TPTP Problem File: NUM924^3.p
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%------------------------------------------------------------------------------
% File : NUM924^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 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_l102 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.44 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v6.0.0, 0.57 v5.5.0, 0.33 v5.4.0, 0.60 v5.3.0
% Syntax : Number of formulae : 1259 ( 463 unt; 60 typ; 0 def)
% Number of atoms : 2694 ( 807 equ; 1 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 11285 ( 263 ~; 71 |; 163 &;9489 @)
% ( 255 <=>;1044 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 101 ( 101 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 56 usr; 13 con; 0-3 aty)
% Number of variables : 2562 ( 1 ^;2544 !; 17 ?;2562 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:43:58
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
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 ).
thf(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
%----Explicit typings (56)
thf(sy_c_Divides_Odiv__class_Omod_000tc__Int__Oint,type,
div_mod_int: int > int > int ).
thf(sy_c_Divides_Odiv__class_Omod_000tc__Nat__Onat,type,
div_mod_nat: nat > nat > nat ).
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_Int2_OMultInv,type,
multInv: int > int > int ).
thf(sy_c_IntFact_Od22set,type,
d22set: int > int > $o ).
thf(sy_c_IntFact_Ozfact,type,
zfact: int > int ).
thf(sy_c_IntPrimes_Ozcong,type,
zcong: int > int > int > $o ).
thf(sy_c_IntPrimes_Ozprime,type,
zprime: int > $o ).
thf(sy_c_Int_OBit0,type,
bit0: int > int ).
thf(sy_c_Int_OBit1,type,
bit1: int > int ).
thf(sy_c_Int_OMin,type,
min: int ).
thf(sy_c_Int_OPls,type,
pls: int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
number267125858f_real: int > real ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Residues_OLegendre,type,
legendre: int > int > int ).
thf(sy_c_Residues_OQuadRes,type,
quadRes: int > int > $o ).
thf(sy_c_Residues_OSR,type,
sr: int > int > $o ).
thf(sy_c_Residues_OStandardRes,type,
standardRes: int > int > int ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__RealDef__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Set_OCollect_000tc__Int__Oint,type,
collect_int: ( int > $o ) > int > $o ).
thf(sy_c_TwoSquares__Mirabelle__dzzvbppuls_Ois__sum2sq,type,
twoSqu919416604sum2sq: int > $o ).
thf(sy_c_TwoSquares__Mirabelle__dzzvbppuls_Osum2sq,type,
twoSqu2057625106sum2sq: product_prod_int_int > int ).
thf(sy_c_WilsonRuss_Oinv,type,
inv: int > int > int ).
thf(sy_c_WilsonRuss_Owset,type,
wset: int > int > int > $o ).
thf(sy_c_member_000tc__Int__Oint,type,
member_int: int > ( int > $o ) > $o ).
thf(sy_v_m,type,
m: int ).
thf(sy_v_s1____,type,
s1: int ).
thf(sy_v_s____,type,
s: int ).
thf(sy_v_t____,type,
t: int ).
%----Relevant facts (1198)
thf(fact_0__096t_A_060_A0_096,axiom,
ord_less_int @ t @ zero_zero_int ).
thf(fact_1_calculation_I1_J,axiom,
ord_less_int @ t @ one_one_int ).
thf(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ zero_zero_int ) ).
thf(fact_3_t,axiom,
( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ) ).
thf(fact_4_calculation_I2_J,axiom,
( ( t = zero_zero_int )
=> ( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
= zero_zero_int ) ) ).
thf(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ( ord_less_eq_int @ one_one_int @ t ) ).
thf(fact_6_p0,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_7_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ) ).
thf(fact_8_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int ) ).
thf(fact_9_sum__power2__gt__zero__iff,axiom,
! [X_39: real,Y_31: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_31 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_39 != zero_zero_real )
| ( Y_31 != zero_zero_real ) ) ) ).
thf(fact_10_sum__power2__gt__zero__iff,axiom,
! [X_39: int,Y_31: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_31 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_39 != zero_zero_int )
| ( Y_31 != zero_zero_int ) ) ) ).
thf(fact_11_sum__power2__eq__zero__iff,axiom,
! [X_38: real,Y_30: real] :
( ( ( plus_plus_real @ ( power_power_real @ X_38 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_30 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X_38 = zero_zero_real )
& ( Y_30 = zero_zero_real ) ) ) ).
thf(fact_12_sum__power2__eq__zero__iff,axiom,
! [X_38: int,Y_30: int] :
( ( ( plus_plus_int @ ( power_power_int @ X_38 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_30 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_int )
<=> ( ( X_38 = zero_zero_int )
& ( Y_30 = zero_zero_int ) ) ) ).
thf(fact_13_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_14_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_15_zero__less__power2,axiom,
! [A_128: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_128 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_128 != zero_zero_real ) ) ).
thf(fact_16_zero__less__power2,axiom,
! [A_128: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_128 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_128 != zero_zero_int ) ) ).
thf(fact_17_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_real ) ).
thf(fact_18_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_19_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_20_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real ) ).
thf(fact_21_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_22_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(fact_23_zero__eq__power2,axiom,
! [A_127: real] :
( ( ( power_power_real @ A_127 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real )
<=> ( A_127 = zero_zero_real ) ) ).
thf(fact_24_zero__eq__power2,axiom,
! [A_127: int] :
( ( ( power_power_int @ A_127 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int )
<=> ( A_127 = zero_zero_int ) ) ).
thf(fact_25_add__special_I2_J,axiom,
! [W_17: int] :
( ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_17 ) )
= ( number267125858f_real @ ( plus_plus_int @ ( bit1 @ pls ) @ W_17 ) ) ) ).
thf(fact_26_add__special_I2_J,axiom,
! [W_17: int] :
( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_17 ) )
= ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_17 ) ) ) ).
thf(fact_27_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_28_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_29_t__l__p,axiom,
ord_less_int @ t @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_30__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T_1: int] :
( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
!= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ T_1 ) ) ).
thf(fact_31_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_32_qf1pt,axiom,
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_33_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_34_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_35_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_36_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_37_times__numeral__code_I5_J,axiom,
! [V: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
thf(fact_38_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_39_le__number__of,axiom,
! [X_37: int,Y_29: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_37 ) @ ( number267125858f_real @ Y_29 ) )
<=> ( ord_less_eq_int @ X_37 @ Y_29 ) ) ).
thf(fact_40_le__number__of,axiom,
! [X_37: int,Y_29: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_37 ) @ ( number_number_of_int @ Y_29 ) )
<=> ( ord_less_eq_int @ X_37 @ Y_29 ) ) ).
thf(fact_41_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
= ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
thf(fact_42_zle__trans,axiom,
! [K: int,I_1: int,J_1: int] :
( ( ord_less_eq_int @ I_1 @ J_1 )
=> ( ( ord_less_eq_int @ J_1 @ K )
=> ( ord_less_eq_int @ I_1 @ K ) ) ) ).
thf(fact_43_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_44_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z: nat] :
( ( power_power_int @ X @ ( plus_plus_nat @ Y @ Z ) )
= ( times_times_int @ ( power_power_int @ X @ Y ) @ ( power_power_int @ X @ Z ) ) ) ).
thf(fact_45_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_46_rel__simps_I34_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_47_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_48_less__eq__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_49_rel__simps_I31_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_50_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_51_zadd__left__mono,axiom,
! [K: int,I_1: int,J_1: int] :
( ( ord_less_eq_int @ I_1 @ J_1 )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I_1 ) @ ( plus_plus_int @ K @ J_1 ) ) ) ).
thf(fact_52_eq__number__of__0,axiom,
! [V: int] :
( ( ( number_number_of_nat @ V )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_53_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V ) )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_54_semiring__mult__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( times_times_real @ ( number267125858f_real @ V_14 ) @ ( number267125858f_real @ V_15 ) )
= ( number267125858f_real @ ( times_times_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_55_semiring__mult__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_14 ) @ ( number_number_of_nat @ V_15 ) )
= ( number_number_of_nat @ ( times_times_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_56_semiring__mult__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( times_times_int @ ( number_number_of_int @ V_14 ) @ ( number_number_of_int @ V_15 ) )
= ( number_number_of_int @ ( times_times_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_57_mult__number__of__left,axiom,
! [V_13: int,W_16: int,Z_9: real] :
( ( times_times_real @ ( number267125858f_real @ V_13 ) @ ( times_times_real @ ( number267125858f_real @ W_16 ) @ Z_9 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_13 @ W_16 ) ) @ Z_9 ) ) ).
thf(fact_58_mult__number__of__left,axiom,
! [V_13: int,W_16: int,Z_9: int] :
( ( times_times_int @ ( number_number_of_int @ V_13 ) @ ( times_times_int @ ( number_number_of_int @ W_16 ) @ Z_9 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_13 @ W_16 ) ) @ Z_9 ) ) ).
thf(fact_59_arith__simps_I32_J,axiom,
! [V_12: int,W_15: int] :
( ( times_times_real @ ( number267125858f_real @ V_12 ) @ ( number267125858f_real @ W_15 ) )
= ( number267125858f_real @ ( times_times_int @ V_12 @ W_15 ) ) ) ).
thf(fact_60_arith__simps_I32_J,axiom,
! [V_12: int,W_15: int] :
( ( times_times_int @ ( number_number_of_int @ V_12 ) @ ( number_number_of_int @ W_15 ) )
= ( number_number_of_int @ ( times_times_int @ V_12 @ W_15 ) ) ) ).
thf(fact_61_number__of__mult,axiom,
! [V_11: int,W_14: int] :
( ( number267125858f_real @ ( times_times_int @ V_11 @ W_14 ) )
= ( times_times_real @ ( number267125858f_real @ V_11 ) @ ( number267125858f_real @ W_14 ) ) ) ).
thf(fact_62_number__of__mult,axiom,
! [V_11: int,W_14: int] :
( ( number_number_of_int @ ( times_times_int @ V_11 @ W_14 ) )
= ( times_times_int @ ( number_number_of_int @ V_11 ) @ ( number_number_of_int @ W_14 ) ) ) ).
thf(fact_63_sum__squares__le__zero__iff,axiom,
! [X_36: real,Y_28: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_36 @ X_36 ) @ ( times_times_real @ Y_28 @ Y_28 ) ) @ zero_zero_real )
<=> ( ( X_36 = zero_zero_real )
& ( Y_28 = zero_zero_real ) ) ) ).
thf(fact_64_sum__squares__le__zero__iff,axiom,
! [X_36: int,Y_28: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_36 @ X_36 ) @ ( times_times_int @ Y_28 @ Y_28 ) ) @ zero_zero_int )
<=> ( ( X_36 = zero_zero_int )
& ( Y_28 = zero_zero_int ) ) ) ).
thf(fact_65_sum__squares__ge__zero,axiom,
! [X_35: real,Y_27: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_35 @ X_35 ) @ ( times_times_real @ Y_27 @ Y_27 ) ) ) ).
thf(fact_66_sum__squares__ge__zero,axiom,
! [X_35: int,Y_27: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_35 @ X_35 ) @ ( times_times_int @ Y_27 @ Y_27 ) ) ) ).
thf(fact_67_le__special_I3_J,axiom,
! [X_34: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_34 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ X_34 @ pls ) ) ).
thf(fact_68_le__special_I3_J,axiom,
! [X_34: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_34 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_34 @ pls ) ) ).
thf(fact_69_le__special_I1_J,axiom,
! [Y_26: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( number267125858f_real @ Y_26 ) )
<=> ( ord_less_eq_int @ pls @ Y_26 ) ) ).
thf(fact_70_le__special_I1_J,axiom,
! [Y_26: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_26 ) )
<=> ( ord_less_eq_int @ pls @ Y_26 ) ) ).
thf(fact_71_less__0__number__of,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
<=> ( ord_less_int @ pls @ V ) ) ).
thf(fact_72_le__number__of__eq__not__less,axiom,
! [V_10: int,W_13: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_10 ) @ ( number267125858f_real @ W_13 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_13 ) @ ( number267125858f_real @ V_10 ) ) ) ).
thf(fact_73_le__number__of__eq__not__less,axiom,
! [V_10: int,W_13: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_10 ) @ ( number_number_of_nat @ W_13 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_13 ) @ ( number_number_of_nat @ V_10 ) ) ) ).
thf(fact_74_le__number__of__eq__not__less,axiom,
! [V_10: int,W_13: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_10 ) @ ( number_number_of_int @ W_13 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_13 ) @ ( number_number_of_int @ V_10 ) ) ) ).
thf(fact_75_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_76_less__eq__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit0 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_77_rel__simps_I32_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_78_rel__simps_I27_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_eq_int @ K @ pls ) ) ).
thf(fact_79_rel__simps_I21_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_80_zadd__zless__mono,axiom,
! [Z_8: int,Z: int,W_12: int,W: int] :
( ( ord_less_int @ W_12 @ W )
=> ( ( ord_less_eq_int @ Z_8 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_12 @ Z_8 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_81_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_82_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_83_le__special_I4_J,axiom,
! [X_33: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_33 ) @ one_one_real )
<=> ( ord_less_eq_int @ X_33 @ ( bit1 @ pls ) ) ) ).
thf(fact_84_le__special_I4_J,axiom,
! [X_33: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_33 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_33 @ ( bit1 @ pls ) ) ) ).
thf(fact_85_le__special_I2_J,axiom,
! [Y_25: int] :
( ( ord_less_eq_real @ one_one_real @ ( number267125858f_real @ Y_25 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_25 ) ) ).
thf(fact_86_le__special_I2_J,axiom,
! [Y_25: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_25 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_25 ) ) ).
thf(fact_87_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_88_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_89_mult__Bit0,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit0 @ K ) @ L )
= ( bit0 @ ( times_times_int @ K @ L ) ) ) ).
thf(fact_90_less__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_91_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_92_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_93_plus__numeral__code_I9_J,axiom,
! [V: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
thf(fact_94_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_95_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_96_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_97_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_98_less__eq__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_99_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_100_less__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_101_mem__def,axiom,
! [X_32: int,A_126: int > $o] :
( ( member_int @ X_32 @ A_126 )
<=> ( A_126 @ X_32 ) ) ).
thf(fact_102_Collect__def,axiom,
! [P_6: int > $o] :
( ( collect_int @ P_6 )
= P_6 ) ).
thf(fact_103_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_104_less__nat__number__of,axiom,
! [V: int,V_2: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
<=> ( ( ( ord_less_int @ V @ V_2 )
=> ( ord_less_int @ pls @ V_2 ) )
& ( ord_less_int @ V @ V_2 ) ) ) ).
thf(fact_105_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_106_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_107_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_108_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_109_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_110_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_111_semiring__add__number__of,axiom,
! [V_9: int,V_8: int] :
( ( ord_less_eq_int @ pls @ V_8 )
=> ( ( ord_less_eq_int @ pls @ V_9 )
=> ( ( plus_plus_real @ ( number267125858f_real @ V_8 ) @ ( number267125858f_real @ V_9 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_8 @ V_9 ) ) ) ) ) ).
thf(fact_112_semiring__add__number__of,axiom,
! [V_9: int,V_8: int] :
( ( ord_less_eq_int @ pls @ V_8 )
=> ( ( ord_less_eq_int @ pls @ V_9 )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_8 ) @ ( number_number_of_nat @ V_9 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_8 @ V_9 ) ) ) ) ) ).
thf(fact_113_semiring__add__number__of,axiom,
! [V_9: int,V_8: int] :
( ( ord_less_eq_int @ pls @ V_8 )
=> ( ( ord_less_eq_int @ pls @ V_9 )
=> ( ( plus_plus_int @ ( number_number_of_int @ V_8 ) @ ( number_number_of_int @ V_9 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_8 @ V_9 ) ) ) ) ) ).
thf(fact_114_add__nat__number__of,axiom,
! [V_2: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ V_2 ) ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( ( ord_less_int @ V_2 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ V ) ) )
& ( ~ ( ord_less_int @ V_2 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V @ V_2 ) ) ) ) ) ) ) ).
thf(fact_115_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_116_eq__number__of,axiom,
! [X_31: int,Y_24: int] :
( ( ( number267125858f_real @ X_31 )
= ( number267125858f_real @ Y_24 ) )
<=> ( X_31 = Y_24 ) ) ).
thf(fact_117_eq__number__of,axiom,
! [X_31: int,Y_24: int] :
( ( ( number_number_of_int @ X_31 )
= ( number_number_of_int @ Y_24 ) )
<=> ( X_31 = Y_24 ) ) ).
thf(fact_118_number__of__reorient,axiom,
! [W_11: int,X_30: real] :
( ( ( number267125858f_real @ W_11 )
= X_30 )
<=> ( X_30
= ( number267125858f_real @ W_11 ) ) ) ).
thf(fact_119_number__of__reorient,axiom,
! [W_11: int,X_30: int] :
( ( ( number_number_of_int @ W_11 )
= X_30 )
<=> ( X_30
= ( number_number_of_int @ W_11 ) ) ) ).
thf(fact_120_number__of__reorient,axiom,
! [W_11: int,X_30: nat] :
( ( ( number_number_of_nat @ W_11 )
= X_30 )
<=> ( X_30
= ( number_number_of_nat @ W_11 ) ) ) ).
thf(fact_121_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_122_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( ( bit0 @ K )
= ( bit0 @ L ) )
<=> ( K = L ) ) ).
thf(fact_123_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_124_sum__squares__eq__zero__iff,axiom,
! [X_29: real,Y_23: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_29 @ X_29 ) @ ( times_times_real @ Y_23 @ Y_23 ) )
= zero_zero_real )
<=> ( ( X_29 = zero_zero_real )
& ( Y_23 = zero_zero_real ) ) ) ).
thf(fact_125_sum__squares__eq__zero__iff,axiom,
! [X_29: int,Y_23: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_29 @ X_29 ) @ ( times_times_int @ Y_23 @ Y_23 ) )
= zero_zero_int )
<=> ( ( X_29 = zero_zero_int )
& ( Y_23 = zero_zero_int ) ) ) ).
thf(fact_126_left__distrib__number__of,axiom,
! [A_125: real,B_80: real,V_7: int] :
( ( times_times_real @ ( plus_plus_real @ A_125 @ B_80 ) @ ( number267125858f_real @ V_7 ) )
= ( plus_plus_real @ ( times_times_real @ A_125 @ ( number267125858f_real @ V_7 ) ) @ ( times_times_real @ B_80 @ ( number267125858f_real @ V_7 ) ) ) ) ).
thf(fact_127_left__distrib__number__of,axiom,
! [A_125: nat,B_80: nat,V_7: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_125 @ B_80 ) @ ( number_number_of_nat @ V_7 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_125 @ ( number_number_of_nat @ V_7 ) ) @ ( times_times_nat @ B_80 @ ( number_number_of_nat @ V_7 ) ) ) ) ).
thf(fact_128_left__distrib__number__of,axiom,
! [A_125: int,B_80: int,V_7: int] :
( ( times_times_int @ ( plus_plus_int @ A_125 @ B_80 ) @ ( number_number_of_int @ V_7 ) )
= ( plus_plus_int @ ( times_times_int @ A_125 @ ( number_number_of_int @ V_7 ) ) @ ( times_times_int @ B_80 @ ( number_number_of_int @ V_7 ) ) ) ) ).
thf(fact_129_right__distrib__number__of,axiom,
! [V_6: int,B_79: real,C_50: real] :
( ( times_times_real @ ( number267125858f_real @ V_6 ) @ ( plus_plus_real @ B_79 @ C_50 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_6 ) @ B_79 ) @ ( times_times_real @ ( number267125858f_real @ V_6 ) @ C_50 ) ) ) ).
thf(fact_130_right__distrib__number__of,axiom,
! [V_6: int,B_79: nat,C_50: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_6 ) @ ( plus_plus_nat @ B_79 @ C_50 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_6 ) @ B_79 ) @ ( times_times_nat @ ( number_number_of_nat @ V_6 ) @ C_50 ) ) ) ).
thf(fact_131_right__distrib__number__of,axiom,
! [V_6: int,B_79: int,C_50: int] :
( ( times_times_int @ ( number_number_of_int @ V_6 ) @ ( plus_plus_int @ B_79 @ C_50 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_6 ) @ B_79 ) @ ( times_times_int @ ( number_number_of_int @ V_6 ) @ C_50 ) ) ) ).
thf(fact_132_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_133_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_134_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_135_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_136_zmult__zless__mono2,axiom,
! [K: int,I_1: int,J_1: int] :
( ( ord_less_int @ I_1 @ J_1 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I_1 ) @ ( times_times_int @ K @ J_1 ) ) ) ) ).
thf(fact_137_power2__eq__imp__eq,axiom,
! [X_28: real,Y_22: real] :
( ( ( power_power_real @ X_28 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_real @ Y_22 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X_28 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_22 )
=> ( X_28 = Y_22 ) ) ) ) ).
thf(fact_138_power2__eq__imp__eq,axiom,
! [X_28: nat,Y_22: nat] :
( ( ( power_power_nat @ X_28 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_nat @ Y_22 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X_28 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_22 )
=> ( X_28 = Y_22 ) ) ) ) ).
thf(fact_139_power2__eq__imp__eq,axiom,
! [X_28: int,Y_22: int] :
( ( ( power_power_int @ X_28 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ Y_22 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X_28 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_22 )
=> ( X_28 = Y_22 ) ) ) ) ).
thf(fact_140_power2__le__imp__le,axiom,
! [X_27: real,Y_21: real] :
( ( ord_less_eq_real @ ( power_power_real @ X_27 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_21 )
=> ( ord_less_eq_real @ X_27 @ Y_21 ) ) ) ).
thf(fact_141_power2__le__imp__le,axiom,
! [X_27: nat,Y_21: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X_27 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_21 )
=> ( ord_less_eq_nat @ X_27 @ Y_21 ) ) ) ).
thf(fact_142_power2__le__imp__le,axiom,
! [X_27: int,Y_21: int] :
( ( ord_less_eq_int @ ( power_power_int @ X_27 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_21 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_21 )
=> ( ord_less_eq_int @ X_27 @ Y_21 ) ) ) ).
thf(fact_143_zero__le__power2,axiom,
! [A_124: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_124 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_144_zero__le__power2,axiom,
! [A_124: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_124 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_145_power2__less__imp__less,axiom,
! [X_26: real,Y_20: real] :
( ( ord_less_real @ ( power_power_real @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_20 )
=> ( ord_less_real @ X_26 @ Y_20 ) ) ) ).
thf(fact_146_power2__less__imp__less,axiom,
! [X_26: nat,Y_20: nat] :
( ( ord_less_nat @ ( power_power_nat @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_20 )
=> ( ord_less_nat @ X_26 @ Y_20 ) ) ) ).
thf(fact_147_power2__less__imp__less,axiom,
! [X_26: int,Y_20: int] :
( ( ord_less_int @ ( power_power_int @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_20 )
=> ( ord_less_int @ X_26 @ Y_20 ) ) ) ).
thf(fact_148_sum__power2__le__zero__iff,axiom,
! [X_25: real,Y_19: real] :
( ( ord_less_eq_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 ) ) ) ) ) @ zero_zero_real )
<=> ( ( X_25 = zero_zero_real )
& ( Y_19 = zero_zero_real ) ) ) ).
thf(fact_149_sum__power2__le__zero__iff,axiom,
! [X_25: int,Y_19: int] :
( ( ord_less_eq_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 ) ) ) ) ) @ zero_zero_int )
<=> ( ( X_25 = zero_zero_int )
& ( Y_19 = zero_zero_int ) ) ) ).
thf(fact_150_sum__power2__ge__zero,axiom,
! [X_24: real,Y_18: real] : ( ord_less_eq_real @ zero_zero_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 ) ) ) ) ) ) ).
thf(fact_151_sum__power2__ge__zero,axiom,
! [X_24: int,Y_18: int] : ( ord_less_eq_int @ zero_zero_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 ) ) ) ) ) ) ).
thf(fact_152_sum__squares__gt__zero__iff,axiom,
! [X_23: real,Y_17: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_23 @ X_23 ) @ ( times_times_real @ Y_17 @ Y_17 ) ) )
<=> ( ( X_23 != zero_zero_real )
| ( Y_17 != zero_zero_real ) ) ) ).
thf(fact_153_sum__squares__gt__zero__iff,axiom,
! [X_23: int,Y_17: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_23 @ X_23 ) @ ( times_times_int @ Y_17 @ Y_17 ) ) )
<=> ( ( X_23 != zero_zero_int )
| ( Y_17 != zero_zero_int ) ) ) ).
thf(fact_154_not__sum__squares__lt__zero,axiom,
! [X_22: real,Y_16: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X_22 @ X_22 ) @ ( times_times_real @ Y_16 @ Y_16 ) ) @ zero_zero_real ) ).
thf(fact_155_not__sum__squares__lt__zero,axiom,
! [X_22: int,Y_16: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_22 @ X_22 ) @ ( times_times_int @ Y_16 @ Y_16 ) ) @ zero_zero_int ) ).
thf(fact_156_mult__numeral__1,axiom,
! [A_123: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit1 @ pls ) ) @ A_123 )
= A_123 ) ).
thf(fact_157_mult__numeral__1,axiom,
! [A_123: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit1 @ pls ) ) @ A_123 )
= A_123 ) ).
thf(fact_158_mult__numeral__1__right,axiom,
! [A_122: real] :
( ( times_times_real @ A_122 @ ( number267125858f_real @ ( bit1 @ pls ) ) )
= A_122 ) ).
thf(fact_159_mult__numeral__1__right,axiom,
! [A_122: int] :
( ( times_times_int @ A_122 @ ( number_number_of_int @ ( bit1 @ pls ) ) )
= A_122 ) ).
thf(fact_160_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_161_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_162_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_163_double__eq__0__iff,axiom,
! [A_121: real] :
( ( ( plus_plus_real @ A_121 @ A_121 )
= zero_zero_real )
<=> ( A_121 = zero_zero_real ) ) ).
thf(fact_164_double__eq__0__iff,axiom,
! [A_121: int] :
( ( ( plus_plus_int @ A_121 @ A_121 )
= zero_zero_int )
<=> ( A_121 = zero_zero_int ) ) ).
thf(fact_165_rel__simps_I46_J,axiom,
! [K: int] :
( ( bit1 @ K )
!= pls ) ).
thf(fact_166_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_167_rel__simps_I50_J,axiom,
! [K: int,L: int] :
( ( bit1 @ K )
!= ( bit0 @ L ) ) ).
thf(fact_168_rel__simps_I49_J,axiom,
! [K: int,L: int] :
( ( bit0 @ K )
!= ( bit1 @ L ) ) ).
thf(fact_169_rel__simps_I44_J,axiom,
! [K: int] :
( ( ( bit0 @ K )
= pls )
<=> ( K = pls ) ) ).
thf(fact_170_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_171_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_172_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_173_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_174_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_175_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_176_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_177_rel__simps_I14_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_178_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_179_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_180_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_181_add__Bit0__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_182_Bit0__def,axiom,
! [K: int] :
( ( bit0 @ K )
= ( plus_plus_int @ K @ K ) ) ).
thf(fact_183_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_184_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_185_zadd__strict__right__mono,axiom,
! [K: int,I_1: int,J_1: int] :
( ( ord_less_int @ I_1 @ J_1 )
=> ( ord_less_int @ ( plus_plus_int @ I_1 @ K ) @ ( plus_plus_int @ J_1 @ K ) ) ) ).
thf(fact_186_double__number__of__Bit0,axiom,
! [W_10: int] :
( ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ ( number267125858f_real @ W_10 ) )
= ( number267125858f_real @ ( bit0 @ W_10 ) ) ) ).
thf(fact_187_double__number__of__Bit0,axiom,
! [W_10: int] :
( ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ ( number_number_of_int @ W_10 ) )
= ( number_number_of_int @ ( bit0 @ W_10 ) ) ) ).
thf(fact_188_power3__eq__cube,axiom,
! [A_120: real] :
( ( power_power_real @ A_120 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( times_times_real @ A_120 @ A_120 ) @ A_120 ) ) ).
thf(fact_189_power3__eq__cube,axiom,
! [A_120: nat] :
( ( power_power_nat @ A_120 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( times_times_nat @ A_120 @ A_120 ) @ A_120 ) ) ).
thf(fact_190_power3__eq__cube,axiom,
! [A_120: int] :
( ( power_power_int @ A_120 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( times_times_int @ A_120 @ A_120 ) @ A_120 ) ) ).
thf(fact_191_semiring__mult__2,axiom,
! [Z_7: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_7 )
= ( plus_plus_real @ Z_7 @ Z_7 ) ) ).
thf(fact_192_semiring__mult__2,axiom,
! [Z_7: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_7 )
= ( plus_plus_nat @ Z_7 @ Z_7 ) ) ).
thf(fact_193_semiring__mult__2,axiom,
! [Z_7: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_7 )
= ( plus_plus_int @ Z_7 @ Z_7 ) ) ).
thf(fact_194_mult__2,axiom,
! [Z_6: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_6 )
= ( plus_plus_real @ Z_6 @ Z_6 ) ) ).
thf(fact_195_mult__2,axiom,
! [Z_6: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_6 )
= ( plus_plus_int @ Z_6 @ Z_6 ) ) ).
thf(fact_196_semiring__mult__2__right,axiom,
! [Z_5: real] :
( ( times_times_real @ Z_5 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_5 @ Z_5 ) ) ).
thf(fact_197_semiring__mult__2__right,axiom,
! [Z_5: nat] :
( ( times_times_nat @ Z_5 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ Z_5 @ Z_5 ) ) ).
thf(fact_198_semiring__mult__2__right,axiom,
! [Z_5: int] :
( ( times_times_int @ Z_5 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_5 @ Z_5 ) ) ).
thf(fact_199_mult__2__right,axiom,
! [Z_4: real] :
( ( times_times_real @ Z_4 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_4 @ Z_4 ) ) ).
thf(fact_200_mult__2__right,axiom,
! [Z_4: int] :
( ( times_times_int @ Z_4 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_4 @ Z_4 ) ) ).
thf(fact_201_power2__eq__square,axiom,
! [A_119: real] :
( ( power_power_real @ A_119 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ A_119 @ A_119 ) ) ).
thf(fact_202_power2__eq__square,axiom,
! [A_119: nat] :
( ( power_power_nat @ A_119 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ A_119 @ A_119 ) ) ).
thf(fact_203_power2__eq__square,axiom,
! [A_119: int] :
( ( power_power_int @ A_119 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ A_119 @ A_119 ) ) ).
thf(fact_204_even__less__0__iff,axiom,
! [A_118: real] :
( ( ord_less_real @ ( plus_plus_real @ A_118 @ A_118 ) @ zero_zero_real )
<=> ( ord_less_real @ A_118 @ zero_zero_real ) ) ).
thf(fact_205_even__less__0__iff,axiom,
! [A_118: int] :
( ( ord_less_int @ ( plus_plus_int @ A_118 @ A_118 ) @ zero_zero_int )
<=> ( ord_less_int @ A_118 @ zero_zero_int ) ) ).
thf(fact_206_semiring__numeral__0__eq__0,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_207_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_208_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_209_number__of__Pls,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_210_number__of__Pls,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_211_semiring__norm_I112_J,axiom,
( zero_zero_real
= ( number267125858f_real @ pls ) ) ).
thf(fact_212_semiring__norm_I112_J,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_213_add__numeral__0,axiom,
! [A_117: real] :
( ( plus_plus_real @ ( number267125858f_real @ pls ) @ A_117 )
= A_117 ) ).
thf(fact_214_add__numeral__0,axiom,
! [A_117: int] :
( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_117 )
= A_117 ) ).
thf(fact_215_add__numeral__0__right,axiom,
! [A_116: real] :
( ( plus_plus_real @ A_116 @ ( number267125858f_real @ pls ) )
= A_116 ) ).
thf(fact_216_add__numeral__0__right,axiom,
! [A_116: int] :
( ( plus_plus_int @ A_116 @ ( number_number_of_int @ pls ) )
= A_116 ) ).
thf(fact_217_less__number__of,axiom,
! [X_21: int,Y_15: int] :
( ( ord_less_real @ ( number267125858f_real @ X_21 ) @ ( number267125858f_real @ Y_15 ) )
<=> ( ord_less_int @ X_21 @ Y_15 ) ) ).
thf(fact_218_less__number__of,axiom,
! [X_21: int,Y_15: int] :
( ( ord_less_int @ ( number_number_of_int @ X_21 ) @ ( number_number_of_int @ Y_15 ) )
<=> ( ord_less_int @ X_21 @ Y_15 ) ) ).
thf(fact_219_add__number__of__left,axiom,
! [V_5: int,W_9: int,Z_3: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_5 ) @ ( plus_plus_real @ ( number267125858f_real @ W_9 ) @ Z_3 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_5 @ W_9 ) ) @ Z_3 ) ) ).
thf(fact_220_add__number__of__left,axiom,
! [V_5: int,W_9: int,Z_3: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_5 ) @ ( plus_plus_int @ ( number_number_of_int @ W_9 ) @ Z_3 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_5 @ W_9 ) ) @ Z_3 ) ) ).
thf(fact_221_add__number__of__eq,axiom,
! [V_4: int,W_8: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_4 ) @ ( number267125858f_real @ W_8 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_4 @ W_8 ) ) ) ).
thf(fact_222_add__number__of__eq,axiom,
! [V_4: int,W_8: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_4 ) @ ( number_number_of_int @ W_8 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_4 @ W_8 ) ) ) ).
thf(fact_223_number__of__add,axiom,
! [V_3: int,W_7: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_3 @ W_7 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_3 ) @ ( number267125858f_real @ W_7 ) ) ) ).
thf(fact_224_number__of__add,axiom,
! [V_3: int,W_7: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_3 @ W_7 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_3 ) @ ( number_number_of_int @ W_7 ) ) ) ).
thf(fact_225_rel__simps_I12_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_226_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_227_rel__simps_I16_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_228_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_229_rel__simps_I10_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_230_rel__simps_I4_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_int @ pls @ K ) ) ).
thf(fact_231_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_232_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_233_add__Bit1__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_234_add__Bit0__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_235_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_236_Bit1__def,axiom,
! [K: int] :
( ( bit1 @ K )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
thf(fact_237_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_238_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_239_power2__eq__square__number__of,axiom,
! [W_6: int] :
( ( power_power_real @ ( number267125858f_real @ W_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( number267125858f_real @ W_6 ) @ ( number267125858f_real @ W_6 ) ) ) ).
thf(fact_240_power2__eq__square__number__of,axiom,
! [W_6: int] :
( ( power_power_nat @ ( number_number_of_nat @ W_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ W_6 ) @ ( number_number_of_nat @ W_6 ) ) ) ).
thf(fact_241_power2__eq__square__number__of,axiom,
! [W_6: int] :
( ( power_power_int @ ( number_number_of_int @ W_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( number_number_of_int @ W_6 ) @ ( number_number_of_int @ W_6 ) ) ) ).
thf(fact_242_power2__sum,axiom,
! [X_20: real,Y_14: real] :
( ( power_power_real @ ( plus_plus_real @ X_20 @ Y_14 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_20 ) @ Y_14 ) ) ) ).
thf(fact_243_power2__sum,axiom,
! [X_20: nat,Y_14: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X_20 @ Y_14 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_20 ) @ Y_14 ) ) ) ).
thf(fact_244_power2__sum,axiom,
! [X_20: int,Y_14: int] :
( ( power_power_int @ ( plus_plus_int @ X_20 @ Y_14 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X_20 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_20 ) @ Y_14 ) ) ) ).
thf(fact_245_number__of__Bit0,axiom,
! [W_5: int] :
( ( number267125858f_real @ ( bit0 @ W_5 ) )
= ( plus_plus_real @ ( plus_plus_real @ zero_zero_real @ ( number267125858f_real @ W_5 ) ) @ ( number267125858f_real @ W_5 ) ) ) ).
thf(fact_246_number__of__Bit0,axiom,
! [W_5: int] :
( ( number_number_of_int @ ( bit0 @ W_5 ) )
= ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ ( number_number_of_int @ W_5 ) ) @ ( number_number_of_int @ W_5 ) ) ) ).
thf(fact_247_number__of__Bit1,axiom,
! [W_4: int] :
( ( number267125858f_real @ ( bit1 @ W_4 ) )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_4 ) ) @ ( number267125858f_real @ W_4 ) ) ) ).
thf(fact_248_number__of__Bit1,axiom,
! [W_4: int] :
( ( number_number_of_int @ ( bit1 @ W_4 ) )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_4 ) ) @ ( number_number_of_int @ W_4 ) ) ) ).
thf(fact_249_semiring__numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_250_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_251_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_252_numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_253_numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_254_semiring__norm_I110_J,axiom,
( one_one_real
= ( number267125858f_real @ ( bit1 @ pls ) ) ) ).
thf(fact_255_semiring__norm_I110_J,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_256_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_257_less__special_I3_J,axiom,
! [X_19: int] :
( ( ord_less_real @ ( number267125858f_real @ X_19 ) @ zero_zero_real )
<=> ( ord_less_int @ X_19 @ pls ) ) ).
thf(fact_258_less__special_I3_J,axiom,
! [X_19: int] :
( ( ord_less_int @ ( number_number_of_int @ X_19 ) @ zero_zero_int )
<=> ( ord_less_int @ X_19 @ pls ) ) ).
thf(fact_259_less__special_I1_J,axiom,
! [Y_13: int] :
( ( ord_less_real @ zero_zero_real @ ( number267125858f_real @ Y_13 ) )
<=> ( ord_less_int @ pls @ Y_13 ) ) ).
thf(fact_260_less__special_I1_J,axiom,
! [Y_13: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_13 ) )
<=> ( ord_less_int @ pls @ Y_13 ) ) ).
thf(fact_261_semiring__one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_262_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_263_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_264_one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_265_one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_266_less__special_I4_J,axiom,
! [X_18: int] :
( ( ord_less_real @ ( number267125858f_real @ X_18 ) @ one_one_real )
<=> ( ord_less_int @ X_18 @ ( bit1 @ pls ) ) ) ).
thf(fact_267_less__special_I4_J,axiom,
! [X_18: int] :
( ( ord_less_int @ ( number_number_of_int @ X_18 ) @ one_one_int )
<=> ( ord_less_int @ X_18 @ ( bit1 @ pls ) ) ) ).
thf(fact_268_less__special_I2_J,axiom,
! [Y_12: int] :
( ( ord_less_real @ one_one_real @ ( number267125858f_real @ Y_12 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_12 ) ) ).
thf(fact_269_less__special_I2_J,axiom,
! [Y_12: int] :
( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_12 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_12 ) ) ).
thf(fact_270__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_271_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_272_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_273_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_274_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_275_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_276_power2__ge__self,axiom,
! [X: int] : ( ord_less_eq_int @ X @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_277_self__quotient__aux1,axiom,
! [R: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_int @ R @ A )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_278_self__quotient__aux2,axiom,
! [R: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_279_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_280_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: real,N_25: nat] :
( ( power_power_real @ X_17 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_25 ) )
= ( times_times_real @ ( power_power_real @ X_17 @ N_25 ) @ ( power_power_real @ X_17 @ N_25 ) ) ) ).
thf(fact_281_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: nat,N_25: nat] :
( ( power_power_nat @ X_17 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_25 ) )
= ( times_times_nat @ ( power_power_nat @ X_17 @ N_25 ) @ ( power_power_nat @ X_17 @ N_25 ) ) ) ).
thf(fact_282_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: int,N_25: nat] :
( ( power_power_int @ X_17 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_25 ) )
= ( times_times_int @ ( power_power_int @ X_17 @ N_25 ) @ ( power_power_int @ X_17 @ N_25 ) ) ) ).
thf(fact_283_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_16: real] :
( ( times_times_real @ X_16 @ X_16 )
= ( power_power_real @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_284_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_16: nat] :
( ( times_times_nat @ X_16 @ X_16 )
= ( power_power_nat @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_285_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_16: int] :
( ( times_times_int @ X_16 @ X_16 )
= ( power_power_int @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_286__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ s1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_287__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ X_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ Y_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_288__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_1: int] :
~ ( ( ord_less_eq_int @ zero_zero_int @ S_1 )
& ( ord_less_int @ S_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ S_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_289_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_290_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_291_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_292_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: real,P_5: nat,Q_6: nat] :
( ( power_power_real @ ( power_power_real @ X_15 @ P_5 ) @ Q_6 )
= ( power_power_real @ X_15 @ ( times_times_nat @ P_5 @ Q_6 ) ) ) ).
thf(fact_293_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: nat,P_5: nat,Q_6: nat] :
( ( power_power_nat @ ( power_power_nat @ X_15 @ P_5 ) @ Q_6 )
= ( power_power_nat @ X_15 @ ( times_times_nat @ P_5 @ Q_6 ) ) ) ).
thf(fact_294_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: int,P_5: nat,Q_6: nat] :
( ( power_power_int @ ( power_power_int @ X_15 @ P_5 ) @ Q_6 )
= ( power_power_int @ X_15 @ ( times_times_nat @ P_5 @ Q_6 ) ) ) ).
thf(fact_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_304_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_305_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( ( twoSqu919416604sum2sq @ X )
=> ( ( twoSqu919416604sum2sq @ Y )
=> ( twoSqu919416604sum2sq @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_306_le__nat__number__of,axiom,
! [V: int,V_2: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
<=> ( ~ ( ord_less_eq_int @ V @ V_2 )
=> ( ord_less_eq_int @ V @ pls ) ) ) ).
thf(fact_307_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: real,Ly_4: real,Rx_6: real,Ry_4: real] :
( ( times_times_real @ ( times_times_real @ Lx_6 @ Ly_4 ) @ ( times_times_real @ Rx_6 @ Ry_4 ) )
= ( times_times_real @ ( times_times_real @ Lx_6 @ Rx_6 ) @ ( times_times_real @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_6 @ Ly_4 ) @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_6 @ Rx_6 ) @ ( times_times_nat @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_309_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_6 @ Ly_4 ) @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ ( times_times_int @ Lx_6 @ Rx_6 ) @ ( times_times_int @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_310_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: real,Ly_3: real,Rx_5: real,Ry_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_5 @ Ly_3 ) @ ( times_times_real @ Rx_5 @ Ry_3 ) )
= ( times_times_real @ Rx_5 @ ( times_times_real @ ( times_times_real @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_311_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ Rx_5 @ ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_312_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ Rx_5 @ ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: real,Ly_2: real,Rx_4: real,Ry_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_4 @ Ly_2 ) @ ( times_times_real @ Rx_4 @ Ry_2 ) )
= ( times_times_real @ Lx_4 @ ( times_times_real @ Ly_2 @ ( times_times_real @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_314_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_2 ) @ ( times_times_nat @ Rx_4 @ Ry_2 ) )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_315_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_2 ) @ ( times_times_int @ Rx_4 @ Ry_2 ) )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_316_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_1: real,Rx_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_real @ ( times_times_real @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_317_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_1: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_318_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: real,Ly: real,Rx_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_real @ Lx_2 @ ( times_times_real @ Ly @ Rx_2 ) ) ) ).
thf(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly @ Rx_2 ) ) ) ).
thf(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly @ Rx_2 ) ) ) ).
thf(fact_322_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: real,Rx_1: real,Ry_1: real] :
( ( times_times_real @ Lx_1 @ ( times_times_real @ Rx_1 @ Ry_1 ) )
= ( times_times_real @ ( times_times_real @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ Lx_1 @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_324_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ Lx_1 @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ ( times_times_int @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: real,Rx: real,Ry: real] :
( ( times_times_real @ Lx @ ( times_times_real @ Rx @ Ry ) )
= ( times_times_real @ Rx @ ( times_times_real @ Lx @ Ry ) ) ) ).
thf(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ Lx @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ Rx @ ( times_times_nat @ Lx @ Ry ) ) ) ).
thf(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] :
( ( times_times_int @ Lx @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ Rx @ ( times_times_int @ Lx @ Ry ) ) ) ).
thf(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_115: real,B_78: real] :
( ( times_times_real @ A_115 @ B_78 )
= ( times_times_real @ B_78 @ A_115 ) ) ).
thf(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_115: nat,B_78: nat] :
( ( times_times_nat @ A_115 @ B_78 )
= ( times_times_nat @ B_78 @ A_115 ) ) ).
thf(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_115: int,B_78: int] :
( ( times_times_int @ A_115 @ B_78 )
= ( times_times_int @ B_78 @ A_115 ) ) ).
thf(fact_331_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_114: real,C_49: real] :
( ( plus_plus_real @ A_114 @ C_49 )
= ( plus_plus_real @ C_49 @ A_114 ) ) ).
thf(fact_332_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_114: nat,C_49: nat] :
( ( plus_plus_nat @ A_114 @ C_49 )
= ( plus_plus_nat @ C_49 @ A_114 ) ) ).
thf(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_114: int,C_49: int] :
( ( plus_plus_int @ A_114 @ C_49 )
= ( plus_plus_int @ C_49 @ A_114 ) ) ).
thf(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_113: real,C_48: real,D_14: real] :
( ( plus_plus_real @ A_113 @ ( plus_plus_real @ C_48 @ D_14 ) )
= ( plus_plus_real @ C_48 @ ( plus_plus_real @ A_113 @ D_14 ) ) ) ).
thf(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_113: nat,C_48: nat,D_14: nat] :
( ( plus_plus_nat @ A_113 @ ( plus_plus_nat @ C_48 @ D_14 ) )
= ( plus_plus_nat @ C_48 @ ( plus_plus_nat @ A_113 @ D_14 ) ) ) ).
thf(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_113: int,C_48: int,D_14: int] :
( ( plus_plus_int @ A_113 @ ( plus_plus_int @ C_48 @ D_14 ) )
= ( plus_plus_int @ C_48 @ ( plus_plus_int @ A_113 @ D_14 ) ) ) ).
thf(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_112: real,C_47: real,D_13: real] :
( ( plus_plus_real @ A_112 @ ( plus_plus_real @ C_47 @ D_13 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_112 @ C_47 ) @ D_13 ) ) ).
thf(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_112: nat,C_47: nat,D_13: nat] :
( ( plus_plus_nat @ A_112 @ ( plus_plus_nat @ C_47 @ D_13 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_112 @ C_47 ) @ D_13 ) ) ).
thf(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_112: int,C_47: int,D_13: int] :
( ( plus_plus_int @ A_112 @ ( plus_plus_int @ C_47 @ D_13 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_112 @ C_47 ) @ D_13 ) ) ).
thf(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_111: real,B_77: real,C_46: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_111 @ B_77 ) @ C_46 )
= ( plus_plus_real @ A_111 @ ( plus_plus_real @ B_77 @ C_46 ) ) ) ).
thf(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_111: nat,B_77: nat,C_46: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_111 @ B_77 ) @ C_46 )
= ( plus_plus_nat @ A_111 @ ( plus_plus_nat @ B_77 @ C_46 ) ) ) ).
thf(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_111: int,B_77: int,C_46: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_111 @ B_77 ) @ C_46 )
= ( plus_plus_int @ A_111 @ ( plus_plus_int @ B_77 @ C_46 ) ) ) ).
thf(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_110: real,B_76: real,C_45: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_110 @ B_76 ) @ C_45 )
= ( plus_plus_real @ ( plus_plus_real @ A_110 @ C_45 ) @ B_76 ) ) ).
thf(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_110: nat,B_76: nat,C_45: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_110 @ B_76 ) @ C_45 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_110 @ C_45 ) @ B_76 ) ) ).
thf(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_110: int,B_76: int,C_45: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_110 @ B_76 ) @ C_45 )
= ( plus_plus_int @ ( plus_plus_int @ A_110 @ C_45 ) @ B_76 ) ) ).
thf(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_109: real,B_75: real,C_44: real,D_12: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_109 @ B_75 ) @ ( plus_plus_real @ C_44 @ D_12 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_109 @ C_44 ) @ ( plus_plus_real @ B_75 @ D_12 ) ) ) ).
thf(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_109: nat,B_75: nat,C_44: nat,D_12: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_109 @ B_75 ) @ ( plus_plus_nat @ C_44 @ D_12 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_109 @ C_44 ) @ ( plus_plus_nat @ B_75 @ D_12 ) ) ) ).
thf(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_109: int,B_75: int,C_44: int,D_12: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_109 @ B_75 ) @ ( plus_plus_int @ C_44 @ D_12 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_109 @ C_44 ) @ ( plus_plus_int @ B_75 @ D_12 ) ) ) ).
thf(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_14: real] :
( ( power_power_real @ X_14 @ one_one_nat )
= X_14 ) ).
thf(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_14: nat] :
( ( power_power_nat @ X_14 @ one_one_nat )
= X_14 ) ).
thf(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_14: int] :
( ( power_power_int @ X_14 @ one_one_nat )
= X_14 ) ).
thf(fact_352_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_353_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_354_nat__mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
thf(fact_355_mult__nat__number__of,axiom,
! [V_2: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_nat @ ( times_times_int @ V @ V_2 ) ) ) ) ) ).
thf(fact_356_nat__number__of__mult__left,axiom,
! [V_2: int,K: nat,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_2 ) @ K ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_2 ) @ K ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V @ V_2 ) ) @ K ) ) ) ) ).
thf(fact_357_power__even__eq,axiom,
! [A_108: real,N_24: nat] :
( ( power_power_real @ A_108 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( power_power_real @ ( power_power_real @ A_108 @ N_24 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_358_power__even__eq,axiom,
! [A_108: nat,N_24: nat] :
( ( power_power_nat @ A_108 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( power_power_nat @ ( power_power_nat @ A_108 @ N_24 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_359_power__even__eq,axiom,
! [A_108: int,N_24: nat] :
( ( power_power_int @ A_108 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( power_power_int @ ( power_power_int @ A_108 @ N_24 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_360_even__power__le__0__imp__0,axiom,
! [A_107: real,K_3: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A_107 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_3 ) ) @ zero_zero_real )
=> ( A_107 = zero_zero_real ) ) ).
thf(fact_361_even__power__le__0__imp__0,axiom,
! [A_107: int,K_3: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A_107 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_3 ) ) @ zero_zero_int )
=> ( A_107 = zero_zero_int ) ) ).
thf(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_106: real] :
( ( times_times_real @ zero_zero_real @ A_106 )
= zero_zero_real ) ).
thf(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_106: nat] :
( ( times_times_nat @ zero_zero_nat @ A_106 )
= zero_zero_nat ) ).
thf(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_106: int] :
( ( times_times_int @ zero_zero_int @ A_106 )
= zero_zero_int ) ).
thf(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_105: real] :
( ( times_times_real @ A_105 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_105: nat] :
( ( times_times_nat @ A_105 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_367_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_105: int] :
( ( times_times_int @ A_105 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_368_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_104: real] :
( ( plus_plus_real @ zero_zero_real @ A_104 )
= A_104 ) ).
thf(fact_369_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_104: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_104 )
= A_104 ) ).
thf(fact_370_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_104: int] :
( ( plus_plus_int @ zero_zero_int @ A_104 )
= A_104 ) ).
thf(fact_371_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_103: real] :
( ( plus_plus_real @ A_103 @ zero_zero_real )
= A_103 ) ).
thf(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_103: nat] :
( ( plus_plus_nat @ A_103 @ zero_zero_nat )
= A_103 ) ).
thf(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_103: int] :
( ( plus_plus_int @ A_103 @ zero_zero_int )
= A_103 ) ).
thf(fact_374_add__0__iff,axiom,
! [B_74: real,A_102: real] :
( ( B_74
= ( plus_plus_real @ B_74 @ A_102 ) )
<=> ( A_102 = zero_zero_real ) ) ).
thf(fact_375_add__0__iff,axiom,
! [B_74: nat,A_102: nat] :
( ( B_74
= ( plus_plus_nat @ B_74 @ A_102 ) )
<=> ( A_102 = zero_zero_nat ) ) ).
thf(fact_376_add__0__iff,axiom,
! [B_74: int,A_102: int] :
( ( B_74
= ( plus_plus_int @ B_74 @ A_102 ) )
<=> ( A_102 = zero_zero_int ) ) ).
thf(fact_377_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_13: real,Y_11: real,Z_2: real] :
( ( times_times_real @ X_13 @ ( plus_plus_real @ Y_11 @ Z_2 ) )
= ( plus_plus_real @ ( times_times_real @ X_13 @ Y_11 ) @ ( times_times_real @ X_13 @ Z_2 ) ) ) ).
thf(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_13: nat,Y_11: nat,Z_2: nat] :
( ( times_times_nat @ X_13 @ ( plus_plus_nat @ Y_11 @ Z_2 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_13 @ Y_11 ) @ ( times_times_nat @ X_13 @ Z_2 ) ) ) ).
thf(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_13: int,Y_11: int,Z_2: int] :
( ( times_times_int @ X_13 @ ( plus_plus_int @ Y_11 @ Z_2 ) )
= ( plus_plus_int @ ( times_times_int @ X_13 @ Y_11 ) @ ( times_times_int @ X_13 @ Z_2 ) ) ) ).
thf(fact_380_crossproduct__noteq,axiom,
! [C_43: real,D_11: real,A_101: real,B_73: real] :
( ( ( A_101 != B_73 )
& ( C_43 != D_11 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_101 @ C_43 ) @ ( times_times_real @ B_73 @ D_11 ) )
!= ( plus_plus_real @ ( times_times_real @ A_101 @ D_11 ) @ ( times_times_real @ B_73 @ C_43 ) ) ) ) ).
thf(fact_381_crossproduct__noteq,axiom,
! [C_43: nat,D_11: nat,A_101: nat,B_73: nat] :
( ( ( A_101 != B_73 )
& ( C_43 != D_11 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_101 @ C_43 ) @ ( times_times_nat @ B_73 @ D_11 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_101 @ D_11 ) @ ( times_times_nat @ B_73 @ C_43 ) ) ) ) ).
thf(fact_382_crossproduct__noteq,axiom,
! [C_43: int,D_11: int,A_101: int,B_73: int] :
( ( ( A_101 != B_73 )
& ( C_43 != D_11 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_101 @ C_43 ) @ ( times_times_int @ B_73 @ D_11 ) )
!= ( plus_plus_int @ ( times_times_int @ A_101 @ D_11 ) @ ( times_times_int @ B_73 @ C_43 ) ) ) ) ).
thf(fact_383_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_100: real,B_72: real,C_42: real] :
( ( times_times_real @ ( plus_plus_real @ A_100 @ B_72 ) @ C_42 )
= ( plus_plus_real @ ( times_times_real @ A_100 @ C_42 ) @ ( times_times_real @ B_72 @ C_42 ) ) ) ).
thf(fact_384_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_100: nat,B_72: nat,C_42: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_100 @ B_72 ) @ C_42 )
= ( plus_plus_nat @ ( times_times_nat @ A_100 @ C_42 ) @ ( times_times_nat @ B_72 @ C_42 ) ) ) ).
thf(fact_385_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_100: int,B_72: int,C_42: int] :
( ( times_times_int @ ( plus_plus_int @ A_100 @ B_72 ) @ C_42 )
= ( plus_plus_int @ ( times_times_int @ A_100 @ C_42 ) @ ( times_times_int @ B_72 @ C_42 ) ) ) ).
thf(fact_386_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99: real,M_13: real,B_71: real] :
( ( plus_plus_real @ ( times_times_real @ A_99 @ M_13 ) @ ( times_times_real @ B_71 @ M_13 ) )
= ( times_times_real @ ( plus_plus_real @ A_99 @ B_71 ) @ M_13 ) ) ).
thf(fact_387_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99: nat,M_13: nat,B_71: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_99 @ M_13 ) @ ( times_times_nat @ B_71 @ M_13 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_99 @ B_71 ) @ M_13 ) ) ).
thf(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99: int,M_13: int,B_71: int] :
( ( plus_plus_int @ ( times_times_int @ A_99 @ M_13 ) @ ( times_times_int @ B_71 @ M_13 ) )
= ( times_times_int @ ( plus_plus_int @ A_99 @ B_71 ) @ M_13 ) ) ).
thf(fact_389_crossproduct__eq,axiom,
! [W_3: real,Y_10: real,X_12: real,Z_1: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_3 @ Y_10 ) @ ( times_times_real @ X_12 @ Z_1 ) )
= ( plus_plus_real @ ( times_times_real @ W_3 @ Z_1 ) @ ( times_times_real @ X_12 @ Y_10 ) ) )
<=> ( ( W_3 = X_12 )
| ( Y_10 = Z_1 ) ) ) ).
thf(fact_390_crossproduct__eq,axiom,
! [W_3: nat,Y_10: nat,X_12: nat,Z_1: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_3 @ Y_10 ) @ ( times_times_nat @ X_12 @ Z_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_3 @ Z_1 ) @ ( times_times_nat @ X_12 @ Y_10 ) ) )
<=> ( ( W_3 = X_12 )
| ( Y_10 = Z_1 ) ) ) ).
thf(fact_391_crossproduct__eq,axiom,
! [W_3: int,Y_10: int,X_12: int,Z_1: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_3 @ Y_10 ) @ ( times_times_int @ X_12 @ Z_1 ) )
= ( plus_plus_int @ ( times_times_int @ W_3 @ Z_1 ) @ ( times_times_int @ X_12 @ Y_10 ) ) )
<=> ( ( W_3 = X_12 )
| ( Y_10 = Z_1 ) ) ) ).
thf(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_98: real] :
( ( times_times_real @ one_one_real @ A_98 )
= A_98 ) ).
thf(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_98: nat] :
( ( times_times_nat @ one_one_nat @ A_98 )
= A_98 ) ).
thf(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_98: int] :
( ( times_times_int @ one_one_int @ A_98 )
= A_98 ) ).
thf(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_97: real] :
( ( times_times_real @ A_97 @ one_one_real )
= A_97 ) ).
thf(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_97: nat] :
( ( times_times_nat @ A_97 @ one_one_nat )
= A_97 ) ).
thf(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_97: int] :
( ( times_times_int @ A_97 @ one_one_int )
= A_97 ) ).
thf(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_11: real,Y_9: real,Q_5: nat] :
( ( power_power_real @ ( times_times_real @ X_11 @ Y_9 ) @ Q_5 )
= ( times_times_real @ ( power_power_real @ X_11 @ Q_5 ) @ ( power_power_real @ Y_9 @ Q_5 ) ) ) ).
thf(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_11: nat,Y_9: nat,Q_5: nat] :
( ( power_power_nat @ ( times_times_nat @ X_11 @ Y_9 ) @ Q_5 )
= ( times_times_nat @ ( power_power_nat @ X_11 @ Q_5 ) @ ( power_power_nat @ Y_9 @ Q_5 ) ) ) ).
thf(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_11: int,Y_9: int,Q_5: nat] :
( ( power_power_int @ ( times_times_int @ X_11 @ Y_9 ) @ Q_5 )
= ( times_times_int @ ( power_power_int @ X_11 @ Q_5 ) @ ( power_power_int @ Y_9 @ Q_5 ) ) ) ).
thf(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_10: real,P_4: nat,Q_4: nat] :
( ( times_times_real @ ( power_power_real @ X_10 @ P_4 ) @ ( power_power_real @ X_10 @ Q_4 ) )
= ( power_power_real @ X_10 @ ( plus_plus_nat @ P_4 @ Q_4 ) ) ) ).
thf(fact_402_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_10: nat,P_4: nat,Q_4: nat] :
( ( times_times_nat @ ( power_power_nat @ X_10 @ P_4 ) @ ( power_power_nat @ X_10 @ Q_4 ) )
= ( power_power_nat @ X_10 @ ( plus_plus_nat @ P_4 @ Q_4 ) ) ) ).
thf(fact_403_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_10: int,P_4: nat,Q_4: nat] :
( ( times_times_int @ ( power_power_int @ X_10 @ P_4 ) @ ( power_power_int @ X_10 @ Q_4 ) )
= ( power_power_int @ X_10 @ ( plus_plus_nat @ P_4 @ Q_4 ) ) ) ).
thf(fact_404_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_9: real] :
( ( power_power_real @ X_9 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_405_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_9: nat] :
( ( power_power_nat @ X_9 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_406_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_9: int] :
( ( power_power_int @ X_9 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_407_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_408_zero__le__even__power_H,axiom,
! [A_96: real,N_23: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_96 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_23 ) ) ) ).
thf(fact_409_zero__le__even__power_H,axiom,
! [A_96: int,N_23: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_96 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_23 ) ) ) ).
thf(fact_410_add__scale__eq__noteq,axiom,
! [C_41: real,D_10: real,A_95: real,B_70: real,R_3: real] :
( ( R_3 != zero_zero_real )
=> ( ( ( A_95 = B_70 )
& ( C_41 != D_10 ) )
=> ( ( plus_plus_real @ A_95 @ ( times_times_real @ R_3 @ C_41 ) )
!= ( plus_plus_real @ B_70 @ ( times_times_real @ R_3 @ D_10 ) ) ) ) ) ).
thf(fact_411_add__scale__eq__noteq,axiom,
! [C_41: nat,D_10: nat,A_95: nat,B_70: nat,R_3: nat] :
( ( R_3 != zero_zero_nat )
=> ( ( ( A_95 = B_70 )
& ( C_41 != D_10 ) )
=> ( ( plus_plus_nat @ A_95 @ ( times_times_nat @ R_3 @ C_41 ) )
!= ( plus_plus_nat @ B_70 @ ( times_times_nat @ R_3 @ D_10 ) ) ) ) ) ).
thf(fact_412_add__scale__eq__noteq,axiom,
! [C_41: int,D_10: int,A_95: int,B_70: int,R_3: int] :
( ( R_3 != zero_zero_int )
=> ( ( ( A_95 = B_70 )
& ( C_41 != D_10 ) )
=> ( ( plus_plus_int @ A_95 @ ( times_times_int @ R_3 @ C_41 ) )
!= ( plus_plus_int @ B_70 @ ( times_times_int @ R_3 @ D_10 ) ) ) ) ) ).
thf(fact_413_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_12: real] :
( ( plus_plus_real @ M_12 @ M_12 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_12 ) ) ).
thf(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_12: nat] :
( ( plus_plus_nat @ M_12 @ M_12 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_12 ) ) ).
thf(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_12: int] :
( ( plus_plus_int @ M_12 @ M_12 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_12 ) ) ).
thf(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: real,A_94: real] :
( ( plus_plus_real @ M_11 @ ( times_times_real @ A_94 @ M_11 ) )
= ( times_times_real @ ( plus_plus_real @ A_94 @ one_one_real ) @ M_11 ) ) ).
thf(fact_417_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: nat,A_94: nat] :
( ( plus_plus_nat @ M_11 @ ( times_times_nat @ A_94 @ M_11 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_94 @ one_one_nat ) @ M_11 ) ) ).
thf(fact_418_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: int,A_94: int] :
( ( plus_plus_int @ M_11 @ ( times_times_int @ A_94 @ M_11 ) )
= ( times_times_int @ ( plus_plus_int @ A_94 @ one_one_int ) @ M_11 ) ) ).
thf(fact_419_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_93: real,M_10: real] :
( ( plus_plus_real @ ( times_times_real @ A_93 @ M_10 ) @ M_10 )
= ( times_times_real @ ( plus_plus_real @ A_93 @ one_one_real ) @ M_10 ) ) ).
thf(fact_420_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_93: nat,M_10: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_93 @ M_10 ) @ M_10 )
= ( times_times_nat @ ( plus_plus_nat @ A_93 @ one_one_nat ) @ M_10 ) ) ).
thf(fact_421_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_93: int,M_10: int] :
( ( plus_plus_int @ ( times_times_int @ A_93 @ M_10 ) @ M_10 )
= ( times_times_int @ ( plus_plus_int @ A_93 @ one_one_int ) @ M_10 ) ) ).
thf(fact_422_power__eq__0__iff__number__of,axiom,
! [A_92: real,W_2: int] :
( ( ( power_power_real @ A_92 @ ( number_number_of_nat @ W_2 ) )
= zero_zero_real )
<=> ( ( A_92 = zero_zero_real )
& ( ( number_number_of_nat @ W_2 )
!= zero_zero_nat ) ) ) ).
thf(fact_423_power__eq__0__iff__number__of,axiom,
! [A_92: nat,W_2: int] :
( ( ( power_power_nat @ A_92 @ ( number_number_of_nat @ W_2 ) )
= zero_zero_nat )
<=> ( ( A_92 = zero_zero_nat )
& ( ( number_number_of_nat @ W_2 )
!= zero_zero_nat ) ) ) ).
thf(fact_424_power__eq__0__iff__number__of,axiom,
! [A_92: int,W_2: int] :
( ( ( power_power_int @ A_92 @ ( number_number_of_nat @ W_2 ) )
= zero_zero_int )
<=> ( ( A_92 = zero_zero_int )
& ( ( number_number_of_nat @ W_2 )
!= zero_zero_nat ) ) ) ).
thf(fact_425_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_426_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_427_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_428_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_429_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_430_power__0__left__number__of,axiom,
! [W_1: int] :
( ( ( ( number_number_of_nat @ W_1 )
= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_1 ) )
= one_one_real ) )
& ( ( ( number_number_of_nat @ W_1 )
!= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_1 ) )
= zero_zero_real ) ) ) ).
thf(fact_431_power__0__left__number__of,axiom,
! [W_1: int] :
( ( ( ( number_number_of_nat @ W_1 )
= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_1 ) )
= one_one_nat ) )
& ( ( ( number_number_of_nat @ W_1 )
!= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_1 ) )
= zero_zero_nat ) ) ) ).
thf(fact_432_power__0__left__number__of,axiom,
! [W_1: int] :
( ( ( ( number_number_of_nat @ W_1 )
= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_1 ) )
= one_one_int ) )
& ( ( ( number_number_of_nat @ W_1 )
!= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_1 ) )
= zero_zero_int ) ) ) ).
thf(fact_433_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_434_q__pos__lemma,axiom,
! [B_69: int,Q_3: int,R_2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_69 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_69 )
=> ( ( ord_less_int @ zero_zero_int @ B_69 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_3 ) ) ) ) ).
thf(fact_435_q__neg__lemma,axiom,
! [B_69: int,Q_3: int,R_2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_69 @ Q_3 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_69 )
=> ( ord_less_eq_int @ Q_3 @ zero_zero_int ) ) ) ) ).
thf(fact_436_unique__quotient__lemma,axiom,
! [B: int,Q_3: int,R_2: int,Q: int,R: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_3 ) @ R_2 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ R_2 @ B )
=> ( ( ord_less_int @ R @ B )
=> ( ord_less_eq_int @ Q_3 @ Q ) ) ) ) ) ).
thf(fact_437_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R: int,B_69: int,Q_3: int,R_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_69 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_69 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_69 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B_69 )
=> ( ( ord_less_eq_int @ B_69 @ B )
=> ( ord_less_eq_int @ Q @ Q_3 ) ) ) ) ) ) ) ).
thf(fact_438_unique__quotient__lemma__neg,axiom,
! [B: int,Q_3: int,R_2: int,Q: int,R: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_3 ) @ R_2 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( ord_less_int @ B @ R_2 )
=> ( ord_less_eq_int @ Q @ Q_3 ) ) ) ) ) ).
thf(fact_439_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R: int,B_69: int,Q_3: int,R_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_69 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_69 @ Q_3 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_int @ R @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_69 )
=> ( ( ord_less_eq_int @ B_69 @ B )
=> ( ord_less_eq_int @ Q_3 @ Q ) ) ) ) ) ) ) ).
thf(fact_440_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_441_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_442_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_443_zprime__def,axiom,
! [P_1: int] :
( ( zprime @ P_1 )
<=> ( ( ord_less_int @ one_one_int @ P_1 )
& ! [M_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M_2 )
& ( dvd_dvd_int @ M_2 @ P_1 ) )
=> ( ( M_2 = one_one_int )
| ( M_2 = P_1 ) ) ) ) ) ).
thf(fact_444_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_445_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_446_power__strict__mono,axiom,
! [N_22: nat,A_91: real,B_68: real] :
( ( ord_less_real @ A_91 @ B_68 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_91 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_22 )
=> ( ord_less_real @ ( power_power_real @ A_91 @ N_22 ) @ ( power_power_real @ B_68 @ N_22 ) ) ) ) ) ).
thf(fact_447_power__strict__mono,axiom,
! [N_22: nat,A_91: nat,B_68: nat] :
( ( ord_less_nat @ A_91 @ B_68 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_91 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_22 )
=> ( ord_less_nat @ ( power_power_nat @ A_91 @ N_22 ) @ ( power_power_nat @ B_68 @ N_22 ) ) ) ) ) ).
thf(fact_448_power__strict__mono,axiom,
! [N_22: nat,A_91: int,B_68: int] :
( ( ord_less_int @ A_91 @ B_68 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_91 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_22 )
=> ( ord_less_int @ ( power_power_int @ A_91 @ N_22 ) @ ( power_power_int @ B_68 @ N_22 ) ) ) ) ) ).
thf(fact_449_convex__bound__lt,axiom,
! [V_1: real,U_1: real,Y_8: real,X_8: real,A_90: real] :
( ( ord_less_real @ X_8 @ A_90 )
=> ( ( ord_less_real @ Y_8 @ A_90 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_1 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_1 )
=> ( ( ( plus_plus_real @ U_1 @ V_1 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U_1 @ X_8 ) @ ( times_times_real @ V_1 @ Y_8 ) ) @ A_90 ) ) ) ) ) ) ).
thf(fact_450_convex__bound__lt,axiom,
! [V_1: int,U_1: int,Y_8: int,X_8: int,A_90: int] :
( ( ord_less_int @ X_8 @ A_90 )
=> ( ( ord_less_int @ Y_8 @ A_90 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_1 )
=> ( ( ( plus_plus_int @ U_1 @ V_1 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U_1 @ X_8 ) @ ( times_times_int @ V_1 @ Y_8 ) ) @ A_90 ) ) ) ) ) ) ).
thf(fact_451_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_452_dvd__0__right,axiom,
! [A_89: nat] : ( dvd_dvd_nat @ A_89 @ zero_zero_nat ) ).
thf(fact_453_dvd__0__right,axiom,
! [A_89: int] : ( dvd_dvd_int @ A_89 @ zero_zero_int ) ).
thf(fact_454_dvd__0__right,axiom,
! [A_89: real] : ( dvd_dvd_real @ A_89 @ zero_zero_real ) ).
thf(fact_455__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_456__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_457_rel__simps_I47_J,axiom,
! [K: int] :
( ( ( bit1 @ K )
= min )
<=> ( K = min ) ) ).
thf(fact_458_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_459_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_460_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_461_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_462_rel__simps_I45_J,axiom,
! [K: int] :
( ( bit0 @ K )
!= min ) ).
thf(fact_463_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_464_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_465_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_466_rel__simps_I13_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_int @ K @ min ) ) ).
thf(fact_467_rel__simps_I9_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_468_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_469_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_470_rel__simps_I8_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_471_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_472_rel__simps_I26_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ min @ K ) ) ).
thf(fact_473_rel__simps_I30_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_474_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_475_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_476_rel__simps_I28_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_477_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_478_rel__simps_I11_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_479_rel__simps_I25_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_480_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_481_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_482_linorder__neqE__linordered__idom,axiom,
! [X_7: real,Y_7: real] :
( ( X_7 != Y_7 )
=> ( ~ ( ord_less_real @ X_7 @ Y_7 )
=> ( ord_less_real @ Y_7 @ X_7 ) ) ) ).
thf(fact_483_linorder__neqE__linordered__idom,axiom,
! [X_7: int,Y_7: int] :
( ( X_7 != Y_7 )
=> ( ~ ( ord_less_int @ X_7 @ Y_7 )
=> ( ord_less_int @ Y_7 @ X_7 ) ) ) ).
thf(fact_484_dvd__refl,axiom,
! [A_88: nat] : ( dvd_dvd_nat @ A_88 @ A_88 ) ).
thf(fact_485_dvd__refl,axiom,
! [A_88: int] : ( dvd_dvd_int @ A_88 @ A_88 ) ).
thf(fact_486_dvd__trans,axiom,
! [C_40: nat,A_87: nat,B_67: nat] :
( ( dvd_dvd_nat @ A_87 @ B_67 )
=> ( ( dvd_dvd_nat @ B_67 @ C_40 )
=> ( dvd_dvd_nat @ A_87 @ C_40 ) ) ) ).
thf(fact_487_dvd__trans,axiom,
! [C_40: int,A_87: int,B_67: int] :
( ( dvd_dvd_int @ A_87 @ B_67 )
=> ( ( dvd_dvd_int @ B_67 @ C_40 )
=> ( dvd_dvd_int @ A_87 @ C_40 ) ) ) ).
thf(fact_488_zcong__sym,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ B @ A @ M ) ) ).
thf(fact_489_zcong__refl,axiom,
! [K: int,M: int] : ( zcong @ K @ K @ M ) ).
thf(fact_490_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_491_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_492_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_493_order__le__neq__implies__less,axiom,
! [X_6: real,Y_6: real] :
( ( ord_less_eq_real @ X_6 @ Y_6 )
=> ( ( X_6 != Y_6 )
=> ( ord_less_real @ X_6 @ Y_6 ) ) ) ).
thf(fact_494_order__le__neq__implies__less,axiom,
! [X_6: nat,Y_6: nat] :
( ( ord_less_eq_nat @ X_6 @ Y_6 )
=> ( ( X_6 != Y_6 )
=> ( ord_less_nat @ X_6 @ Y_6 ) ) ) ).
thf(fact_495_order__le__neq__implies__less,axiom,
! [X_6: int,Y_6: int] :
( ( ord_less_eq_int @ X_6 @ Y_6 )
=> ( ( X_6 != Y_6 )
=> ( ord_less_int @ X_6 @ Y_6 ) ) ) ).
thf(fact_496_divisors__zero,axiom,
! [A_86: real,B_66: real] :
( ( ( times_times_real @ A_86 @ B_66 )
= zero_zero_real )
=> ( ( A_86 = zero_zero_real )
| ( B_66 = zero_zero_real ) ) ) ).
thf(fact_497_divisors__zero,axiom,
! [A_86: nat,B_66: nat] :
( ( ( times_times_nat @ A_86 @ B_66 )
= zero_zero_nat )
=> ( ( A_86 = zero_zero_nat )
| ( B_66 = zero_zero_nat ) ) ) ).
thf(fact_498_divisors__zero,axiom,
! [A_86: int,B_66: int] :
( ( ( times_times_int @ A_86 @ B_66 )
= zero_zero_int )
=> ( ( A_86 = zero_zero_int )
| ( B_66 = zero_zero_int ) ) ) ).
thf(fact_499_no__zero__divisors,axiom,
! [B_65: real,A_85: real] :
( ( A_85 != zero_zero_real )
=> ( ( B_65 != zero_zero_real )
=> ( ( times_times_real @ A_85 @ B_65 )
!= zero_zero_real ) ) ) ).
thf(fact_500_no__zero__divisors,axiom,
! [B_65: nat,A_85: nat] :
( ( A_85 != zero_zero_nat )
=> ( ( B_65 != zero_zero_nat )
=> ( ( times_times_nat @ A_85 @ B_65 )
!= zero_zero_nat ) ) ) ).
thf(fact_501_no__zero__divisors,axiom,
! [B_65: int,A_85: int] :
( ( A_85 != zero_zero_int )
=> ( ( B_65 != zero_zero_int )
=> ( ( times_times_int @ A_85 @ B_65 )
!= zero_zero_int ) ) ) ).
thf(fact_502_mult__eq__0__iff,axiom,
! [A_84: real,B_64: real] :
( ( ( times_times_real @ A_84 @ B_64 )
= zero_zero_real )
<=> ( ( A_84 = zero_zero_real )
| ( B_64 = zero_zero_real ) ) ) ).
thf(fact_503_mult__eq__0__iff,axiom,
! [A_84: int,B_64: int] :
( ( ( times_times_int @ A_84 @ B_64 )
= zero_zero_int )
<=> ( ( A_84 = zero_zero_int )
| ( B_64 = zero_zero_int ) ) ) ).
thf(fact_504_mult__zero__right,axiom,
! [A_83: real] :
( ( times_times_real @ A_83 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_505_mult__zero__right,axiom,
! [A_83: nat] :
( ( times_times_nat @ A_83 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_506_mult__zero__right,axiom,
! [A_83: int] :
( ( times_times_int @ A_83 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_507_mult__zero__left,axiom,
! [A_82: real] :
( ( times_times_real @ zero_zero_real @ A_82 )
= zero_zero_real ) ).
thf(fact_508_mult__zero__left,axiom,
! [A_82: nat] :
( ( times_times_nat @ zero_zero_nat @ A_82 )
= zero_zero_nat ) ).
thf(fact_509_mult__zero__left,axiom,
! [A_82: int] :
( ( times_times_int @ zero_zero_int @ A_82 )
= zero_zero_int ) ).
thf(fact_510_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_511_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_512_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_513_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_514_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_515_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_516_combine__common__factor,axiom,
! [A_81: real,E: real,B_63: real,C_39: real] :
( ( plus_plus_real @ ( times_times_real @ A_81 @ E ) @ ( plus_plus_real @ ( times_times_real @ B_63 @ E ) @ C_39 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_81 @ B_63 ) @ E ) @ C_39 ) ) ).
thf(fact_517_combine__common__factor,axiom,
! [A_81: nat,E: nat,B_63: nat,C_39: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_81 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B_63 @ E ) @ C_39 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_81 @ B_63 ) @ E ) @ C_39 ) ) ).
thf(fact_518_combine__common__factor,axiom,
! [A_81: int,E: int,B_63: int,C_39: int] :
( ( plus_plus_int @ ( times_times_int @ A_81 @ E ) @ ( plus_plus_int @ ( times_times_int @ B_63 @ E ) @ C_39 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_81 @ B_63 ) @ E ) @ C_39 ) ) ).
thf(fact_519_comm__semiring__class_Odistrib,axiom,
! [A_80: real,B_62: real,C_38: real] :
( ( times_times_real @ ( plus_plus_real @ A_80 @ B_62 ) @ C_38 )
= ( plus_plus_real @ ( times_times_real @ A_80 @ C_38 ) @ ( times_times_real @ B_62 @ C_38 ) ) ) ).
thf(fact_520_comm__semiring__class_Odistrib,axiom,
! [A_80: nat,B_62: nat,C_38: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_80 @ B_62 ) @ C_38 )
= ( plus_plus_nat @ ( times_times_nat @ A_80 @ C_38 ) @ ( times_times_nat @ B_62 @ C_38 ) ) ) ).
thf(fact_521_comm__semiring__class_Odistrib,axiom,
! [A_80: int,B_62: int,C_38: int] :
( ( times_times_int @ ( plus_plus_int @ A_80 @ B_62 ) @ C_38 )
= ( plus_plus_int @ ( times_times_int @ A_80 @ C_38 ) @ ( times_times_int @ B_62 @ C_38 ) ) ) ).
thf(fact_522_field__power__not__zero,axiom,
! [N_21: nat,A_79: real] :
( ( A_79 != zero_zero_real )
=> ( ( power_power_real @ A_79 @ N_21 )
!= zero_zero_real ) ) ).
thf(fact_523_field__power__not__zero,axiom,
! [N_21: nat,A_79: int] :
( ( A_79 != zero_zero_int )
=> ( ( power_power_int @ A_79 @ N_21 )
!= zero_zero_int ) ) ).
thf(fact_524_dvd__0__left,axiom,
! [A_78: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A_78 )
=> ( A_78 = zero_zero_nat ) ) ).
thf(fact_525_dvd__0__left,axiom,
! [A_78: int] :
( ( dvd_dvd_int @ zero_zero_int @ A_78 )
=> ( A_78 = zero_zero_int ) ) ).
thf(fact_526_dvd__0__left,axiom,
! [A_78: real] :
( ( dvd_dvd_real @ zero_zero_real @ A_78 )
=> ( A_78 = zero_zero_real ) ) ).
thf(fact_527_power__mult__distrib,axiom,
! [A_77: real,B_61: real,N_20: nat] :
( ( power_power_real @ ( times_times_real @ A_77 @ B_61 ) @ N_20 )
= ( times_times_real @ ( power_power_real @ A_77 @ N_20 ) @ ( power_power_real @ B_61 @ N_20 ) ) ) ).
thf(fact_528_power__mult__distrib,axiom,
! [A_77: nat,B_61: nat,N_20: nat] :
( ( power_power_nat @ ( times_times_nat @ A_77 @ B_61 ) @ N_20 )
= ( times_times_nat @ ( power_power_nat @ A_77 @ N_20 ) @ ( power_power_nat @ B_61 @ N_20 ) ) ) ).
thf(fact_529_power__mult__distrib,axiom,
! [A_77: int,B_61: int,N_20: nat] :
( ( power_power_int @ ( times_times_int @ A_77 @ B_61 ) @ N_20 )
= ( times_times_int @ ( power_power_int @ A_77 @ N_20 ) @ ( power_power_int @ B_61 @ N_20 ) ) ) ).
thf(fact_530_power__commutes,axiom,
! [A_76: real,N_19: nat] :
( ( times_times_real @ ( power_power_real @ A_76 @ N_19 ) @ A_76 )
= ( times_times_real @ A_76 @ ( power_power_real @ A_76 @ N_19 ) ) ) ).
thf(fact_531_power__commutes,axiom,
! [A_76: nat,N_19: nat] :
( ( times_times_nat @ ( power_power_nat @ A_76 @ N_19 ) @ A_76 )
= ( times_times_nat @ A_76 @ ( power_power_nat @ A_76 @ N_19 ) ) ) ).
thf(fact_532_power__commutes,axiom,
! [A_76: int,N_19: nat] :
( ( times_times_int @ ( power_power_int @ A_76 @ N_19 ) @ A_76 )
= ( times_times_int @ A_76 @ ( power_power_int @ A_76 @ N_19 ) ) ) ).
thf(fact_533_dvd__triv__left,axiom,
! [A_75: nat,B_60: nat] : ( dvd_dvd_nat @ A_75 @ ( times_times_nat @ A_75 @ B_60 ) ) ).
thf(fact_534_dvd__triv__left,axiom,
! [A_75: int,B_60: int] : ( dvd_dvd_int @ A_75 @ ( times_times_int @ A_75 @ B_60 ) ) ).
thf(fact_535_dvd__triv__left,axiom,
! [A_75: real,B_60: real] : ( dvd_dvd_real @ A_75 @ ( times_times_real @ A_75 @ B_60 ) ) ).
thf(fact_536_dvd__triv__right,axiom,
! [A_74: nat,B_59: nat] : ( dvd_dvd_nat @ A_74 @ ( times_times_nat @ B_59 @ A_74 ) ) ).
thf(fact_537_dvd__triv__right,axiom,
! [A_74: int,B_59: int] : ( dvd_dvd_int @ A_74 @ ( times_times_int @ B_59 @ A_74 ) ) ).
thf(fact_538_dvd__triv__right,axiom,
! [A_74: real,B_59: real] : ( dvd_dvd_real @ A_74 @ ( times_times_real @ B_59 @ A_74 ) ) ).
thf(fact_539_dvd__mult2,axiom,
! [C_37: nat,A_73: nat,B_58: nat] :
( ( dvd_dvd_nat @ A_73 @ B_58 )
=> ( dvd_dvd_nat @ A_73 @ ( times_times_nat @ B_58 @ C_37 ) ) ) ).
thf(fact_540_dvd__mult2,axiom,
! [C_37: int,A_73: int,B_58: int] :
( ( dvd_dvd_int @ A_73 @ B_58 )
=> ( dvd_dvd_int @ A_73 @ ( times_times_int @ B_58 @ C_37 ) ) ) ).
thf(fact_541_dvd__mult2,axiom,
! [C_37: real,A_73: real,B_58: real] :
( ( dvd_dvd_real @ A_73 @ B_58 )
=> ( dvd_dvd_real @ A_73 @ ( times_times_real @ B_58 @ C_37 ) ) ) ).
thf(fact_542_dvd__mult,axiom,
! [B_57: nat,A_72: nat,C_36: nat] :
( ( dvd_dvd_nat @ A_72 @ C_36 )
=> ( dvd_dvd_nat @ A_72 @ ( times_times_nat @ B_57 @ C_36 ) ) ) ).
thf(fact_543_dvd__mult,axiom,
! [B_57: int,A_72: int,C_36: int] :
( ( dvd_dvd_int @ A_72 @ C_36 )
=> ( dvd_dvd_int @ A_72 @ ( times_times_int @ B_57 @ C_36 ) ) ) ).
thf(fact_544_dvd__mult,axiom,
! [B_57: real,A_72: real,C_36: real] :
( ( dvd_dvd_real @ A_72 @ C_36 )
=> ( dvd_dvd_real @ A_72 @ ( times_times_real @ B_57 @ C_36 ) ) ) ).
thf(fact_545_mult__dvd__mono,axiom,
! [C_35: nat,D_9: nat,A_71: nat,B_56: nat] :
( ( dvd_dvd_nat @ A_71 @ B_56 )
=> ( ( dvd_dvd_nat @ C_35 @ D_9 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A_71 @ C_35 ) @ ( times_times_nat @ B_56 @ D_9 ) ) ) ) ).
thf(fact_546_mult__dvd__mono,axiom,
! [C_35: int,D_9: int,A_71: int,B_56: int] :
( ( dvd_dvd_int @ A_71 @ B_56 )
=> ( ( dvd_dvd_int @ C_35 @ D_9 )
=> ( dvd_dvd_int @ ( times_times_int @ A_71 @ C_35 ) @ ( times_times_int @ B_56 @ D_9 ) ) ) ) ).
thf(fact_547_mult__dvd__mono,axiom,
! [C_35: real,D_9: real,A_71: real,B_56: real] :
( ( dvd_dvd_real @ A_71 @ B_56 )
=> ( ( dvd_dvd_real @ C_35 @ D_9 )
=> ( dvd_dvd_real @ ( times_times_real @ A_71 @ C_35 ) @ ( times_times_real @ B_56 @ D_9 ) ) ) ) ).
thf(fact_548_dvdI,axiom,
! [A_70: nat,B_55: nat,K_2: nat] :
( ( A_70
= ( times_times_nat @ B_55 @ K_2 ) )
=> ( dvd_dvd_nat @ B_55 @ A_70 ) ) ).
thf(fact_549_dvdI,axiom,
! [A_70: int,B_55: int,K_2: int] :
( ( A_70
= ( times_times_int @ B_55 @ K_2 ) )
=> ( dvd_dvd_int @ B_55 @ A_70 ) ) ).
thf(fact_550_dvdI,axiom,
! [A_70: real,B_55: real,K_2: real] :
( ( A_70
= ( times_times_real @ B_55 @ K_2 ) )
=> ( dvd_dvd_real @ B_55 @ A_70 ) ) ).
thf(fact_551_dvd__mult__left,axiom,
! [A_69: nat,B_54: nat,C_34: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_69 @ B_54 ) @ C_34 )
=> ( dvd_dvd_nat @ A_69 @ C_34 ) ) ).
thf(fact_552_dvd__mult__left,axiom,
! [A_69: int,B_54: int,C_34: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_69 @ B_54 ) @ C_34 )
=> ( dvd_dvd_int @ A_69 @ C_34 ) ) ).
thf(fact_553_dvd__mult__left,axiom,
! [A_69: real,B_54: real,C_34: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_69 @ B_54 ) @ C_34 )
=> ( dvd_dvd_real @ A_69 @ C_34 ) ) ).
thf(fact_554_dvd__mult__right,axiom,
! [A_68: nat,B_53: nat,C_33: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_68 @ B_53 ) @ C_33 )
=> ( dvd_dvd_nat @ B_53 @ C_33 ) ) ).
thf(fact_555_dvd__mult__right,axiom,
! [A_68: int,B_53: int,C_33: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_68 @ B_53 ) @ C_33 )
=> ( dvd_dvd_int @ B_53 @ C_33 ) ) ).
thf(fact_556_dvd__mult__right,axiom,
! [A_68: real,B_53: real,C_33: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_68 @ B_53 ) @ C_33 )
=> ( dvd_dvd_real @ B_53 @ C_33 ) ) ).
thf(fact_557_dvd__add,axiom,
! [C_32: nat,A_67: nat,B_52: nat] :
( ( dvd_dvd_nat @ A_67 @ B_52 )
=> ( ( dvd_dvd_nat @ A_67 @ C_32 )
=> ( dvd_dvd_nat @ A_67 @ ( plus_plus_nat @ B_52 @ C_32 ) ) ) ) ).
thf(fact_558_dvd__add,axiom,
! [C_32: int,A_67: int,B_52: int] :
( ( dvd_dvd_int @ A_67 @ B_52 )
=> ( ( dvd_dvd_int @ A_67 @ C_32 )
=> ( dvd_dvd_int @ A_67 @ ( plus_plus_int @ B_52 @ C_32 ) ) ) ) ).
thf(fact_559_dvd__add,axiom,
! [C_32: real,A_67: real,B_52: real] :
( ( dvd_dvd_real @ A_67 @ B_52 )
=> ( ( dvd_dvd_real @ A_67 @ C_32 )
=> ( dvd_dvd_real @ A_67 @ ( plus_plus_real @ B_52 @ C_32 ) ) ) ) ).
thf(fact_560_power__one,axiom,
! [N_18: nat] :
( ( power_power_real @ one_one_real @ N_18 )
= one_one_real ) ).
thf(fact_561_power__one,axiom,
! [N_18: nat] :
( ( power_power_nat @ one_one_nat @ N_18 )
= one_one_nat ) ).
thf(fact_562_power__one,axiom,
! [N_18: nat] :
( ( power_power_int @ one_one_int @ N_18 )
= one_one_int ) ).
thf(fact_563_one__dvd,axiom,
! [A_66: nat] : ( dvd_dvd_nat @ one_one_nat @ A_66 ) ).
thf(fact_564_one__dvd,axiom,
! [A_66: int] : ( dvd_dvd_int @ one_one_int @ A_66 ) ).
thf(fact_565_one__dvd,axiom,
! [A_66: real] : ( dvd_dvd_real @ one_one_real @ A_66 ) ).
thf(fact_566_dvd__power__same,axiom,
! [N_17: nat,X_5: nat,Y_5: nat] :
( ( dvd_dvd_nat @ X_5 @ Y_5 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_5 @ N_17 ) @ ( power_power_nat @ Y_5 @ N_17 ) ) ) ).
thf(fact_567_dvd__power__same,axiom,
! [N_17: nat,X_5: int,Y_5: int] :
( ( dvd_dvd_int @ X_5 @ Y_5 )
=> ( dvd_dvd_int @ ( power_power_int @ X_5 @ N_17 ) @ ( power_power_int @ Y_5 @ N_17 ) ) ) ).
thf(fact_568_dvd__power__same,axiom,
! [N_17: nat,X_5: real,Y_5: real] :
( ( dvd_dvd_real @ X_5 @ Y_5 )
=> ( dvd_dvd_real @ ( power_power_real @ X_5 @ N_17 ) @ ( power_power_real @ Y_5 @ N_17 ) ) ) ).
thf(fact_569_power__le__dvd,axiom,
! [M_9: nat,A_65: nat,N_16: nat,B_51: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A_65 @ N_16 ) @ B_51 )
=> ( ( ord_less_eq_nat @ M_9 @ N_16 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_65 @ M_9 ) @ B_51 ) ) ) ).
thf(fact_570_power__le__dvd,axiom,
! [M_9: nat,A_65: int,N_16: nat,B_51: int] :
( ( dvd_dvd_int @ ( power_power_int @ A_65 @ N_16 ) @ B_51 )
=> ( ( ord_less_eq_nat @ M_9 @ N_16 )
=> ( dvd_dvd_int @ ( power_power_int @ A_65 @ M_9 ) @ B_51 ) ) ) ).
thf(fact_571_power__le__dvd,axiom,
! [M_9: nat,A_65: real,N_16: nat,B_51: real] :
( ( dvd_dvd_real @ ( power_power_real @ A_65 @ N_16 ) @ B_51 )
=> ( ( ord_less_eq_nat @ M_9 @ N_16 )
=> ( dvd_dvd_real @ ( power_power_real @ A_65 @ M_9 ) @ B_51 ) ) ) ).
thf(fact_572_dvd__power__le,axiom,
! [N_15: nat,M_8: nat,X_4: nat,Y_4: nat] :
( ( dvd_dvd_nat @ X_4 @ Y_4 )
=> ( ( ord_less_eq_nat @ N_15 @ M_8 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_4 @ N_15 ) @ ( power_power_nat @ Y_4 @ M_8 ) ) ) ) ).
thf(fact_573_dvd__power__le,axiom,
! [N_15: nat,M_8: nat,X_4: int,Y_4: int] :
( ( dvd_dvd_int @ X_4 @ Y_4 )
=> ( ( ord_less_eq_nat @ N_15 @ M_8 )
=> ( dvd_dvd_int @ ( power_power_int @ X_4 @ N_15 ) @ ( power_power_int @ Y_4 @ M_8 ) ) ) ) ).
thf(fact_574_dvd__power__le,axiom,
! [N_15: nat,M_8: nat,X_4: real,Y_4: real] :
( ( dvd_dvd_real @ X_4 @ Y_4 )
=> ( ( ord_less_eq_nat @ N_15 @ M_8 )
=> ( dvd_dvd_real @ ( power_power_real @ X_4 @ N_15 ) @ ( power_power_real @ Y_4 @ M_8 ) ) ) ) ).
thf(fact_575_le__imp__power__dvd,axiom,
! [A_64: nat,M_7: nat,N_14: nat] :
( ( ord_less_eq_nat @ M_7 @ N_14 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_64 @ M_7 ) @ ( power_power_nat @ A_64 @ N_14 ) ) ) ).
thf(fact_576_le__imp__power__dvd,axiom,
! [A_64: int,M_7: nat,N_14: nat] :
( ( ord_less_eq_nat @ M_7 @ N_14 )
=> ( dvd_dvd_int @ ( power_power_int @ A_64 @ M_7 ) @ ( power_power_int @ A_64 @ N_14 ) ) ) ).
thf(fact_577_le__imp__power__dvd,axiom,
! [A_64: real,M_7: nat,N_14: nat] :
( ( ord_less_eq_nat @ M_7 @ N_14 )
=> ( dvd_dvd_real @ ( power_power_real @ A_64 @ M_7 ) @ ( power_power_real @ A_64 @ N_14 ) ) ) ).
thf(fact_578_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_579_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_580_power__mult,axiom,
! [A_63: real,M_6: nat,N_13: nat] :
( ( power_power_real @ A_63 @ ( times_times_nat @ M_6 @ N_13 ) )
= ( power_power_real @ ( power_power_real @ A_63 @ M_6 ) @ N_13 ) ) ).
thf(fact_581_power__mult,axiom,
! [A_63: nat,M_6: nat,N_13: nat] :
( ( power_power_nat @ A_63 @ ( times_times_nat @ M_6 @ N_13 ) )
= ( power_power_nat @ ( power_power_nat @ A_63 @ M_6 ) @ N_13 ) ) ).
thf(fact_582_power__mult,axiom,
! [A_63: int,M_6: nat,N_13: nat] :
( ( power_power_int @ A_63 @ ( times_times_nat @ M_6 @ N_13 ) )
= ( power_power_int @ ( power_power_int @ A_63 @ M_6 ) @ N_13 ) ) ).
thf(fact_583_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_584_power__one__right,axiom,
! [A_62: real] :
( ( power_power_real @ A_62 @ one_one_nat )
= A_62 ) ).
thf(fact_585_power__one__right,axiom,
! [A_62: nat] :
( ( power_power_nat @ A_62 @ one_one_nat )
= A_62 ) ).
thf(fact_586_power__one__right,axiom,
! [A_62: int] :
( ( power_power_int @ A_62 @ one_one_nat )
= A_62 ) ).
thf(fact_587_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_588_IntPrimes_Ozcong__zero,axiom,
! [A: int,B: int] :
( ( zcong @ A @ B @ zero_zero_int )
<=> ( A = B ) ) ).
thf(fact_589_zcong__1,axiom,
! [A: int,B: int] : ( zcong @ A @ B @ one_one_int ) ).
thf(fact_590_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : ( zcong @ ( times_times_int @ A @ M ) @ ( times_times_int @ B @ M ) @ M ) ).
thf(fact_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_power__m1__even,axiom,
! [N_12: nat] :
( ( power_power_real @ ( number267125858f_real @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_12 ) )
= one_one_real ) ).
thf(fact_600_power__m1__even,axiom,
! [N_12: nat] :
( ( power_power_int @ ( number_number_of_int @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_12 ) )
= one_one_int ) ).
thf(fact_601_split__mult__neg__le,axiom,
! [B_50: real,A_61: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_61 )
& ( ord_less_eq_real @ B_50 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_61 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_50 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_61 @ B_50 ) @ zero_zero_real ) ) ).
thf(fact_602_split__mult__neg__le,axiom,
! [B_50: nat,A_61: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_61 )
& ( ord_less_eq_nat @ B_50 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_61 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_50 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_61 @ B_50 ) @ zero_zero_nat ) ) ).
thf(fact_603_split__mult__neg__le,axiom,
! [B_50: int,A_61: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_61 )
& ( ord_less_eq_int @ B_50 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_61 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_50 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_61 @ B_50 ) @ zero_zero_int ) ) ).
thf(fact_604_split__mult__pos__le,axiom,
! [B_49: real,A_60: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_60 )
& ( ord_less_eq_real @ zero_zero_real @ B_49 ) )
| ( ( ord_less_eq_real @ A_60 @ zero_zero_real )
& ( ord_less_eq_real @ B_49 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_60 @ B_49 ) ) ) ).
thf(fact_605_split__mult__pos__le,axiom,
! [B_49: int,A_60: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_60 )
& ( ord_less_eq_int @ zero_zero_int @ B_49 ) )
| ( ( ord_less_eq_int @ A_60 @ zero_zero_int )
& ( ord_less_eq_int @ B_49 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_60 @ B_49 ) ) ) ).
thf(fact_606_mult__mono,axiom,
! [C_31: real,D_8: real,A_59: real,B_48: real] :
( ( ord_less_eq_real @ A_59 @ B_48 )
=> ( ( ord_less_eq_real @ C_31 @ D_8 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_48 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_31 )
=> ( ord_less_eq_real @ ( times_times_real @ A_59 @ C_31 ) @ ( times_times_real @ B_48 @ D_8 ) ) ) ) ) ) ).
thf(fact_607_mult__mono,axiom,
! [C_31: nat,D_8: nat,A_59: nat,B_48: nat] :
( ( ord_less_eq_nat @ A_59 @ B_48 )
=> ( ( ord_less_eq_nat @ C_31 @ D_8 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_48 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_31 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_59 @ C_31 ) @ ( times_times_nat @ B_48 @ D_8 ) ) ) ) ) ) ).
thf(fact_608_mult__mono,axiom,
! [C_31: int,D_8: int,A_59: int,B_48: int] :
( ( ord_less_eq_int @ A_59 @ B_48 )
=> ( ( ord_less_eq_int @ C_31 @ D_8 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_48 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_31 )
=> ( ord_less_eq_int @ ( times_times_int @ A_59 @ C_31 ) @ ( times_times_int @ B_48 @ D_8 ) ) ) ) ) ) ).
thf(fact_609_mult__mono_H,axiom,
! [C_30: real,D_7: real,A_58: real,B_47: real] :
( ( ord_less_eq_real @ A_58 @ B_47 )
=> ( ( ord_less_eq_real @ C_30 @ D_7 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_58 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_30 )
=> ( ord_less_eq_real @ ( times_times_real @ A_58 @ C_30 ) @ ( times_times_real @ B_47 @ D_7 ) ) ) ) ) ) ).
thf(fact_610_mult__mono_H,axiom,
! [C_30: nat,D_7: nat,A_58: nat,B_47: nat] :
( ( ord_less_eq_nat @ A_58 @ B_47 )
=> ( ( ord_less_eq_nat @ C_30 @ D_7 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_58 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_30 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_58 @ C_30 ) @ ( times_times_nat @ B_47 @ D_7 ) ) ) ) ) ) ).
thf(fact_611_mult__mono_H,axiom,
! [C_30: int,D_7: int,A_58: int,B_47: int] :
( ( ord_less_eq_int @ A_58 @ B_47 )
=> ( ( ord_less_eq_int @ C_30 @ D_7 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_58 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_30 )
=> ( ord_less_eq_int @ ( times_times_int @ A_58 @ C_30 ) @ ( times_times_int @ B_47 @ D_7 ) ) ) ) ) ) ).
thf(fact_612_mult__left__mono__neg,axiom,
! [C_29: real,B_46: real,A_57: real] :
( ( ord_less_eq_real @ B_46 @ A_57 )
=> ( ( ord_less_eq_real @ C_29 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_29 @ A_57 ) @ ( times_times_real @ C_29 @ B_46 ) ) ) ) ).
thf(fact_613_mult__left__mono__neg,axiom,
! [C_29: int,B_46: int,A_57: int] :
( ( ord_less_eq_int @ B_46 @ A_57 )
=> ( ( ord_less_eq_int @ C_29 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_29 @ A_57 ) @ ( times_times_int @ C_29 @ B_46 ) ) ) ) ).
thf(fact_614_mult__right__mono__neg,axiom,
! [C_28: real,B_45: real,A_56: real] :
( ( ord_less_eq_real @ B_45 @ A_56 )
=> ( ( ord_less_eq_real @ C_28 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_56 @ C_28 ) @ ( times_times_real @ B_45 @ C_28 ) ) ) ) ).
thf(fact_615_mult__right__mono__neg,axiom,
! [C_28: int,B_45: int,A_56: int] :
( ( ord_less_eq_int @ B_45 @ A_56 )
=> ( ( ord_less_eq_int @ C_28 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_56 @ C_28 ) @ ( times_times_int @ B_45 @ C_28 ) ) ) ) ).
thf(fact_616_comm__mult__left__mono,axiom,
! [C_27: real,A_55: real,B_44: real] :
( ( ord_less_eq_real @ A_55 @ B_44 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_27 )
=> ( ord_less_eq_real @ ( times_times_real @ C_27 @ A_55 ) @ ( times_times_real @ C_27 @ B_44 ) ) ) ) ).
thf(fact_617_comm__mult__left__mono,axiom,
! [C_27: nat,A_55: nat,B_44: nat] :
( ( ord_less_eq_nat @ A_55 @ B_44 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_27 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_27 @ A_55 ) @ ( times_times_nat @ C_27 @ B_44 ) ) ) ) ).
thf(fact_618_comm__mult__left__mono,axiom,
! [C_27: int,A_55: int,B_44: int] :
( ( ord_less_eq_int @ A_55 @ B_44 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_27 )
=> ( ord_less_eq_int @ ( times_times_int @ C_27 @ A_55 ) @ ( times_times_int @ C_27 @ B_44 ) ) ) ) ).
thf(fact_619_mult__left__mono,axiom,
! [C_26: real,A_54: real,B_43: real] :
( ( ord_less_eq_real @ A_54 @ B_43 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_26 )
=> ( ord_less_eq_real @ ( times_times_real @ C_26 @ A_54 ) @ ( times_times_real @ C_26 @ B_43 ) ) ) ) ).
thf(fact_620_mult__left__mono,axiom,
! [C_26: nat,A_54: nat,B_43: nat] :
( ( ord_less_eq_nat @ A_54 @ B_43 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_26 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_26 @ A_54 ) @ ( times_times_nat @ C_26 @ B_43 ) ) ) ) ).
thf(fact_621_mult__left__mono,axiom,
! [C_26: int,A_54: int,B_43: int] :
( ( ord_less_eq_int @ A_54 @ B_43 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_26 )
=> ( ord_less_eq_int @ ( times_times_int @ C_26 @ A_54 ) @ ( times_times_int @ C_26 @ B_43 ) ) ) ) ).
thf(fact_622_mult__right__mono,axiom,
! [C_25: real,A_53: real,B_42: real] :
( ( ord_less_eq_real @ A_53 @ B_42 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_25 )
=> ( ord_less_eq_real @ ( times_times_real @ A_53 @ C_25 ) @ ( times_times_real @ B_42 @ C_25 ) ) ) ) ).
thf(fact_623_mult__right__mono,axiom,
! [C_25: nat,A_53: nat,B_42: nat] :
( ( ord_less_eq_nat @ A_53 @ B_42 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_25 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_53 @ C_25 ) @ ( times_times_nat @ B_42 @ C_25 ) ) ) ) ).
thf(fact_624_mult__right__mono,axiom,
! [C_25: int,A_53: int,B_42: int] :
( ( ord_less_eq_int @ A_53 @ B_42 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_25 )
=> ( ord_less_eq_int @ ( times_times_int @ A_53 @ C_25 ) @ ( times_times_int @ B_42 @ C_25 ) ) ) ) ).
thf(fact_625_mult__nonpos__nonpos,axiom,
! [B_41: real,A_52: real] :
( ( ord_less_eq_real @ A_52 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_41 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_52 @ B_41 ) ) ) ) ).
thf(fact_626_mult__nonpos__nonpos,axiom,
! [B_41: int,A_52: int] :
( ( ord_less_eq_int @ A_52 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_41 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_52 @ B_41 ) ) ) ) ).
thf(fact_627_mult__nonpos__nonneg,axiom,
! [B_40: real,A_51: real] :
( ( ord_less_eq_real @ A_51 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_40 )
=> ( ord_less_eq_real @ ( times_times_real @ A_51 @ B_40 ) @ zero_zero_real ) ) ) ).
thf(fact_628_mult__nonpos__nonneg,axiom,
! [B_40: nat,A_51: nat] :
( ( ord_less_eq_nat @ A_51 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_40 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_51 @ B_40 ) @ zero_zero_nat ) ) ) ).
thf(fact_629_mult__nonpos__nonneg,axiom,
! [B_40: int,A_51: int] :
( ( ord_less_eq_int @ A_51 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_40 )
=> ( ord_less_eq_int @ ( times_times_int @ A_51 @ B_40 ) @ zero_zero_int ) ) ) ).
thf(fact_630_mult__nonneg__nonpos2,axiom,
! [B_39: real,A_50: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_50 )
=> ( ( ord_less_eq_real @ B_39 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_39 @ A_50 ) @ zero_zero_real ) ) ) ).
thf(fact_631_mult__nonneg__nonpos2,axiom,
! [B_39: nat,A_50: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_50 )
=> ( ( ord_less_eq_nat @ B_39 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_39 @ A_50 ) @ zero_zero_nat ) ) ) ).
thf(fact_632_mult__nonneg__nonpos2,axiom,
! [B_39: int,A_50: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_50 )
=> ( ( ord_less_eq_int @ B_39 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_39 @ A_50 ) @ zero_zero_int ) ) ) ).
thf(fact_633_mult__nonneg__nonpos,axiom,
! [B_38: real,A_49: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_49 )
=> ( ( ord_less_eq_real @ B_38 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_49 @ B_38 ) @ zero_zero_real ) ) ) ).
thf(fact_634_mult__nonneg__nonpos,axiom,
! [B_38: nat,A_49: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_49 )
=> ( ( ord_less_eq_nat @ B_38 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_49 @ B_38 ) @ zero_zero_nat ) ) ) ).
thf(fact_635_mult__nonneg__nonpos,axiom,
! [B_38: int,A_49: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_49 )
=> ( ( ord_less_eq_int @ B_38 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_49 @ B_38 ) @ zero_zero_int ) ) ) ).
thf(fact_636_mult__nonneg__nonneg,axiom,
! [B_37: real,A_48: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_48 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_37 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_48 @ B_37 ) ) ) ) ).
thf(fact_637_mult__nonneg__nonneg,axiom,
! [B_37: nat,A_48: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_48 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_37 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_48 @ B_37 ) ) ) ) ).
thf(fact_638_mult__nonneg__nonneg,axiom,
! [B_37: int,A_48: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_48 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_37 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_48 @ B_37 ) ) ) ) ).
thf(fact_639_mult__le__0__iff,axiom,
! [A_47: real,B_36: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_47 @ B_36 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_47 )
& ( ord_less_eq_real @ B_36 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_47 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_36 ) ) ) ) ).
thf(fact_640_mult__le__0__iff,axiom,
! [A_47: int,B_36: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_47 @ B_36 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_47 )
& ( ord_less_eq_int @ B_36 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_47 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_36 ) ) ) ) ).
thf(fact_641_zero__le__mult__iff,axiom,
! [A_46: real,B_35: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_46 @ B_35 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_46 )
& ( ord_less_eq_real @ zero_zero_real @ B_35 ) )
| ( ( ord_less_eq_real @ A_46 @ zero_zero_real )
& ( ord_less_eq_real @ B_35 @ zero_zero_real ) ) ) ) ).
thf(fact_642_zero__le__mult__iff,axiom,
! [A_46: int,B_35: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_46 @ B_35 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_46 )
& ( ord_less_eq_int @ zero_zero_int @ B_35 ) )
| ( ( ord_less_eq_int @ A_46 @ zero_zero_int )
& ( ord_less_eq_int @ B_35 @ zero_zero_int ) ) ) ) ).
thf(fact_643_zero__le__square,axiom,
! [A_45: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_45 @ A_45 ) ) ).
thf(fact_644_zero__le__square,axiom,
! [A_45: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_45 @ A_45 ) ) ).
thf(fact_645_mult__strict__left__mono__neg,axiom,
! [C_24: real,B_34: real,A_44: real] :
( ( ord_less_real @ B_34 @ A_44 )
=> ( ( ord_less_real @ C_24 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C_24 @ A_44 ) @ ( times_times_real @ C_24 @ B_34 ) ) ) ) ).
thf(fact_646_mult__strict__left__mono__neg,axiom,
! [C_24: int,B_34: int,A_44: int] :
( ( ord_less_int @ B_34 @ A_44 )
=> ( ( ord_less_int @ C_24 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_24 @ A_44 ) @ ( times_times_int @ C_24 @ B_34 ) ) ) ) ).
thf(fact_647_mult__strict__right__mono__neg,axiom,
! [C_23: real,B_33: real,A_43: real] :
( ( ord_less_real @ B_33 @ A_43 )
=> ( ( ord_less_real @ C_23 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_43 @ C_23 ) @ ( times_times_real @ B_33 @ C_23 ) ) ) ) ).
thf(fact_648_mult__strict__right__mono__neg,axiom,
! [C_23: int,B_33: int,A_43: int] :
( ( ord_less_int @ B_33 @ A_43 )
=> ( ( ord_less_int @ C_23 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_43 @ C_23 ) @ ( times_times_int @ B_33 @ C_23 ) ) ) ) ).
thf(fact_649_comm__mult__strict__left__mono,axiom,
! [C_22: real,A_42: real,B_32: real] :
( ( ord_less_real @ A_42 @ B_32 )
=> ( ( ord_less_real @ zero_zero_real @ C_22 )
=> ( ord_less_real @ ( times_times_real @ C_22 @ A_42 ) @ ( times_times_real @ C_22 @ B_32 ) ) ) ) ).
thf(fact_650_comm__mult__strict__left__mono,axiom,
! [C_22: nat,A_42: nat,B_32: nat] :
( ( ord_less_nat @ A_42 @ B_32 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_22 )
=> ( ord_less_nat @ ( times_times_nat @ C_22 @ A_42 ) @ ( times_times_nat @ C_22 @ B_32 ) ) ) ) ).
thf(fact_651_comm__mult__strict__left__mono,axiom,
! [C_22: int,A_42: int,B_32: int] :
( ( ord_less_int @ A_42 @ B_32 )
=> ( ( ord_less_int @ zero_zero_int @ C_22 )
=> ( ord_less_int @ ( times_times_int @ C_22 @ A_42 ) @ ( times_times_int @ C_22 @ B_32 ) ) ) ) ).
thf(fact_652_mult__strict__left__mono,axiom,
! [C_21: real,A_41: real,B_31: real] :
( ( ord_less_real @ A_41 @ B_31 )
=> ( ( ord_less_real @ zero_zero_real @ C_21 )
=> ( ord_less_real @ ( times_times_real @ C_21 @ A_41 ) @ ( times_times_real @ C_21 @ B_31 ) ) ) ) ).
thf(fact_653_mult__strict__left__mono,axiom,
! [C_21: nat,A_41: nat,B_31: nat] :
( ( ord_less_nat @ A_41 @ B_31 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_21 )
=> ( ord_less_nat @ ( times_times_nat @ C_21 @ A_41 ) @ ( times_times_nat @ C_21 @ B_31 ) ) ) ) ).
thf(fact_654_mult__strict__left__mono,axiom,
! [C_21: int,A_41: int,B_31: int] :
( ( ord_less_int @ A_41 @ B_31 )
=> ( ( ord_less_int @ zero_zero_int @ C_21 )
=> ( ord_less_int @ ( times_times_int @ C_21 @ A_41 ) @ ( times_times_int @ C_21 @ B_31 ) ) ) ) ).
thf(fact_655_mult__strict__right__mono,axiom,
! [C_20: real,A_40: real,B_30: real] :
( ( ord_less_real @ A_40 @ B_30 )
=> ( ( ord_less_real @ zero_zero_real @ C_20 )
=> ( ord_less_real @ ( times_times_real @ A_40 @ C_20 ) @ ( times_times_real @ B_30 @ C_20 ) ) ) ) ).
thf(fact_656_mult__strict__right__mono,axiom,
! [C_20: nat,A_40: nat,B_30: nat] :
( ( ord_less_nat @ A_40 @ B_30 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_20 )
=> ( ord_less_nat @ ( times_times_nat @ A_40 @ C_20 ) @ ( times_times_nat @ B_30 @ C_20 ) ) ) ) ).
thf(fact_657_mult__strict__right__mono,axiom,
! [C_20: int,A_40: int,B_30: int] :
( ( ord_less_int @ A_40 @ B_30 )
=> ( ( ord_less_int @ zero_zero_int @ C_20 )
=> ( ord_less_int @ ( times_times_int @ A_40 @ C_20 ) @ ( times_times_int @ B_30 @ C_20 ) ) ) ) ).
thf(fact_658_mult__neg__neg,axiom,
! [B_29: real,A_39: real] :
( ( ord_less_real @ A_39 @ zero_zero_real )
=> ( ( ord_less_real @ B_29 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_39 @ B_29 ) ) ) ) ).
thf(fact_659_mult__neg__neg,axiom,
! [B_29: int,A_39: int] :
( ( ord_less_int @ A_39 @ zero_zero_int )
=> ( ( ord_less_int @ B_29 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_39 @ B_29 ) ) ) ) ).
thf(fact_660_mult__neg__pos,axiom,
! [B_28: real,A_38: real] :
( ( ord_less_real @ A_38 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B_28 )
=> ( ord_less_real @ ( times_times_real @ A_38 @ B_28 ) @ zero_zero_real ) ) ) ).
thf(fact_661_mult__neg__pos,axiom,
! [B_28: nat,A_38: nat] :
( ( ord_less_nat @ A_38 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_28 )
=> ( ord_less_nat @ ( times_times_nat @ A_38 @ B_28 ) @ zero_zero_nat ) ) ) ).
thf(fact_662_mult__neg__pos,axiom,
! [B_28: int,A_38: int] :
( ( ord_less_int @ A_38 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_28 )
=> ( ord_less_int @ ( times_times_int @ A_38 @ B_28 ) @ zero_zero_int ) ) ) ).
thf(fact_663_mult__less__cancel__left__neg,axiom,
! [A_37: real,B_27: real,C_19: real] :
( ( ord_less_real @ C_19 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C_19 @ A_37 ) @ ( times_times_real @ C_19 @ B_27 ) )
<=> ( ord_less_real @ B_27 @ A_37 ) ) ) ).
thf(fact_664_mult__less__cancel__left__neg,axiom,
! [A_37: int,B_27: int,C_19: int] :
( ( ord_less_int @ C_19 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_19 @ A_37 ) @ ( times_times_int @ C_19 @ B_27 ) )
<=> ( ord_less_int @ B_27 @ A_37 ) ) ) ).
thf(fact_665_zero__less__mult__pos2,axiom,
! [B_26: real,A_36: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B_26 @ A_36 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_36 )
=> ( ord_less_real @ zero_zero_real @ B_26 ) ) ) ).
thf(fact_666_zero__less__mult__pos2,axiom,
! [B_26: nat,A_36: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_26 @ A_36 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_36 )
=> ( ord_less_nat @ zero_zero_nat @ B_26 ) ) ) ).
thf(fact_667_zero__less__mult__pos2,axiom,
! [B_26: int,A_36: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_26 @ A_36 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_36 )
=> ( ord_less_int @ zero_zero_int @ B_26 ) ) ) ).
thf(fact_668_zero__less__mult__pos,axiom,
! [A_35: real,B_25: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_35 @ B_25 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_35 )
=> ( ord_less_real @ zero_zero_real @ B_25 ) ) ) ).
thf(fact_669_zero__less__mult__pos,axiom,
! [A_35: nat,B_25: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_35 @ B_25 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_35 )
=> ( ord_less_nat @ zero_zero_nat @ B_25 ) ) ) ).
thf(fact_670_zero__less__mult__pos,axiom,
! [A_35: int,B_25: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_35 @ B_25 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_35 )
=> ( ord_less_int @ zero_zero_int @ B_25 ) ) ) ).
thf(fact_671_mult__pos__neg2,axiom,
! [B_24: real,A_34: real] :
( ( ord_less_real @ zero_zero_real @ A_34 )
=> ( ( ord_less_real @ B_24 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B_24 @ A_34 ) @ zero_zero_real ) ) ) ).
thf(fact_672_mult__pos__neg2,axiom,
! [B_24: nat,A_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_34 )
=> ( ( ord_less_nat @ B_24 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_24 @ A_34 ) @ zero_zero_nat ) ) ) ).
thf(fact_673_mult__pos__neg2,axiom,
! [B_24: int,A_34: int] :
( ( ord_less_int @ zero_zero_int @ A_34 )
=> ( ( ord_less_int @ B_24 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_24 @ A_34 ) @ zero_zero_int ) ) ) ).
thf(fact_674_mult__pos__neg,axiom,
! [B_23: real,A_33: real] :
( ( ord_less_real @ zero_zero_real @ A_33 )
=> ( ( ord_less_real @ B_23 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_33 @ B_23 ) @ zero_zero_real ) ) ) ).
thf(fact_675_mult__pos__neg,axiom,
! [B_23: nat,A_33: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_33 )
=> ( ( ord_less_nat @ B_23 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_33 @ B_23 ) @ zero_zero_nat ) ) ) ).
thf(fact_676_mult__pos__neg,axiom,
! [B_23: int,A_33: int] :
( ( ord_less_int @ zero_zero_int @ A_33 )
=> ( ( ord_less_int @ B_23 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_33 @ B_23 ) @ zero_zero_int ) ) ) ).
thf(fact_677_mult__pos__pos,axiom,
! [B_22: real,A_32: real] :
( ( ord_less_real @ zero_zero_real @ A_32 )
=> ( ( ord_less_real @ zero_zero_real @ B_22 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_32 @ B_22 ) ) ) ) ).
thf(fact_678_mult__pos__pos,axiom,
! [B_22: nat,A_32: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_32 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_22 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_32 @ B_22 ) ) ) ) ).
thf(fact_679_mult__pos__pos,axiom,
! [B_22: int,A_32: int] :
( ( ord_less_int @ zero_zero_int @ A_32 )
=> ( ( ord_less_int @ zero_zero_int @ B_22 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_32 @ B_22 ) ) ) ) ).
thf(fact_680_mult__less__cancel__left__pos,axiom,
! [A_31: real,B_21: real,C_18: real] :
( ( ord_less_real @ zero_zero_real @ C_18 )
=> ( ( ord_less_real @ ( times_times_real @ C_18 @ A_31 ) @ ( times_times_real @ C_18 @ B_21 ) )
<=> ( ord_less_real @ A_31 @ B_21 ) ) ) ).
thf(fact_681_mult__less__cancel__left__pos,axiom,
! [A_31: int,B_21: int,C_18: int] :
( ( ord_less_int @ zero_zero_int @ C_18 )
=> ( ( ord_less_int @ ( times_times_int @ C_18 @ A_31 ) @ ( times_times_int @ C_18 @ B_21 ) )
<=> ( ord_less_int @ A_31 @ B_21 ) ) ) ).
thf(fact_682_mult__less__cancel__left__disj,axiom,
! [C_17: real,A_30: real,B_20: real] :
( ( ord_less_real @ ( times_times_real @ C_17 @ A_30 ) @ ( times_times_real @ C_17 @ B_20 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_17 )
& ( ord_less_real @ A_30 @ B_20 ) )
| ( ( ord_less_real @ C_17 @ zero_zero_real )
& ( ord_less_real @ B_20 @ A_30 ) ) ) ) ).
thf(fact_683_mult__less__cancel__left__disj,axiom,
! [C_17: int,A_30: int,B_20: int] :
( ( ord_less_int @ ( times_times_int @ C_17 @ A_30 ) @ ( times_times_int @ C_17 @ B_20 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_17 )
& ( ord_less_int @ A_30 @ B_20 ) )
| ( ( ord_less_int @ C_17 @ zero_zero_int )
& ( ord_less_int @ B_20 @ A_30 ) ) ) ) ).
thf(fact_684_mult__less__cancel__right__disj,axiom,
! [A_29: real,C_16: real,B_19: real] :
( ( ord_less_real @ ( times_times_real @ A_29 @ C_16 ) @ ( times_times_real @ B_19 @ C_16 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_16 )
& ( ord_less_real @ A_29 @ B_19 ) )
| ( ( ord_less_real @ C_16 @ zero_zero_real )
& ( ord_less_real @ B_19 @ A_29 ) ) ) ) ).
thf(fact_685_mult__less__cancel__right__disj,axiom,
! [A_29: int,C_16: int,B_19: int] :
( ( ord_less_int @ ( times_times_int @ A_29 @ C_16 ) @ ( times_times_int @ B_19 @ C_16 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_16 )
& ( ord_less_int @ A_29 @ B_19 ) )
| ( ( ord_less_int @ C_16 @ zero_zero_int )
& ( ord_less_int @ B_19 @ A_29 ) ) ) ) ).
thf(fact_686_not__square__less__zero,axiom,
! [A_28: real] :
~ ( ord_less_real @ ( times_times_real @ A_28 @ A_28 ) @ zero_zero_real ) ).
thf(fact_687_not__square__less__zero,axiom,
! [A_28: int] :
~ ( ord_less_int @ ( times_times_int @ A_28 @ A_28 ) @ zero_zero_int ) ).
thf(fact_688_pos__add__strict,axiom,
! [B_18: real,C_15: real,A_27: real] :
( ( ord_less_real @ zero_zero_real @ A_27 )
=> ( ( ord_less_real @ B_18 @ C_15 )
=> ( ord_less_real @ B_18 @ ( plus_plus_real @ A_27 @ C_15 ) ) ) ) ).
thf(fact_689_pos__add__strict,axiom,
! [B_18: nat,C_15: nat,A_27: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_27 )
=> ( ( ord_less_nat @ B_18 @ C_15 )
=> ( ord_less_nat @ B_18 @ ( plus_plus_nat @ A_27 @ C_15 ) ) ) ) ).
thf(fact_690_pos__add__strict,axiom,
! [B_18: int,C_15: int,A_27: int] :
( ( ord_less_int @ zero_zero_int @ A_27 )
=> ( ( ord_less_int @ B_18 @ C_15 )
=> ( ord_less_int @ B_18 @ ( plus_plus_int @ A_27 @ C_15 ) ) ) ) ).
thf(fact_691_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
thf(fact_692_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_693_zero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_694_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
thf(fact_695_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_696_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
thf(fact_697_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
thf(fact_698_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_699_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_700_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
thf(fact_701_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_702_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
thf(fact_703_power__mono,axiom,
! [N_11: nat,A_26: real,B_17: real] :
( ( ord_less_eq_real @ A_26 @ B_17 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_26 )
=> ( ord_less_eq_real @ ( power_power_real @ A_26 @ N_11 ) @ ( power_power_real @ B_17 @ N_11 ) ) ) ) ).
thf(fact_704_power__mono,axiom,
! [N_11: nat,A_26: nat,B_17: nat] :
( ( ord_less_eq_nat @ A_26 @ B_17 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_26 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_26 @ N_11 ) @ ( power_power_nat @ B_17 @ N_11 ) ) ) ) ).
thf(fact_705_power__mono,axiom,
! [N_11: nat,A_26: int,B_17: int] :
( ( ord_less_eq_int @ A_26 @ B_17 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_26 )
=> ( ord_less_eq_int @ ( power_power_int @ A_26 @ N_11 ) @ ( power_power_int @ B_17 @ N_11 ) ) ) ) ).
thf(fact_706_zero__le__power,axiom,
! [N_10: nat,A_25: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_25 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_25 @ N_10 ) ) ) ).
thf(fact_707_zero__le__power,axiom,
! [N_10: nat,A_25: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_25 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A_25 @ N_10 ) ) ) ).
thf(fact_708_zero__le__power,axiom,
! [N_10: nat,A_25: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_25 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_25 @ N_10 ) ) ) ).
thf(fact_709_less__1__mult,axiom,
! [N_9: real,M_5: real] :
( ( ord_less_real @ one_one_real @ M_5 )
=> ( ( ord_less_real @ one_one_real @ N_9 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M_5 @ N_9 ) ) ) ) ).
thf(fact_710_less__1__mult,axiom,
! [N_9: nat,M_5: nat] :
( ( ord_less_nat @ one_one_nat @ M_5 )
=> ( ( ord_less_nat @ one_one_nat @ N_9 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_5 @ N_9 ) ) ) ) ).
thf(fact_711_less__1__mult,axiom,
! [N_9: int,M_5: int] :
( ( ord_less_int @ one_one_int @ M_5 )
=> ( ( ord_less_int @ one_one_int @ N_9 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_5 @ N_9 ) ) ) ) ).
thf(fact_712_zero__less__power,axiom,
! [N_8: nat,A_24: real] :
( ( ord_less_real @ zero_zero_real @ A_24 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_24 @ N_8 ) ) ) ).
thf(fact_713_zero__less__power,axiom,
! [N_8: nat,A_24: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_24 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A_24 @ N_8 ) ) ) ).
thf(fact_714_zero__less__power,axiom,
! [N_8: nat,A_24: int] :
( ( ord_less_int @ zero_zero_int @ A_24 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_24 @ N_8 ) ) ) ).
thf(fact_715_less__add__one,axiom,
! [A_23: real] : ( ord_less_real @ A_23 @ ( plus_plus_real @ A_23 @ one_one_real ) ) ).
thf(fact_716_less__add__one,axiom,
! [A_23: nat] : ( ord_less_nat @ A_23 @ ( plus_plus_nat @ A_23 @ one_one_nat ) ) ).
thf(fact_717_less__add__one,axiom,
! [A_23: int] : ( ord_less_int @ A_23 @ ( plus_plus_int @ A_23 @ one_one_int ) ) ).
thf(fact_718_dvd__mult__cancel__left,axiom,
! [C_14: int,A_22: int,B_16: int] :
( ( dvd_dvd_int @ ( times_times_int @ C_14 @ A_22 ) @ ( times_times_int @ C_14 @ B_16 ) )
<=> ( ( C_14 = zero_zero_int )
| ( dvd_dvd_int @ A_22 @ B_16 ) ) ) ).
thf(fact_719_dvd__mult__cancel__left,axiom,
! [C_14: real,A_22: real,B_16: real] :
( ( dvd_dvd_real @ ( times_times_real @ C_14 @ A_22 ) @ ( times_times_real @ C_14 @ B_16 ) )
<=> ( ( C_14 = zero_zero_real )
| ( dvd_dvd_real @ A_22 @ B_16 ) ) ) ).
thf(fact_720_dvd__mult__cancel__right,axiom,
! [A_21: int,C_13: int,B_15: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_21 @ C_13 ) @ ( times_times_int @ B_15 @ C_13 ) )
<=> ( ( C_13 = zero_zero_int )
| ( dvd_dvd_int @ A_21 @ B_15 ) ) ) ).
thf(fact_721_dvd__mult__cancel__right,axiom,
! [A_21: real,C_13: real,B_15: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_21 @ C_13 ) @ ( times_times_real @ B_15 @ C_13 ) )
<=> ( ( C_13 = zero_zero_real )
| ( dvd_dvd_real @ A_21 @ B_15 ) ) ) ).
thf(fact_722_power__increasing,axiom,
! [A_20: real,N_7: nat,N_6: nat] :
( ( ord_less_eq_nat @ N_7 @ N_6 )
=> ( ( ord_less_eq_real @ one_one_real @ A_20 )
=> ( ord_less_eq_real @ ( power_power_real @ A_20 @ N_7 ) @ ( power_power_real @ A_20 @ N_6 ) ) ) ) ).
thf(fact_723_power__increasing,axiom,
! [A_20: nat,N_7: nat,N_6: nat] :
( ( ord_less_eq_nat @ N_7 @ N_6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A_20 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_20 @ N_7 ) @ ( power_power_nat @ A_20 @ N_6 ) ) ) ) ).
thf(fact_724_power__increasing,axiom,
! [A_20: int,N_7: nat,N_6: nat] :
( ( ord_less_eq_nat @ N_7 @ N_6 )
=> ( ( ord_less_eq_int @ one_one_int @ A_20 )
=> ( ord_less_eq_int @ ( power_power_int @ A_20 @ N_7 ) @ ( power_power_int @ A_20 @ N_6 ) ) ) ) ).
thf(fact_725_one__le__power,axiom,
! [N_5: nat,A_19: real] :
( ( ord_less_eq_real @ one_one_real @ A_19 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A_19 @ N_5 ) ) ) ).
thf(fact_726_one__le__power,axiom,
! [N_5: nat,A_19: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A_19 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A_19 @ N_5 ) ) ) ).
thf(fact_727_one__le__power,axiom,
! [N_5: nat,A_19: int] :
( ( ord_less_eq_int @ one_one_int @ A_19 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A_19 @ N_5 ) ) ) ).
thf(fact_728_power__inject__exp,axiom,
! [M_4: nat,N_4: nat,A_18: real] :
( ( ord_less_real @ one_one_real @ A_18 )
=> ( ( ( power_power_real @ A_18 @ M_4 )
= ( power_power_real @ A_18 @ N_4 ) )
<=> ( M_4 = N_4 ) ) ) ).
thf(fact_729_power__inject__exp,axiom,
! [M_4: nat,N_4: nat,A_18: nat] :
( ( ord_less_nat @ one_one_nat @ A_18 )
=> ( ( ( power_power_nat @ A_18 @ M_4 )
= ( power_power_nat @ A_18 @ N_4 ) )
<=> ( M_4 = N_4 ) ) ) ).
thf(fact_730_power__inject__exp,axiom,
! [M_4: nat,N_4: nat,A_18: int] :
( ( ord_less_int @ one_one_int @ A_18 )
=> ( ( ( power_power_int @ A_18 @ M_4 )
= ( power_power_int @ A_18 @ N_4 ) )
<=> ( M_4 = N_4 ) ) ) ).
thf(fact_731_power__eq__0__iff,axiom,
! [A_17: real,N_3: nat] :
( ( ( power_power_real @ A_17 @ N_3 )
= zero_zero_real )
<=> ( ( A_17 = zero_zero_real )
& ( N_3 != zero_zero_nat ) ) ) ).
thf(fact_732_power__eq__0__iff,axiom,
! [A_17: nat,N_3: nat] :
( ( ( power_power_nat @ A_17 @ N_3 )
= zero_zero_nat )
<=> ( ( A_17 = zero_zero_nat )
& ( N_3 != zero_zero_nat ) ) ) ).
thf(fact_733_power__eq__0__iff,axiom,
! [A_17: int,N_3: nat] :
( ( ( power_power_int @ A_17 @ N_3 )
= zero_zero_int )
<=> ( ( A_17 = zero_zero_int )
& ( N_3 != zero_zero_nat ) ) ) ).
thf(fact_734_power__0,axiom,
! [A_16: real] :
( ( power_power_real @ A_16 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_735_power__0,axiom,
! [A_16: nat] :
( ( power_power_nat @ A_16 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_736_power__0,axiom,
! [A_16: int] :
( ( power_power_int @ A_16 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_737_power__add,axiom,
! [A_15: real,M_3: nat,N_2: nat] :
( ( power_power_real @ A_15 @ ( plus_plus_nat @ M_3 @ N_2 ) )
= ( times_times_real @ ( power_power_real @ A_15 @ M_3 ) @ ( power_power_real @ A_15 @ N_2 ) ) ) ).
thf(fact_738_power__add,axiom,
! [A_15: nat,M_3: nat,N_2: nat] :
( ( power_power_nat @ A_15 @ ( plus_plus_nat @ M_3 @ N_2 ) )
= ( times_times_nat @ ( power_power_nat @ A_15 @ M_3 ) @ ( power_power_nat @ A_15 @ N_2 ) ) ) ).
thf(fact_739_power__add,axiom,
! [A_15: int,M_3: nat,N_2: nat] :
( ( power_power_int @ A_15 @ ( plus_plus_nat @ M_3 @ N_2 ) )
= ( times_times_int @ ( power_power_int @ A_15 @ M_3 ) @ ( power_power_int @ A_15 @ N_2 ) ) ) ).
thf(fact_740_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_741_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_742_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_743_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_744_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_745_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_746_zcong__zero__equiv__div,axiom,
! [A: int,M: int] :
( ( zcong @ A @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A ) ) ).
thf(fact_747_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_748_mult__left__le__imp__le,axiom,
! [C_12: real,A_14: real,B_14: real] :
( ( ord_less_eq_real @ ( times_times_real @ C_12 @ A_14 ) @ ( times_times_real @ C_12 @ B_14 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_12 )
=> ( ord_less_eq_real @ A_14 @ B_14 ) ) ) ).
thf(fact_749_mult__left__le__imp__le,axiom,
! [C_12: nat,A_14: nat,B_14: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C_12 @ A_14 ) @ ( times_times_nat @ C_12 @ B_14 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_12 )
=> ( ord_less_eq_nat @ A_14 @ B_14 ) ) ) ).
thf(fact_750_mult__left__le__imp__le,axiom,
! [C_12: int,A_14: int,B_14: int] :
( ( ord_less_eq_int @ ( times_times_int @ C_12 @ A_14 ) @ ( times_times_int @ C_12 @ B_14 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_12 )
=> ( ord_less_eq_int @ A_14 @ B_14 ) ) ) ).
thf(fact_751_mult__right__le__imp__le,axiom,
! [A_13: real,C_11: real,B_13: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_13 @ C_11 ) @ ( times_times_real @ B_13 @ C_11 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_11 )
=> ( ord_less_eq_real @ A_13 @ B_13 ) ) ) ).
thf(fact_752_mult__right__le__imp__le,axiom,
! [A_13: nat,C_11: nat,B_13: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A_13 @ C_11 ) @ ( times_times_nat @ B_13 @ C_11 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_11 )
=> ( ord_less_eq_nat @ A_13 @ B_13 ) ) ) ).
thf(fact_753_mult__right__le__imp__le,axiom,
! [A_13: int,C_11: int,B_13: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_13 @ C_11 ) @ ( times_times_int @ B_13 @ C_11 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_11 )
=> ( ord_less_eq_int @ A_13 @ B_13 ) ) ) ).
thf(fact_754_mult__less__imp__less__left,axiom,
! [C_10: real,A_12: real,B_12: real] :
( ( ord_less_real @ ( times_times_real @ C_10 @ A_12 ) @ ( times_times_real @ C_10 @ B_12 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_10 )
=> ( ord_less_real @ A_12 @ B_12 ) ) ) ).
thf(fact_755_mult__less__imp__less__left,axiom,
! [C_10: nat,A_12: nat,B_12: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_10 @ A_12 ) @ ( times_times_nat @ C_10 @ B_12 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_10 )
=> ( ord_less_nat @ A_12 @ B_12 ) ) ) ).
thf(fact_756_mult__less__imp__less__left,axiom,
! [C_10: int,A_12: int,B_12: int] :
( ( ord_less_int @ ( times_times_int @ C_10 @ A_12 ) @ ( times_times_int @ C_10 @ B_12 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_10 )
=> ( ord_less_int @ A_12 @ B_12 ) ) ) ).
thf(fact_757_mult__left__less__imp__less,axiom,
! [C_9: real,A_11: real,B_11: real] :
( ( ord_less_real @ ( times_times_real @ C_9 @ A_11 ) @ ( times_times_real @ C_9 @ B_11 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_9 )
=> ( ord_less_real @ A_11 @ B_11 ) ) ) ).
thf(fact_758_mult__left__less__imp__less,axiom,
! [C_9: nat,A_11: nat,B_11: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_9 @ A_11 ) @ ( times_times_nat @ C_9 @ B_11 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_9 )
=> ( ord_less_nat @ A_11 @ B_11 ) ) ) ).
thf(fact_759_mult__left__less__imp__less,axiom,
! [C_9: int,A_11: int,B_11: int] :
( ( ord_less_int @ ( times_times_int @ C_9 @ A_11 ) @ ( times_times_int @ C_9 @ B_11 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_9 )
=> ( ord_less_int @ A_11 @ B_11 ) ) ) ).
thf(fact_760_mult__less__imp__less__right,axiom,
! [A_10: real,C_8: real,B_10: real] :
( ( ord_less_real @ ( times_times_real @ A_10 @ C_8 ) @ ( times_times_real @ B_10 @ C_8 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_8 )
=> ( ord_less_real @ A_10 @ B_10 ) ) ) ).
thf(fact_761_mult__less__imp__less__right,axiom,
! [A_10: nat,C_8: nat,B_10: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_10 @ C_8 ) @ ( times_times_nat @ B_10 @ C_8 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_8 )
=> ( ord_less_nat @ A_10 @ B_10 ) ) ) ).
thf(fact_762_mult__less__imp__less__right,axiom,
! [A_10: int,C_8: int,B_10: int] :
( ( ord_less_int @ ( times_times_int @ A_10 @ C_8 ) @ ( times_times_int @ B_10 @ C_8 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_8 )
=> ( ord_less_int @ A_10 @ B_10 ) ) ) ).
thf(fact_763_mult__right__less__imp__less,axiom,
! [A_9: real,C_7: real,B_9: real] :
( ( ord_less_real @ ( times_times_real @ A_9 @ C_7 ) @ ( times_times_real @ B_9 @ C_7 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_7 )
=> ( ord_less_real @ A_9 @ B_9 ) ) ) ).
thf(fact_764_mult__right__less__imp__less,axiom,
! [A_9: nat,C_7: nat,B_9: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_9 @ C_7 ) @ ( times_times_nat @ B_9 @ C_7 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_7 )
=> ( ord_less_nat @ A_9 @ B_9 ) ) ) ).
thf(fact_765_mult__right__less__imp__less,axiom,
! [A_9: int,C_7: int,B_9: int] :
( ( ord_less_int @ ( times_times_int @ A_9 @ C_7 ) @ ( times_times_int @ B_9 @ C_7 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_7 )
=> ( ord_less_int @ A_9 @ B_9 ) ) ) ).
thf(fact_766_mult__le__less__imp__less,axiom,
! [C_6: real,D_6: real,A_8: real,B_8: real] :
( ( ord_less_eq_real @ A_8 @ B_8 )
=> ( ( ord_less_real @ C_6 @ D_6 )
=> ( ( ord_less_real @ zero_zero_real @ A_8 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_6 )
=> ( ord_less_real @ ( times_times_real @ A_8 @ C_6 ) @ ( times_times_real @ B_8 @ D_6 ) ) ) ) ) ) ).
thf(fact_767_mult__le__less__imp__less,axiom,
! [C_6: nat,D_6: nat,A_8: nat,B_8: nat] :
( ( ord_less_eq_nat @ A_8 @ B_8 )
=> ( ( ord_less_nat @ C_6 @ D_6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_8 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_6 )
=> ( ord_less_nat @ ( times_times_nat @ A_8 @ C_6 ) @ ( times_times_nat @ B_8 @ D_6 ) ) ) ) ) ) ).
thf(fact_768_mult__le__less__imp__less,axiom,
! [C_6: int,D_6: int,A_8: int,B_8: int] :
( ( ord_less_eq_int @ A_8 @ B_8 )
=> ( ( ord_less_int @ C_6 @ D_6 )
=> ( ( ord_less_int @ zero_zero_int @ A_8 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_6 )
=> ( ord_less_int @ ( times_times_int @ A_8 @ C_6 ) @ ( times_times_int @ B_8 @ D_6 ) ) ) ) ) ) ).
thf(fact_769_mult__less__le__imp__less,axiom,
! [C_5: real,D_5: real,A_7: real,B_7: real] :
( ( ord_less_real @ A_7 @ B_7 )
=> ( ( ord_less_eq_real @ C_5 @ D_5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_7 )
=> ( ( ord_less_real @ zero_zero_real @ C_5 )
=> ( ord_less_real @ ( times_times_real @ A_7 @ C_5 ) @ ( times_times_real @ B_7 @ D_5 ) ) ) ) ) ) ).
thf(fact_770_mult__less__le__imp__less,axiom,
! [C_5: nat,D_5: nat,A_7: nat,B_7: nat] :
( ( ord_less_nat @ A_7 @ B_7 )
=> ( ( ord_less_eq_nat @ C_5 @ D_5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_7 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_5 )
=> ( ord_less_nat @ ( times_times_nat @ A_7 @ C_5 ) @ ( times_times_nat @ B_7 @ D_5 ) ) ) ) ) ) ).
thf(fact_771_mult__less__le__imp__less,axiom,
! [C_5: int,D_5: int,A_7: int,B_7: int] :
( ( ord_less_int @ A_7 @ B_7 )
=> ( ( ord_less_eq_int @ C_5 @ D_5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_7 )
=> ( ( ord_less_int @ zero_zero_int @ C_5 )
=> ( ord_less_int @ ( times_times_int @ A_7 @ C_5 ) @ ( times_times_int @ B_7 @ D_5 ) ) ) ) ) ) ).
thf(fact_772_mult__strict__mono_H,axiom,
! [C_4: real,D_4: real,A_6: real,B_6: real] :
( ( ord_less_real @ A_6 @ B_6 )
=> ( ( ord_less_real @ C_4 @ D_4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_4 )
=> ( ord_less_real @ ( times_times_real @ A_6 @ C_4 ) @ ( times_times_real @ B_6 @ D_4 ) ) ) ) ) ) ).
thf(fact_773_mult__strict__mono_H,axiom,
! [C_4: nat,D_4: nat,A_6: nat,B_6: nat] :
( ( ord_less_nat @ A_6 @ B_6 )
=> ( ( ord_less_nat @ C_4 @ D_4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_4 )
=> ( ord_less_nat @ ( times_times_nat @ A_6 @ C_4 ) @ ( times_times_nat @ B_6 @ D_4 ) ) ) ) ) ) ).
thf(fact_774_mult__strict__mono_H,axiom,
! [C_4: int,D_4: int,A_6: int,B_6: int] :
( ( ord_less_int @ A_6 @ B_6 )
=> ( ( ord_less_int @ C_4 @ D_4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_4 )
=> ( ord_less_int @ ( times_times_int @ A_6 @ C_4 ) @ ( times_times_int @ B_6 @ D_4 ) ) ) ) ) ) ).
thf(fact_775_mult__strict__mono,axiom,
! [C_3: real,D_3: real,A_5: real,B_5: real] :
( ( ord_less_real @ A_5 @ B_5 )
=> ( ( ord_less_real @ C_3 @ D_3 )
=> ( ( ord_less_real @ zero_zero_real @ B_5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_3 )
=> ( ord_less_real @ ( times_times_real @ A_5 @ C_3 ) @ ( times_times_real @ B_5 @ D_3 ) ) ) ) ) ) ).
thf(fact_776_mult__strict__mono,axiom,
! [C_3: nat,D_3: nat,A_5: nat,B_5: nat] :
( ( ord_less_nat @ A_5 @ B_5 )
=> ( ( ord_less_nat @ C_3 @ D_3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_3 )
=> ( ord_less_nat @ ( times_times_nat @ A_5 @ C_3 ) @ ( times_times_nat @ B_5 @ D_3 ) ) ) ) ) ) ).
thf(fact_777_mult__strict__mono,axiom,
! [C_3: int,D_3: int,A_5: int,B_5: int] :
( ( ord_less_int @ A_5 @ B_5 )
=> ( ( ord_less_int @ C_3 @ D_3 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_3 )
=> ( ord_less_int @ ( times_times_int @ A_5 @ C_3 ) @ ( times_times_int @ B_5 @ D_3 ) ) ) ) ) ) ).
thf(fact_778_mult__le__cancel__left__neg,axiom,
! [A_4: real,B_4: real,C_2: real] :
( ( ord_less_real @ C_2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_2 @ A_4 ) @ ( times_times_real @ C_2 @ B_4 ) )
<=> ( ord_less_eq_real @ B_4 @ A_4 ) ) ) ).
thf(fact_779_mult__le__cancel__left__neg,axiom,
! [A_4: int,B_4: int,C_2: int] :
( ( ord_less_int @ C_2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_2 @ A_4 ) @ ( times_times_int @ C_2 @ B_4 ) )
<=> ( ord_less_eq_int @ B_4 @ A_4 ) ) ) ).
thf(fact_780_mult__le__cancel__left__pos,axiom,
! [A_3: real,B_3: real,C_1: real] :
( ( ord_less_real @ zero_zero_real @ C_1 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_1 @ A_3 ) @ ( times_times_real @ C_1 @ B_3 ) )
<=> ( ord_less_eq_real @ A_3 @ B_3 ) ) ) ).
thf(fact_781_mult__le__cancel__left__pos,axiom,
! [A_3: int,B_3: int,C_1: int] :
( ( ord_less_int @ zero_zero_int @ C_1 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_1 @ A_3 ) @ ( times_times_int @ C_1 @ B_3 ) )
<=> ( ord_less_eq_int @ A_3 @ B_3 ) ) ) ).
thf(fact_782_mult__left__le__one__le,axiom,
! [Y_3: real,X_3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_3 )
=> ( ( ord_less_eq_real @ Y_3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y_3 @ X_3 ) @ X_3 ) ) ) ) ).
thf(fact_783_mult__left__le__one__le,axiom,
! [Y_3: int,X_3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_3 )
=> ( ( ord_less_eq_int @ Y_3 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y_3 @ X_3 ) @ X_3 ) ) ) ) ).
thf(fact_784_mult__right__le__one__le,axiom,
! [Y_2: real,X_2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_2 )
=> ( ( ord_less_eq_real @ Y_2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X_2 @ Y_2 ) @ X_2 ) ) ) ) ).
thf(fact_785_mult__right__le__one__le,axiom,
! [Y_2: int,X_2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_2 )
=> ( ( ord_less_eq_int @ Y_2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X_2 @ Y_2 ) @ X_2 ) ) ) ) ).
thf(fact_786_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
thf(fact_787_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_788_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799__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_800__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_801_neg__one__power__eq__mod__m,axiom,
! [J_1: 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_1 ) @ ( power_power_int @ ( number_number_of_int @ min ) @ K ) @ M )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ J_1 )
= ( power_power_int @ ( number_number_of_int @ min ) @ K ) ) ) ) ).
thf(fact_802_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_803__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_804__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
( ~ ( quadRes @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( number_number_of_int @ min ) )
=> ( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
!= one_one_int ) ) ).
thf(fact_805_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_806_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_807_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_808_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_809_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_810_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_811_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_812_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_813_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_814_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_815_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_816_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_817_xzgcda__linear__aux1,axiom,
! [A: int,R: int,B: int,M: int,C: int,D: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ ( times_times_int @ R @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R @ D ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A @ M ) @ ( times_times_int @ C @ N ) ) @ ( times_times_int @ R @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D @ N ) ) ) ) ) ).
thf(fact_818_zcong__def,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A @ B ) ) ) ).
thf(fact_819_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_820_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_821_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_822_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_823_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_824_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_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_divides__antisym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
<=> ( X = Y ) ) ).
thf(fact_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_divides__ge,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = zero_zero_nat )
| ( ord_less_eq_nat @ A @ B ) ) ) ).
thf(fact_846_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_847_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_848_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_849_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_850_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_851_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_852_divides__div__not,axiom,
! [X: nat,Q: nat,N: nat,R: nat] :
( ( X
= ( plus_plus_nat @ ( times_times_nat @ Q @ N ) @ R ) )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( ord_less_nat @ R @ N )
=> ~ ( dvd_dvd_nat @ N @ X ) ) ) ) ).
thf(fact_853_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_854_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_855_realpow__pos__nth,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R_1: real] :
( ( ord_less_real @ zero_zero_real @ R_1 )
& ( ( power_power_real @ R_1 @ N )
= A ) ) ) ) ).
thf(fact_856_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_857_dvd_Oorder__refl,axiom,
! [X: nat] : ( dvd_dvd_nat @ X @ X ) ).
thf(fact_858_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_859_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_860_diff__commute,axiom,
! [I_1: nat,J_1: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K ) @ J_1 ) ) ).
thf(fact_861_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_862_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_863_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_864_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_865_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_866_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_867_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_868_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_869_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_870_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_871_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_872_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_873_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_874_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_875_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_876_less__imp__diff__less,axiom,
! [N: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ J_1 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J_1 @ N ) @ K ) ) ).
thf(fact_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_dvd_Oless__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( Y != X ) ) ).
thf(fact_885_dvd_Oless__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_886_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_887_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_888_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_889_dvd_Oless__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_890_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_891_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_892_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_893_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_894_dvd_Oantisym,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ X )
=> ( X = Y ) ) ) ).
thf(fact_895_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_896_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_897_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_898_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_899_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_900_dvd_Oantisym__conv,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( dvd_dvd_nat @ X @ Y )
<=> ( X = Y ) ) ) ).
thf(fact_901_dvd_Oeq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_902_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_903_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_904_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_905_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_906_dvd_Oeq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_907_dvd_Oless__irrefl,axiom,
! [X: nat] :
~ ( ( dvd_dvd_nat @ X @ X )
& ~ ( dvd_dvd_nat @ X @ X ) ) ).
thf(fact_908_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_909_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_910_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_911_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_912_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_913_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_914_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_915_diff__diff__left,axiom,
! [I_1: nat,J_1: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ K )
= ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_916_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_917_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_918_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_919_real__mult__commute,axiom,
! [Z: real,W: real] :
( ( times_times_real @ Z @ W )
= ( times_times_real @ W @ Z ) ) ).
thf(fact_920_real__mult__1,axiom,
! [Z: real] :
( ( times_times_real @ one_one_real @ Z )
= Z ) ).
thf(fact_921_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_922_le__trans,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ J_1 @ K )
=> ( ord_less_eq_nat @ I_1 @ K ) ) ) ).
thf(fact_923_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_924_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_925_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_926_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_927_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_928_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_929_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_930_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_931_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_932_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_933_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_934_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_935_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_936_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_937_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_938_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_939_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_940_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_941_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_942_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_943_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_944_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_945_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_946_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_947_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_948_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_949_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_950_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_951_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_952_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_953_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_954_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_955_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_956_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_957_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_958_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_959_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_960_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_961_not__add__less1,axiom,
! [I_1: nat,J_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ I_1 ) ).
thf(fact_962_not__add__less2,axiom,
! [J_1: nat,I_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ I_1 ) ).
thf(fact_963_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_964_trans__less__add1,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J_1 @ M ) ) ) ).
thf(fact_965_trans__less__add2,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J_1 ) ) ) ).
thf(fact_966_add__less__mono1,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_967_add__less__mono,axiom,
! [K: nat,L: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_968_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_969_add__lessD1,axiom,
! [I_1: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ K )
=> ( ord_less_nat @ I_1 @ K ) ) ).
thf(fact_970_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_971_less__diff__conv,axiom,
! [I_1: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ J_1 ) ) ).
thf(fact_972_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_973_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_974_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_975_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_976_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_977_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_978_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_979_dvd__reduce,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( plus_plus_nat @ N @ K ) )
<=> ( dvd_dvd_nat @ K @ N ) ) ).
thf(fact_980_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_981_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_982_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_983_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_984_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_985_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_986_trans__le__add1,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J_1 @ M ) ) ) ).
thf(fact_987_trans__le__add2,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J_1 ) ) ) ).
thf(fact_988_add__le__mono1,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_989_add__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_990_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_991_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_992_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_993_diff__diff__right,axiom,
! [I_1: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K ) @ J_1 ) ) ) ).
thf(fact_994_le__diff__conv,axiom,
! [J_1: nat,K: nat,I_1: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J_1 @ K ) @ I_1 )
<=> ( ord_less_eq_nat @ J_1 @ ( plus_plus_nat @ I_1 @ K ) ) ) ).
thf(fact_995_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_996_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_997_add__diff__assoc,axiom,
! [I_1: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ K ) ) ) ).
thf(fact_998_le__diff__conv2,axiom,
! [I_1: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ J_1 ) ) ) ).
thf(fact_999_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_1000_le__imp__diff__is__add,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ( minus_minus_nat @ J_1 @ I_1 )
= K )
<=> ( J_1
= ( plus_plus_nat @ K @ I_1 ) ) ) ) ).
thf(fact_1001_diff__add__assoc,axiom,
! [I_1: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ K )
= ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K ) ) ) ) ).
thf(fact_1002_add__diff__assoc2,axiom,
! [I_1: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J_1 @ K ) @ I_1 )
= ( minus_minus_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ K ) ) ) ).
thf(fact_1003_diff__add__assoc2,axiom,
! [I_1: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J_1 @ K ) @ I_1 ) ) ) ).
thf(fact_1004_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_1005_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_1006_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
<=> ( M = one_one_nat ) ) ).
thf(fact_1007_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_1008_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_1009_mult__le__mono1,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J_1 @ K ) ) ) ).
thf(fact_1010_mult__le__mono2,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J_1 ) ) ) ).
thf(fact_1011_mult__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J_1 @ L ) ) ) ) ).
thf(fact_1012_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_1013_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_1014_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_1015_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_1016_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_1017_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_1018_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_1019_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_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
& ~ ( P @ D_2 ) ) ) ) ).
thf(fact_1020_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_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
=> ( P @ D_2 ) ) ) ) ).
thf(fact_1021_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_1022_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_1023_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_1024_mult__less__mono1,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J_1 @ K ) ) ) ) ).
thf(fact_1025_mult__less__mono2,axiom,
! [K: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J_1 ) ) ) ) ).
thf(fact_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_nat__less__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1041_nat__less__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1042_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_1043_real__le__linear,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
| ( ord_less_eq_real @ W @ Z ) ) ).
thf(fact_1044_real__le__trans,axiom,
! [K: real,I_1: real,J_1: real] :
( ( ord_less_eq_real @ I_1 @ J_1 )
=> ( ( ord_less_eq_real @ J_1 @ K )
=> ( ord_less_eq_real @ I_1 @ K ) ) ) ).
thf(fact_1045_real__le__antisym,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_1046_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_1047_left__add__mult__distrib,axiom,
! [I_1: nat,U: nat,J_1: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ U ) @ K ) ) ).
thf(fact_1048_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_1049_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_1050_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_1051_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_1052_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_1053_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1054_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1055_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M )
= N ) ) ) ).
thf(fact_1056_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1057_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1058_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1059_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_1060_zprime__factor__exists,axiom,
! [A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ? [P_3: int] :
( ( zprime @ P_3 )
& ( dvd_dvd_int @ P_3 @ A ) ) ) ).
thf(fact_1061_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_1062_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_1063_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_1064_number__of2,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ pls ) ).
thf(fact_1065_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_1066_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_1067_zcong__zless__unique,axiom,
! [A: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ M )
& ( zcong @ A @ X_1 @ M )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ M )
& ( zcong @ A @ Y_1 @ M ) )
=> ( Y_1 = X_1 ) ) ) ) ).
thf(fact_1068__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
( ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ s @ 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_1069_mult__sum2sq,axiom,
! [A: int,B: int,P_1: int,Q: int] :
( ( times_times_int @ ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ A @ B ) ) @ ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ P_1 @ Q ) ) )
= ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A @ P_1 ) @ ( times_times_int @ B @ Q ) ) @ ( minus_minus_int @ ( times_times_int @ A @ Q ) @ ( times_times_int @ B @ P_1 ) ) ) ) ) ).
thf(fact_1070_is__sum2sq__def,axiom,
! [X: int] :
( ( twoSqu919416604sum2sq @ X )
<=> ? [A_2: int,B_2: int] :
( ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ A_2 @ B_2 ) )
= X ) ) ).
thf(fact_1071_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_1072_Wilson__Russ,axiom,
! [P_1: int] :
( ( zprime @ P_1 )
=> ( zcong @ ( zfact @ ( minus_minus_int @ P_1 @ one_one_int ) ) @ ( number_number_of_int @ min ) @ P_1 ) ) ).
thf(fact_1073_zfact_Osimps,axiom,
! [N: int] :
( ( ( ord_less_eq_int @ N @ zero_zero_int )
=> ( ( zfact @ N )
= one_one_int ) )
& ( ~ ( ord_less_eq_int @ N @ zero_zero_int )
=> ( ( zfact @ N )
= ( times_times_int @ N @ ( zfact @ ( minus_minus_int @ N @ one_one_int ) ) ) ) ) ) ).
thf(fact_1074_inv__inv,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_1 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_1 )
=> ( ( inv @ P_1 @ ( inv @ P_1 @ A ) )
= A ) ) ) ) ) ).
thf(fact_1075_inv__less__p__minus__1,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ord_less_int @ ( inv @ P_1 @ A ) @ ( minus_minus_int @ P_1 @ one_one_int ) ) ) ) ) ).
thf(fact_1076_inv__g__1,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ord_less_int @ one_one_int @ ( inv @ P_1 @ A ) ) ) ) ) ).
thf(fact_1077_inv__not__p__minus__1,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( inv @ P_1 @ A )
!= ( minus_minus_int @ P_1 @ one_one_int ) ) ) ) ) ).
thf(fact_1078_inv__not__1,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( inv @ P_1 @ A )
!= one_one_int ) ) ) ) ).
thf(fact_1079_inv__distinct,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( A
!= ( inv @ P_1 @ A ) ) ) ) ) ).
thf(fact_1080_inv__not__0,axiom,
! [A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( inv @ P_1 @ A )
!= zero_zero_int ) ) ) ) ).
thf(fact_1081_inv__is__inv,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 @ ( inv @ P_1 @ A ) ) @ one_one_int @ P_1 ) ) ) ) ).
thf(fact_1082_wset__mem__inv__mem,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_1 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_1 ) )
=> ( member_int @ ( inv @ P_1 @ B ) @ ( wset @ A @ P_1 ) ) ) ) ) ) ).
thf(fact_1083_wset__inv__mem__mem,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_1 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ B @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( member_int @ ( inv @ P_1 @ B ) @ ( wset @ A @ P_1 ) )
=> ( member_int @ B @ ( wset @ A @ P_1 ) ) ) ) ) ) ) ) ).
thf(fact_1084_wset__mem__mem,axiom,
! [P_1: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( member_int @ A @ ( wset @ A @ P_1 ) ) ) ).
thf(fact_1085_wset__subset,axiom,
! [B: int,P_1: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P_1 ) )
=> ( member_int @ B @ ( wset @ A @ P_1 ) ) ) ) ).
thf(fact_1086_wset__g__1,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_1 ) )
=> ( ord_less_int @ one_one_int @ B ) ) ) ) ).
thf(fact_1087_wset__less,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_1 ) )
=> ( ord_less_int @ B @ ( minus_minus_int @ P_1 @ one_one_int ) ) ) ) ) ).
thf(fact_1088_wset__mem__imp__or,axiom,
! [B: int,P_1: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ~ ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P_1 ) )
=> ( ( member_int @ B @ ( wset @ A @ P_1 ) )
=> ( ( B = A )
| ( B
= ( inv @ P_1 @ A ) ) ) ) ) ) ).
thf(fact_1089_wset__mem,axiom,
! [B: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_1 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( wset @ A @ P_1 ) ) ) ) ) ) ).
thf(fact_1090_MultInvPair__distinct,axiom,
! [J_1: int,A: int,P_1: int] :
( ( zprime @ P_1 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_1 )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P_1 )
=> ( ~ ( quadRes @ P_1 @ A )
=> ~ ( zcong @ J_1 @ ( times_times_int @ A @ ( multInv @ P_1 @ J_1 ) ) @ P_1 ) ) ) ) ) ) ).
thf(fact_1091_int__le__induct,axiom,
! [P: int > $o,I_1: int,K: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1092_aux______1,axiom,
! [J_1: int,A: int,P_1: int,K: int] :
( ( zcong @ J_1 @ ( times_times_int @ A @ ( multInv @ P_1 @ K ) ) @ P_1 )
=> ( zcong @ ( times_times_int @ J_1 @ K ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P_1 @ K ) ) @ K ) @ P_1 ) ) ).
thf(fact_1093_aux______3,axiom,
! [J_1: int,K: int,A: int,P_1: int] :
( ( zcong @ ( times_times_int @ J_1 @ K ) @ A @ P_1 )
=> ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_1 @ J_1 ) @ J_1 ) @ K ) @ ( times_times_int @ ( multInv @ P_1 @ J_1 ) @ A ) @ P_1 ) ) ).
thf(fact_1094_MultInv__prop1,axiom,
! [X: int,Y: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zcong @ X @ Y @ P_1 )
=> ( zcong @ ( multInv @ P_1 @ X ) @ ( multInv @ P_1 @ Y ) @ P_1 ) ) ) ).
thf(fact_1095_MultInv__zcong__prop1,axiom,
! [A: int,J_1: int,K: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zcong @ J_1 @ K @ P_1 )
=> ( zcong @ ( times_times_int @ A @ ( multInv @ P_1 @ J_1 ) ) @ ( times_times_int @ A @ ( multInv @ P_1 @ K ) ) @ P_1 ) ) ) ).
thf(fact_1096_MultInv__prop5,axiom,
! [Y: int,X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( ~ ( zcong @ Y @ zero_zero_int @ P_1 )
=> ( ( zcong @ ( multInv @ P_1 @ X ) @ ( multInv @ P_1 @ Y ) @ P_1 )
=> ( zcong @ X @ Y @ P_1 ) ) ) ) ) ) ).
thf(fact_1097_MultInv__prop4,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( zcong @ ( multInv @ P_1 @ ( multInv @ P_1 @ X ) ) @ X @ P_1 ) ) ) ) ).
thf(fact_1098_MultInv__prop3,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ~ ( zcong @ ( multInv @ P_1 @ X ) @ zero_zero_int @ P_1 ) ) ) ) ).
thf(fact_1099_aux______4,axiom,
! [K: int,A: int,J_1: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P_1 )
=> ( ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_1 @ J_1 ) @ J_1 ) @ K ) @ ( times_times_int @ ( multInv @ P_1 @ J_1 ) @ A ) @ P_1 )
=> ( zcong @ K @ ( times_times_int @ A @ ( multInv @ P_1 @ J_1 ) ) @ P_1 ) ) ) ) ) ).
thf(fact_1100_aux______2,axiom,
! [J_1: int,A: int,K: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ K @ zero_zero_int @ P_1 )
=> ( ( zcong @ ( times_times_int @ J_1 @ K ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P_1 @ K ) ) @ K ) @ P_1 )
=> ( zcong @ ( times_times_int @ J_1 @ K ) @ A @ P_1 ) ) ) ) ) ).
thf(fact_1101_MultInv__zcong__prop2,axiom,
! [A: int,J_1: int,K: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ K @ zero_zero_int @ P_1 )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P_1 )
=> ( ( zcong @ J_1 @ ( times_times_int @ A @ ( multInv @ P_1 @ K ) ) @ P_1 )
=> ( zcong @ K @ ( times_times_int @ A @ ( multInv @ P_1 @ J_1 ) ) @ P_1 ) ) ) ) ) ) ).
thf(fact_1102_MultInv__zcong__prop3,axiom,
! [J_1: int,K: int,A: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_1 )
=> ( ~ ( zcong @ K @ zero_zero_int @ P_1 )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P_1 )
=> ( ( zcong @ ( times_times_int @ A @ ( multInv @ P_1 @ J_1 ) ) @ ( times_times_int @ A @ ( multInv @ P_1 @ K ) ) @ P_1 )
=> ( zcong @ J_1 @ K @ P_1 ) ) ) ) ) ) ) ).
thf(fact_1103_Int2_Oaux____2,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( zcong @ ( times_times_int @ ( times_times_int @ X @ ( multInv @ P_1 @ X ) ) @ ( multInv @ P_1 @ ( multInv @ P_1 @ X ) ) ) @ X @ P_1 ) ) ) ) ).
thf(fact_1104_Int2_Oaux____1,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( zcong @ ( multInv @ P_1 @ ( multInv @ P_1 @ X ) ) @ ( times_times_int @ ( times_times_int @ X @ ( multInv @ P_1 @ X ) ) @ ( multInv @ P_1 @ ( multInv @ P_1 @ X ) ) ) @ P_1 ) ) ) ) ).
thf(fact_1105_MultInv__prop2a,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( zcong @ ( times_times_int @ ( multInv @ P_1 @ X ) @ X ) @ one_one_int @ P_1 ) ) ) ) ).
thf(fact_1106_MultInv__prop2,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_1 )
=> ( ( zprime @ P_1 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_1 )
=> ( zcong @ ( times_times_int @ X @ ( multInv @ P_1 @ X ) ) @ one_one_int @ P_1 ) ) ) ) ).
thf(fact_1107_d22set__induct__old,axiom,
! [X: int,P: int > $o] :
( ! [A_2: int] :
( ( ( ord_less_int @ one_one_int @ A_2 )
=> ( P @ ( minus_minus_int @ A_2 @ one_one_int ) ) )
=> ( P @ A_2 ) )
=> ( P @ X ) ) ).
thf(fact_1108_int__less__induct,axiom,
! [P: int > $o,I_1: int,K: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1109_int__ge__induct,axiom,
! [P: int > $o,K: int,I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1110_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_1111_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: nat > nat] :
( ! [M_2: nat,N_1: nat] :
( ( ord_less_nat @ M_2 @ N_1 )
=> ( ord_less_nat @ ( F @ M_2 ) @ ( F @ N_1 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_1112_d22set__eq__wset,axiom,
! [P_1: int] :
( ( zprime @ P_1 )
=> ( ( d22set @ ( minus_minus_int @ P_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( wset @ ( minus_minus_int @ P_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_1 ) ) ) ).
thf(fact_1113_d22set__le__swap,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( member_int @ B @ ( d22set @ A ) ) ) ).
thf(fact_1114_d22set__le,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
thf(fact_1115_d22set__g__1,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_int @ one_one_int @ B ) ) ).
thf(fact_1116_d22set__mem,axiom,
! [A: int,B: int] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( d22set @ A ) ) ) ) ).
thf(fact_1117_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K_1 )
=> ~ ( P @ I ) )
& ( P @ K_1 ) ) ) ) ).
thf(fact_1118_less__imp__add__positive,axiom,
! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I_1 @ K_1 )
= J_1 ) ) ) ).
thf(fact_1119_pow__divides__eq__int,axiom,
! [A: int,B: int,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
<=> ( dvd_dvd_int @ A @ B ) ) ) ).
thf(fact_1120_pow__divides__pow__int,axiom,
! [A: int,N: nat,B: int] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_int @ A @ B ) ) ) ).
thf(fact_1121_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
thf(fact_1122_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
<=> ( A = zero_zero_nat ) ) ).
thf(fact_1123_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ~ ( dvd_dvd_nat @ zero_zero_nat @ A ) ) ) ).
thf(fact_1124_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
thf(fact_1125_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ~ ( dvd_dvd_nat @ A @ zero_zero_nat ) ) ).
thf(fact_1126_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( A = one_one_nat ) ) ).
thf(fact_1127_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
<=> ( A = one_one_nat ) ) ).
thf(fact_1128_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A: nat] :
( ( A != one_one_nat )
<=> ( ( dvd_dvd_nat @ one_one_nat @ A )
& ~ ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
thf(fact_1129_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
thf(fact_1130_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ one_one_nat )
& ~ ( dvd_dvd_nat @ one_one_nat @ A ) ) ).
thf(fact_1131_dvd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
thf(fact_1132_pow__divides__eq__nat,axiom,
! [A: nat,B: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
<=> ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_1133_divides__le,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( ord_less_eq_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1134_mult__left__cancel,axiom,
! [N: nat,M: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ N )
= ( times_times_nat @ K @ M ) )
=> ( N = M ) ) ) ).
thf(fact_1135_SR__def,axiom,
! [P_1: int] :
( ( sr @ P_1 )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ P_1 ) ) ) ) ).
thf(fact_1136_zmod__number__of__Bit1,axiom,
! [V: int,W: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_mod_int @ ( number_number_of_int @ ( bit1 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) @ one_one_int ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_mod_int @ ( number_number_of_int @ ( bit1 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( plus_plus_int @ ( number_number_of_int @ V ) @ one_one_int ) @ ( number_number_of_int @ W ) ) ) @ one_one_int ) ) ) ) ).
thf(fact_1137_neg__zmod__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
thf(fact_1138_zmod__simps_I3_J,axiom,
! [A: int,B: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C )
= ( div_mod_int @ ( times_times_int @ A @ B ) @ C ) ) ).
thf(fact_1139_zmod__zmult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A @ B ) @ C )
= ( div_mod_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C ) ) ).
thf(fact_1140_zcong__zmod,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ ( div_mod_int @ A @ M ) @ ( div_mod_int @ B @ M ) @ M ) ) ).
thf(fact_1141_zpower__zmod,axiom,
! [X: int,M: int,Y: nat] :
( ( div_mod_int @ ( power_power_int @ ( div_mod_int @ X @ M ) @ Y ) @ M )
= ( div_mod_int @ ( power_power_int @ X @ Y ) @ M ) ) ).
thf(fact_1142_mod__mod__is__mod,axiom,
! [X: int,M: int] : ( zcong @ X @ ( div_mod_int @ X @ M ) @ M ) ).
thf(fact_1143_Residues_Oaux,axiom,
! [X: int,M: int,Y: int] :
( ( ( div_mod_int @ X @ M )
= ( div_mod_int @ Y @ M ) )
=> ( zcong @ X @ Y @ M ) ) ).
thf(fact_1144_zdvd__zmod,axiom,
! [N: int,F: int,M: int] :
( ( dvd_dvd_int @ F @ M )
=> ( ( dvd_dvd_int @ F @ N )
=> ( dvd_dvd_int @ F @ ( div_mod_int @ M @ N ) ) ) ) ).
thf(fact_1145_zdvd__zmod__imp__zdvd,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( div_mod_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_1146_zmod__self,axiom,
! [A: int] :
( ( div_mod_int @ A @ A )
= zero_zero_int ) ).
thf(fact_1147_zmod__zero,axiom,
! [B: int] :
( ( div_mod_int @ zero_zero_int @ B )
= zero_zero_int ) ).
thf(fact_1148_zmod__eq__dvd__iff,axiom,
! [X: int,N: int,Y: int] :
( ( ( div_mod_int @ X @ N )
= ( div_mod_int @ Y @ N ) )
<=> ( dvd_dvd_int @ N @ ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_1149_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( ( div_mod_int @ M @ D )
= zero_zero_int )
<=> ? [Q_1: int] :
( M
= ( times_times_int @ D @ Q_1 ) ) ) ).
thf(fact_1150_Divides_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 @ ( div_mod_int @ X @ Y ) ) ) ) ).
thf(fact_1151_zmod__le__nonneg__dividend,axiom,
! [K: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( div_mod_int @ M @ K ) @ M ) ) ).
thf(fact_1152_pos__mod__bound,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( div_mod_int @ A @ B ) @ B ) ) ).
thf(fact_1153_neg__mod__bound,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ B @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_1154_zdiff__zmod__left,axiom,
! [X: int,M: int,Y: int] :
( ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ X @ M ) @ Y ) @ M )
= ( div_mod_int @ ( minus_minus_int @ X @ Y ) @ M ) ) ).
thf(fact_1155_zdiff__zmod__right,axiom,
! [X: int,Y: int,M: int] :
( ( div_mod_int @ ( minus_minus_int @ X @ ( div_mod_int @ Y @ M ) ) @ M )
= ( div_mod_int @ ( minus_minus_int @ X @ Y ) @ M ) ) ).
thf(fact_1156_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( number_number_of_int @ X ) @ ( number_number_of_int @ Y ) )
<=> ( ( div_mod_int @ ( number_number_of_int @ Y ) @ ( number_number_of_int @ X ) )
= zero_zero_int ) ) ).
thf(fact_1157_zmod__minus1__right,axiom,
! [A: int] :
( ( div_mod_int @ A @ ( number_number_of_int @ min ) )
= zero_zero_int ) ).
thf(fact_1158_zmod__zdvd__zmod,axiom,
! [A: int,B: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( dvd_dvd_int @ M @ B )
=> ( ( div_mod_int @ ( div_mod_int @ A @ B ) @ M )
= ( div_mod_int @ A @ M ) ) ) ) ).
thf(fact_1159_zcong__zmod__eq,axiom,
! [A: int,B: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ A @ B @ M )
<=> ( ( div_mod_int @ A @ M )
= ( div_mod_int @ B @ M ) ) ) ) ).
thf(fact_1160_pos__mod__sign,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_1161_pos__mod__conj,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A @ B ) )
& ( ord_less_int @ ( div_mod_int @ A @ B ) @ B ) ) ) ).
thf(fact_1162_mod__pos__pos__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( div_mod_int @ A @ B )
= A ) ) ) ).
thf(fact_1163_neg__mod__sign,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_mod_int @ A @ B ) @ zero_zero_int ) ) ).
thf(fact_1164_neg__mod__conj,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( div_mod_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_1165_mod__neg__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ A )
=> ( ( div_mod_int @ A @ B )
= A ) ) ) ).
thf(fact_1166_mod__pos__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( plus_plus_int @ A @ B ) ) ) ) ).
thf(fact_1167_zmod__number__of__Bit0,axiom,
! [V: int,W: int] :
( ( div_mod_int @ ( number_number_of_int @ ( bit0 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) ) ).
thf(fact_1168_divmod__int__rel__mod__eq,axiom,
! [A_1: int,B_1: int,Q_2: int,Y: int] :
( ( A_1
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_2 ) @ Y ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
& ( ord_less_int @ Y @ B_1 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_int @ B_1 @ Y )
& ( ord_less_eq_int @ Y @ zero_zero_int ) ) ) )
=> ( ( B_1 != zero_zero_int )
=> ( ( div_mod_int @ A_1 @ B_1 )
= Y ) ) ) ) ).
thf(fact_1169_zmult2__lemma__aux2,axiom,
! [Q: int,B: int,R: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B @ R )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R ) @ zero_zero_int ) ) ) ) ).
thf(fact_1170_zmult2__lemma__aux1,axiom,
! [Q: int,B: int,R: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B @ R )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R ) ) ) ) ) ).
thf(fact_1171_zmult2__lemma__aux4,axiom,
! [Q: int,B: int,R: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R ) @ ( times_times_int @ B @ C ) ) ) ) ) ).
thf(fact_1172_zmult2__lemma__aux3,axiom,
! [Q: int,B: int,R: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R ) ) ) ) ) ).
thf(fact_1173_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( div_mod_int @ N @ K ) )
<=> ( ( ( K = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I: int,J: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J )
& ( ord_less_int @ J @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I ) @ J ) ) )
=> ( P @ J ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I: int,J: int] :
( ( ( ord_less_int @ K @ J )
& ( ord_less_eq_int @ J @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I ) @ J ) ) )
=> ( P @ J ) ) ) ) ) ).
thf(fact_1174_neq__one__mod__two,axiom,
! [X: int] :
( ( ( div_mod_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_int )
<=> ( ( div_mod_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ) ).
thf(fact_1175_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ ( number_number_of_int @ min ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
thf(fact_1176_pos__zmod__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ B @ A ) ) ) ) ) ).
thf(fact_1177_zmod__eq__0D,axiom,
! [M_1: int,D_1: int] :
( ( ( div_mod_int @ M_1 @ D_1 )
= zero_zero_int )
=> ? [Q_1: int] :
( M_1
= ( times_times_int @ D_1 @ Q_1 ) ) ) ).
thf(fact_1178_StandardRes__prop4,axiom,
! [X: int,Y: int,M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ( zcong @ ( times_times_int @ ( standardRes @ M @ X ) @ ( standardRes @ M @ Y ) ) @ ( times_times_int @ X @ Y ) @ M ) ) ).
thf(fact_1179_mod__le__divisor,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ N ) ) ).
thf(fact_1180_mod__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( div_mod_nat @ M @ N ) )
= ( div_mod_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_1181_mod__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( div_mod_nat @ M @ N ) @ K )
= ( div_mod_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_1182_StandardRes__def,axiom,
! [M: int,X: int] :
( ( standardRes @ M @ X )
= ( div_mod_int @ X @ M ) ) ).
thf(fact_1183_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) ) ).
thf(fact_1184_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( div_mod_nat @ M @ N )
= ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
thf(fact_1185_mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
= ( div_mod_nat @ M @ N ) ) ).
thf(fact_1186_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
thf(fact_1187_mod__if,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) )
& ( ~ ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ).
thf(fact_1188_mod__eq__0__iff,axiom,
! [M: nat,D: nat] :
( ( ( div_mod_nat @ M @ D )
= zero_zero_nat )
<=> ? [Q_1: nat] :
( M
= ( times_times_nat @ D @ Q_1 ) ) ) ).
thf(fact_1189_mod__less__divisor,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( div_mod_nat @ M @ N ) @ N ) ) ).
thf(fact_1190_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ M ) ).
thf(fact_1191_StandardRes__eq__zcong,axiom,
! [M: int,X: int] :
( ( ( standardRes @ M @ X )
= zero_zero_int )
<=> ( zcong @ X @ zero_zero_int @ M ) ) ).
thf(fact_1192_StandardRes__prop3,axiom,
! [X: int,P_1: int] :
( ~ ( zcong @ X @ zero_zero_int @ P_1 )
<=> ( ( standardRes @ P_1 @ X )
!= zero_zero_int ) ) ).
thf(fact_1193_StandardRes__prop1,axiom,
! [X: int,M: int] : ( zcong @ X @ ( standardRes @ M @ X ) @ M ) ).
thf(fact_1194_StandardRes__ubound,axiom,
! [X: int,P_1: int] :
( ( ord_less_int @ zero_zero_int @ P_1 )
=> ( ord_less_int @ ( standardRes @ P_1 @ X ) @ P_1 ) ) ).
thf(fact_1195_StandardRes__SR__prop,axiom,
! [X: int,P_1: int] :
( ( member_int @ X @ ( sr @ P_1 ) )
=> ( ( standardRes @ P_1 @ X )
= X ) ) ).
thf(fact_1196_split__mod,axiom,
! [P: nat > $o,N: nat,K: nat] :
( ( P @ ( div_mod_nat @ N @ K ) )
<=> ( ( ( K = zero_zero_nat )
=> ( P @ N ) )
& ( ( K != zero_zero_nat )
=> ! [I: nat,J: nat] :
( ( ord_less_nat @ J @ K )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K @ I ) @ J ) )
=> ( P @ J ) ) ) ) ) ) ).
thf(fact_1197_mod__lemma,axiom,
! [Q: nat,R: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ R @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ Q @ C ) ) @ R ) @ ( times_times_nat @ B @ C ) ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
ord_less_int @ ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) @ zero_zero_int ).
%------------------------------------------------------------------------------