( ord_less_eq_int @ A_11 @ B_7 ) ) ).
thf(fact_689_add__le__cancel__right,axiom,
! [A_10: real,C_2: real,B_6: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_10 @ C_2 ) @ ( plus_plus_real @ B_6 @ C_2 ) )
<=> ( ord_less_eq_real @ A_10 @ B_6 ) ) ).
thf(fact_690_add__le__cancel__right,axiom,
! [A_10: nat,C_2: nat,B_6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_10 @ C_2 ) @ ( plus_plus_nat @ B_6 @ C_2 ) )
<=> ( ord_less_eq_nat @ A_10 @ B_6 ) ) ).
thf(fact_691_add__le__cancel__right,axiom,
! [A_10: int,C_2: int,B_6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_10 @ C_2 ) @ ( plus_plus_int @ B_6 @ C_2 ) )
<=> ( ord_less_eq_int @ A_10 @ B_6 ) ) ).
thf(fact_692_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
thf(fact_693_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_694_power__commutes,axiom,
! [A_9: real,N_7: nat] :
( ( times_times_real @ ( power_power_real @ A_9 @ N_7 ) @ A_9 )
= ( times_times_real @ A_9 @ ( power_power_real @ A_9 @ N_7 ) ) ) ).
thf(fact_695_power__commutes,axiom,
! [A_9: nat,N_7: nat] :
( ( times_times_nat @ ( power_power_nat @ A_9 @ N_7 ) @ A_9 )
= ( times_times_nat @ A_9 @ ( power_power_nat @ A_9 @ N_7 ) ) ) ).
thf(fact_696_power__commutes,axiom,
! [A_9: int,N_7: nat] :
( ( times_times_int @ ( power_power_int @ A_9 @ N_7 ) @ A_9 )
= ( times_times_int @ A_9 @ ( power_power_int @ A_9 @ N_7 ) ) ) ).
thf(fact_697_power__mult__distrib,axiom,
! [A_8: real,B_5: real,N_6: nat] :
( ( power_power_real @ ( times_times_real @ A_8 @ B_5 ) @ N_6 )
= ( times_times_real @ ( power_power_real @ A_8 @ N_6 ) @ ( power_power_real @ B_5 @ N_6 ) ) ) ).
thf(fact_698_power__mult__distrib,axiom,
! [A_8: nat,B_5: nat,N_6: nat] :
( ( power_power_nat @ ( times_times_nat @ A_8 @ B_5 ) @ N_6 )
= ( times_times_nat @ ( power_power_nat @ A_8 @ N_6 ) @ ( power_power_nat @ B_5 @ N_6 ) ) ) ).
thf(fact_699_power__mult__distrib,axiom,
! [A_8: int,B_5: int,N_6: nat] :
( ( power_power_int @ ( times_times_int @ A_8 @ B_5 ) @ N_6 )
= ( times_times_int @ ( power_power_int @ A_8 @ N_6 ) @ ( power_power_int @ B_5 @ N_6 ) ) ) ).
thf(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_8: real,Y_7: real,Q_2: nat] :
( ( power_power_real @ ( times_times_real @ X_8 @ Y_7 ) @ Q_2 )
= ( times_times_real @ ( power_power_real @ X_8 @ Q_2 ) @ ( power_power_real @ Y_7 @ Q_2 ) ) ) ).
thf(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_8: nat,Y_7: nat,Q_2: nat] :
( ( power_power_nat @ ( times_times_nat @ X_8 @ Y_7 ) @ Q_2 )
= ( times_times_nat @ ( power_power_nat @ X_8 @ Q_2 ) @ ( power_power_nat @ Y_7 @ Q_2 ) ) ) ).
thf(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_8: int,Y_7: int,Q_2: nat] :
( ( power_power_int @ ( times_times_int @ X_8 @ Y_7 ) @ Q_2 )
= ( times_times_int @ ( power_power_int @ X_8 @ Q_2 ) @ ( power_power_int @ Y_7 @ Q_2 ) ) ) ).
thf(fact_703_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_704_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_705_diff__diff__left,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
= ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_706_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_707_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_708_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_709_mult__Bit0,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit0 @ K ) @ L )
= ( bit0 @ ( times_times_int @ K @ L ) ) ) ).
thf(fact_710_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_711_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_712_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_713_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_714_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_715_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_716_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_717_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_718_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_719_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_720_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_721_zadd__left__mono,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_eq_int @ I_1 @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I_1 ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_722_times__numeral__code_I5_J,axiom,
! [V_1: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V_1 ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V_1 @ W ) ) ) ).
thf(fact_723_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_724_convex__bound__le,axiom,
! [V_4: real,U_2: real,Y_6: real,X_7: real,A_7: real] :
( ( ord_less_eq_real @ X_7 @ A_7 )
=> ( ( ord_less_eq_real @ Y_6 @ A_7 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_4 )
=> ( ( ( plus_plus_real @ U_2 @ V_4 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U_2 @ X_7 ) @ ( times_times_real @ V_4 @ Y_6 ) ) @ A_7 ) ) ) ) ) ) ).
thf(fact_725_convex__bound__le,axiom,
! [V_4: int,U_2: int,Y_6: int,X_7: int,A_7: int] :
( ( ord_less_eq_int @ X_7 @ A_7 )
=> ( ( ord_less_eq_int @ Y_6 @ A_7 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_4 )
=> ( ( ( plus_plus_int @ U_2 @ V_4 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U_2 @ X_7 ) @ ( times_times_int @ V_4 @ Y_6 ) ) @ A_7 ) ) ) ) ) ) ).
thf(fact_726_transfer__nat__int__relations_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_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
<=> ( ord_less_int @ X @ Y ) ) ) ) ).
thf(fact_727_nat__less__eq__zless,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z ) )
<=> ( ord_less_int @ W @ Z ) ) ) ).
thf(fact_728_le__special_I1_J,axiom,
! [Y_5: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( number267125858f_real @ Y_5 ) )
<=> ( ord_less_eq_int @ pls @ Y_5 ) ) ).
thf(fact_729_le__special_I1_J,axiom,
! [Y_5: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_5 ) )
<=> ( ord_less_eq_int @ pls @ Y_5 ) ) ).
thf(fact_730_le__special_I3_J,axiom,
! [X_6: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_6 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ X_6 @ pls ) ) ).
thf(fact_731_le__special_I3_J,axiom,
! [X_6: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_6 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_6 @ pls ) ) ).
thf(fact_732_power__decreasing,axiom,
! [A_6: real,N_5: nat,N_4: nat] :
( ( ord_less_eq_nat @ N_5 @ N_4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_6 )
=> ( ( ord_less_eq_real @ A_6 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A_6 @ N_4 ) @ ( power_power_real @ A_6 @ N_5 ) ) ) ) ) ).
thf(fact_733_power__decreasing,axiom,
! [A_6: nat,N_5: nat,N_4: nat] :
( ( ord_less_eq_nat @ N_5 @ N_4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_6 )
=> ( ( ord_less_eq_nat @ A_6 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_6 @ N_4 ) @ ( power_power_nat @ A_6 @ N_5 ) ) ) ) ) ).
thf(fact_734_power__decreasing,axiom,
! [A_6: int,N_5: nat,N_4: nat] :
( ( ord_less_eq_nat @ N_5 @ N_4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_6 )
=> ( ( ord_less_eq_int @ A_6 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A_6 @ N_4 ) @ ( power_power_int @ A_6 @ N_5 ) ) ) ) ) ).
thf(fact_735_power__increasing__iff,axiom,
! [X_5: nat,Y_4: nat,B_4: real] :
( ( ord_less_real @ one_one_real @ B_4 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B_4 @ X_5 ) @ ( power_power_real @ B_4 @ Y_4 ) )
<=> ( ord_less_eq_nat @ X_5 @ Y_4 ) ) ) ).
thf(fact_736_power__increasing__iff,axiom,
! [X_5: nat,Y_4: nat,B_4: nat] :
( ( ord_less_nat @ one_one_nat @ B_4 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B_4 @ X_5 ) @ ( power_power_nat @ B_4 @ Y_4 ) )
<=> ( ord_less_eq_nat @ X_5 @ Y_4 ) ) ) ).
thf(fact_737_power__increasing__iff,axiom,
! [X_5: nat,Y_4: nat,B_4: int] :
( ( ord_less_int @ one_one_int @ B_4 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B_4 @ X_5 ) @ ( power_power_int @ B_4 @ Y_4 ) )
<=> ( ord_less_eq_nat @ X_5 @ Y_4 ) ) ) ).
thf(fact_738_power__le__imp__le__exp,axiom,
! [M_2: nat,N_3: nat,A_5: real] :
( ( ord_less_real @ one_one_real @ A_5 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A_5 @ M_2 ) @ ( power_power_real @ A_5 @ N_3 ) )
=> ( ord_less_eq_nat @ M_2 @ N_3 ) ) ) ).
thf(fact_739_power__le__imp__le__exp,axiom,
! [M_2: nat,N_3: nat,A_5: nat] :
( ( ord_less_nat @ one_one_nat @ A_5 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A_5 @ M_2 ) @ ( power_power_nat @ A_5 @ N_3 ) )
=> ( ord_less_eq_nat @ M_2 @ N_3 ) ) ) ).
thf(fact_740_power__le__imp__le__exp,axiom,
! [M_2: nat,N_3: nat,A_5: int] :
( ( ord_less_int @ one_one_int @ A_5 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A_5 @ M_2 ) @ ( power_power_int @ A_5 @ N_3 ) )
=> ( ord_less_eq_nat @ M_2 @ N_3 ) ) ) ).
thf(fact_741_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat_1 @ W ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1621563631at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
thf(fact_742_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat_1 @ W )
= M )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1621563631at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
thf(fact_743_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( plus_plus_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( plus_plus_int @ X @ Y ) ) ) ) ) ).
thf(fact_744_nat__add__distrib,axiom,
! [Z_1: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( nat_1 @ ( plus_plus_int @ Z @ Z_1 ) )
= ( plus_plus_nat @ ( nat_1 @ Z ) @ ( nat_1 @ Z_1 ) ) ) ) ) ).
thf(fact_745_int__eq__iff__number__of,axiom,
! [M: nat,V_1: int] :
( ( ( semiri1621563631at_int @ M )
= ( number_number_of_int @ V_1 ) )
<=> ( ( M
= ( nat_1 @ ( number_number_of_int @ V_1 ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_1 ) ) ) ) ).
thf(fact_746_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N: nat,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_nat @ ( nat_1 @ X ) @ N )
= ( nat_1 @ ( power_power_int @ X @ N ) ) ) ) ).
thf(fact_747_nat__power__eq,axiom,
! [N: nat,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat_1 @ ( power_power_int @ Z @ N ) )
= ( power_power_nat @ ( nat_1 @ Z ) @ N ) ) ) ).
thf(fact_748_convex__bound__lt,axiom,
! [V_3: real,U_1: real,Y_3: real,X_4: real,A_4: real] :
( ( ord_less_real @ X_4 @ A_4 )
=> ( ( ord_less_real @ Y_3 @ A_4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_1 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_3 )
=> ( ( ( plus_plus_real @ U_1 @ V_3 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U_1 @ X_4 ) @ ( times_times_real @ V_3 @ Y_3 ) ) @ A_4 ) ) ) ) ) ) ).
thf(fact_749_convex__bound__lt,axiom,
! [V_3: int,U_1: int,Y_3: int,X_4: int,A_4: int] :
( ( ord_less_int @ X_4 @ A_4 )
=> ( ( ord_less_int @ Y_3 @ A_4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_3 )
=> ( ( ( plus_plus_int @ U_1 @ V_3 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U_1 @ X_4 ) @ ( times_times_int @ V_3 @ Y_3 ) ) @ A_4 ) ) ) ) ) ) ).
thf(fact_750_nat__less__iff,axiom,
! [M: nat,W: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat_1 @ W ) @ M )
<=> ( ord_less_int @ W @ ( semiri1621563631at_int @ M ) ) ) ) ).
thf(fact_751_le__special_I2_J,axiom,
! [Y_2: int] :
( ( ord_less_eq_real @ one_one_real @ ( number267125858f_real @ Y_2 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_2 ) ) ).
thf(fact_752_le__special_I2_J,axiom,
! [Y_2: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_2 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_2 ) ) ).
thf(fact_753_le__special_I4_J,axiom,
! [X_3: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_3 ) @ one_one_real )
<=> ( ord_less_eq_int @ X_3 @ ( bit1 @ pls ) ) ) ).
thf(fact_754_le__special_I4_J,axiom,
! [X_3: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_3 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_3 @ ( bit1 @ pls ) ) ) ).
thf(fact_755_realpow__minus__mult,axiom,
! [X_2: real,N_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_2 )
=> ( ( times_times_real @ ( power_power_real @ X_2 @ ( minus_minus_nat @ N_2 @ one_one_nat ) ) @ X_2 )
= ( power_power_real @ X_2 @ N_2 ) ) ) ).
thf(fact_756_realpow__minus__mult,axiom,
! [X_2: nat,N_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_2 )
=> ( ( times_times_nat @ ( power_power_nat @ X_2 @ ( minus_minus_nat @ N_2 @ one_one_nat ) ) @ X_2 )
= ( power_power_nat @ X_2 @ N_2 ) ) ) ).
thf(fact_757_realpow__minus__mult,axiom,
! [X_2: int,N_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_2 )
=> ( ( times_times_int @ ( power_power_int @ X_2 @ ( minus_minus_nat @ N_2 @ one_one_nat ) ) @ X_2 )
= ( power_power_int @ X_2 @ N_2 ) ) ) ).
thf(fact_758_less__iff__diff__less__0,axiom,
! [A_3: real,B_3: real] :
( ( ord_less_real @ A_3 @ B_3 )
<=> ( ord_less_real @ ( minus_minus_real @ A_3 @ B_3 ) @ zero_zero_real ) ) ).
thf(fact_759_less__iff__diff__less__0,axiom,
! [A_3: int,B_3: int] :
( ( ord_less_int @ A_3 @ B_3 )
<=> ( ord_less_int @ ( minus_minus_int @ A_3 @ B_3 ) @ zero_zero_int ) ) ).
thf(fact_760_transfer__nat__int__numerals_I1_J,axiom,
( zero_zero_nat
= ( nat_1 @ zero_zero_int ) ) ).
thf(fact_761_nat__0,axiom,
( ( nat_1 @ zero_zero_int )
= zero_zero_nat ) ).
thf(fact_762_not__square__less__zero,axiom,
! [A_2: real] :
~ ( ord_less_real @ ( times_times_real @ A_2 @ A_2 ) @ zero_zero_real ) ).
thf(fact_763_not__square__less__zero,axiom,
! [A_2: int] :
~ ( ord_less_int @ ( times_times_int @ A_2 @ A_2 ) @ zero_zero_int ) ).
thf(fact_764_mult__less__cancel__right__disj,axiom,
! [A_1: real,C_1: real,B_2: real] :
( ( ord_less_real @ ( times_times_real @ A_1 @ C_1 ) @ ( times_times_real @ B_2 @ C_1 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_1 )
& ( ord_less_real @ A_1 @ B_2 ) )
| ( ( ord_less_real @ C_1 @ zero_zero_real )
& ( ord_less_real @ B_2 @ A_1 ) ) ) ) ).
thf(fact_765_mult__less__cancel__right__disj,axiom,
! [A_1: int,C_1: int,B_2: int] :
( ( ord_less_int @ ( times_times_int @ A_1 @ C_1 ) @ ( times_times_int @ B_2 @ C_1 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_1 )
& ( ord_less_int @ A_1 @ B_2 ) )
| ( ( ord_less_int @ C_1 @ zero_zero_int )
& ( ord_less_int @ B_2 @ A_1 ) ) ) ) ).
thf(fact_766_nat__number__of,axiom,
! [W: int] :
( ( nat_1 @ ( number_number_of_int @ W ) )
= ( number_number_of_nat @ W ) ) ).
thf(fact_767_nat__number__of__def,axiom,
! [V_1: int] :
( ( number_number_of_nat @ V_1 )
= ( nat_1 @ ( number_number_of_int @ V_1 ) ) ) ).
thf(fact_768_transfer__nat__int__numerals_I2_J,axiom,
( one_one_nat
= ( nat_1 @ one_one_int ) ) ).
thf(fact_769_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_770_diff__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
thf(fact_771_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_772_less__diff__conv,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ).
thf(fact_773_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_774_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_775_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_776_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_777_rel__simps_I21_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_778_rel__simps_I27_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_eq_int @ K @ pls ) ) ).
thf(fact_779_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_780_zmult__zless__mono2,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I_1 ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_781_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_782_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_783_zadd__zless__mono,axiom,
! [Z_1: int,Z: int,W_2: int,W: int] :
( ( ord_less_int @ W_2 @ W )
=> ( ( ord_less_eq_int @ Z_1 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_2 @ Z_1 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_784_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z ) ) ).
thf(fact_785_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ X_1 ) )
<=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_786_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ X_1 ) )
<=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_787_zero__zle__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_788_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_789_zle__iff__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ Z )
<=> ? [N_1: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_790_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_791_nat__mono__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z ) )
<=> ( ord_less_int @ W @ Z ) ) ) ).
thf(fact_792_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z ) )
<=> ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
thf(fact_793_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat_1 @ Z ) )
<=> ( ord_less_int @ ( semiri1621563631at_int @ M ) @ Z ) ) ).
thf(fact_794_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D_1: nat] :
( ( A
= ( plus_plus_nat @ B @ D_1 ) )
& ~ ( P @ D_1 ) ) ) ) ).
thf(fact_795_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D_1: nat] :
( ( A
= ( plus_plus_nat @ B @ D_1 ) )
=> ( P @ D_1 ) ) ) ) ).
thf(fact_796_eq__0__number__of,axiom,
! [V_1: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V_1 ) )
<=> ( ord_less_eq_int @ V_1 @ pls ) ) ).
thf(fact_797_eq__number__of__0,axiom,
! [V_1: int] :
( ( ( number_number_of_nat @ V_1 )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V_1 @ pls ) ) ).
thf(fact_798_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_799_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_800_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_801_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_802_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_803_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_804_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_805_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_806_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_807_int__le__0__conv,axiom,
! [N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_808_succ__Pls,axiom,
( ( succ @ pls )
= ( bit1 @ pls ) ) ).
thf(fact_809_succ__Bit0,axiom,
! [K: int] :
( ( succ @ ( bit0 @ K ) )
= ( bit1 @ K ) ) ).
thf(fact_810_succ__Bit1,axiom,
! [K: int] :
( ( succ @ ( bit1 @ K ) )
= ( bit0 @ ( succ @ K ) ) ) ).
thf(fact_811_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_812_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_813_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_814_succ__def,axiom,
! [K: int] :
( ( succ @ K )
= ( plus_plus_int @ K @ one_one_int ) ) ).
thf(fact_815_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat_1 @ Z ) )
<=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
thf(fact_816_transfer__nat__int__numerals_I4_J,axiom,
( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
= ( nat_1 @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_817_split__nat,axiom,
! [P: nat > $o,I_1: int] :
( ( P @ ( nat_1 @ I_1 ) )
<=> ( ! [N_1: nat] :
( ( I_1
= ( semiri1621563631at_int @ N_1 ) )
=> ( P @ N_1 ) )
& ( ( ord_less_int @ I_1 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
thf(fact_818_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_819_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_820_zmult__zless__mono2__lemma,axiom,
! [K: nat,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I_1 ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
thf(fact_821_add__Bit1__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ ( succ @ L ) ) ) ) ).
thf(fact_822_transfer__nat__int__numerals_I3_J,axiom,
( ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) )
= ( nat_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_823_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_824_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_825_power2__ge__self,axiom,
! [X: int] : ( ord_less_eq_int @ X @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_826_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_827_Euler_Oaux__1,axiom,
! [A: int,P_1: int] :
( ( ord_less_int @ zero_zero_int @ P_1 )
=> ( ( power_power_int @ A @ ( nat_1 @ P_1 ) )
= ( times_times_int @ A @ ( power_power_int @ A @ ( minus_minus_nat @ ( nat_1 @ P_1 ) @ one_one_nat ) ) ) ) ) ).
thf(fact_828_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_829_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_int @ R_1 @ A )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_830_smaller_I2_J,axiom,
~ ( ( ord_less_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
=> ~ ( twoSqu919416604sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ) ).
thf(fact_831_q__pos__lemma,axiom,
! [B_1: int,Q_1: int,R_2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_1 ) ) ) ) ).
thf(fact_832_q__neg__lemma,axiom,
! [B_1: int,Q_1: int,R_2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ Q_1 @ zero_zero_int ) ) ) ) ).
thf(fact_833_nQ1,axiom,
~ ( twoSqu919416604sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ zero_zero_nat ) ) ) ) ).
thf(fact_834_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_835_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_836_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_837_qf1pt,axiom,
twoSqu919416604sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ).
thf(fact_838_IH,axiom,
( ( ord_less_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( twoSqu919416604sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ) ).
thf(fact_839__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A,axiom,
~ ! [X_1: int,Y_1: int] :
( ( plus_plus_int @ ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
!= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_840_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_841_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_842__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_,axiom,
( ( t = one_one_int )
=> ? [X_1: int,Y_1: int] :
( ( plus_plus_int @ ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_1 @ ( 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_843_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_844_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_845_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_846_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_847_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_848_nat__mult__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K )
= ( times_times_nat @ M @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_849_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_850_mult__le__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
thf(fact_851_mult__le__mono2,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ).
thf(fact_852_le__trans,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I_1 @ K ) ) ) ).
thf(fact_853_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_854_mult__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_855_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_856_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_857_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_858_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_859_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_860_zdiff__int,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_861_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_862_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_863_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_864_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_865_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_866_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_867_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_868_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_869_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_870_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_871_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_872_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_873_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_874_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_875_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_876_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_877_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_878_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_879_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_880_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_881_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_882_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_883_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_884_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_885_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
thf(fact_886_add__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_887_add__le__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_888_trans__le__add2,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_889_trans__le__add1,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_890_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_891_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( plus_plus_nat @ M @ K_1 ) ) ) ).
thf(fact_892_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_893_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_894_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_895_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_896_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_897_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_898_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_899_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_900_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_901_real__mult__1,axiom,
! [Z: real] :
( ( times_times_real @ one_one_real @ Z )
= Z ) ).
thf(fact_902_le__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_903_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_904_eq__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
<=> ( M = N ) ) ) ) ).
thf(fact_905_diff__diff__cancel,axiom,
! [I_1: nat,N: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I_1 ) )
= I_1 ) ) ).
thf(fact_906_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
thf(fact_907_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_908_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_909_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_910_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_911_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_912_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_913_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_914_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_915_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_916_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_917_mult__less__mono2,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
thf(fact_918_mult__less__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
thf(fact_919_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_920_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_921_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_922_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_923_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_924_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_925_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_926_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
thf(fact_927_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_928_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_929_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_930_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
thf(fact_931_less__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_932_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
thf(fact_933_zmult__int,axiom,
! [M: nat,N: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_934_int__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_935_diff__diff__right,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
thf(fact_936_le__diff__conv,axiom,
! [J: nat,K: nat,I_1: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I_1 @ K ) ) ) ).
thf(fact_937_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
thf(fact_938_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_939_add__diff__assoc,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K ) ) ) ).
thf(fact_940_le__diff__conv2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
thf(fact_941_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
= M ) ) ).
thf(fact_942_le__imp__diff__is__add,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ( minus_minus_nat @ J @ I_1 )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I_1 ) ) ) ) ).
thf(fact_943_diff__add__assoc,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
= ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_944_add__diff__assoc2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K ) ) ) ).
thf(fact_945_diff__add__assoc2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 ) ) ) ).
thf(fact_946_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_947_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_948_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( ( twoSqu919416604sum2sq @ X )
=> ( ( twoSqu919416604sum2sq @ Y )
=> ( twoSqu919416604sum2sq @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_949_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_950_le__nat__number__of,axiom,
! [V_1: int,V_2: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
<=> ( ~ ( ord_less_eq_int @ V_1 @ V_2 )
=> ( ord_less_eq_int @ V_1 @ pls ) ) ) ).
thf(fact_951_transfer__nat__int__relations_I3_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_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
<=> ( ord_less_eq_int @ X @ Y ) ) ) ) ).
thf(fact_952_nat__diff__distrib,axiom,
! [Z: int,Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( ord_less_eq_int @ Z_1 @ Z )
=> ( ( nat_1 @ ( minus_minus_int @ Z @ Z_1 ) )
= ( minus_minus_nat @ ( nat_1 @ Z ) @ ( nat_1 @ Z_1 ) ) ) ) ) ).
thf(fact_953_nat__mult__distrib,axiom,
! [Z_1: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat_1 @ ( times_times_int @ Z @ Z_1 ) )
= ( times_times_nat @ ( nat_1 @ Z ) @ ( nat_1 @ Z_1 ) ) ) ) ).
thf(fact_954_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( times_times_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( times_times_int @ X @ Y ) ) ) ) ) ).
thf(fact_955_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_956_nat__le__eq__zle,axiom,
! [Z: int,W: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z ) )
<=> ( ord_less_eq_int @ W @ Z ) ) ) ).
thf(fact_957_power__eq__if,axiom,
! [P_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( power_power_nat @ P_1 @ M )
= one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( power_power_nat @ P_1 @ M )
= ( times_times_nat @ P_1 @ ( power_power_nat @ P_1 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).
thf(fact_958_nat__mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
thf(fact_959_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_960_nat__number__of__mult__left,axiom,
! [V_2: int,K: nat,V_1: int] :
( ( ( ord_less_int @ V_1 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ ( times_times_nat @ ( number_number_of_nat @ V_2 ) @ K ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ ( times_times_nat @ ( number_number_of_nat @ V_2 ) @ K ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V_1 @ V_2 ) ) @ K ) ) ) ) ).
thf(fact_961_mult__nat__number__of,axiom,
! [V_2: int,V_1: int] :
( ( ( ord_less_int @ V_1 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ ( times_times_int @ V_1 @ V_2 ) ) ) ) ) ).
thf(fact_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R_1: int,B_1: int,Q_1: int,R_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_2 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
thf(fact_970_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_1 ) @ R_2 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_int @ B @ R_2 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
thf(fact_971_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R_1: int,B_1: int,Q_1: int,R_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ B )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
thf(fact_972_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_1 ) @ R_2 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ R_2 @ B )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
thf(fact_973_xy,axiom,
( ( 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 ) ) ) ) )
= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_974_Int2_Oaux__1,axiom,
! [P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( minus_minus_nat @ ( nat_1 @ P_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( nat_1 @ ( minus_minus_int @ P_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_975_Int2_Oaux__2,axiom,
! [P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ord_less_nat @ zero_zero_nat @ ( nat_1 @ ( minus_minus_int @ P_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_976__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_,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_977_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_978_real__mult__commute,axiom,
! [Z: real,W: real] :
( ( times_times_real @ Z @ W )
= ( times_times_real @ W @ Z ) ) ).
thf(fact_979_real__le__linear,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
| ( ord_less_eq_real @ W @ Z ) ) ).
thf(fact_980_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_981_real__le__trans,axiom,
! [K: real,I_1: real,J: real] :
( ( ord_less_eq_real @ I_1 @ J )
=> ( ( ord_less_eq_real @ J @ K )
=> ( ord_less_eq_real @ I_1 @ K ) ) ) ).
thf(fact_982_real__le__antisym,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_983_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_984_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_985__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_986_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_987__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_988__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_989__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_990_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_991_zprime__zdvd__zmult__better,axiom,
! [M: int,N: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( dvd_dvd_int @ P_1 @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P_1 @ M )
| ( dvd_dvd_int @ P_1 @ N ) ) ) ) ).
thf(fact_992_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_993_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_994_zprime__zdvd__power,axiom,
! [A: int,N: nat,P_1: int] :
( ( zprime @ P_1 )
=> ( ( dvd_dvd_int @ P_1 @ ( power_power_int @ A @ N ) )
=> ( dvd_dvd_int @ P_1 @ A ) ) ) ).
thf(fact_995_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_996_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_997_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_998_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_999_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_1000_rel__simps_I9_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_1001_rel__simps_I13_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_int @ K @ min ) ) ).
thf(fact_1002_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_1003_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_1004_rel__simps_I8_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_1005_rel__simps_I26_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ min @ K ) ) ).
thf(fact_1006_rel__simps_I30_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_1007_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_1008_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_1009_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_1010_rel__simps_I28_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_1011_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_1012_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_1013_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_1014_rel__simps_I45_J,axiom,
! [K: int] :
( ( bit0 @ K )
!= min ) ).
thf(fact_1015_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_1016_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_1017_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_1018_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_1019_rel__simps_I47_J,axiom,
! [K: int] :
( ( ( bit1 @ K )
= min )
<=> ( K = min ) ) ).
thf(fact_1020_succ__Min,axiom,
( ( succ @ min )
= pls ) ).
thf(fact_1021_diff__bin__simps_I2_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ min )
= ( succ @ K ) ) ).
thf(fact_1022_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_1023_zpower__zdvd__prop1,axiom,
! [P_1: int,Y: int,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P_1 @ Y )
=> ( dvd_dvd_int @ P_1 @ ( power_power_int @ Y @ N ) ) ) ) ).
thf(fact_1024_rel__simps_I25_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_1025_rel__simps_I11_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_1026_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_1027_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_1028_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_1029_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_1030_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_1031_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ~ ( dvd_dvd_int @ P_1 @ B )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_1 @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_1 @ N ) @ A ) ) ) ) ).
thf(fact_1032_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ~ ( dvd_dvd_int @ P_1 @ A )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_1 @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_1 @ N ) @ B ) ) ) ) ).
thf(fact_1033_zpower__zdvd__prop2,axiom,
! [Y: int,N: nat,P_1: int] :
( ( zprime @ P_1 )
=> ( ( dvd_dvd_int @ P_1 @ ( power_power_int @ Y @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( dvd_dvd_int @ P_1 @ Y ) ) ) ) ).
thf(fact_1034__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_1035_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_1036__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_,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_1037_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_1038_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_1039__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_1040_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_1041_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_1042_dvd__diff__nat,axiom,
! [N: nat,K: nat,M: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_1043_zcong__eq__zdvd__prop,axiom,
! [X: int,P_1: int] :
( ( zcong @ X @ zero_zero_int @ P_1 )
<=> ( dvd_dvd_int @ P_1 @ X ) ) ).
thf(fact_1044_zcong__zero__equiv__div,axiom,
! [A: int,M: int] :
( ( zcong @ A @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A ) ) ).
thf(fact_1045_nat__dvd__not__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
thf(fact_1046_divides__ge,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = zero_zero_nat )
| ( ord_less_eq_nat @ A @ B ) ) ) ).
thf(fact_1047_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_1048_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
thf(fact_1049_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
thf(fact_1050_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_1051_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_1052_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_1053_zdvd__int,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( dvd_dvd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) ).
thf(fact_1054_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_1055_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
<=> ( M = one_one_nat ) ) ).
thf(fact_1056_dvd__reduce,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( plus_plus_nat @ N @ K ) )
<=> ( dvd_dvd_nat @ K @ N ) ) ).
thf(fact_1057_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_1058_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_zcong__zmult__prop3,axiom,
! [Y: int,X: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( ~ ( zcong @ Y @ zero_zero_int @ P_1 )
=> ~ ( zcong @ ( times_times_int @ X @ Y ) @ zero_zero_int @ P_1 ) ) ) ) ).
thf(fact_1068_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_1069_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1070_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P_1 )
=> ( ( zcong @ A @ zero_zero_int @ P_1 )
| ( zcong @ B @ zero_zero_int @ P_1 ) ) ) ) ) ).
thf(fact_1071_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ~ ( zcong @ A @ zero_zero_int @ P_1 )
& ~ ( zcong @ B @ zero_zero_int @ P_1 ) )
=> ~ ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P_1 ) ) ) ) ).
thf(fact_1072_divides__div__not,axiom,
! [X: nat,Q: nat,N: nat,R_1: nat] :
( ( X
= ( plus_plus_nat @ ( times_times_nat @ Q @ N ) @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ R_1 )
=> ( ( ord_less_nat @ R_1 @ N )
=> ~ ( dvd_dvd_nat @ N @ X ) ) ) ) ).
thf(fact_1073_dvd__mult__cancel1,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
<=> ( N = one_one_nat ) ) ) ).
thf(fact_1074_dvd__mult__cancel2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
<=> ( N = one_one_nat ) ) ) ).
thf(fact_1075_transfer__nat__int__relations_I4_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( dvd_dvd_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
<=> ( dvd_dvd_int @ X @ Y ) ) ) ) ).
thf(fact_1076_power__dvd__imp__le,axiom,
! [I_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I_1 @ M ) @ ( power_power_nat @ I_1 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I_1 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1077_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd_nat @ ( nat_1 @ Z ) @ M )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( dvd_dvd_int @ Z @ ( semiri1621563631at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( M = zero_zero_nat ) ) ) ) ).
thf(fact_1078_zcong__neg__1__impl__ne__1,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zcong @ X @ ( number_number_of_int @ min ) @ P_1 )
=> ~ ( zcong @ X @ one_one_int @ P_1 ) ) ) ).
thf(fact_1079_Euler_Oaux____1,axiom,
! [Y: int,X: int,P_1: int] :
( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( ( zcong @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X @ P_1 )
=> ~ ( dvd_dvd_int @ P_1 @ Y ) ) ) ).
thf(fact_1080_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 ) ) ) ).
thf(fact_1081_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_1082__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_,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_1083__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s,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_1084_dvd_Oorder__refl,axiom,
! [X: nat] : ( dvd_dvd_nat @ X @ X ) ).
thf(fact_1085__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_1086_dvd_Oless__asym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ~ ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1087_dvd_Oless__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( ( ( dvd_dvd_nat @ Y @ Z )
& ~ ( dvd_dvd_nat @ Z @ Y ) )
=> ( ( dvd_dvd_nat @ X @ Z )
& ~ ( dvd_dvd_nat @ Z @ X ) ) ) ) ).
thf(fact_1088_dvd_Oless__asym_H,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ~ ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_1089_dvd_Oless__le__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( ( dvd_dvd_nat @ Y @ Z )
=> ( ( dvd_dvd_nat @ X @ Z )
& ~ ( dvd_dvd_nat @ Z @ X ) ) ) ) ).
thf(fact_1090_dvd_Oord__less__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) )
=> ( ( B = C )
=> ( ( dvd_dvd_nat @ A @ C )
& ~ ( dvd_dvd_nat @ C @ A ) ) ) ) ).
thf(fact_1091_dvd_Oless__imp__triv,axiom,
! [P: $o,X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) )
=> P ) ) ).
thf(fact_1092_dvd_Oless__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( Y != X ) ) ).
thf(fact_1093_dvd_Oless__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_1094_dvd_Oless__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ~ ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1095_dvd_Oless__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_1096_dvd_Oless__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ~ ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1097_dvd_Oless__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_1098_dvd_Ole__less__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ( dvd_dvd_nat @ Y @ Z )
& ~ ( dvd_dvd_nat @ Z @ Y ) )
=> ( ( dvd_dvd_nat @ X @ Z )
& ~ ( dvd_dvd_nat @ Z @ X ) ) ) ) ).
thf(fact_1099_dvd_Oord__eq__less__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ~ ( dvd_dvd_nat @ C @ B ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ~ ( dvd_dvd_nat @ C @ A ) ) ) ) ).
thf(fact_1100_dvd_Oorder__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ Z )
=> ( dvd_dvd_nat @ X @ Z ) ) ) ).
thf(fact_1101_dvd_Oantisym,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ X )
=> ( X = Y ) ) ) ).
thf(fact_1102_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_1103_dvd_Oord__le__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
thf(fact_1104_dvd_Oord__eq__le__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
thf(fact_1105_dvd_Ole__neq__trans,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ) ).
thf(fact_1106_dvd_Ole__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
| ( X = Y ) ) ) ).
thf(fact_1107_dvd_Oantisym__conv,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( dvd_dvd_nat @ X @ Y )
<=> ( X = Y ) ) ) ).
thf(fact_1108_dvd_Oeq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_1109_dvd_Oneq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ) ).
thf(fact_1110_dvd_Oless__le__not__le,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_1111_dvd_Oless__le,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_1112_dvd_Ole__less,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
| ( X = Y ) ) ) ).
thf(fact_1113_dvd_Oeq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_1114_dvd_Oless__irrefl,axiom,
! [X: nat] :
~ ( ( dvd_dvd_nat @ X @ X )
& ~ ( dvd_dvd_nat @ X @ X ) ) ).
thf(fact_1115_divides__antisym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
<=> ( X = Y ) ) ).
thf(fact_1116_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_1117_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_1118_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_1119_Legendre__def,axiom,
! [A: int,P_1: int] :
( ( ( zcong @ A @ zero_zero_int @ P_1 )
=> ( ( legendre @ A @ P_1 )
= zero_zero_int ) )
& ( ~ ( zcong @ A @ zero_zero_int @ P_1 )
=> ( ( ( quadRes @ P_1 @ A )
=> ( ( legendre @ A @ P_1 )
= one_one_int ) )
& ( ~ ( quadRes @ P_1 @ A )
=> ( ( legendre @ A @ P_1 )
= ( number_number_of_int @ min ) ) ) ) ) ) ).
thf(fact_1120_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_1121_Little__Fermat,axiom,
! [X: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ~ ( dvd_dvd_int @ P_1 @ X )
=> ( zcong @ ( power_power_int @ X @ ( nat_1 @ ( minus_minus_int @ P_1 @ one_one_int ) ) ) @ one_one_int @ P_1 ) ) ) ).
thf(fact_1122_zcong__square__zless,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_1 )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P_1 )
=> ( ( A = one_one_int )
| ( A
= ( minus_minus_int @ P_1 @ one_one_int ) ) ) ) ) ) ) ).
thf(fact_1123_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_1124_zcong__refl,axiom,
! [K: int,M: int] : ( zcong @ K @ K @ M ) ).
thf(fact_1125_zcong__sym,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ B @ A @ M ) ) ).
thf(fact_1126_IntPrimes_Ozcong__zero,axiom,
! [A: int,B: int] :
( ( zcong @ A @ B @ zero_zero_int )
<=> ( A = B ) ) ).
thf(fact_1127_zcong__1,axiom,
! [A: int,B: int] : ( zcong @ A @ B @ one_one_int ) ).
thf(fact_1128_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : ( zcong @ ( times_times_int @ A @ M ) @ ( times_times_int @ B @ M ) @ M ) ).
thf(fact_1129_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_1130_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_1131_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_1132_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_1133_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_1134_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_1135_xzgcda__linear__aux1,axiom,
! [A: int,R_1: 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_1 @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R_1 @ D ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A @ M ) @ ( times_times_int @ C @ N ) ) @ ( times_times_int @ R_1 @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D @ N ) ) ) ) ) ).
thf(fact_1136_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_1137_zcong__def,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A @ B ) ) ) ).
thf(fact_1138_norR__mem__unique__aux,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ B @ one_one_int ) )
=> ( ord_less_int @ A @ B ) ) ).
thf(fact_1139_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_1140_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_1141_zprime__zdvd__zmult,axiom,
! [N: int,P_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( zprime @ P_1 )
=> ( ( dvd_dvd_int @ P_1 @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P_1 @ M )
| ( dvd_dvd_int @ P_1 @ N ) ) ) ) ) ).
thf(fact_1142_zprime__def,axiom,
! [P_1: int] :
( ( zprime @ P_1 )
<=> ( ( ord_less_int @ one_one_int @ P_1 )
& ! [M_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M_1 )
& ( dvd_dvd_int @ M_1 @ P_1 ) )
=> ( ( M_1 = one_one_int )
| ( M_1 = P_1 ) ) ) ) ) ).
thf(fact_1143_zcong__square,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P_1 )
=> ( ( zcong @ A @ one_one_int @ P_1 )
| ( zcong @ A @ ( minus_minus_int @ P_1 @ one_one_int ) @ P_1 ) ) ) ) ) ).
thf(fact_1144_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
<=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
thf(fact_1145_prime__g__5,axiom,
! [P_1: int] :
( ( zprime @ P_1 )
=> ( ( P_1
!= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( P_1
!= ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_1 ) ) ) ) ).
thf(fact_1146_conj__le__cong,axiom,
! [P_2: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
& P_2 ) ) ) ).
thf(fact_1147_imp__le__cong,axiom,
! [P_2: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P_2 ) ) ) ).
thf(fact_1148_zdvd__mono,axiom,
! [M: int,T: int,K: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
<=> ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
thf(fact_1149_number__of2,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ pls ) ).
thf(fact_1150_inv__not__p__minus__1__aux,axiom,
! [A: int,P_1: int] :
( ( zcong @ ( times_times_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) ) @ one_one_int @ P_1 )
<=> ( zcong @ A @ ( minus_minus_int @ P_1 @ one_one_int ) @ P_1 ) ) ).
thf(fact_1151_zcong__zpower__zmult,axiom,
! [Z: nat,X: int,Y: nat,P_1: int] :
( ( zcong @ ( power_power_int @ X @ Y ) @ one_one_int @ P_1 )
=> ( zcong @ ( power_power_int @ X @ ( times_times_nat @ Y @ Z ) ) @ one_one_int @ P_1 ) ) ).
thf(fact_1152_number__of1,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit0 @ N ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ N ) ) ) ) ) ).
thf(fact_1153__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [S: int,W_1: int] :
~ ( ( W_1
= ( minus_minus_int @ y @ ( times_times_int @ S @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) )
& ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ W_1 ) ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_1154__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [R: int,V: int] :
~ ( ( V
= ( minus_minus_int @ x @ ( times_times_int @ R @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) )
& ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ V ) ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_1155_dvd__imp__le__int,axiom,
! [D: int,I_1: int] :
( ( I_1 != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I_1 )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I_1 ) ) ) ) ).
thf(fact_1156_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat_1 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
= ( times_times_nat @ ( nat_1 @ ( abs_abs_int @ W ) ) @ ( nat_1 @ ( abs_abs_int @ Z ) ) ) ) ).
thf(fact_1157_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ) ).
thf(fact_1158_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
<=> ( ( abs_abs_int @ X )
= one_one_int ) ) ).
thf(fact_1159_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_1160_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
thf(fact_1161_zero__le__zpower__abs,axiom,
! [X: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ X ) @ N ) ) ).
thf(fact_1162_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
<=> ( Z = zero_zero_int ) ) ).
thf(fact_1163_abs__eq__1__iff,axiom,
! [Z: int] :
( ( ( abs_abs_int @ Z )
= one_one_int )
<=> ( ( Z = one_one_int )
| ( Z
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_1164_abs__power3__distrib,axiom,
! [X: int] :
( ( abs_abs_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ ( abs_abs_int @ X ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1165_zero__less__zpower__abs__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ X ) @ N ) )
<=> ( ( X != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1166_zdvd__mult__cancel1,axiom,
! [N: int,M: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
<=> ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
thf(fact_1167_int__dvd__iff,axiom,
! [M: nat,Z: int] :
( ( dvd_dvd_int @ ( semiri1621563631at_int @ M ) @ Z )
<=> ( dvd_dvd_nat @ M @ ( nat_1 @ ( abs_abs_int @ Z ) ) ) ) ).
thf(fact_1168_dvd__int__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd_int @ Z @ ( semiri1621563631at_int @ M ) )
<=> ( dvd_dvd_nat @ ( nat_1 @ ( abs_abs_int @ Z ) ) @ M ) ) ).
thf(fact_1169_abs__power2__distrib,axiom,
! [A: int] :
( ( abs_abs_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1170_power2__eq__iff__abs__eq,axiom,
! [A: int,B: int] :
( ( ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ).
thf(fact_1171_power2__eq1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int )
=> ( ( abs_abs_int @ A )
= one_one_int ) ) ).
thf(fact_1172_incr__lemma,axiom,
! [Z: int,X: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
thf(fact_1173_decr__lemma,axiom,
! [X: int,Z: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
thf(fact_1174_best__division__abs,axiom,
! [Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ? [N_1: int] : ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ Y @ ( times_times_int @ N_1 @ X ) ) ) ) @ X ) ) ).
thf(fact_1175_decr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1176_abs__add__one__not__less__self,axiom,
! [X: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( abs_abs_real @ X ) @ one_one_real ) @ X ) ).
thf(fact_1177_abs__add__one__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
thf(fact_1178_less__one__imp__sqr__less__one,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_real ) ) ).
thf(fact_1179_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K_1 )
=> ~ ( P @ I ) )
& ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_1180_incr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1181_nat__less__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1182_nat__less__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1183_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
thf(fact_1184_left__add__mult__distrib,axiom,
! [I_1: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J ) @ U ) @ K ) ) ).
thf(fact_1185_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1186_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( M = N ) ) ) ).
thf(fact_1187_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1188_nat__mult__dvd__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1189_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1190_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1191_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1192_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M )
= N ) ) ) ).
thf(fact_1193_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1194_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1195_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1196_nat0__intermed__int__val,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
thf(fact_1197_int__val__lemma,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
%----Helper facts (6)
thf(help_If_1_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Int__Oint_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_int ) ).
%------------------------------------------------------------------------------