TPTP Problem File: NUM926^3.p
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%------------------------------------------------------------------------------
% File : NUM926^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 258, 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_l258 [Bla11]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.54 v8.1.0, 0.64 v7.5.0, 0.43 v7.4.0, 0.56 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 1259 ( 455 unt; 60 typ; 0 def)
% Number of atoms : 2686 ( 842 equ; 1 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 11416 ( 259 ~; 70 |; 166 &;9632 @)
% ( 254 <=>;1035 =>; 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 : 2591 ( 1 ^;2567 !; 23 ?;2591 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:54:07
%------------------------------------------------------------------------------
%----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_tpos,axiom,
ord_less_eq_int @ one_one_int @ t ).
thf(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( ( t = one_one_int )
=> ? [X: int,Y: int] :
( ( plus_plus_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( ( ord_less_int @ one_one_int @ t )
=> ? [X: int,Y: int] :
( ( plus_plus_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_3_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_4_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_5_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_6_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_7_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_8_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_9_power2__sum,axiom,
! [X_44: int,Y_34: int] :
( ( power_power_int @ ( plus_plus_int @ X_44 @ Y_34 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X_44 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_44 ) @ Y_34 ) ) ) ).
thf(fact_10_power2__sum,axiom,
! [X_44: nat,Y_34: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X_44 @ Y_34 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X_44 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_44 ) @ Y_34 ) ) ) ).
thf(fact_11_power2__sum,axiom,
! [X_44: real,Y_34: real] :
( ( power_power_real @ ( plus_plus_real @ X_44 @ Y_34 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X_44 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_44 ) @ Y_34 ) ) ) ).
thf(fact_12_power2__eq__square__number__of,axiom,
! [W_19: int] :
( ( power_power_int @ ( number_number_of_int @ W_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( number_number_of_int @ W_19 ) @ ( number_number_of_int @ W_19 ) ) ) ).
thf(fact_13_power2__eq__square__number__of,axiom,
! [W_19: int] :
( ( power_power_real @ ( number267125858f_real @ W_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( number267125858f_real @ W_19 ) @ ( number267125858f_real @ W_19 ) ) ) ).
thf(fact_14_power2__eq__square__number__of,axiom,
! [W_19: int] :
( ( power_power_nat @ ( number_number_of_nat @ W_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ W_19 ) @ ( number_number_of_nat @ W_19 ) ) ) ).
thf(fact_15_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_16_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_17_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_18_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_real ) ).
thf(fact_19_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_43: int] :
( ( times_times_int @ X_43 @ X_43 )
= ( power_power_int @ X_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_20_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_43: real] :
( ( times_times_real @ X_43 @ X_43 )
= ( power_power_real @ X_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_43: nat] :
( ( times_times_nat @ X_43 @ X_43 )
= ( power_power_nat @ X_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_22_power2__eq__square,axiom,
! [A_127: int] :
( ( power_power_int @ A_127 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ A_127 @ A_127 ) ) ).
thf(fact_23_power2__eq__square,axiom,
! [A_127: real] :
( ( power_power_real @ A_127 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ A_127 @ A_127 ) ) ).
thf(fact_24_power2__eq__square,axiom,
! [A_127: nat] :
( ( power_power_nat @ A_127 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ A_127 @ A_127 ) ) ).
thf(fact_25_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_42: int,N_41: nat] :
( ( power_power_int @ X_42 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_41 ) )
= ( times_times_int @ ( power_power_int @ X_42 @ N_41 ) @ ( power_power_int @ X_42 @ N_41 ) ) ) ).
thf(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_42: real,N_41: nat] :
( ( power_power_real @ X_42 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_41 ) )
= ( times_times_real @ ( power_power_real @ X_42 @ N_41 ) @ ( power_power_real @ X_42 @ N_41 ) ) ) ).
thf(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_42: nat,N_41: nat] :
( ( power_power_nat @ X_42 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_41 ) )
= ( times_times_nat @ ( power_power_nat @ X_42 @ N_41 ) @ ( power_power_nat @ X_42 @ N_41 ) ) ) ).
thf(fact_28_add__special_I2_J,axiom,
! [W_18: int] :
( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_18 ) )
= ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_18 ) ) ) ).
thf(fact_29_add__special_I2_J,axiom,
! [W_18: int] :
( ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_18 ) )
= ( number267125858f_real @ ( plus_plus_int @ ( bit1 @ pls ) @ W_18 ) ) ) ).
thf(fact_30_add__special_I3_J,axiom,
! [V_20: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_20 ) @ one_one_int )
= ( number_number_of_int @ ( plus_plus_int @ V_20 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_31_add__special_I3_J,axiom,
! [V_20: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_20 ) @ one_one_real )
= ( number267125858f_real @ ( plus_plus_int @ V_20 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_32_one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_33_one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_34__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_35_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
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_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_38_zless__linear,axiom,
! [X_1: int,Y_1: int] :
( ( ord_less_int @ X_1 @ Y_1 )
| ( X_1 = Y_1 )
| ( ord_less_int @ Y_1 @ X_1 ) ) ).
thf(fact_39_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_40_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_41: int,P_6: nat,Q_6: nat] :
( ( power_power_int @ ( power_power_int @ X_41 @ P_6 ) @ Q_6 )
= ( power_power_int @ X_41 @ ( times_times_nat @ P_6 @ Q_6 ) ) ) ).
thf(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_41: real,P_6: nat,Q_6: nat] :
( ( power_power_real @ ( power_power_real @ X_41 @ P_6 ) @ Q_6 )
= ( power_power_real @ X_41 @ ( times_times_nat @ P_6 @ Q_6 ) ) ) ).
thf(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_41: nat,P_6: nat,Q_6: nat] :
( ( power_power_nat @ ( power_power_nat @ X_41 @ P_6 ) @ Q_6 )
= ( power_power_nat @ X_41 @ ( times_times_nat @ P_6 @ Q_6 ) ) ) ).
thf(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_40: int] :
( ( power_power_int @ X_40 @ one_one_nat )
= X_40 ) ).
thf(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_40: real] :
( ( power_power_real @ X_40 @ one_one_nat )
= X_40 ) ).
thf(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_40: nat] :
( ( power_power_nat @ X_40 @ one_one_nat )
= X_40 ) ).
thf(fact_47_zpower__zpower,axiom,
! [X_1: int,Y_1: nat,Z: nat] :
( ( power_power_int @ ( power_power_int @ X_1 @ Y_1 ) @ Z )
= ( power_power_int @ X_1 @ ( times_times_nat @ Y_1 @ Z ) ) ) ).
thf(fact_48_le__number__of__eq__not__less,axiom,
! [V_19: int,W_17: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_19 ) @ ( number_number_of_int @ W_17 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_17 ) @ ( number_number_of_int @ V_19 ) ) ) ).
thf(fact_49_le__number__of__eq__not__less,axiom,
! [V_19: int,W_17: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_19 ) @ ( number_number_of_nat @ W_17 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_17 ) @ ( number_number_of_nat @ V_19 ) ) ) ).
thf(fact_50_le__number__of__eq__not__less,axiom,
! [V_19: int,W_17: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_19 ) @ ( number267125858f_real @ W_17 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_17 ) @ ( number267125858f_real @ V_19 ) ) ) ).
thf(fact_51_less__number__of,axiom,
! [X_39: int,Y_33: int] :
( ( ord_less_int @ ( number_number_of_int @ X_39 ) @ ( number_number_of_int @ Y_33 ) )
<=> ( ord_less_int @ X_39 @ Y_33 ) ) ).
thf(fact_52_less__number__of,axiom,
! [X_39: int,Y_33: int] :
( ( ord_less_real @ ( number267125858f_real @ X_39 ) @ ( number267125858f_real @ Y_33 ) )
<=> ( ord_less_int @ X_39 @ Y_33 ) ) ).
thf(fact_53_le__number__of,axiom,
! [X_38: int,Y_32: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_38 ) @ ( number_number_of_int @ Y_32 ) )
<=> ( ord_less_eq_int @ X_38 @ Y_32 ) ) ).
thf(fact_54_le__number__of,axiom,
! [X_38: int,Y_32: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_38 ) @ ( number267125858f_real @ Y_32 ) )
<=> ( ord_less_eq_int @ X_38 @ Y_32 ) ) ).
thf(fact_55_zadd__zless__mono,axiom,
! [Z_10: int,Z: int,W_16: int,W: int] :
( ( ord_less_int @ W_16 @ W )
=> ( ( ord_less_eq_int @ Z_10 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_16 @ Z_10 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_37: int,P_5: nat,Q_5: nat] :
( ( times_times_int @ ( power_power_int @ X_37 @ P_5 ) @ ( power_power_int @ X_37 @ Q_5 ) )
= ( power_power_int @ X_37 @ ( plus_plus_nat @ P_5 @ Q_5 ) ) ) ).
thf(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_37: real,P_5: nat,Q_5: nat] :
( ( times_times_real @ ( power_power_real @ X_37 @ P_5 ) @ ( power_power_real @ X_37 @ Q_5 ) )
= ( power_power_real @ X_37 @ ( plus_plus_nat @ P_5 @ Q_5 ) ) ) ).
thf(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_37: nat,P_5: nat,Q_5: nat] :
( ( times_times_nat @ ( power_power_nat @ X_37 @ P_5 ) @ ( power_power_nat @ X_37 @ Q_5 ) )
= ( power_power_nat @ X_37 @ ( plus_plus_nat @ P_5 @ Q_5 ) ) ) ).
thf(fact_59_zpower__zadd__distrib,axiom,
! [X_1: int,Y_1: nat,Z: nat] :
( ( power_power_int @ X_1 @ ( plus_plus_nat @ Y_1 @ Z ) )
= ( times_times_int @ ( power_power_int @ X_1 @ Y_1 ) @ ( power_power_int @ X_1 @ Z ) ) ) ).
thf(fact_60_nat__mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
thf(fact_61_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_62_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_63_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_64_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_65_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_66_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_67_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_68_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_69_rel__simps_I14_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_70_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_71_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_72_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_73_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_74_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_75_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_76_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_77_add__nat__number__of,axiom,
! [V_6: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
= ( number_number_of_nat @ V_6 ) ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( ( ord_less_int @ V_6 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
= ( number_number_of_nat @ V ) ) )
& ( ~ ( ord_less_int @ V_6 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V @ V_6 ) ) ) ) ) ) ) ).
thf(fact_78_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_79_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_80_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_81_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_82_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_83_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_84_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_85_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_86_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_87_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_88_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_89_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_90_is__mult__sum2sq,axiom,
! [Y_1: int,X_1: int] :
( ( twoSqu919416604sum2sq @ X_1 )
=> ( ( twoSqu919416604sum2sq @ Y_1 )
=> ( twoSqu919416604sum2sq @ ( times_times_int @ X_1 @ Y_1 ) ) ) ) ).
thf(fact_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_126: int,B_73: int] :
( ( times_times_int @ A_126 @ B_73 )
= ( times_times_int @ B_73 @ A_126 ) ) ).
thf(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_126: nat,B_73: nat] :
( ( times_times_nat @ A_126 @ B_73 )
= ( times_times_nat @ B_73 @ A_126 ) ) ).
thf(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_126: real,B_73: real] :
( ( times_times_real @ A_126 @ B_73 )
= ( times_times_real @ B_73 @ A_126 ) ) ).
thf(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_125: int,B_72: int,C_39: int,D_11: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_125 @ B_72 ) @ ( plus_plus_int @ C_39 @ D_11 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_125 @ C_39 ) @ ( plus_plus_int @ B_72 @ D_11 ) ) ) ).
thf(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_125: nat,B_72: nat,C_39: nat,D_11: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_125 @ B_72 ) @ ( plus_plus_nat @ C_39 @ D_11 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_125 @ C_39 ) @ ( plus_plus_nat @ B_72 @ D_11 ) ) ) ).
thf(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_125: real,B_72: real,C_39: real,D_11: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_125 @ B_72 ) @ ( plus_plus_real @ C_39 @ D_11 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_125 @ C_39 ) @ ( plus_plus_real @ B_72 @ D_11 ) ) ) ).
thf(fact_118_mem__def,axiom,
! [X_36: int,A_124: int > $o] :
( ( member_int @ X_36 @ A_124 )
<=> ( A_124 @ X_36 ) ) ).
thf(fact_119_Collect__def,axiom,
! [P_4: int > $o] :
( ( collect_int @ P_4 )
= P_4 ) ).
thf(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_123: int,B_71: int,C_38: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_123 @ B_71 ) @ C_38 )
= ( plus_plus_int @ ( plus_plus_int @ A_123 @ C_38 ) @ B_71 ) ) ).
thf(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_123: nat,B_71: nat,C_38: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_123 @ B_71 ) @ C_38 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_123 @ C_38 ) @ B_71 ) ) ).
thf(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_123: real,B_71: real,C_38: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_123 @ B_71 ) @ C_38 )
= ( plus_plus_real @ ( plus_plus_real @ A_123 @ C_38 ) @ B_71 ) ) ).
thf(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_122: int,B_70: int,C_37: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_122 @ B_70 ) @ C_37 )
= ( plus_plus_int @ A_122 @ ( plus_plus_int @ B_70 @ C_37 ) ) ) ).
thf(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_122: nat,B_70: nat,C_37: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_122 @ B_70 ) @ C_37 )
= ( plus_plus_nat @ A_122 @ ( plus_plus_nat @ B_70 @ C_37 ) ) ) ).
thf(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_122: real,B_70: real,C_37: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_122 @ B_70 ) @ C_37 )
= ( plus_plus_real @ A_122 @ ( plus_plus_real @ B_70 @ C_37 ) ) ) ).
thf(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_121: int,C_36: int,D_10: int] :
( ( plus_plus_int @ A_121 @ ( plus_plus_int @ C_36 @ D_10 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_121 @ C_36 ) @ D_10 ) ) ).
thf(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_121: nat,C_36: nat,D_10: nat] :
( ( plus_plus_nat @ A_121 @ ( plus_plus_nat @ C_36 @ D_10 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_121 @ C_36 ) @ D_10 ) ) ).
thf(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_121: real,C_36: real,D_10: real] :
( ( plus_plus_real @ A_121 @ ( plus_plus_real @ C_36 @ D_10 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_121 @ C_36 ) @ D_10 ) ) ).
thf(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_120: int,C_35: int,D_9: int] :
( ( plus_plus_int @ A_120 @ ( plus_plus_int @ C_35 @ D_9 ) )
= ( plus_plus_int @ C_35 @ ( plus_plus_int @ A_120 @ D_9 ) ) ) ).
thf(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_120: nat,C_35: nat,D_9: nat] :
( ( plus_plus_nat @ A_120 @ ( plus_plus_nat @ C_35 @ D_9 ) )
= ( plus_plus_nat @ C_35 @ ( plus_plus_nat @ A_120 @ D_9 ) ) ) ).
thf(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_120: real,C_35: real,D_9: real] :
( ( plus_plus_real @ A_120 @ ( plus_plus_real @ C_35 @ D_9 ) )
= ( plus_plus_real @ C_35 @ ( plus_plus_real @ A_120 @ D_9 ) ) ) ).
thf(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_119: int,C_34: int] :
( ( plus_plus_int @ A_119 @ C_34 )
= ( plus_plus_int @ C_34 @ A_119 ) ) ).
thf(fact_133_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_119: nat,C_34: nat] :
( ( plus_plus_nat @ A_119 @ C_34 )
= ( plus_plus_nat @ C_34 @ A_119 ) ) ).
thf(fact_134_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_119: real,C_34: real] :
( ( plus_plus_real @ A_119 @ C_34 )
= ( plus_plus_real @ C_34 @ A_119 ) ) ).
thf(fact_135_eq__number__of,axiom,
! [X_35: int,Y_31: int] :
( ( ( number_number_of_int @ X_35 )
= ( number_number_of_int @ Y_31 ) )
<=> ( X_35 = Y_31 ) ) ).
thf(fact_136_eq__number__of,axiom,
! [X_35: int,Y_31: int] :
( ( ( number267125858f_real @ X_35 )
= ( number267125858f_real @ Y_31 ) )
<=> ( X_35 = Y_31 ) ) ).
thf(fact_137_number__of__reorient,axiom,
! [W_15: int,X_34: nat] :
( ( ( number_number_of_nat @ W_15 )
= X_34 )
<=> ( X_34
= ( number_number_of_nat @ W_15 ) ) ) ).
thf(fact_138_number__of__reorient,axiom,
! [W_15: int,X_34: int] :
( ( ( number_number_of_int @ W_15 )
= X_34 )
<=> ( X_34
= ( number_number_of_int @ W_15 ) ) ) ).
thf(fact_139_number__of__reorient,axiom,
! [W_15: int,X_34: real] :
( ( ( number267125858f_real @ W_15 )
= X_34 )
<=> ( X_34
= ( number267125858f_real @ W_15 ) ) ) ).
thf(fact_140_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_141_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( ( bit0 @ K )
= ( bit0 @ L ) )
<=> ( K = L ) ) ).
thf(fact_142_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_143_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_144_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_145_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_146_zadd__left__commute,axiom,
! [X_1: int,Y_1: int,Z: int] :
( ( plus_plus_int @ X_1 @ ( plus_plus_int @ Y_1 @ Z ) )
= ( plus_plus_int @ Y_1 @ ( plus_plus_int @ X_1 @ Z ) ) ) ).
thf(fact_147_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_148_rel__simps_I12_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_149_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_150_rel__simps_I16_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_151_rel__simps_I10_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_152_rel__simps_I4_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_int @ pls @ K ) ) ).
thf(fact_153_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_154_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_155_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_156_rel__simps_I27_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_eq_int @ K @ pls ) ) ).
thf(fact_157_rel__simps_I21_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_158_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_159_power__even__eq,axiom,
! [A_118: int,N_40: nat] :
( ( power_power_int @ A_118 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_40 ) )
= ( power_power_int @ ( power_power_int @ A_118 @ N_40 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_160_power__even__eq,axiom,
! [A_118: real,N_40: nat] :
( ( power_power_real @ A_118 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_40 ) )
= ( power_power_real @ ( power_power_real @ A_118 @ N_40 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_161_power__even__eq,axiom,
! [A_118: nat,N_40: nat] :
( ( power_power_nat @ A_118 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_40 ) )
= ( power_power_nat @ ( power_power_nat @ A_118 @ N_40 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_162_less__special_I4_J,axiom,
! [X_33: int] :
( ( ord_less_int @ ( number_number_of_int @ X_33 ) @ one_one_int )
<=> ( ord_less_int @ X_33 @ ( bit1 @ pls ) ) ) ).
thf(fact_163_less__special_I4_J,axiom,
! [X_33: int] :
( ( ord_less_real @ ( number267125858f_real @ X_33 ) @ one_one_real )
<=> ( ord_less_int @ X_33 @ ( bit1 @ pls ) ) ) ).
thf(fact_164_less__special_I2_J,axiom,
! [Y_30: int] :
( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_30 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_30 ) ) ).
thf(fact_165_less__special_I2_J,axiom,
! [Y_30: int] :
( ( ord_less_real @ one_one_real @ ( number267125858f_real @ Y_30 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_30 ) ) ).
thf(fact_166_le__special_I4_J,axiom,
! [X_32: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_32 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_32 @ ( bit1 @ pls ) ) ) ).
thf(fact_167_le__special_I4_J,axiom,
! [X_32: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_32 ) @ one_one_real )
<=> ( ord_less_eq_int @ X_32 @ ( bit1 @ pls ) ) ) ).
thf(fact_168_le__special_I2_J,axiom,
! [Y_29: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_29 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_29 ) ) ).
thf(fact_169_le__special_I2_J,axiom,
! [Y_29: int] :
( ( ord_less_eq_real @ one_one_real @ ( number267125858f_real @ Y_29 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_29 ) ) ).
thf(fact_170_crossproduct__eq,axiom,
! [W_14: int,Y_28: int,X_31: int,Z_9: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_14 @ Y_28 ) @ ( times_times_int @ X_31 @ Z_9 ) )
= ( plus_plus_int @ ( times_times_int @ W_14 @ Z_9 ) @ ( times_times_int @ X_31 @ Y_28 ) ) )
<=> ( ( W_14 = X_31 )
| ( Y_28 = Z_9 ) ) ) ).
thf(fact_171_crossproduct__eq,axiom,
! [W_14: nat,Y_28: nat,X_31: nat,Z_9: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_14 @ Y_28 ) @ ( times_times_nat @ X_31 @ Z_9 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_14 @ Z_9 ) @ ( times_times_nat @ X_31 @ Y_28 ) ) )
<=> ( ( W_14 = X_31 )
| ( Y_28 = Z_9 ) ) ) ).
thf(fact_172_crossproduct__eq,axiom,
! [W_14: real,Y_28: real,X_31: real,Z_9: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_14 @ Y_28 ) @ ( times_times_real @ X_31 @ Z_9 ) )
= ( plus_plus_real @ ( times_times_real @ W_14 @ Z_9 ) @ ( times_times_real @ X_31 @ Y_28 ) ) )
<=> ( ( W_14 = X_31 )
| ( Y_28 = Z_9 ) ) ) ).
thf(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_117: int,M_14: int,B_69: int] :
( ( plus_plus_int @ ( times_times_int @ A_117 @ M_14 ) @ ( times_times_int @ B_69 @ M_14 ) )
= ( times_times_int @ ( plus_plus_int @ A_117 @ B_69 ) @ M_14 ) ) ).
thf(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_117: nat,M_14: nat,B_69: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_117 @ M_14 ) @ ( times_times_nat @ B_69 @ M_14 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_117 @ B_69 ) @ M_14 ) ) ).
thf(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_117: real,M_14: real,B_69: real] :
( ( plus_plus_real @ ( times_times_real @ A_117 @ M_14 ) @ ( times_times_real @ B_69 @ M_14 ) )
= ( times_times_real @ ( plus_plus_real @ A_117 @ B_69 ) @ M_14 ) ) ).
thf(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_116: int,B_68: int,C_33: int] :
( ( times_times_int @ ( plus_plus_int @ A_116 @ B_68 ) @ C_33 )
= ( plus_plus_int @ ( times_times_int @ A_116 @ C_33 ) @ ( times_times_int @ B_68 @ C_33 ) ) ) ).
thf(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_116: nat,B_68: nat,C_33: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_116 @ B_68 ) @ C_33 )
= ( plus_plus_nat @ ( times_times_nat @ A_116 @ C_33 ) @ ( times_times_nat @ B_68 @ C_33 ) ) ) ).
thf(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_116: real,B_68: real,C_33: real] :
( ( times_times_real @ ( plus_plus_real @ A_116 @ B_68 ) @ C_33 )
= ( plus_plus_real @ ( times_times_real @ A_116 @ C_33 ) @ ( times_times_real @ B_68 @ C_33 ) ) ) ).
thf(fact_179_crossproduct__noteq,axiom,
! [C_32: int,D_8: int,A_115: int,B_67: int] :
( ( ( A_115 != B_67 )
& ( C_32 != D_8 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_115 @ C_32 ) @ ( times_times_int @ B_67 @ D_8 ) )
!= ( plus_plus_int @ ( times_times_int @ A_115 @ D_8 ) @ ( times_times_int @ B_67 @ C_32 ) ) ) ) ).
thf(fact_180_crossproduct__noteq,axiom,
! [C_32: nat,D_8: nat,A_115: nat,B_67: nat] :
( ( ( A_115 != B_67 )
& ( C_32 != D_8 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_115 @ C_32 ) @ ( times_times_nat @ B_67 @ D_8 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_115 @ D_8 ) @ ( times_times_nat @ B_67 @ C_32 ) ) ) ) ).
thf(fact_181_crossproduct__noteq,axiom,
! [C_32: real,D_8: real,A_115: real,B_67: real] :
( ( ( A_115 != B_67 )
& ( C_32 != D_8 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_115 @ C_32 ) @ ( times_times_real @ B_67 @ D_8 ) )
!= ( plus_plus_real @ ( times_times_real @ A_115 @ D_8 ) @ ( times_times_real @ B_67 @ C_32 ) ) ) ) ).
thf(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_30: int,Y_27: int,Z_8: int] :
( ( times_times_int @ X_30 @ ( plus_plus_int @ Y_27 @ Z_8 ) )
= ( plus_plus_int @ ( times_times_int @ X_30 @ Y_27 ) @ ( times_times_int @ X_30 @ Z_8 ) ) ) ).
thf(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_30: nat,Y_27: nat,Z_8: nat] :
( ( times_times_nat @ X_30 @ ( plus_plus_nat @ Y_27 @ Z_8 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_30 @ Y_27 ) @ ( times_times_nat @ X_30 @ Z_8 ) ) ) ).
thf(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_30: real,Y_27: real,Z_8: real] :
( ( times_times_real @ X_30 @ ( plus_plus_real @ Y_27 @ Z_8 ) )
= ( plus_plus_real @ ( times_times_real @ X_30 @ Y_27 ) @ ( times_times_real @ X_30 @ Z_8 ) ) ) ).
thf(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114: int] :
( ( times_times_int @ A_114 @ one_one_int )
= A_114 ) ).
thf(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114: nat] :
( ( times_times_nat @ A_114 @ one_one_nat )
= A_114 ) ).
thf(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114: real] :
( ( times_times_real @ A_114 @ one_one_real )
= A_114 ) ).
thf(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_113: int] :
( ( times_times_int @ one_one_int @ A_113 )
= A_113 ) ).
thf(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_113: nat] :
( ( times_times_nat @ one_one_nat @ A_113 )
= A_113 ) ).
thf(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_113: real] :
( ( times_times_real @ one_one_real @ A_113 )
= A_113 ) ).
thf(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_29: int,Y_26: int,Q_4: nat] :
( ( power_power_int @ ( times_times_int @ X_29 @ Y_26 ) @ Q_4 )
= ( times_times_int @ ( power_power_int @ X_29 @ Q_4 ) @ ( power_power_int @ Y_26 @ Q_4 ) ) ) ).
thf(fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_29: real,Y_26: real,Q_4: nat] :
( ( power_power_real @ ( times_times_real @ X_29 @ Y_26 ) @ Q_4 )
= ( times_times_real @ ( power_power_real @ X_29 @ Q_4 ) @ ( power_power_real @ Y_26 @ Q_4 ) ) ) ).
thf(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_29: nat,Y_26: nat,Q_4: nat] :
( ( power_power_nat @ ( times_times_nat @ X_29 @ Y_26 ) @ Q_4 )
= ( times_times_nat @ ( power_power_nat @ X_29 @ Q_4 ) @ ( power_power_nat @ Y_26 @ Q_4 ) ) ) ).
thf(fact_194_rel__simps_I46_J,axiom,
! [K: int] :
( ( bit1 @ K )
!= pls ) ).
thf(fact_195_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_196_rel__simps_I50_J,axiom,
! [K: int,L: int] :
( ( bit1 @ K )
!= ( bit0 @ L ) ) ).
thf(fact_197_rel__simps_I49_J,axiom,
! [K: int,L: int] :
( ( bit0 @ K )
!= ( bit1 @ L ) ) ).
thf(fact_198_rel__simps_I44_J,axiom,
! [K: int] :
( ( ( bit0 @ K )
= pls )
<=> ( K = pls ) ) ).
thf(fact_199_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_200_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_201_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_202_mult__Bit0,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit0 @ K ) @ L )
= ( bit0 @ ( times_times_int @ K @ L ) ) ) ).
thf(fact_203_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_204_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_205_add__Bit0__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_206_Bit0__def,axiom,
! [K: int] :
( ( bit0 @ K )
= ( plus_plus_int @ K @ K ) ) ).
thf(fact_207_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_208_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_209_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_210_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_211_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_212_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_213_semiring__mult__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( times_times_int @ ( number_number_of_int @ V_17 ) @ ( number_number_of_int @ V_18 ) )
= ( number_number_of_int @ ( times_times_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_214_semiring__mult__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_17 ) @ ( number_number_of_nat @ V_18 ) )
= ( number_number_of_nat @ ( times_times_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_215_semiring__mult__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( times_times_real @ ( number267125858f_real @ V_17 ) @ ( number267125858f_real @ V_18 ) )
= ( number267125858f_real @ ( times_times_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_216_semiring__add__number__of,axiom,
! [V_16: int,V_15: int] :
( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( ord_less_eq_int @ pls @ V_16 )
=> ( ( plus_plus_int @ ( number_number_of_int @ V_15 ) @ ( number_number_of_int @ V_16 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_15 @ V_16 ) ) ) ) ) ).
thf(fact_217_semiring__add__number__of,axiom,
! [V_16: int,V_15: int] :
( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( ord_less_eq_int @ pls @ V_16 )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_15 ) @ ( number_number_of_nat @ V_16 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_15 @ V_16 ) ) ) ) ) ).
thf(fact_218_semiring__add__number__of,axiom,
! [V_16: int,V_15: int] :
( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( ord_less_eq_int @ pls @ V_16 )
=> ( ( plus_plus_real @ ( number267125858f_real @ V_15 ) @ ( number267125858f_real @ V_16 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_15 @ V_16 ) ) ) ) ) ).
thf(fact_219_power2__ge__self,axiom,
! [X_1: int] : ( ord_less_eq_int @ X_1 @ ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_220_left__distrib__number__of,axiom,
! [A_112: int,B_66: int,V_14: int] :
( ( times_times_int @ ( plus_plus_int @ A_112 @ B_66 ) @ ( number_number_of_int @ V_14 ) )
= ( plus_plus_int @ ( times_times_int @ A_112 @ ( number_number_of_int @ V_14 ) ) @ ( times_times_int @ B_66 @ ( number_number_of_int @ V_14 ) ) ) ) ).
thf(fact_221_left__distrib__number__of,axiom,
! [A_112: nat,B_66: nat,V_14: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_112 @ B_66 ) @ ( number_number_of_nat @ V_14 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_112 @ ( number_number_of_nat @ V_14 ) ) @ ( times_times_nat @ B_66 @ ( number_number_of_nat @ V_14 ) ) ) ) ).
thf(fact_222_left__distrib__number__of,axiom,
! [A_112: real,B_66: real,V_14: int] :
( ( times_times_real @ ( plus_plus_real @ A_112 @ B_66 ) @ ( number267125858f_real @ V_14 ) )
= ( plus_plus_real @ ( times_times_real @ A_112 @ ( number267125858f_real @ V_14 ) ) @ ( times_times_real @ B_66 @ ( number267125858f_real @ V_14 ) ) ) ) ).
thf(fact_223_right__distrib__number__of,axiom,
! [V_13: int,B_65: int,C_31: int] :
( ( times_times_int @ ( number_number_of_int @ V_13 ) @ ( plus_plus_int @ B_65 @ C_31 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_13 ) @ B_65 ) @ ( times_times_int @ ( number_number_of_int @ V_13 ) @ C_31 ) ) ) ).
thf(fact_224_right__distrib__number__of,axiom,
! [V_13: int,B_65: nat,C_31: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_13 ) @ ( plus_plus_nat @ B_65 @ C_31 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_13 ) @ B_65 ) @ ( times_times_nat @ ( number_number_of_nat @ V_13 ) @ C_31 ) ) ) ).
thf(fact_225_right__distrib__number__of,axiom,
! [V_13: int,B_65: real,C_31: real] :
( ( times_times_real @ ( number267125858f_real @ V_13 ) @ ( plus_plus_real @ B_65 @ C_31 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_13 ) @ B_65 ) @ ( times_times_real @ ( number267125858f_real @ V_13 ) @ C_31 ) ) ) ).
thf(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_111: int,M_13: int] :
( ( plus_plus_int @ ( times_times_int @ A_111 @ M_13 ) @ M_13 )
= ( times_times_int @ ( plus_plus_int @ A_111 @ one_one_int ) @ M_13 ) ) ).
thf(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_111: nat,M_13: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_111 @ M_13 ) @ M_13 )
= ( times_times_nat @ ( plus_plus_nat @ A_111 @ one_one_nat ) @ M_13 ) ) ).
thf(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_111: real,M_13: real] :
( ( plus_plus_real @ ( times_times_real @ A_111 @ M_13 ) @ M_13 )
= ( times_times_real @ ( plus_plus_real @ A_111 @ one_one_real ) @ M_13 ) ) ).
thf(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_12: int,A_110: int] :
( ( plus_plus_int @ M_12 @ ( times_times_int @ A_110 @ M_12 ) )
= ( times_times_int @ ( plus_plus_int @ A_110 @ one_one_int ) @ M_12 ) ) ).
thf(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_12: nat,A_110: nat] :
( ( plus_plus_nat @ M_12 @ ( times_times_nat @ A_110 @ M_12 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_110 @ one_one_nat ) @ M_12 ) ) ).
thf(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_12: real,A_110: real] :
( ( plus_plus_real @ M_12 @ ( times_times_real @ A_110 @ M_12 ) )
= ( times_times_real @ ( plus_plus_real @ A_110 @ one_one_real ) @ M_12 ) ) ).
thf(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: int] :
( ( plus_plus_int @ M_11 @ M_11 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_11 ) ) ).
thf(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: nat] :
( ( plus_plus_nat @ M_11 @ M_11 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_11 ) ) ).
thf(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: real] :
( ( plus_plus_real @ M_11 @ M_11 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_11 ) ) ).
thf(fact_235_add__numeral__0,axiom,
! [A_109: int] :
( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_109 )
= A_109 ) ).
thf(fact_236_add__numeral__0,axiom,
! [A_109: real] :
( ( plus_plus_real @ ( number267125858f_real @ pls ) @ A_109 )
= A_109 ) ).
thf(fact_237_add__numeral__0__right,axiom,
! [A_108: int] :
( ( plus_plus_int @ A_108 @ ( number_number_of_int @ pls ) )
= A_108 ) ).
thf(fact_238_add__numeral__0__right,axiom,
! [A_108: real] :
( ( plus_plus_real @ A_108 @ ( number267125858f_real @ pls ) )
= A_108 ) ).
thf(fact_239_mult__number__of__left,axiom,
! [V_12: int,W_13: int,Z_7: int] :
( ( times_times_int @ ( number_number_of_int @ V_12 ) @ ( times_times_int @ ( number_number_of_int @ W_13 ) @ Z_7 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_12 @ W_13 ) ) @ Z_7 ) ) ).
thf(fact_240_mult__number__of__left,axiom,
! [V_12: int,W_13: int,Z_7: real] :
( ( times_times_real @ ( number267125858f_real @ V_12 ) @ ( times_times_real @ ( number267125858f_real @ W_13 ) @ Z_7 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_12 @ W_13 ) ) @ Z_7 ) ) ).
thf(fact_241_arith__simps_I32_J,axiom,
! [V_11: int,W_12: int] :
( ( times_times_int @ ( number_number_of_int @ V_11 ) @ ( number_number_of_int @ W_12 ) )
= ( number_number_of_int @ ( times_times_int @ V_11 @ W_12 ) ) ) ).
thf(fact_242_arith__simps_I32_J,axiom,
! [V_11: int,W_12: int] :
( ( times_times_real @ ( number267125858f_real @ V_11 ) @ ( number267125858f_real @ W_12 ) )
= ( number267125858f_real @ ( times_times_int @ V_11 @ W_12 ) ) ) ).
thf(fact_243_number__of__mult,axiom,
! [V_10: int,W_11: int] :
( ( number_number_of_int @ ( times_times_int @ V_10 @ W_11 ) )
= ( times_times_int @ ( number_number_of_int @ V_10 ) @ ( number_number_of_int @ W_11 ) ) ) ).
thf(fact_244_number__of__mult,axiom,
! [V_10: int,W_11: int] :
( ( number267125858f_real @ ( times_times_int @ V_10 @ W_11 ) )
= ( times_times_real @ ( number267125858f_real @ V_10 ) @ ( number267125858f_real @ W_11 ) ) ) ).
thf(fact_245_add__number__of__left,axiom,
! [V_9: int,W_10: int,Z_6: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_9 ) @ ( plus_plus_int @ ( number_number_of_int @ W_10 ) @ Z_6 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_9 @ W_10 ) ) @ Z_6 ) ) ).
thf(fact_246_add__number__of__left,axiom,
! [V_9: int,W_10: int,Z_6: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_9 ) @ ( plus_plus_real @ ( number267125858f_real @ W_10 ) @ Z_6 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_9 @ W_10 ) ) @ Z_6 ) ) ).
thf(fact_247_add__number__of__eq,axiom,
! [V_8: int,W_9: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_8 ) @ ( number_number_of_int @ W_9 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_8 @ W_9 ) ) ) ).
thf(fact_248_add__number__of__eq,axiom,
! [V_8: int,W_9: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_8 ) @ ( number267125858f_real @ W_9 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_8 @ W_9 ) ) ) ).
thf(fact_249_number__of__add,axiom,
! [V_7: int,W_8: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_7 @ W_8 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_7 ) @ ( number_number_of_int @ W_8 ) ) ) ).
thf(fact_250_number__of__add,axiom,
! [V_7: int,W_8: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_7 @ W_8 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_7 ) @ ( number267125858f_real @ W_8 ) ) ) ).
thf(fact_251_add__Bit1__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_252_add__Bit0__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_253_Bit1__def,axiom,
! [K: int] :
( ( bit1 @ K )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
thf(fact_254_number__of__Bit1,axiom,
! [W_7: int] :
( ( number_number_of_int @ ( bit1 @ W_7 ) )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_7 ) ) @ ( number_number_of_int @ W_7 ) ) ) ).
thf(fact_255_number__of__Bit1,axiom,
! [W_7: int] :
( ( number267125858f_real @ ( bit1 @ W_7 ) )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_7 ) ) @ ( number267125858f_real @ W_7 ) ) ) ).
thf(fact_256_mult__numeral__1,axiom,
! [A_107: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit1 @ pls ) ) @ A_107 )
= A_107 ) ).
thf(fact_257_mult__numeral__1,axiom,
! [A_107: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit1 @ pls ) ) @ A_107 )
= A_107 ) ).
thf(fact_258_mult__numeral__1__right,axiom,
! [A_106: int] :
( ( times_times_int @ A_106 @ ( number_number_of_int @ ( bit1 @ pls ) ) )
= A_106 ) ).
thf(fact_259_mult__numeral__1__right,axiom,
! [A_106: real] :
( ( times_times_real @ A_106 @ ( number267125858f_real @ ( bit1 @ pls ) ) )
= A_106 ) ).
thf(fact_260_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_261_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_262_semiring__numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_263_numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_264_numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_265_semiring__norm_I110_J,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_266_semiring__norm_I110_J,axiom,
( one_one_real
= ( number267125858f_real @ ( bit1 @ pls ) ) ) ).
thf(fact_267_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_268_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_269_double__number__of__Bit0,axiom,
! [W_6: int] :
( ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ ( number_number_of_int @ W_6 ) )
= ( number_number_of_int @ ( bit0 @ W_6 ) ) ) ).
thf(fact_270_double__number__of__Bit0,axiom,
! [W_6: int] :
( ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ ( number267125858f_real @ W_6 ) )
= ( number267125858f_real @ ( bit0 @ W_6 ) ) ) ).
thf(fact_271_power3__eq__cube,axiom,
! [A_105: int] :
( ( power_power_int @ A_105 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( times_times_int @ A_105 @ A_105 ) @ A_105 ) ) ).
thf(fact_272_power3__eq__cube,axiom,
! [A_105: real] :
( ( power_power_real @ A_105 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( times_times_real @ A_105 @ A_105 ) @ A_105 ) ) ).
thf(fact_273_power3__eq__cube,axiom,
! [A_105: nat] :
( ( power_power_nat @ A_105 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( times_times_nat @ A_105 @ A_105 ) @ A_105 ) ) ).
thf(fact_274_quartic__square__square,axiom,
! [X_1: int] :
( ( power_power_int @ ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_275_semiring__mult__2,axiom,
! [Z_5: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_5 )
= ( plus_plus_int @ Z_5 @ Z_5 ) ) ).
thf(fact_276_semiring__mult__2,axiom,
! [Z_5: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_5 )
= ( plus_plus_nat @ Z_5 @ Z_5 ) ) ).
thf(fact_277_semiring__mult__2,axiom,
! [Z_5: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_5 )
= ( plus_plus_real @ Z_5 @ Z_5 ) ) ).
thf(fact_278_mult__2,axiom,
! [Z_4: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_4 )
= ( plus_plus_int @ Z_4 @ Z_4 ) ) ).
thf(fact_279_mult__2,axiom,
! [Z_4: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_4 )
= ( plus_plus_real @ Z_4 @ Z_4 ) ) ).
thf(fact_280_semiring__mult__2__right,axiom,
! [Z_3: int] :
( ( times_times_int @ Z_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_3 @ Z_3 ) ) ).
thf(fact_281_semiring__mult__2__right,axiom,
! [Z_3: nat] :
( ( times_times_nat @ Z_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ Z_3 @ Z_3 ) ) ).
thf(fact_282_semiring__mult__2__right,axiom,
! [Z_3: real] :
( ( times_times_real @ Z_3 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_3 @ Z_3 ) ) ).
thf(fact_283_mult__2__right,axiom,
! [Z_2: int] :
( ( times_times_int @ Z_2 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_2 @ Z_2 ) ) ).
thf(fact_284_mult__2__right,axiom,
! [Z_2: real] :
( ( times_times_real @ Z_2 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_2 @ Z_2 ) ) ).
thf(fact_285_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_286_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_287_semiring__one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_288_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_289__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_290_prime__g__5,axiom,
! [P: int] :
( ( zprime @ P )
=> ( ( P
!= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( P
!= ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P ) ) ) ) ).
thf(fact_291__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_292_real__sum__squared__expand,axiom,
! [X_1: real,Y_1: real] :
( ( power_power_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_1 ) @ Y_1 ) ) ) ).
thf(fact_293_four__x__squared,axiom,
! [X_1: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_294_power__less__power__Suc,axiom,
! [N_39: nat,A_104: int] :
( ( ord_less_int @ one_one_int @ A_104 )
=> ( ord_less_int @ ( power_power_int @ A_104 @ N_39 ) @ ( times_times_int @ A_104 @ ( power_power_int @ A_104 @ N_39 ) ) ) ) ).
thf(fact_295_power__less__power__Suc,axiom,
! [N_39: nat,A_104: nat] :
( ( ord_less_nat @ one_one_nat @ A_104 )
=> ( ord_less_nat @ ( power_power_nat @ A_104 @ N_39 ) @ ( times_times_nat @ A_104 @ ( power_power_nat @ A_104 @ N_39 ) ) ) ) ).
thf(fact_296_power__less__power__Suc,axiom,
! [N_39: nat,A_104: real] :
( ( ord_less_real @ one_one_real @ A_104 )
=> ( ord_less_real @ ( power_power_real @ A_104 @ N_39 ) @ ( times_times_real @ A_104 @ ( power_power_real @ A_104 @ N_39 ) ) ) ) ).
thf(fact_297_power__gt1__lemma,axiom,
! [N_38: nat,A_103: int] :
( ( ord_less_int @ one_one_int @ A_103 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A_103 @ ( power_power_int @ A_103 @ N_38 ) ) ) ) ).
thf(fact_298_power__gt1__lemma,axiom,
! [N_38: nat,A_103: nat] :
( ( ord_less_nat @ one_one_nat @ A_103 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A_103 @ ( power_power_nat @ A_103 @ N_38 ) ) ) ) ).
thf(fact_299_power__gt1__lemma,axiom,
! [N_38: nat,A_103: real] :
( ( ord_less_real @ one_one_real @ A_103 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A_103 @ ( power_power_real @ A_103 @ N_38 ) ) ) ) ).
thf(fact_300_power__le__imp__le__exp,axiom,
! [M_10: nat,N_37: nat,A_102: int] :
( ( ord_less_int @ one_one_int @ A_102 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A_102 @ M_10 ) @ ( power_power_int @ A_102 @ N_37 ) )
=> ( ord_less_eq_nat @ M_10 @ N_37 ) ) ) ).
thf(fact_301_power__le__imp__le__exp,axiom,
! [M_10: nat,N_37: nat,A_102: nat] :
( ( ord_less_nat @ one_one_nat @ A_102 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A_102 @ M_10 ) @ ( power_power_nat @ A_102 @ N_37 ) )
=> ( ord_less_eq_nat @ M_10 @ N_37 ) ) ) ).
thf(fact_302_power__le__imp__le__exp,axiom,
! [M_10: nat,N_37: nat,A_102: real] :
( ( ord_less_real @ one_one_real @ A_102 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A_102 @ M_10 ) @ ( power_power_real @ A_102 @ N_37 ) )
=> ( ord_less_eq_nat @ M_10 @ N_37 ) ) ) ).
thf(fact_303_power__increasing__iff,axiom,
! [X_28: nat,Y_25: nat,B_64: int] :
( ( ord_less_int @ one_one_int @ B_64 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B_64 @ X_28 ) @ ( power_power_int @ B_64 @ Y_25 ) )
<=> ( ord_less_eq_nat @ X_28 @ Y_25 ) ) ) ).
thf(fact_304_power__increasing__iff,axiom,
! [X_28: nat,Y_25: nat,B_64: nat] :
( ( ord_less_nat @ one_one_nat @ B_64 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B_64 @ X_28 ) @ ( power_power_nat @ B_64 @ Y_25 ) )
<=> ( ord_less_eq_nat @ X_28 @ Y_25 ) ) ) ).
thf(fact_305_power__increasing__iff,axiom,
! [X_28: nat,Y_25: nat,B_64: real] :
( ( ord_less_real @ one_one_real @ B_64 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B_64 @ X_28 ) @ ( power_power_real @ B_64 @ Y_25 ) )
<=> ( ord_less_eq_nat @ X_28 @ Y_25 ) ) ) ).
thf(fact_306__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_307_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_308__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: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ X @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ! [Y: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y )
& ( ord_less_int @ Y @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ Y @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) )
=> ( Y = X ) ) ) ).
thf(fact_309__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_310_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_311_power__eq__0__iff,axiom,
! [A_101: int,N_36: nat] :
( ( ( power_power_int @ A_101 @ N_36 )
= zero_zero_int )
<=> ( ( A_101 = zero_zero_int )
& ( N_36 != zero_zero_nat ) ) ) ).
thf(fact_312_power__eq__0__iff,axiom,
! [A_101: nat,N_36: nat] :
( ( ( power_power_nat @ A_101 @ N_36 )
= zero_zero_nat )
<=> ( ( A_101 = zero_zero_nat )
& ( N_36 != zero_zero_nat ) ) ) ).
thf(fact_313_power__eq__0__iff,axiom,
! [A_101: real,N_36: nat] :
( ( ( power_power_real @ A_101 @ N_36 )
= zero_zero_real )
<=> ( ( A_101 = zero_zero_real )
& ( N_36 != zero_zero_nat ) ) ) ).
thf(fact_314_le__imp__power__dvd,axiom,
! [A_100: real,M_9: nat,N_35: nat] :
( ( ord_less_eq_nat @ M_9 @ N_35 )
=> ( dvd_dvd_real @ ( power_power_real @ A_100 @ M_9 ) @ ( power_power_real @ A_100 @ N_35 ) ) ) ).
thf(fact_315_le__imp__power__dvd,axiom,
! [A_100: int,M_9: nat,N_35: nat] :
( ( ord_less_eq_nat @ M_9 @ N_35 )
=> ( dvd_dvd_int @ ( power_power_int @ A_100 @ M_9 ) @ ( power_power_int @ A_100 @ N_35 ) ) ) ).
thf(fact_316_le__imp__power__dvd,axiom,
! [A_100: nat,M_9: nat,N_35: nat] :
( ( ord_less_eq_nat @ M_9 @ N_35 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_100 @ M_9 ) @ ( power_power_nat @ A_100 @ N_35 ) ) ) ).
thf(fact_317_dvd__power__le,axiom,
! [N_34: nat,M_8: nat,X_27: real,Y_24: real] :
( ( dvd_dvd_real @ X_27 @ Y_24 )
=> ( ( ord_less_eq_nat @ N_34 @ M_8 )
=> ( dvd_dvd_real @ ( power_power_real @ X_27 @ N_34 ) @ ( power_power_real @ Y_24 @ M_8 ) ) ) ) ).
thf(fact_318_dvd__power__le,axiom,
! [N_34: nat,M_8: nat,X_27: int,Y_24: int] :
( ( dvd_dvd_int @ X_27 @ Y_24 )
=> ( ( ord_less_eq_nat @ N_34 @ M_8 )
=> ( dvd_dvd_int @ ( power_power_int @ X_27 @ N_34 ) @ ( power_power_int @ Y_24 @ M_8 ) ) ) ) ).
thf(fact_319_dvd__power__le,axiom,
! [N_34: nat,M_8: nat,X_27: nat,Y_24: nat] :
( ( dvd_dvd_nat @ X_27 @ Y_24 )
=> ( ( ord_less_eq_nat @ N_34 @ M_8 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_27 @ N_34 ) @ ( power_power_nat @ Y_24 @ M_8 ) ) ) ) ).
thf(fact_320_power__le__dvd,axiom,
! [M_7: nat,A_99: real,N_33: nat,B_63: real] :
( ( dvd_dvd_real @ ( power_power_real @ A_99 @ N_33 ) @ B_63 )
=> ( ( ord_less_eq_nat @ M_7 @ N_33 )
=> ( dvd_dvd_real @ ( power_power_real @ A_99 @ M_7 ) @ B_63 ) ) ) ).
thf(fact_321_power__le__dvd,axiom,
! [M_7: nat,A_99: int,N_33: nat,B_63: int] :
( ( dvd_dvd_int @ ( power_power_int @ A_99 @ N_33 ) @ B_63 )
=> ( ( ord_less_eq_nat @ M_7 @ N_33 )
=> ( dvd_dvd_int @ ( power_power_int @ A_99 @ M_7 ) @ B_63 ) ) ) ).
thf(fact_322_power__le__dvd,axiom,
! [M_7: nat,A_99: nat,N_33: nat,B_63: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A_99 @ N_33 ) @ B_63 )
=> ( ( ord_less_eq_nat @ M_7 @ N_33 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_99 @ M_7 ) @ B_63 ) ) ) ).
thf(fact_323_power__eq__imp__eq__base,axiom,
! [A_98: int,N_32: nat,B_62: int] :
( ( ( power_power_int @ A_98 @ N_32 )
= ( power_power_int @ B_62 @ N_32 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_98 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_62 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_32 )
=> ( A_98 = B_62 ) ) ) ) ) ).
thf(fact_324_power__eq__imp__eq__base,axiom,
! [A_98: nat,N_32: nat,B_62: nat] :
( ( ( power_power_nat @ A_98 @ N_32 )
= ( power_power_nat @ B_62 @ N_32 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_98 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_62 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_32 )
=> ( A_98 = B_62 ) ) ) ) ) ).
thf(fact_325_power__eq__imp__eq__base,axiom,
! [A_98: real,N_32: nat,B_62: real] :
( ( ( power_power_real @ A_98 @ N_32 )
= ( power_power_real @ B_62 @ N_32 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_98 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_62 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_32 )
=> ( A_98 = B_62 ) ) ) ) ) ).
thf(fact_326_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_327_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_328_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_329_dvd__power__same,axiom,
! [N_31: nat,X_26: real,Y_23: real] :
( ( dvd_dvd_real @ X_26 @ Y_23 )
=> ( dvd_dvd_real @ ( power_power_real @ X_26 @ N_31 ) @ ( power_power_real @ Y_23 @ N_31 ) ) ) ).
thf(fact_330_dvd__power__same,axiom,
! [N_31: nat,X_26: int,Y_23: int] :
( ( dvd_dvd_int @ X_26 @ Y_23 )
=> ( dvd_dvd_int @ ( power_power_int @ X_26 @ N_31 ) @ ( power_power_int @ Y_23 @ N_31 ) ) ) ).
thf(fact_331_dvd__power__same,axiom,
! [N_31: nat,X_26: nat,Y_23: nat] :
( ( dvd_dvd_nat @ X_26 @ Y_23 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_26 @ N_31 ) @ ( power_power_nat @ Y_23 @ N_31 ) ) ) ).
thf(fact_332_field__power__not__zero,axiom,
! [N_30: nat,A_97: int] :
( ( A_97 != zero_zero_int )
=> ( ( power_power_int @ A_97 @ N_30 )
!= zero_zero_int ) ) ).
thf(fact_333_field__power__not__zero,axiom,
! [N_30: nat,A_97: real] :
( ( A_97 != zero_zero_real )
=> ( ( power_power_real @ A_97 @ N_30 )
!= zero_zero_real ) ) ).
thf(fact_334_power__0__left,axiom,
! [N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_29 )
= one_one_int ) )
& ( ( N_29 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_29 )
= zero_zero_int ) ) ) ).
thf(fact_335_power__0__left,axiom,
! [N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_29 )
= one_one_nat ) )
& ( ( N_29 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_29 )
= zero_zero_nat ) ) ) ).
thf(fact_336_power__0__left,axiom,
! [N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_29 )
= one_one_real ) )
& ( ( N_29 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_29 )
= zero_zero_real ) ) ) ).
thf(fact_337_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_338_power__strict__mono,axiom,
! [N_28: nat,A_96: int,B_61: int] :
( ( ord_less_int @ A_96 @ B_61 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_96 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_28 )
=> ( ord_less_int @ ( power_power_int @ A_96 @ N_28 ) @ ( power_power_int @ B_61 @ N_28 ) ) ) ) ) ).
thf(fact_339_power__strict__mono,axiom,
! [N_28: nat,A_96: nat,B_61: nat] :
( ( ord_less_nat @ A_96 @ B_61 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_96 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_28 )
=> ( ord_less_nat @ ( power_power_nat @ A_96 @ N_28 ) @ ( power_power_nat @ B_61 @ N_28 ) ) ) ) ) ).
thf(fact_340_power__strict__mono,axiom,
! [N_28: nat,A_96: real,B_61: real] :
( ( ord_less_real @ A_96 @ B_61 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_96 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_28 )
=> ( ord_less_real @ ( power_power_real @ A_96 @ N_28 ) @ ( power_power_real @ B_61 @ N_28 ) ) ) ) ) ).
thf(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_95: int] :
( ( times_times_int @ zero_zero_int @ A_95 )
= zero_zero_int ) ).
thf(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_95: nat] :
( ( times_times_nat @ zero_zero_nat @ A_95 )
= zero_zero_nat ) ).
thf(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_95: real] :
( ( times_times_real @ zero_zero_real @ A_95 )
= zero_zero_real ) ).
thf(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_94: int] :
( ( times_times_int @ A_94 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_94: nat] :
( ( times_times_nat @ A_94 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_94: real] :
( ( times_times_real @ A_94 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_93: int] :
( ( plus_plus_int @ zero_zero_int @ A_93 )
= A_93 ) ).
thf(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_93: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_93 )
= A_93 ) ).
thf(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_93: real] :
( ( plus_plus_real @ zero_zero_real @ A_93 )
= A_93 ) ).
thf(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_92: int] :
( ( plus_plus_int @ A_92 @ zero_zero_int )
= A_92 ) ).
thf(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_92: nat] :
( ( plus_plus_nat @ A_92 @ zero_zero_nat )
= A_92 ) ).
thf(fact_352_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_92: real] :
( ( plus_plus_real @ A_92 @ zero_zero_real )
= A_92 ) ).
thf(fact_353_add__0__iff,axiom,
! [B_60: int,A_91: int] :
( ( B_60
= ( plus_plus_int @ B_60 @ A_91 ) )
<=> ( A_91 = zero_zero_int ) ) ).
thf(fact_354_add__0__iff,axiom,
! [B_60: nat,A_91: nat] :
( ( B_60
= ( plus_plus_nat @ B_60 @ A_91 ) )
<=> ( A_91 = zero_zero_nat ) ) ).
thf(fact_355_add__0__iff,axiom,
! [B_60: real,A_91: real] :
( ( B_60
= ( plus_plus_real @ B_60 @ A_91 ) )
<=> ( A_91 = zero_zero_real ) ) ).
thf(fact_356_double__eq__0__iff,axiom,
! [A_90: int] :
( ( ( plus_plus_int @ A_90 @ A_90 )
= zero_zero_int )
<=> ( A_90 = zero_zero_int ) ) ).
thf(fact_357_double__eq__0__iff,axiom,
! [A_90: real] :
( ( ( plus_plus_real @ A_90 @ A_90 )
= zero_zero_real )
<=> ( A_90 = zero_zero_real ) ) ).
thf(fact_358_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_359_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_360_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_361_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_362_zero__le__power,axiom,
! [N_27: nat,A_89: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_89 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_89 @ N_27 ) ) ) ).
thf(fact_363_zero__le__power,axiom,
! [N_27: nat,A_89: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_89 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A_89 @ N_27 ) ) ) ).
thf(fact_364_zero__le__power,axiom,
! [N_27: nat,A_89: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_89 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_89 @ N_27 ) ) ) ).
thf(fact_365_power__mono,axiom,
! [N_26: nat,A_88: int,B_59: int] :
( ( ord_less_eq_int @ A_88 @ B_59 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_88 )
=> ( ord_less_eq_int @ ( power_power_int @ A_88 @ N_26 ) @ ( power_power_int @ B_59 @ N_26 ) ) ) ) ).
thf(fact_366_power__mono,axiom,
! [N_26: nat,A_88: nat,B_59: nat] :
( ( ord_less_eq_nat @ A_88 @ B_59 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_88 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_88 @ N_26 ) @ ( power_power_nat @ B_59 @ N_26 ) ) ) ) ).
thf(fact_367_power__mono,axiom,
! [N_26: nat,A_88: real,B_59: real] :
( ( ord_less_eq_real @ A_88 @ B_59 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_88 )
=> ( ord_less_eq_real @ ( power_power_real @ A_88 @ N_26 ) @ ( power_power_real @ B_59 @ N_26 ) ) ) ) ).
thf(fact_368_zero__less__power,axiom,
! [N_25: nat,A_87: int] :
( ( ord_less_int @ zero_zero_int @ A_87 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_87 @ N_25 ) ) ) ).
thf(fact_369_zero__less__power,axiom,
! [N_25: nat,A_87: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_87 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A_87 @ N_25 ) ) ) ).
thf(fact_370_zero__less__power,axiom,
! [N_25: nat,A_87: real] :
( ( ord_less_real @ zero_zero_real @ A_87 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_87 @ N_25 ) ) ) ).
thf(fact_371_zcong__zpower__zmult,axiom,
! [Z: nat,X_1: int,Y_1: nat,P: int] :
( ( zcong @ ( power_power_int @ X_1 @ Y_1 ) @ one_one_int @ P )
=> ( zcong @ ( power_power_int @ X_1 @ ( times_times_nat @ Y_1 @ Z ) ) @ one_one_int @ P ) ) ).
thf(fact_372_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_373_zdvd__period,axiom,
! [C: int,X_1: int,T: int,A: int,D: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X_1 @ T ) )
<=> ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
thf(fact_374_power__less__imp__less__base,axiom,
! [A_86: int,N_24: nat,B_58: int] :
( ( ord_less_int @ ( power_power_int @ A_86 @ N_24 ) @ ( power_power_int @ B_58 @ N_24 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_58 )
=> ( ord_less_int @ A_86 @ B_58 ) ) ) ).
thf(fact_375_power__less__imp__less__base,axiom,
! [A_86: nat,N_24: nat,B_58: nat] :
( ( ord_less_nat @ ( power_power_nat @ A_86 @ N_24 ) @ ( power_power_nat @ B_58 @ N_24 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_58 )
=> ( ord_less_nat @ A_86 @ B_58 ) ) ) ).
thf(fact_376_power__less__imp__less__base,axiom,
! [A_86: real,N_24: nat,B_58: real] :
( ( ord_less_real @ ( power_power_real @ A_86 @ N_24 ) @ ( power_power_real @ B_58 @ N_24 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_58 )
=> ( ord_less_real @ A_86 @ B_58 ) ) ) ).
thf(fact_377_power__decreasing,axiom,
! [A_85: int,N_23: nat,N_22: nat] :
( ( ord_less_eq_nat @ N_23 @ N_22 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_85 )
=> ( ( ord_less_eq_int @ A_85 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A_85 @ N_22 ) @ ( power_power_int @ A_85 @ N_23 ) ) ) ) ) ).
thf(fact_378_power__decreasing,axiom,
! [A_85: nat,N_23: nat,N_22: nat] :
( ( ord_less_eq_nat @ N_23 @ N_22 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_85 )
=> ( ( ord_less_eq_nat @ A_85 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_85 @ N_22 ) @ ( power_power_nat @ A_85 @ N_23 ) ) ) ) ) ).
thf(fact_379_power__decreasing,axiom,
! [A_85: real,N_23: nat,N_22: nat] :
( ( ord_less_eq_nat @ N_23 @ N_22 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_85 )
=> ( ( ord_less_eq_real @ A_85 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A_85 @ N_22 ) @ ( power_power_real @ A_85 @ N_23 ) ) ) ) ) ).
thf(fact_380_power__strict__decreasing,axiom,
! [A_84: int,N_21: nat,N_20: nat] :
( ( ord_less_nat @ N_21 @ N_20 )
=> ( ( ord_less_int @ zero_zero_int @ A_84 )
=> ( ( ord_less_int @ A_84 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A_84 @ N_20 ) @ ( power_power_int @ A_84 @ N_21 ) ) ) ) ) ).
thf(fact_381_power__strict__decreasing,axiom,
! [A_84: nat,N_21: nat,N_20: nat] :
( ( ord_less_nat @ N_21 @ N_20 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_84 )
=> ( ( ord_less_nat @ A_84 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A_84 @ N_20 ) @ ( power_power_nat @ A_84 @ N_21 ) ) ) ) ) ).
thf(fact_382_power__strict__decreasing,axiom,
! [A_84: real,N_21: nat,N_20: nat] :
( ( ord_less_nat @ N_21 @ N_20 )
=> ( ( ord_less_real @ zero_zero_real @ A_84 )
=> ( ( ord_less_real @ A_84 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A_84 @ N_20 ) @ ( power_power_real @ A_84 @ N_21 ) ) ) ) ) ).
thf(fact_383_even__less__0__iff,axiom,
! [A_83: int] :
( ( ord_less_int @ ( plus_plus_int @ A_83 @ A_83 ) @ zero_zero_int )
<=> ( ord_less_int @ A_83 @ zero_zero_int ) ) ).
thf(fact_384_even__less__0__iff,axiom,
! [A_83: real] :
( ( ord_less_real @ ( plus_plus_real @ A_83 @ A_83 ) @ zero_zero_real )
<=> ( ord_less_real @ A_83 @ zero_zero_real ) ) ).
thf(fact_385_sum__squares__eq__zero__iff,axiom,
! [X_25: int,Y_22: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_25 @ X_25 ) @ ( times_times_int @ Y_22 @ Y_22 ) )
= zero_zero_int )
<=> ( ( X_25 = zero_zero_int )
& ( Y_22 = zero_zero_int ) ) ) ).
thf(fact_386_sum__squares__eq__zero__iff,axiom,
! [X_25: real,Y_22: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_25 @ X_25 ) @ ( times_times_real @ Y_22 @ Y_22 ) )
= zero_zero_real )
<=> ( ( X_25 = zero_zero_real )
& ( Y_22 = zero_zero_real ) ) ) ).
thf(fact_387_add__scale__eq__noteq,axiom,
! [C_30: int,D_7: int,A_82: int,B_57: int,R_3: int] :
( ( R_3 != zero_zero_int )
=> ( ( ( A_82 = B_57 )
& ( C_30 != D_7 ) )
=> ( ( plus_plus_int @ A_82 @ ( times_times_int @ R_3 @ C_30 ) )
!= ( plus_plus_int @ B_57 @ ( times_times_int @ R_3 @ D_7 ) ) ) ) ) ).
thf(fact_388_add__scale__eq__noteq,axiom,
! [C_30: nat,D_7: nat,A_82: nat,B_57: nat,R_3: nat] :
( ( R_3 != zero_zero_nat )
=> ( ( ( A_82 = B_57 )
& ( C_30 != D_7 ) )
=> ( ( plus_plus_nat @ A_82 @ ( times_times_nat @ R_3 @ C_30 ) )
!= ( plus_plus_nat @ B_57 @ ( times_times_nat @ R_3 @ D_7 ) ) ) ) ) ).
thf(fact_389_add__scale__eq__noteq,axiom,
! [C_30: real,D_7: real,A_82: real,B_57: real,R_3: real] :
( ( R_3 != zero_zero_real )
=> ( ( ( A_82 = B_57 )
& ( C_30 != D_7 ) )
=> ( ( plus_plus_real @ A_82 @ ( times_times_real @ R_3 @ C_30 ) )
!= ( plus_plus_real @ B_57 @ ( times_times_real @ R_3 @ D_7 ) ) ) ) ) ).
thf(fact_390_zprime__zdvd__power,axiom,
! [A: int,N: nat,P: int] :
( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ A @ N ) )
=> ( dvd_dvd_int @ P @ A ) ) ) ).
thf(fact_391_semiring__norm_I112_J,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_392_semiring__norm_I112_J,axiom,
( zero_zero_real
= ( number267125858f_real @ pls ) ) ).
thf(fact_393_number__of__Pls,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_394_number__of__Pls,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_395_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_396_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_397_semiring__numeral__0__eq__0,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_398_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_399_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_400_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_401_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_402_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_403_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_404_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_405_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_406_power__Suc__less,axiom,
! [N_19: nat,A_81: int] :
( ( ord_less_int @ zero_zero_int @ A_81 )
=> ( ( ord_less_int @ A_81 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A_81 @ ( power_power_int @ A_81 @ N_19 ) ) @ ( power_power_int @ A_81 @ N_19 ) ) ) ) ).
thf(fact_407_power__Suc__less,axiom,
! [N_19: nat,A_81: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_81 )
=> ( ( ord_less_nat @ A_81 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_81 @ ( power_power_nat @ A_81 @ N_19 ) ) @ ( power_power_nat @ A_81 @ N_19 ) ) ) ) ).
thf(fact_408_power__Suc__less,axiom,
! [N_19: nat,A_81: real] :
( ( ord_less_real @ zero_zero_real @ A_81 )
=> ( ( ord_less_real @ A_81 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A_81 @ ( power_power_real @ A_81 @ N_19 ) ) @ ( power_power_real @ A_81 @ N_19 ) ) ) ) ).
thf(fact_409_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ~ ( dvd_dvd_int @ P @ A )
=> ( ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ B ) ) ) ) ).
thf(fact_410_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B: int,P: int] :
( ( zprime @ P )
=> ( ~ ( dvd_dvd_int @ P @ B )
=> ( ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P @ N ) @ A ) ) ) ) ).
thf(fact_411_sum__squares__ge__zero,axiom,
! [X_24: int,Y_21: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_24 @ X_24 ) @ ( times_times_int @ Y_21 @ Y_21 ) ) ) ).
thf(fact_412_sum__squares__ge__zero,axiom,
! [X_24: real,Y_21: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_24 @ X_24 ) @ ( times_times_real @ Y_21 @ Y_21 ) ) ) ).
thf(fact_413_sum__squares__le__zero__iff,axiom,
! [X_23: int,Y_20: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_23 @ X_23 ) @ ( times_times_int @ Y_20 @ Y_20 ) ) @ zero_zero_int )
<=> ( ( X_23 = zero_zero_int )
& ( Y_20 = zero_zero_int ) ) ) ).
thf(fact_414_sum__squares__le__zero__iff,axiom,
! [X_23: real,Y_20: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_23 @ X_23 ) @ ( times_times_real @ Y_20 @ Y_20 ) ) @ zero_zero_real )
<=> ( ( X_23 = zero_zero_real )
& ( Y_20 = zero_zero_real ) ) ) ).
thf(fact_415_less__nat__number__of,axiom,
! [V: int,V_6: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
<=> ( ( ( ord_less_int @ V @ V_6 )
=> ( ord_less_int @ pls @ V_6 ) )
& ( ord_less_int @ V @ V_6 ) ) ) ).
thf(fact_416_not__sum__squares__lt__zero,axiom,
! [X_22: int,Y_19: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_22 @ X_22 ) @ ( times_times_int @ Y_19 @ Y_19 ) ) @ zero_zero_int ) ).
thf(fact_417_not__sum__squares__lt__zero,axiom,
! [X_22: real,Y_19: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X_22 @ X_22 ) @ ( times_times_real @ Y_19 @ Y_19 ) ) @ zero_zero_real ) ).
thf(fact_418_sum__squares__gt__zero__iff,axiom,
! [X_21: int,Y_18: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_21 @ X_21 ) @ ( times_times_int @ Y_18 @ Y_18 ) ) )
<=> ( ( X_21 != zero_zero_int )
| ( Y_18 != zero_zero_int ) ) ) ).
thf(fact_419_sum__squares__gt__zero__iff,axiom,
! [X_21: real,Y_18: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_21 @ X_21 ) @ ( times_times_real @ Y_18 @ Y_18 ) ) )
<=> ( ( X_21 != zero_zero_real )
| ( Y_18 != zero_zero_real ) ) ) ).
thf(fact_420_le__nat__number__of,axiom,
! [V: int,V_6: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
<=> ( ~ ( ord_less_eq_int @ V @ V_6 )
=> ( ord_less_eq_int @ V @ pls ) ) ) ).
thf(fact_421_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_422_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_423_power__one__right,axiom,
! [A_80: int] :
( ( power_power_int @ A_80 @ one_one_nat )
= A_80 ) ).
thf(fact_424_power__one__right,axiom,
! [A_80: real] :
( ( power_power_real @ A_80 @ one_one_nat )
= A_80 ) ).
thf(fact_425_power__one__right,axiom,
! [A_80: nat] :
( ( power_power_nat @ A_80 @ one_one_nat )
= A_80 ) ).
thf(fact_426_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_427_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_428_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_429_less__special_I1_J,axiom,
! [Y_17: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_17 ) )
<=> ( ord_less_int @ pls @ Y_17 ) ) ).
thf(fact_430_less__special_I1_J,axiom,
! [Y_17: int] :
( ( ord_less_real @ zero_zero_real @ ( number267125858f_real @ Y_17 ) )
<=> ( ord_less_int @ pls @ Y_17 ) ) ).
thf(fact_431_less__special_I3_J,axiom,
! [X_20: int] :
( ( ord_less_int @ ( number_number_of_int @ X_20 ) @ zero_zero_int )
<=> ( ord_less_int @ X_20 @ pls ) ) ).
thf(fact_432_less__special_I3_J,axiom,
! [X_20: int] :
( ( ord_less_real @ ( number267125858f_real @ X_20 ) @ zero_zero_real )
<=> ( ord_less_int @ X_20 @ pls ) ) ).
thf(fact_433_le__special_I1_J,axiom,
! [Y_16: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( number267125858f_real @ Y_16 ) )
<=> ( ord_less_eq_int @ pls @ Y_16 ) ) ).
thf(fact_434_le__special_I1_J,axiom,
! [Y_16: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_16 ) )
<=> ( ord_less_eq_int @ pls @ Y_16 ) ) ).
thf(fact_435_le__special_I3_J,axiom,
! [X_19: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_19 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ X_19 @ pls ) ) ).
thf(fact_436_le__special_I3_J,axiom,
! [X_19: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_19 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_19 @ pls ) ) ).
thf(fact_437_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_438_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real ) ).
thf(fact_439_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_440_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(fact_441_zero__eq__power2,axiom,
! [A_79: real] :
( ( ( power_power_real @ A_79 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real )
<=> ( A_79 = zero_zero_real ) ) ).
thf(fact_442_zero__eq__power2,axiom,
! [A_79: int] :
( ( ( power_power_int @ A_79 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int )
<=> ( A_79 = zero_zero_int ) ) ).
thf(fact_443_zero__le__power2,axiom,
! [A_78: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_444_zero__le__power2,axiom,
! [A_78: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_445_power2__le__imp__le,axiom,
! [X_18: real,Y_15: real] :
( ( ord_less_eq_real @ ( power_power_real @ X_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_15 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_15 )
=> ( ord_less_eq_real @ X_18 @ Y_15 ) ) ) ).
thf(fact_446_power2__le__imp__le,axiom,
! [X_18: nat,Y_15: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_15 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_15 )
=> ( ord_less_eq_nat @ X_18 @ Y_15 ) ) ) ).
thf(fact_447_power2__le__imp__le,axiom,
! [X_18: int,Y_15: int] :
( ( ord_less_eq_int @ ( power_power_int @ X_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_15 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_15 )
=> ( ord_less_eq_int @ X_18 @ Y_15 ) ) ) ).
thf(fact_448_power2__eq__imp__eq,axiom,
! [X_17: real,Y_14: real] :
( ( ( power_power_real @ X_17 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_real @ Y_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X_17 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_14 )
=> ( X_17 = Y_14 ) ) ) ) ).
thf(fact_449_power2__eq__imp__eq,axiom,
! [X_17: nat,Y_14: nat] :
( ( ( power_power_nat @ X_17 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_nat @ Y_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X_17 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_14 )
=> ( X_17 = Y_14 ) ) ) ) ).
thf(fact_450_power2__eq__imp__eq,axiom,
! [X_17: int,Y_14: int] :
( ( ( power_power_int @ X_17 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ Y_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X_17 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_14 )
=> ( X_17 = Y_14 ) ) ) ) ).
thf(fact_451_power2__less__0,axiom,
! [A_77: real] :
~ ( ord_less_real @ ( power_power_real @ A_77 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ) ).
thf(fact_452_power2__less__0,axiom,
! [A_77: int] :
~ ( ord_less_int @ ( power_power_int @ A_77 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) ).
thf(fact_453_zero__less__power2,axiom,
! [A_76: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_76 != zero_zero_real ) ) ).
thf(fact_454_zero__less__power2,axiom,
! [A_76: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_76 != zero_zero_int ) ) ).
thf(fact_455_sum__power2__eq__zero__iff,axiom,
! [X_16: real,Y_13: real] :
( ( ( plus_plus_real @ ( power_power_real @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X_16 = zero_zero_real )
& ( Y_13 = zero_zero_real ) ) ) ).
thf(fact_456_sum__power2__eq__zero__iff,axiom,
! [X_16: int,Y_13: int] :
( ( ( plus_plus_int @ ( power_power_int @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_int )
<=> ( ( X_16 = zero_zero_int )
& ( Y_13 = zero_zero_int ) ) ) ).
thf(fact_457_power__commutes,axiom,
! [A_75: nat,N_18: nat] :
( ( times_times_nat @ ( power_power_nat @ A_75 @ N_18 ) @ A_75 )
= ( times_times_nat @ A_75 @ ( power_power_nat @ A_75 @ N_18 ) ) ) ).
thf(fact_458_power__commutes,axiom,
! [A_75: real,N_18: nat] :
( ( times_times_real @ ( power_power_real @ A_75 @ N_18 ) @ A_75 )
= ( times_times_real @ A_75 @ ( power_power_real @ A_75 @ N_18 ) ) ) ).
thf(fact_459_power__commutes,axiom,
! [A_75: int,N_18: nat] :
( ( times_times_int @ ( power_power_int @ A_75 @ N_18 ) @ A_75 )
= ( times_times_int @ A_75 @ ( power_power_int @ A_75 @ N_18 ) ) ) ).
thf(fact_460_power__mult__distrib,axiom,
! [A_74: nat,B_56: nat,N_17: nat] :
( ( power_power_nat @ ( times_times_nat @ A_74 @ B_56 ) @ N_17 )
= ( times_times_nat @ ( power_power_nat @ A_74 @ N_17 ) @ ( power_power_nat @ B_56 @ N_17 ) ) ) ).
thf(fact_461_power__mult__distrib,axiom,
! [A_74: real,B_56: real,N_17: nat] :
( ( power_power_real @ ( times_times_real @ A_74 @ B_56 ) @ N_17 )
= ( times_times_real @ ( power_power_real @ A_74 @ N_17 ) @ ( power_power_real @ B_56 @ N_17 ) ) ) ).
thf(fact_462_power__mult__distrib,axiom,
! [A_74: int,B_56: int,N_17: nat] :
( ( power_power_int @ ( times_times_int @ A_74 @ B_56 ) @ N_17 )
= ( times_times_int @ ( power_power_int @ A_74 @ N_17 ) @ ( power_power_int @ B_56 @ N_17 ) ) ) ).
thf(fact_463_power__add,axiom,
! [A_73: nat,M_6: nat,N_16: nat] :
( ( power_power_nat @ A_73 @ ( plus_plus_nat @ M_6 @ N_16 ) )
= ( times_times_nat @ ( power_power_nat @ A_73 @ M_6 ) @ ( power_power_nat @ A_73 @ N_16 ) ) ) ).
thf(fact_464_power__add,axiom,
! [A_73: real,M_6: nat,N_16: nat] :
( ( power_power_real @ A_73 @ ( plus_plus_nat @ M_6 @ N_16 ) )
= ( times_times_real @ ( power_power_real @ A_73 @ M_6 ) @ ( power_power_real @ A_73 @ N_16 ) ) ) ).
thf(fact_465_power__add,axiom,
! [A_73: int,M_6: nat,N_16: nat] :
( ( power_power_int @ A_73 @ ( plus_plus_nat @ M_6 @ N_16 ) )
= ( times_times_int @ ( power_power_int @ A_73 @ M_6 ) @ ( power_power_int @ A_73 @ N_16 ) ) ) ).
thf(fact_466_power__one,axiom,
! [N_15: nat] :
( ( power_power_real @ one_one_real @ N_15 )
= one_one_real ) ).
thf(fact_467_power__one,axiom,
! [N_15: nat] :
( ( power_power_nat @ one_one_nat @ N_15 )
= one_one_nat ) ).
thf(fact_468_power__one,axiom,
! [N_15: nat] :
( ( power_power_int @ one_one_int @ N_15 )
= one_one_int ) ).
thf(fact_469_power__mult,axiom,
! [A_72: nat,M_5: nat,N_14: nat] :
( ( power_power_nat @ A_72 @ ( times_times_nat @ M_5 @ N_14 ) )
= ( power_power_nat @ ( power_power_nat @ A_72 @ M_5 ) @ N_14 ) ) ).
thf(fact_470_power__mult,axiom,
! [A_72: real,M_5: nat,N_14: nat] :
( ( power_power_real @ A_72 @ ( times_times_nat @ M_5 @ N_14 ) )
= ( power_power_real @ ( power_power_real @ A_72 @ M_5 ) @ N_14 ) ) ).
thf(fact_471_power__mult,axiom,
! [A_72: int,M_5: nat,N_14: nat] :
( ( power_power_int @ A_72 @ ( times_times_nat @ M_5 @ N_14 ) )
= ( power_power_int @ ( power_power_int @ A_72 @ M_5 ) @ N_14 ) ) ).
thf(fact_472_power2__less__imp__less,axiom,
! [X_15: real,Y_12: real] :
( ( ord_less_real @ ( power_power_real @ X_15 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_12 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_12 )
=> ( ord_less_real @ X_15 @ Y_12 ) ) ) ).
thf(fact_473_power2__less__imp__less,axiom,
! [X_15: nat,Y_12: nat] :
( ( ord_less_nat @ ( power_power_nat @ X_15 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_12 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_12 )
=> ( ord_less_nat @ X_15 @ Y_12 ) ) ) ).
thf(fact_474_power2__less__imp__less,axiom,
! [X_15: int,Y_12: int] :
( ( ord_less_int @ ( power_power_int @ X_15 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_12 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_12 )
=> ( ord_less_int @ X_15 @ Y_12 ) ) ) ).
thf(fact_475_sum__power2__ge__zero,axiom,
! [X_14: real,Y_11: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_476_sum__power2__ge__zero,axiom,
! [X_14: int,Y_11: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_14 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_477_sum__power2__le__zero__iff,axiom,
! [X_13: real,Y_10: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_10 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real )
<=> ( ( X_13 = zero_zero_real )
& ( Y_10 = zero_zero_real ) ) ) ).
thf(fact_478_sum__power2__le__zero__iff,axiom,
! [X_13: int,Y_10: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X_13 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_10 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int )
<=> ( ( X_13 = zero_zero_int )
& ( Y_10 = zero_zero_int ) ) ) ).
thf(fact_479_not__sum__power2__lt__zero,axiom,
! [X_12: real,Y_9: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X_12 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_9 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ) ).
thf(fact_480_not__sum__power2__lt__zero,axiom,
! [X_12: int,Y_9: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X_12 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_9 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int ) ).
thf(fact_481_sum__power2__gt__zero__iff,axiom,
! [X_11: real,Y_8: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_11 != zero_zero_real )
| ( Y_8 != zero_zero_real ) ) ) ).
thf(fact_482_sum__power2__gt__zero__iff,axiom,
! [X_11: int,Y_8: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_11 != zero_zero_int )
| ( Y_8 != zero_zero_int ) ) ) ).
thf(fact_483_zero__le__even__power_H,axiom,
! [A_71: real,N_13: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_71 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_13 ) ) ) ).
thf(fact_484_zero__le__even__power_H,axiom,
! [A_71: int,N_13: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_71 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_13 ) ) ) ).
thf(fact_485_one__le__power,axiom,
! [N_12: nat,A_70: real] :
( ( ord_less_eq_real @ one_one_real @ A_70 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A_70 @ N_12 ) ) ) ).
thf(fact_486_one__le__power,axiom,
! [N_12: nat,A_70: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A_70 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A_70 @ N_12 ) ) ) ).
thf(fact_487_one__le__power,axiom,
! [N_12: nat,A_70: int] :
( ( ord_less_eq_int @ one_one_int @ A_70 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A_70 @ N_12 ) ) ) ).
thf(fact_488_power__increasing,axiom,
! [A_69: real,N_11: nat,N_10: nat] :
( ( ord_less_eq_nat @ N_11 @ N_10 )
=> ( ( ord_less_eq_real @ one_one_real @ A_69 )
=> ( ord_less_eq_real @ ( power_power_real @ A_69 @ N_11 ) @ ( power_power_real @ A_69 @ N_10 ) ) ) ) ).
thf(fact_489_power__increasing,axiom,
! [A_69: nat,N_11: nat,N_10: nat] :
( ( ord_less_eq_nat @ N_11 @ N_10 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A_69 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_69 @ N_11 ) @ ( power_power_nat @ A_69 @ N_10 ) ) ) ) ).
thf(fact_490_power__increasing,axiom,
! [A_69: int,N_11: nat,N_10: nat] :
( ( ord_less_eq_nat @ N_11 @ N_10 )
=> ( ( ord_less_eq_int @ one_one_int @ A_69 )
=> ( ord_less_eq_int @ ( power_power_int @ A_69 @ N_11 ) @ ( power_power_int @ A_69 @ N_10 ) ) ) ) ).
thf(fact_491_power__inject__exp,axiom,
! [M_4: nat,N_9: nat,A_68: real] :
( ( ord_less_real @ one_one_real @ A_68 )
=> ( ( ( power_power_real @ A_68 @ M_4 )
= ( power_power_real @ A_68 @ N_9 ) )
<=> ( M_4 = N_9 ) ) ) ).
thf(fact_492_power__inject__exp,axiom,
! [M_4: nat,N_9: nat,A_68: nat] :
( ( ord_less_nat @ one_one_nat @ A_68 )
=> ( ( ( power_power_nat @ A_68 @ M_4 )
= ( power_power_nat @ A_68 @ N_9 ) )
<=> ( M_4 = N_9 ) ) ) ).
thf(fact_493_power__inject__exp,axiom,
! [M_4: nat,N_9: nat,A_68: int] :
( ( ord_less_int @ one_one_int @ A_68 )
=> ( ( ( power_power_int @ A_68 @ M_4 )
= ( power_power_int @ A_68 @ N_9 ) )
<=> ( M_4 = N_9 ) ) ) ).
thf(fact_494_power__strict__increasing__iff,axiom,
! [X_10: nat,Y_7: nat,B_55: real] :
( ( ord_less_real @ one_one_real @ B_55 )
=> ( ( ord_less_real @ ( power_power_real @ B_55 @ X_10 ) @ ( power_power_real @ B_55 @ Y_7 ) )
<=> ( ord_less_nat @ X_10 @ Y_7 ) ) ) ).
thf(fact_495_power__strict__increasing__iff,axiom,
! [X_10: nat,Y_7: nat,B_55: nat] :
( ( ord_less_nat @ one_one_nat @ B_55 )
=> ( ( ord_less_nat @ ( power_power_nat @ B_55 @ X_10 ) @ ( power_power_nat @ B_55 @ Y_7 ) )
<=> ( ord_less_nat @ X_10 @ Y_7 ) ) ) ).
thf(fact_496_power__strict__increasing__iff,axiom,
! [X_10: nat,Y_7: nat,B_55: int] :
( ( ord_less_int @ one_one_int @ B_55 )
=> ( ( ord_less_int @ ( power_power_int @ B_55 @ X_10 ) @ ( power_power_int @ B_55 @ Y_7 ) )
<=> ( ord_less_nat @ X_10 @ Y_7 ) ) ) ).
thf(fact_497_power__less__imp__less__exp,axiom,
! [M_3: nat,N_8: nat,A_67: real] :
( ( ord_less_real @ one_one_real @ A_67 )
=> ( ( ord_less_real @ ( power_power_real @ A_67 @ M_3 ) @ ( power_power_real @ A_67 @ N_8 ) )
=> ( ord_less_nat @ M_3 @ N_8 ) ) ) ).
thf(fact_498_power__less__imp__less__exp,axiom,
! [M_3: nat,N_8: nat,A_67: nat] :
( ( ord_less_nat @ one_one_nat @ A_67 )
=> ( ( ord_less_nat @ ( power_power_nat @ A_67 @ M_3 ) @ ( power_power_nat @ A_67 @ N_8 ) )
=> ( ord_less_nat @ M_3 @ N_8 ) ) ) ).
thf(fact_499_power__less__imp__less__exp,axiom,
! [M_3: nat,N_8: nat,A_67: int] :
( ( ord_less_int @ one_one_int @ A_67 )
=> ( ( ord_less_int @ ( power_power_int @ A_67 @ M_3 ) @ ( power_power_int @ A_67 @ N_8 ) )
=> ( ord_less_nat @ M_3 @ N_8 ) ) ) ).
thf(fact_500_power__strict__increasing,axiom,
! [A_66: real,N_7: nat,N_6: nat] :
( ( ord_less_nat @ N_7 @ N_6 )
=> ( ( ord_less_real @ one_one_real @ A_66 )
=> ( ord_less_real @ ( power_power_real @ A_66 @ N_7 ) @ ( power_power_real @ A_66 @ N_6 ) ) ) ) ).
thf(fact_501_power__strict__increasing,axiom,
! [A_66: nat,N_7: nat,N_6: nat] :
( ( ord_less_nat @ N_7 @ N_6 )
=> ( ( ord_less_nat @ one_one_nat @ A_66 )
=> ( ord_less_nat @ ( power_power_nat @ A_66 @ N_7 ) @ ( power_power_nat @ A_66 @ N_6 ) ) ) ) ).
thf(fact_502_power__strict__increasing,axiom,
! [A_66: int,N_7: nat,N_6: nat] :
( ( ord_less_nat @ N_7 @ N_6 )
=> ( ( ord_less_int @ one_one_int @ A_66 )
=> ( ord_less_int @ ( power_power_int @ A_66 @ N_7 ) @ ( power_power_int @ A_66 @ N_6 ) ) ) ) ).
thf(fact_503_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_504_Euler_Oaux____1,axiom,
! [Y_1: int,X_1: int,P: int] :
( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( ( zcong @ ( power_power_int @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X_1 @ P )
=> ~ ( dvd_dvd_int @ P @ Y_1 ) ) ) ).
thf(fact_505_int__pos__lt__two__imp__zero__or__one,axiom,
! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_int @ X_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( X_1 = zero_zero_int )
| ( X_1 = one_one_int ) ) ) ) ).
thf(fact_506_even__power__le__0__imp__0,axiom,
! [A_65: real,K_3: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A_65 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_3 ) ) @ zero_zero_real )
=> ( A_65 = zero_zero_real ) ) ).
thf(fact_507_even__power__le__0__imp__0,axiom,
! [A_65: int,K_3: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A_65 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_3 ) ) @ zero_zero_int )
=> ( A_65 = zero_zero_int ) ) ).
thf(fact_508_zprime__def,axiom,
! [P: int] :
( ( zprime @ P )
<=> ( ( ord_less_int @ one_one_int @ P )
& ! [M_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M_2 )
& ( dvd_dvd_int @ M_2 @ P ) )
=> ( ( M_2 = one_one_int )
| ( M_2 = P ) ) ) ) ) ).
thf(fact_509_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_510_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_511_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_512__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_513__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_514_nat__zero__less__power__iff,axiom,
! [X_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X_1 @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ X_1 )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_515_zero__less__power__nat__eq,axiom,
! [X_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X_1 @ N ) )
<=> ( ( N = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ X_1 ) ) ) ).
thf(fact_516_zero__less__power__nat__eq__number__of,axiom,
! [X_1: nat,W: int] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X_1 @ ( number_number_of_nat @ W ) ) )
<=> ( ( ( number_number_of_nat @ W )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ X_1 ) ) ) ).
thf(fact_517_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_518_rel__simps_I47_J,axiom,
! [K: int] :
( ( ( bit1 @ K )
= min )
<=> ( K = min ) ) ).
thf(fact_519_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_520_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_521_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_522_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_523_rel__simps_I45_J,axiom,
! [K: int] :
( ( bit0 @ K )
!= min ) ).
thf(fact_524_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_525_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_526_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_527_not__real__square__gt__zero,axiom,
! [X_1: real] :
( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X_1 @ X_1 ) )
<=> ( X_1 = zero_zero_real ) ) ).
thf(fact_528_rel__simps_I13_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_int @ K @ min ) ) ).
thf(fact_529_rel__simps_I9_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_530_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_531_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_532_rel__simps_I8_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_533_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_534_rel__simps_I30_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_535_rel__simps_I26_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ min @ K ) ) ).
thf(fact_536_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_537_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_538_rel__simps_I28_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_539_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_540_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_541_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_542_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_543_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_544_power__0,axiom,
! [A_64: real] :
( ( power_power_real @ A_64 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_545_power__0,axiom,
! [A_64: nat] :
( ( power_power_nat @ A_64 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_546_power__0,axiom,
! [A_64: int] :
( ( power_power_int @ A_64 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_547_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_548_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_549_rel__simps_I25_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_550_rel__simps_I11_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_551_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_552_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_553_one__less__power,axiom,
! [N_5: nat,A_63: real] :
( ( ord_less_real @ one_one_real @ A_63 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A_63 @ N_5 ) ) ) ) ).
thf(fact_554_one__less__power,axiom,
! [N_5: nat,A_63: nat] :
( ( ord_less_nat @ one_one_nat @ A_63 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A_63 @ N_5 ) ) ) ) ).
thf(fact_555_one__less__power,axiom,
! [N_5: nat,A_63: int] :
( ( ord_less_int @ one_one_int @ A_63 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A_63 @ N_5 ) ) ) ) ).
thf(fact_556_dvd__power,axiom,
! [X_8: real,N_4: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_4 )
| ( X_8 = one_one_real ) )
=> ( dvd_dvd_real @ X_8 @ ( power_power_real @ X_8 @ N_4 ) ) ) ).
thf(fact_557_dvd__power,axiom,
! [X_8: nat,N_4: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_4 )
| ( X_8 = one_one_nat ) )
=> ( dvd_dvd_nat @ X_8 @ ( power_power_nat @ X_8 @ N_4 ) ) ) ).
thf(fact_558_dvd__power,axiom,
! [X_8: int,N_4: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_4 )
| ( X_8 = one_one_int ) )
=> ( dvd_dvd_int @ X_8 @ ( power_power_int @ X_8 @ N_4 ) ) ) ).
thf(fact_559_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_560_eq__number__of__0,axiom,
! [V: int] :
( ( ( number_number_of_nat @ V )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_561_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V ) )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_562_zcong__sym,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ B @ A @ M ) ) ).
thf(fact_563_zcong__refl,axiom,
! [K: int,M: int] : ( zcong @ K @ K @ M ) ).
thf(fact_564_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_565_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_566_nat__number__of__mult__left,axiom,
! [V_6: 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_6 ) @ 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_6 ) @ K ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V @ V_6 ) ) @ K ) ) ) ) ).
thf(fact_567_mult__nat__number__of,axiom,
! [V_6: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_6 ) )
= ( number_number_of_nat @ ( times_times_int @ V @ V_6 ) ) ) ) ) ).
thf(fact_568_order__le__neq__implies__less,axiom,
! [X_7: real,Y_6: real] :
( ( ord_less_eq_real @ X_7 @ Y_6 )
=> ( ( X_7 != Y_6 )
=> ( ord_less_real @ X_7 @ Y_6 ) ) ) ).
thf(fact_569_order__le__neq__implies__less,axiom,
! [X_7: nat,Y_6: nat] :
( ( ord_less_eq_nat @ X_7 @ Y_6 )
=> ( ( X_7 != Y_6 )
=> ( ord_less_nat @ X_7 @ Y_6 ) ) ) ).
thf(fact_570_order__le__neq__implies__less,axiom,
! [X_7: int,Y_6: int] :
( ( ord_less_eq_int @ X_7 @ Y_6 )
=> ( ( X_7 != Y_6 )
=> ( ord_less_int @ X_7 @ Y_6 ) ) ) ).
thf(fact_571_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_572_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_573_IntPrimes_Ozcong__zero,axiom,
! [A: int,B: int] :
( ( zcong @ A @ B @ zero_zero_int )
<=> ( A = B ) ) ).
thf(fact_574_zcong__1,axiom,
! [A: int,B: int] : ( zcong @ A @ B @ one_one_int ) ).
thf(fact_575_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_576_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_577_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_578_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : ( zcong @ ( times_times_int @ A @ M ) @ ( times_times_int @ B @ M ) @ M ) ).
thf(fact_579_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_580_power__m1__even,axiom,
! [N_3: nat] :
( ( power_power_real @ ( number267125858f_real @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_3 ) )
= one_one_real ) ).
thf(fact_581_power__m1__even,axiom,
! [N_3: nat] :
( ( power_power_int @ ( number_number_of_int @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_3 ) )
= one_one_int ) ).
thf(fact_582_power__eq__0__iff__number__of,axiom,
! [A_62: real,W_4: int] :
( ( ( power_power_real @ A_62 @ ( number_number_of_nat @ W_4 ) )
= zero_zero_real )
<=> ( ( A_62 = zero_zero_real )
& ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat ) ) ) ).
thf(fact_583_power__eq__0__iff__number__of,axiom,
! [A_62: nat,W_4: int] :
( ( ( power_power_nat @ A_62 @ ( number_number_of_nat @ W_4 ) )
= zero_zero_nat )
<=> ( ( A_62 = zero_zero_nat )
& ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat ) ) ) ).
thf(fact_584_power__eq__0__iff__number__of,axiom,
! [A_62: int,W_4: int] :
( ( ( power_power_int @ A_62 @ ( number_number_of_nat @ W_4 ) )
= zero_zero_int )
<=> ( ( A_62 = zero_zero_int )
& ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat ) ) ) ).
thf(fact_585_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_586_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1: int,X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X_1 @ Y_1 ) ) ) ) ).
thf(fact_587_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y_1: int,X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X_1 @ Y_1 ) ) ) ) ).
thf(fact_588_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_589_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ X_1 @ N ) ) ) ).
thf(fact_590_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_591_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( ( number_number_of_nat @ W_3 )
= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_3 ) )
= one_one_real ) )
& ( ( ( number_number_of_nat @ W_3 )
!= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_3 ) )
= zero_zero_real ) ) ) ).
thf(fact_592_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( ( number_number_of_nat @ W_3 )
= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_3 ) )
= one_one_nat ) )
& ( ( ( number_number_of_nat @ W_3 )
!= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_3 ) )
= zero_zero_nat ) ) ) ).
thf(fact_593_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( ( number_number_of_nat @ W_3 )
= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_3 ) )
= one_one_int ) )
& ( ( ( number_number_of_nat @ W_3 )
!= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_3 ) )
= zero_zero_int ) ) ) ).
thf(fact_594_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_595_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_596_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_597_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R: int,B_54: int,Q_3: int,R_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_54 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_54 @ 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_54 )
=> ( ( ord_less_eq_int @ B_54 @ B )
=> ( ord_less_eq_int @ Q_3 @ Q ) ) ) ) ) ) ) ).
thf(fact_598_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_599_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R: int,B_54: int,Q_3: int,R_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_54 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_54 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_54 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B_54 )
=> ( ( ord_less_eq_int @ B_54 @ B )
=> ( ord_less_eq_int @ Q @ Q_3 ) ) ) ) ) ) ) ).
thf(fact_600_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_601_q__neg__lemma,axiom,
! [B_54: int,Q_3: int,R_2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_54 @ Q_3 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_54 )
=> ( ord_less_eq_int @ Q_3 @ zero_zero_int ) ) ) ) ).
thf(fact_602_q__pos__lemma,axiom,
! [B_54: int,Q_3: int,R_2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_54 @ Q_3 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_54 )
=> ( ( ord_less_int @ zero_zero_int @ B_54 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_3 ) ) ) ) ).
thf(fact_603_zprime__zdvd__zmult,axiom,
! [N: int,P: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P @ M )
| ( dvd_dvd_int @ P @ N ) ) ) ) ) ).
thf(fact_604__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_605__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_606_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_607_zcong__neg__1__impl__ne__1,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zcong @ X_1 @ ( number_number_of_int @ min ) @ P )
=> ~ ( zcong @ X_1 @ one_one_int @ P ) ) ) ).
thf(fact_608_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_609__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_610__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_611_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_612_number__of__diff,axiom,
! [V_5: int,W_2: int] :
( ( number267125858f_real @ ( minus_minus_int @ V_5 @ W_2 ) )
= ( minus_minus_real @ ( number267125858f_real @ V_5 ) @ ( number267125858f_real @ W_2 ) ) ) ).
thf(fact_613_number__of__diff,axiom,
! [V_5: int,W_2: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_5 @ W_2 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_5 ) @ ( number_number_of_int @ W_2 ) ) ) ).
thf(fact_614_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_615_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_616_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_617_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_618_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_619_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_620_right__diff__distrib__number__of,axiom,
! [V_4: int,B_53: real,C_29: real] :
( ( times_times_real @ ( number267125858f_real @ V_4 ) @ ( minus_minus_real @ B_53 @ C_29 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_4 ) @ B_53 ) @ ( times_times_real @ ( number267125858f_real @ V_4 ) @ C_29 ) ) ) ).
thf(fact_621_right__diff__distrib__number__of,axiom,
! [V_4: int,B_53: int,C_29: int] :
( ( times_times_int @ ( number_number_of_int @ V_4 ) @ ( minus_minus_int @ B_53 @ C_29 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_4 ) @ B_53 ) @ ( times_times_int @ ( number_number_of_int @ V_4 ) @ C_29 ) ) ) ).
thf(fact_622_left__diff__distrib__number__of,axiom,
! [A_61: real,B_52: real,V_3: int] :
( ( times_times_real @ ( minus_minus_real @ A_61 @ B_52 ) @ ( number267125858f_real @ V_3 ) )
= ( minus_minus_real @ ( times_times_real @ A_61 @ ( number267125858f_real @ V_3 ) ) @ ( times_times_real @ B_52 @ ( number267125858f_real @ V_3 ) ) ) ) ).
thf(fact_623_left__diff__distrib__number__of,axiom,
! [A_61: int,B_52: int,V_3: int] :
( ( times_times_int @ ( minus_minus_int @ A_61 @ B_52 ) @ ( number_number_of_int @ V_3 ) )
= ( minus_minus_int @ ( times_times_int @ A_61 @ ( number_number_of_int @ V_3 ) ) @ ( times_times_int @ B_52 @ ( number_number_of_int @ V_3 ) ) ) ) ).
thf(fact_624_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_625_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_626_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_627_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_628_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_629_zcong__def,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A @ B ) ) ) ).
thf(fact_630_add__number__of__diff1,axiom,
! [V_2: int,W_1: int,C_28: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_2 ) @ ( minus_minus_real @ ( number267125858f_real @ W_1 ) @ C_28 ) )
= ( minus_minus_real @ ( number267125858f_real @ ( plus_plus_int @ V_2 @ W_1 ) ) @ C_28 ) ) ).
thf(fact_631_add__number__of__diff1,axiom,
! [V_2: int,W_1: int,C_28: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_2 ) @ ( minus_minus_int @ ( number_number_of_int @ W_1 ) @ C_28 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V_2 @ W_1 ) ) @ C_28 ) ) ).
thf(fact_632_Euler_Oaux1,axiom,
! [A: int,X_1: int] :
( ( ord_less_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_int @ X_1 @ A )
=> ( ( X_1
!= ( minus_minus_int @ A @ one_one_int ) )
=> ( ord_less_int @ X_1 @ ( minus_minus_int @ A @ one_one_int ) ) ) ) ) ).
thf(fact_633_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_634_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_635_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_636_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_637_inv__not__p__minus__1__aux,axiom,
! [A: int,P: int] :
( ( zcong @ ( times_times_int @ A @ ( minus_minus_int @ P @ one_one_int ) ) @ one_one_int @ P )
<=> ( zcong @ A @ ( minus_minus_int @ P @ one_one_int ) @ P ) ) ).
thf(fact_638_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_639_mult__sum2sq,axiom,
! [A: int,B: int,P: int,Q: int] :
( ( times_times_int @ ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ A @ B ) ) @ ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ P @ Q ) ) )
= ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A @ P ) @ ( times_times_int @ B @ Q ) ) @ ( minus_minus_int @ ( times_times_int @ A @ Q ) @ ( times_times_int @ B @ P ) ) ) ) ) ).
thf(fact_640_zcong__square,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P )
=> ( ( zcong @ A @ one_one_int @ P )
| ( zcong @ A @ ( minus_minus_int @ P @ one_one_int ) @ P ) ) ) ) ) ).
thf(fact_641_zcong__square__zless,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P )
=> ( ( A = one_one_int )
| ( A
= ( minus_minus_int @ P @ one_one_int ) ) ) ) ) ) ) ).
thf(fact_642_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_643_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_644_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_645_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_646_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_647_zcong__zpower,axiom,
! [Z: nat,X_1: int,Y_1: int,M: int] :
( ( zcong @ X_1 @ Y_1 @ M )
=> ( zcong @ ( power_power_int @ X_1 @ Z ) @ ( power_power_int @ Y_1 @ Z ) @ M ) ) ).
thf(fact_648_power2__diff,axiom,
! [X_6: real,Y_5: real] :
( ( power_power_real @ ( minus_minus_real @ X_6 @ Y_5 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_5 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_6 ) @ Y_5 ) ) ) ).
thf(fact_649_power2__diff,axiom,
! [X_6: int,Y_5: int] :
( ( power_power_int @ ( minus_minus_int @ X_6 @ Y_5 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_5 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_6 ) @ Y_5 ) ) ) ).
thf(fact_650_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_651_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_652_zcong__less__eq,axiom,
! [M: int,Y_1: int,X_1: int] :
( ( ord_less_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_int @ zero_zero_int @ Y_1 )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ X_1 @ Y_1 @ M )
=> ( ( ord_less_int @ X_1 @ M )
=> ( ( ord_less_int @ Y_1 @ M )
=> ( X_1 = Y_1 ) ) ) ) ) ) ) ).
thf(fact_653_zcong__not__zero,axiom,
! [M: int,X_1: int] :
( ( ord_less_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_int @ X_1 @ M )
=> ~ ( zcong @ X_1 @ zero_zero_int @ M ) ) ) ).
thf(fact_654_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_655_zcong__eq__zdvd__prop,axiom,
! [X_1: int,P: int] :
( ( zcong @ X_1 @ zero_zero_int @ P )
<=> ( dvd_dvd_int @ P @ X_1 ) ) ).
thf(fact_656_zcong__zero__equiv__div,axiom,
! [A: int,M: int] :
( ( zcong @ A @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A ) ) ).
thf(fact_657_zprime__zdvd__zmult__better,axiom,
! [M: int,N: int,P: int] :
( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P @ M )
| ( dvd_dvd_int @ P @ N ) ) ) ) ).
thf(fact_658_Int2_Ozcong__zero,axiom,
! [M: int,X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_int @ X_1 @ M )
=> ( ( zcong @ X_1 @ zero_zero_int @ M )
=> ( X_1 = zero_zero_int ) ) ) ) ).
thf(fact_659_zpower__zdvd__prop1,axiom,
! [P: int,Y_1: int,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P @ Y_1 )
=> ( dvd_dvd_int @ P @ ( power_power_int @ Y_1 @ N ) ) ) ) ).
thf(fact_660_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_661_zcong__zmult__prop3,axiom,
! [Y_1: int,X_1: int,P: int] :
( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( ~ ( zcong @ Y_1 @ zero_zero_int @ P )
=> ~ ( zcong @ ( times_times_int @ X_1 @ Y_1 ) @ zero_zero_int @ P ) ) ) ) ).
thf(fact_662_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P )
=> ( ( zcong @ A @ zero_zero_int @ P )
| ( zcong @ B @ zero_zero_int @ P ) ) ) ) ) ).
thf(fact_663_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ~ ( zcong @ A @ zero_zero_int @ P )
& ~ ( zcong @ B @ zero_zero_int @ P ) )
=> ~ ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P ) ) ) ) ).
thf(fact_664_zpower__zdvd__prop2,axiom,
! [Y_1: int,N: nat,P: int] :
( ( zprime @ P )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ Y_1 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( dvd_dvd_int @ P @ Y_1 ) ) ) ) ).
thf(fact_665_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_666_Legendre__def,axiom,
! [A: int,P: int] :
( ( ( zcong @ A @ zero_zero_int @ P )
=> ( ( legendre @ A @ P )
= zero_zero_int ) )
& ( ~ ( zcong @ A @ zero_zero_int @ P )
=> ( ( ( quadRes @ P @ A )
=> ( ( legendre @ A @ P )
= one_one_int ) )
& ( ~ ( quadRes @ P @ A )
=> ( ( legendre @ A @ P )
= ( number_number_of_int @ min ) ) ) ) ) ) ).
thf(fact_667_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_668_QuadRes__def,axiom,
! [M: int,X_1: int] :
( ( quadRes @ M @ X_1 )
<=> ? [Y: int] : ( zcong @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X_1 @ M ) ) ).
thf(fact_669_realpow__two__sum__zero__iff,axiom,
! [X_1: real,Y_1: real] :
( ( ( plus_plus_real @ ( power_power_real @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X_1 = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
thf(fact_670_convex__bound__lt,axiom,
! [V_1: real,U_1: real,Y_4: real,X_5: real,A_60: real] :
( ( ord_less_real @ X_5 @ A_60 )
=> ( ( ord_less_real @ Y_4 @ A_60 )
=> ( ( 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_5 ) @ ( times_times_real @ V_1 @ Y_4 ) ) @ A_60 ) ) ) ) ) ) ).
thf(fact_671_convex__bound__lt,axiom,
! [V_1: int,U_1: int,Y_4: int,X_5: int,A_60: int] :
( ( ord_less_int @ X_5 @ A_60 )
=> ( ( ord_less_int @ Y_4 @ A_60 )
=> ( ( 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_5 ) @ ( times_times_int @ V_1 @ Y_4 ) ) @ A_60 ) ) ) ) ) ) ).
thf(fact_672_dvd__0__right,axiom,
! [A_59: real] : ( dvd_dvd_real @ A_59 @ zero_zero_real ) ).
thf(fact_673_dvd__0__right,axiom,
! [A_59: nat] : ( dvd_dvd_nat @ A_59 @ zero_zero_nat ) ).
thf(fact_674_dvd__0__right,axiom,
! [A_59: int] : ( dvd_dvd_int @ A_59 @ zero_zero_int ) ).
thf(fact_675_real__le__eq__diff,axiom,
! [X_1: real,Y_1: real] :
( ( ord_less_eq_real @ X_1 @ Y_1 )
<=> ( ord_less_eq_real @ ( minus_minus_real @ X_1 @ Y_1 ) @ zero_zero_real ) ) ).
thf(fact_676_linorder__neqE__linordered__idom,axiom,
! [X_4: real,Y_3: real] :
( ( X_4 != Y_3 )
=> ( ~ ( ord_less_real @ X_4 @ Y_3 )
=> ( ord_less_real @ Y_3 @ X_4 ) ) ) ).
thf(fact_677_linorder__neqE__linordered__idom,axiom,
! [X_4: int,Y_3: int] :
( ( X_4 != Y_3 )
=> ( ~ ( ord_less_int @ X_4 @ Y_3 )
=> ( ord_less_int @ Y_3 @ X_4 ) ) ) ).
thf(fact_678_dvd__refl,axiom,
! [A_58: real] : ( dvd_dvd_real @ A_58 @ A_58 ) ).
thf(fact_679_dvd__refl,axiom,
! [A_58: nat] : ( dvd_dvd_nat @ A_58 @ A_58 ) ).
thf(fact_680_dvd__refl,axiom,
! [A_58: int] : ( dvd_dvd_int @ A_58 @ A_58 ) ).
thf(fact_681_dvd__trans,axiom,
! [C_27: real,A_57: real,B_51: real] :
( ( dvd_dvd_real @ A_57 @ B_51 )
=> ( ( dvd_dvd_real @ B_51 @ C_27 )
=> ( dvd_dvd_real @ A_57 @ C_27 ) ) ) ).
thf(fact_682_dvd__trans,axiom,
! [C_27: nat,A_57: nat,B_51: nat] :
( ( dvd_dvd_nat @ A_57 @ B_51 )
=> ( ( dvd_dvd_nat @ B_51 @ C_27 )
=> ( dvd_dvd_nat @ A_57 @ C_27 ) ) ) ).
thf(fact_683_dvd__trans,axiom,
! [C_27: int,A_57: int,B_51: int] :
( ( dvd_dvd_int @ A_57 @ B_51 )
=> ( ( dvd_dvd_int @ B_51 @ C_27 )
=> ( dvd_dvd_int @ A_57 @ C_27 ) ) ) ).
thf(fact_684_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_685_less__eq__real__def,axiom,
! [X_1: real,Y_1: real] :
( ( ord_less_eq_real @ X_1 @ Y_1 )
<=> ( ( ord_less_real @ X_1 @ Y_1 )
| ( X_1 = Y_1 ) ) ) ).
thf(fact_686_real__less__def,axiom,
! [X_1: real,Y_1: real] :
( ( ord_less_real @ X_1 @ Y_1 )
<=> ( ( ord_less_eq_real @ X_1 @ Y_1 )
& ( X_1 != Y_1 ) ) ) ).
thf(fact_687_divides__antisym,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ( dvd_dvd_nat @ Y_1 @ X_1 ) )
<=> ( X_1 = Y_1 ) ) ).
thf(fact_688_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_689_real__mult__commute,axiom,
! [Z: real,W: real] :
( ( times_times_real @ Z @ W )
= ( times_times_real @ W @ Z ) ) ).
thf(fact_690_real__mult__1,axiom,
! [Z: real] :
( ( times_times_real @ one_one_real @ Z )
= Z ) ).
thf(fact_691_real__add__left__mono,axiom,
! [Z: real,X_1: real,Y_1: real] :
( ( ord_less_eq_real @ X_1 @ Y_1 )
=> ( ord_less_eq_real @ ( plus_plus_real @ Z @ X_1 ) @ ( plus_plus_real @ Z @ Y_1 ) ) ) ).
thf(fact_692_realpow__minus__mult,axiom,
! [X_3: nat,N_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_2 )
=> ( ( times_times_nat @ ( power_power_nat @ X_3 @ ( minus_minus_nat @ N_2 @ one_one_nat ) ) @ X_3 )
= ( power_power_nat @ X_3 @ N_2 ) ) ) ).
thf(fact_693_realpow__minus__mult,axiom,
! [X_3: real,N_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_2 )
=> ( ( times_times_real @ ( power_power_real @ X_3 @ ( minus_minus_nat @ N_2 @ one_one_nat ) ) @ X_3 )
= ( power_power_real @ X_3 @ N_2 ) ) ) ).
thf(fact_694_realpow__minus__mult,axiom,
! [X_3: int,N_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_2 )
=> ( ( times_times_int @ ( power_power_int @ X_3 @ ( minus_minus_nat @ N_2 @ one_one_nat ) ) @ X_3 )
= ( power_power_int @ X_3 @ N_2 ) ) ) ).
thf(fact_695_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_696_power__eq__if,axiom,
! [P: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( power_power_nat @ P @ M )
= one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( power_power_nat @ P @ M )
= ( times_times_nat @ P @ ( power_power_nat @ P @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).
thf(fact_697_diff__square,axiom,
! [X_1: nat,Y_1: nat] :
( ( minus_minus_nat @ ( power_power_nat @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( times_times_nat @ ( plus_plus_nat @ X_1 @ Y_1 ) @ ( minus_minus_nat @ X_1 @ Y_1 ) ) ) ).
thf(fact_698_divisors__zero,axiom,
! [A_56: real,B_50: real] :
( ( ( times_times_real @ A_56 @ B_50 )
= zero_zero_real )
=> ( ( A_56 = zero_zero_real )
| ( B_50 = zero_zero_real ) ) ) ).
thf(fact_699_divisors__zero,axiom,
! [A_56: nat,B_50: nat] :
( ( ( times_times_nat @ A_56 @ B_50 )
= zero_zero_nat )
=> ( ( A_56 = zero_zero_nat )
| ( B_50 = zero_zero_nat ) ) ) ).
thf(fact_700_divisors__zero,axiom,
! [A_56: int,B_50: int] :
( ( ( times_times_int @ A_56 @ B_50 )
= zero_zero_int )
=> ( ( A_56 = zero_zero_int )
| ( B_50 = zero_zero_int ) ) ) ).
thf(fact_701_no__zero__divisors,axiom,
! [B_49: real,A_55: real] :
( ( A_55 != zero_zero_real )
=> ( ( B_49 != zero_zero_real )
=> ( ( times_times_real @ A_55 @ B_49 )
!= zero_zero_real ) ) ) ).
thf(fact_702_no__zero__divisors,axiom,
! [B_49: nat,A_55: nat] :
( ( A_55 != zero_zero_nat )
=> ( ( B_49 != zero_zero_nat )
=> ( ( times_times_nat @ A_55 @ B_49 )
!= zero_zero_nat ) ) ) ).
thf(fact_703_no__zero__divisors,axiom,
! [B_49: int,A_55: int] :
( ( A_55 != zero_zero_int )
=> ( ( B_49 != zero_zero_int )
=> ( ( times_times_int @ A_55 @ B_49 )
!= zero_zero_int ) ) ) ).
thf(fact_704_mult__eq__0__iff,axiom,
! [A_54: real,B_48: real] :
( ( ( times_times_real @ A_54 @ B_48 )
= zero_zero_real )
<=> ( ( A_54 = zero_zero_real )
| ( B_48 = zero_zero_real ) ) ) ).
thf(fact_705_mult__eq__0__iff,axiom,
! [A_54: int,B_48: int] :
( ( ( times_times_int @ A_54 @ B_48 )
= zero_zero_int )
<=> ( ( A_54 = zero_zero_int )
| ( B_48 = zero_zero_int ) ) ) ).
thf(fact_706_mult__zero__right,axiom,
! [A_53: real] :
( ( times_times_real @ A_53 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_707_mult__zero__right,axiom,
! [A_53: nat] :
( ( times_times_nat @ A_53 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_708_mult__zero__right,axiom,
! [A_53: int] :
( ( times_times_int @ A_53 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_709_mult__zero__left,axiom,
! [A_52: real] :
( ( times_times_real @ zero_zero_real @ A_52 )
= zero_zero_real ) ).
thf(fact_710_mult__zero__left,axiom,
! [A_52: nat] :
( ( times_times_nat @ zero_zero_nat @ A_52 )
= zero_zero_nat ) ).
thf(fact_711_mult__zero__left,axiom,
! [A_52: int] :
( ( times_times_int @ zero_zero_int @ A_52 )
= zero_zero_int ) ).
thf(fact_712_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_713_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_714_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_715_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_716_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_717_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_718_combine__common__factor,axiom,
! [A_51: real,E: real,B_47: real,C_26: real] :
( ( plus_plus_real @ ( times_times_real @ A_51 @ E ) @ ( plus_plus_real @ ( times_times_real @ B_47 @ E ) @ C_26 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_51 @ B_47 ) @ E ) @ C_26 ) ) ).
thf(fact_719_combine__common__factor,axiom,
! [A_51: nat,E: nat,B_47: nat,C_26: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_51 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B_47 @ E ) @ C_26 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_51 @ B_47 ) @ E ) @ C_26 ) ) ).
thf(fact_720_combine__common__factor,axiom,
! [A_51: int,E: int,B_47: int,C_26: int] :
( ( plus_plus_int @ ( times_times_int @ A_51 @ E ) @ ( plus_plus_int @ ( times_times_int @ B_47 @ E ) @ C_26 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_51 @ B_47 ) @ E ) @ C_26 ) ) ).
thf(fact_721_comm__semiring__class_Odistrib,axiom,
! [A_50: real,B_46: real,C_25: real] :
( ( times_times_real @ ( plus_plus_real @ A_50 @ B_46 ) @ C_25 )
= ( plus_plus_real @ ( times_times_real @ A_50 @ C_25 ) @ ( times_times_real @ B_46 @ C_25 ) ) ) ).
thf(fact_722_comm__semiring__class_Odistrib,axiom,
! [A_50: nat,B_46: nat,C_25: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_50 @ B_46 ) @ C_25 )
= ( plus_plus_nat @ ( times_times_nat @ A_50 @ C_25 ) @ ( times_times_nat @ B_46 @ C_25 ) ) ) ).
thf(fact_723_comm__semiring__class_Odistrib,axiom,
! [A_50: int,B_46: int,C_25: int] :
( ( times_times_int @ ( plus_plus_int @ A_50 @ B_46 ) @ C_25 )
= ( plus_plus_int @ ( times_times_int @ A_50 @ C_25 ) @ ( times_times_int @ B_46 @ C_25 ) ) ) ).
thf(fact_724_dvd__0__left,axiom,
! [A_49: real] :
( ( dvd_dvd_real @ zero_zero_real @ A_49 )
=> ( A_49 = zero_zero_real ) ) ).
thf(fact_725_dvd__0__left,axiom,
! [A_49: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A_49 )
=> ( A_49 = zero_zero_nat ) ) ).
thf(fact_726_dvd__0__left,axiom,
! [A_49: int] :
( ( dvd_dvd_int @ zero_zero_int @ A_49 )
=> ( A_49 = zero_zero_int ) ) ).
thf(fact_727_add__diff__add,axiom,
! [A_48: real,C_24: real,B_45: real,D_6: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_48 @ C_24 ) @ ( plus_plus_real @ B_45 @ D_6 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_48 @ B_45 ) @ ( minus_minus_real @ C_24 @ D_6 ) ) ) ).
thf(fact_728_add__diff__add,axiom,
! [A_48: int,C_24: int,B_45: int,D_6: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_48 @ C_24 ) @ ( plus_plus_int @ B_45 @ D_6 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_48 @ B_45 ) @ ( minus_minus_int @ C_24 @ D_6 ) ) ) ).
thf(fact_729_dvd__mult__right,axiom,
! [A_47: real,B_44: real,C_23: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_47 @ B_44 ) @ C_23 )
=> ( dvd_dvd_real @ B_44 @ C_23 ) ) ).
thf(fact_730_dvd__mult__right,axiom,
! [A_47: nat,B_44: nat,C_23: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_47 @ B_44 ) @ C_23 )
=> ( dvd_dvd_nat @ B_44 @ C_23 ) ) ).
thf(fact_731_dvd__mult__right,axiom,
! [A_47: int,B_44: int,C_23: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_47 @ B_44 ) @ C_23 )
=> ( dvd_dvd_int @ B_44 @ C_23 ) ) ).
thf(fact_732_dvd__mult__left,axiom,
! [A_46: real,B_43: real,C_22: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_46 @ B_43 ) @ C_22 )
=> ( dvd_dvd_real @ A_46 @ C_22 ) ) ).
thf(fact_733_dvd__mult__left,axiom,
! [A_46: nat,B_43: nat,C_22: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_46 @ B_43 ) @ C_22 )
=> ( dvd_dvd_nat @ A_46 @ C_22 ) ) ).
thf(fact_734_dvd__mult__left,axiom,
! [A_46: int,B_43: int,C_22: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_46 @ B_43 ) @ C_22 )
=> ( dvd_dvd_int @ A_46 @ C_22 ) ) ).
thf(fact_735_dvdI,axiom,
! [A_45: real,B_42: real,K_2: real] :
( ( A_45
= ( times_times_real @ B_42 @ K_2 ) )
=> ( dvd_dvd_real @ B_42 @ A_45 ) ) ).
thf(fact_736_dvdI,axiom,
! [A_45: nat,B_42: nat,K_2: nat] :
( ( A_45
= ( times_times_nat @ B_42 @ K_2 ) )
=> ( dvd_dvd_nat @ B_42 @ A_45 ) ) ).
thf(fact_737_dvdI,axiom,
! [A_45: int,B_42: int,K_2: int] :
( ( A_45
= ( times_times_int @ B_42 @ K_2 ) )
=> ( dvd_dvd_int @ B_42 @ A_45 ) ) ).
thf(fact_738_mult__dvd__mono,axiom,
! [C_21: real,D_5: real,A_44: real,B_41: real] :
( ( dvd_dvd_real @ A_44 @ B_41 )
=> ( ( dvd_dvd_real @ C_21 @ D_5 )
=> ( dvd_dvd_real @ ( times_times_real @ A_44 @ C_21 ) @ ( times_times_real @ B_41 @ D_5 ) ) ) ) ).
thf(fact_739_mult__dvd__mono,axiom,
! [C_21: nat,D_5: nat,A_44: nat,B_41: nat] :
( ( dvd_dvd_nat @ A_44 @ B_41 )
=> ( ( dvd_dvd_nat @ C_21 @ D_5 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A_44 @ C_21 ) @ ( times_times_nat @ B_41 @ D_5 ) ) ) ) ).
thf(fact_740_mult__dvd__mono,axiom,
! [C_21: int,D_5: int,A_44: int,B_41: int] :
( ( dvd_dvd_int @ A_44 @ B_41 )
=> ( ( dvd_dvd_int @ C_21 @ D_5 )
=> ( dvd_dvd_int @ ( times_times_int @ A_44 @ C_21 ) @ ( times_times_int @ B_41 @ D_5 ) ) ) ) ).
thf(fact_741_dvd__mult,axiom,
! [B_40: real,A_43: real,C_20: real] :
( ( dvd_dvd_real @ A_43 @ C_20 )
=> ( dvd_dvd_real @ A_43 @ ( times_times_real @ B_40 @ C_20 ) ) ) ).
thf(fact_742_dvd__mult,axiom,
! [B_40: nat,A_43: nat,C_20: nat] :
( ( dvd_dvd_nat @ A_43 @ C_20 )
=> ( dvd_dvd_nat @ A_43 @ ( times_times_nat @ B_40 @ C_20 ) ) ) ).
thf(fact_743_dvd__mult,axiom,
! [B_40: int,A_43: int,C_20: int] :
( ( dvd_dvd_int @ A_43 @ C_20 )
=> ( dvd_dvd_int @ A_43 @ ( times_times_int @ B_40 @ C_20 ) ) ) ).
thf(fact_744_dvd__mult2,axiom,
! [C_19: real,A_42: real,B_39: real] :
( ( dvd_dvd_real @ A_42 @ B_39 )
=> ( dvd_dvd_real @ A_42 @ ( times_times_real @ B_39 @ C_19 ) ) ) ).
thf(fact_745_dvd__mult2,axiom,
! [C_19: nat,A_42: nat,B_39: nat] :
( ( dvd_dvd_nat @ A_42 @ B_39 )
=> ( dvd_dvd_nat @ A_42 @ ( times_times_nat @ B_39 @ C_19 ) ) ) ).
thf(fact_746_dvd__mult2,axiom,
! [C_19: int,A_42: int,B_39: int] :
( ( dvd_dvd_int @ A_42 @ B_39 )
=> ( dvd_dvd_int @ A_42 @ ( times_times_int @ B_39 @ C_19 ) ) ) ).
thf(fact_747_dvd__triv__right,axiom,
! [A_41: real,B_38: real] : ( dvd_dvd_real @ A_41 @ ( times_times_real @ B_38 @ A_41 ) ) ).
thf(fact_748_dvd__triv__right,axiom,
! [A_41: nat,B_38: nat] : ( dvd_dvd_nat @ A_41 @ ( times_times_nat @ B_38 @ A_41 ) ) ).
thf(fact_749_dvd__triv__right,axiom,
! [A_41: int,B_38: int] : ( dvd_dvd_int @ A_41 @ ( times_times_int @ B_38 @ A_41 ) ) ).
thf(fact_750_dvd__triv__left,axiom,
! [A_40: real,B_37: real] : ( dvd_dvd_real @ A_40 @ ( times_times_real @ A_40 @ B_37 ) ) ).
thf(fact_751_dvd__triv__left,axiom,
! [A_40: nat,B_37: nat] : ( dvd_dvd_nat @ A_40 @ ( times_times_nat @ A_40 @ B_37 ) ) ).
thf(fact_752_dvd__triv__left,axiom,
! [A_40: int,B_37: int] : ( dvd_dvd_int @ A_40 @ ( times_times_int @ A_40 @ B_37 ) ) ).
thf(fact_753_dvd__add,axiom,
! [C_18: real,A_39: real,B_36: real] :
( ( dvd_dvd_real @ A_39 @ B_36 )
=> ( ( dvd_dvd_real @ A_39 @ C_18 )
=> ( dvd_dvd_real @ A_39 @ ( plus_plus_real @ B_36 @ C_18 ) ) ) ) ).
thf(fact_754_dvd__add,axiom,
! [C_18: nat,A_39: nat,B_36: nat] :
( ( dvd_dvd_nat @ A_39 @ B_36 )
=> ( ( dvd_dvd_nat @ A_39 @ C_18 )
=> ( dvd_dvd_nat @ A_39 @ ( plus_plus_nat @ B_36 @ C_18 ) ) ) ) ).
thf(fact_755_dvd__add,axiom,
! [C_18: int,A_39: int,B_36: int] :
( ( dvd_dvd_int @ A_39 @ B_36 )
=> ( ( dvd_dvd_int @ A_39 @ C_18 )
=> ( dvd_dvd_int @ A_39 @ ( plus_plus_int @ B_36 @ C_18 ) ) ) ) ).
thf(fact_756_one__dvd,axiom,
! [A_38: real] : ( dvd_dvd_real @ one_one_real @ A_38 ) ).
thf(fact_757_one__dvd,axiom,
! [A_38: nat] : ( dvd_dvd_nat @ one_one_nat @ A_38 ) ).
thf(fact_758_one__dvd,axiom,
! [A_38: int] : ( dvd_dvd_int @ one_one_int @ A_38 ) ).
thf(fact_759_dvd__diff,axiom,
! [Z_1: real,X_2: real,Y_2: real] :
( ( dvd_dvd_real @ X_2 @ Y_2 )
=> ( ( dvd_dvd_real @ X_2 @ Z_1 )
=> ( dvd_dvd_real @ X_2 @ ( minus_minus_real @ Y_2 @ Z_1 ) ) ) ) ).
thf(fact_760_dvd__diff,axiom,
! [Z_1: int,X_2: int,Y_2: int] :
( ( dvd_dvd_int @ X_2 @ Y_2 )
=> ( ( dvd_dvd_int @ X_2 @ Z_1 )
=> ( dvd_dvd_int @ X_2 @ ( minus_minus_int @ Y_2 @ Z_1 ) ) ) ) ).
thf(fact_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_divides__exp,axiom,
! [N: nat,X_1: nat,Y_1: nat] :
( ( dvd_dvd_nat @ X_1 @ Y_1 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_1 @ N ) @ ( power_power_nat @ Y_1 @ N ) ) ) ).
thf(fact_770_split__mult__neg__le,axiom,
! [B_35: real,A_37: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_37 )
& ( ord_less_eq_real @ B_35 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_37 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_35 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_37 @ B_35 ) @ zero_zero_real ) ) ).
thf(fact_771_split__mult__neg__le,axiom,
! [B_35: nat,A_37: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_37 )
& ( ord_less_eq_nat @ B_35 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_37 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_35 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_37 @ B_35 ) @ zero_zero_nat ) ) ).
thf(fact_772_split__mult__neg__le,axiom,
! [B_35: int,A_37: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_37 )
& ( ord_less_eq_int @ B_35 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_37 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_35 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_37 @ B_35 ) @ zero_zero_int ) ) ).
thf(fact_773_split__mult__pos__le,axiom,
! [B_34: real,A_36: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_36 )
& ( ord_less_eq_real @ zero_zero_real @ B_34 ) )
| ( ( ord_less_eq_real @ A_36 @ zero_zero_real )
& ( ord_less_eq_real @ B_34 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_36 @ B_34 ) ) ) ).
thf(fact_774_split__mult__pos__le,axiom,
! [B_34: int,A_36: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_36 )
& ( ord_less_eq_int @ zero_zero_int @ B_34 ) )
| ( ( ord_less_eq_int @ A_36 @ zero_zero_int )
& ( ord_less_eq_int @ B_34 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_36 @ B_34 ) ) ) ).
thf(fact_775_mult__mono,axiom,
! [C_17: real,D_4: real,A_35: real,B_33: real] :
( ( ord_less_eq_real @ A_35 @ B_33 )
=> ( ( ord_less_eq_real @ C_17 @ D_4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_33 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_17 )
=> ( ord_less_eq_real @ ( times_times_real @ A_35 @ C_17 ) @ ( times_times_real @ B_33 @ D_4 ) ) ) ) ) ) ).
thf(fact_776_mult__mono,axiom,
! [C_17: nat,D_4: nat,A_35: nat,B_33: nat] :
( ( ord_less_eq_nat @ A_35 @ B_33 )
=> ( ( ord_less_eq_nat @ C_17 @ D_4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_33 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_17 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_35 @ C_17 ) @ ( times_times_nat @ B_33 @ D_4 ) ) ) ) ) ) ).
thf(fact_777_mult__mono,axiom,
! [C_17: int,D_4: int,A_35: int,B_33: int] :
( ( ord_less_eq_int @ A_35 @ B_33 )
=> ( ( ord_less_eq_int @ C_17 @ D_4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_33 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_17 )
=> ( ord_less_eq_int @ ( times_times_int @ A_35 @ C_17 ) @ ( times_times_int @ B_33 @ D_4 ) ) ) ) ) ) ).
thf(fact_778_mult__mono_H,axiom,
! [C_16: real,D_3: real,A_34: real,B_32: real] :
( ( ord_less_eq_real @ A_34 @ B_32 )
=> ( ( ord_less_eq_real @ C_16 @ D_3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_34 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_16 )
=> ( ord_less_eq_real @ ( times_times_real @ A_34 @ C_16 ) @ ( times_times_real @ B_32 @ D_3 ) ) ) ) ) ) ).
thf(fact_779_mult__mono_H,axiom,
! [C_16: nat,D_3: nat,A_34: nat,B_32: nat] :
( ( ord_less_eq_nat @ A_34 @ B_32 )
=> ( ( ord_less_eq_nat @ C_16 @ D_3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_34 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_16 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_34 @ C_16 ) @ ( times_times_nat @ B_32 @ D_3 ) ) ) ) ) ) ).
thf(fact_780_mult__mono_H,axiom,
! [C_16: int,D_3: int,A_34: int,B_32: int] :
( ( ord_less_eq_int @ A_34 @ B_32 )
=> ( ( ord_less_eq_int @ C_16 @ D_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_34 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_16 )
=> ( ord_less_eq_int @ ( times_times_int @ A_34 @ C_16 ) @ ( times_times_int @ B_32 @ D_3 ) ) ) ) ) ) ).
thf(fact_781_mult__left__mono__neg,axiom,
! [C_15: real,B_31: real,A_33: real] :
( ( ord_less_eq_real @ B_31 @ A_33 )
=> ( ( ord_less_eq_real @ C_15 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_15 @ A_33 ) @ ( times_times_real @ C_15 @ B_31 ) ) ) ) ).
thf(fact_782_mult__left__mono__neg,axiom,
! [C_15: int,B_31: int,A_33: int] :
( ( ord_less_eq_int @ B_31 @ A_33 )
=> ( ( ord_less_eq_int @ C_15 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_15 @ A_33 ) @ ( times_times_int @ C_15 @ B_31 ) ) ) ) ).
thf(fact_783_mult__right__mono__neg,axiom,
! [C_14: real,B_30: real,A_32: real] :
( ( ord_less_eq_real @ B_30 @ A_32 )
=> ( ( ord_less_eq_real @ C_14 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_32 @ C_14 ) @ ( times_times_real @ B_30 @ C_14 ) ) ) ) ).
thf(fact_784_mult__right__mono__neg,axiom,
! [C_14: int,B_30: int,A_32: int] :
( ( ord_less_eq_int @ B_30 @ A_32 )
=> ( ( ord_less_eq_int @ C_14 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_32 @ C_14 ) @ ( times_times_int @ B_30 @ C_14 ) ) ) ) ).
thf(fact_785_comm__mult__left__mono,axiom,
! [C_13: real,A_31: real,B_29: real] :
( ( ord_less_eq_real @ A_31 @ B_29 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_13 )
=> ( ord_less_eq_real @ ( times_times_real @ C_13 @ A_31 ) @ ( times_times_real @ C_13 @ B_29 ) ) ) ) ).
thf(fact_786_comm__mult__left__mono,axiom,
! [C_13: nat,A_31: nat,B_29: nat] :
( ( ord_less_eq_nat @ A_31 @ B_29 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_13 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_13 @ A_31 ) @ ( times_times_nat @ C_13 @ B_29 ) ) ) ) ).
thf(fact_787_comm__mult__left__mono,axiom,
! [C_13: int,A_31: int,B_29: int] :
( ( ord_less_eq_int @ A_31 @ B_29 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_13 )
=> ( ord_less_eq_int @ ( times_times_int @ C_13 @ A_31 ) @ ( times_times_int @ C_13 @ B_29 ) ) ) ) ).
thf(fact_788_mult__left__mono,axiom,
! [C_12: real,A_30: real,B_28: real] :
( ( ord_less_eq_real @ A_30 @ B_28 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_12 )
=> ( ord_less_eq_real @ ( times_times_real @ C_12 @ A_30 ) @ ( times_times_real @ C_12 @ B_28 ) ) ) ) ).
thf(fact_789_mult__left__mono,axiom,
! [C_12: nat,A_30: nat,B_28: nat] :
( ( ord_less_eq_nat @ A_30 @ B_28 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_12 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_12 @ A_30 ) @ ( times_times_nat @ C_12 @ B_28 ) ) ) ) ).
thf(fact_790_mult__left__mono,axiom,
! [C_12: int,A_30: int,B_28: int] :
( ( ord_less_eq_int @ A_30 @ B_28 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_12 )
=> ( ord_less_eq_int @ ( times_times_int @ C_12 @ A_30 ) @ ( times_times_int @ C_12 @ B_28 ) ) ) ) ).
thf(fact_791_mult__right__mono,axiom,
! [C_11: real,A_29: real,B_27: real] :
( ( ord_less_eq_real @ A_29 @ B_27 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_11 )
=> ( ord_less_eq_real @ ( times_times_real @ A_29 @ C_11 ) @ ( times_times_real @ B_27 @ C_11 ) ) ) ) ).
thf(fact_792_mult__right__mono,axiom,
! [C_11: nat,A_29: nat,B_27: nat] :
( ( ord_less_eq_nat @ A_29 @ B_27 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_11 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_29 @ C_11 ) @ ( times_times_nat @ B_27 @ C_11 ) ) ) ) ).
thf(fact_793_mult__right__mono,axiom,
! [C_11: int,A_29: int,B_27: int] :
( ( ord_less_eq_int @ A_29 @ B_27 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_11 )
=> ( ord_less_eq_int @ ( times_times_int @ A_29 @ C_11 ) @ ( times_times_int @ B_27 @ C_11 ) ) ) ) ).
thf(fact_794_mult__nonpos__nonpos,axiom,
! [B_26: real,A_28: real] :
( ( ord_less_eq_real @ A_28 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_26 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_28 @ B_26 ) ) ) ) ).
thf(fact_795_mult__nonpos__nonpos,axiom,
! [B_26: int,A_28: int] :
( ( ord_less_eq_int @ A_28 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_26 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_28 @ B_26 ) ) ) ) ).
thf(fact_796_mult__nonpos__nonneg,axiom,
! [B_25: real,A_27: real] :
( ( ord_less_eq_real @ A_27 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_25 )
=> ( ord_less_eq_real @ ( times_times_real @ A_27 @ B_25 ) @ zero_zero_real ) ) ) ).
thf(fact_797_mult__nonpos__nonneg,axiom,
! [B_25: nat,A_27: nat] :
( ( ord_less_eq_nat @ A_27 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_25 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_27 @ B_25 ) @ zero_zero_nat ) ) ) ).
thf(fact_798_mult__nonpos__nonneg,axiom,
! [B_25: int,A_27: int] :
( ( ord_less_eq_int @ A_27 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_25 )
=> ( ord_less_eq_int @ ( times_times_int @ A_27 @ B_25 ) @ zero_zero_int ) ) ) ).
thf(fact_799_mult__nonneg__nonpos2,axiom,
! [B_24: real,A_26: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_26 )
=> ( ( ord_less_eq_real @ B_24 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_24 @ A_26 ) @ zero_zero_real ) ) ) ).
thf(fact_800_mult__nonneg__nonpos2,axiom,
! [B_24: nat,A_26: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_26 )
=> ( ( ord_less_eq_nat @ B_24 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_24 @ A_26 ) @ zero_zero_nat ) ) ) ).
thf(fact_801_mult__nonneg__nonpos2,axiom,
! [B_24: int,A_26: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_26 )
=> ( ( ord_less_eq_int @ B_24 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_24 @ A_26 ) @ zero_zero_int ) ) ) ).
thf(fact_802_mult__nonneg__nonpos,axiom,
! [B_23: real,A_25: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_25 )
=> ( ( ord_less_eq_real @ B_23 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_25 @ B_23 ) @ zero_zero_real ) ) ) ).
thf(fact_803_mult__nonneg__nonpos,axiom,
! [B_23: nat,A_25: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_25 )
=> ( ( ord_less_eq_nat @ B_23 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_25 @ B_23 ) @ zero_zero_nat ) ) ) ).
thf(fact_804_mult__nonneg__nonpos,axiom,
! [B_23: int,A_25: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_25 )
=> ( ( ord_less_eq_int @ B_23 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_25 @ B_23 ) @ zero_zero_int ) ) ) ).
thf(fact_805_mult__nonneg__nonneg,axiom,
! [B_22: real,A_24: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_24 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_22 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_24 @ B_22 ) ) ) ) ).
thf(fact_806_mult__nonneg__nonneg,axiom,
! [B_22: nat,A_24: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_24 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_22 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_24 @ B_22 ) ) ) ) ).
thf(fact_807_mult__nonneg__nonneg,axiom,
! [B_22: int,A_24: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_24 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_22 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_24 @ B_22 ) ) ) ) ).
thf(fact_808_mult__le__0__iff,axiom,
! [A_23: real,B_21: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_23 @ B_21 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_23 )
& ( ord_less_eq_real @ B_21 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_23 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_21 ) ) ) ) ).
thf(fact_809_mult__le__0__iff,axiom,
! [A_23: int,B_21: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_23 @ B_21 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_23 )
& ( ord_less_eq_int @ B_21 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_23 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_21 ) ) ) ) ).
thf(fact_810_zero__le__mult__iff,axiom,
! [A_22: real,B_20: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_22 @ B_20 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_22 )
& ( ord_less_eq_real @ zero_zero_real @ B_20 ) )
| ( ( ord_less_eq_real @ A_22 @ zero_zero_real )
& ( ord_less_eq_real @ B_20 @ zero_zero_real ) ) ) ) ).
thf(fact_811_zero__le__mult__iff,axiom,
! [A_22: int,B_20: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_22 @ B_20 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_22 )
& ( ord_less_eq_int @ zero_zero_int @ B_20 ) )
| ( ( ord_less_eq_int @ A_22 @ zero_zero_int )
& ( ord_less_eq_int @ B_20 @ zero_zero_int ) ) ) ) ).
thf(fact_812_zero__le__square,axiom,
! [A_21: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_21 @ A_21 ) ) ).
thf(fact_813_zero__le__square,axiom,
! [A_21: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_21 @ A_21 ) ) ).
thf(fact_814_mult__strict__left__mono__neg,axiom,
! [C_10: real,B_19: real,A_20: real] :
( ( ord_less_real @ B_19 @ A_20 )
=> ( ( ord_less_real @ C_10 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C_10 @ A_20 ) @ ( times_times_real @ C_10 @ B_19 ) ) ) ) ).
thf(fact_815_mult__strict__left__mono__neg,axiom,
! [C_10: int,B_19: int,A_20: int] :
( ( ord_less_int @ B_19 @ A_20 )
=> ( ( ord_less_int @ C_10 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_10 @ A_20 ) @ ( times_times_int @ C_10 @ B_19 ) ) ) ) ).
thf(fact_816_mult__strict__right__mono__neg,axiom,
! [C_9: real,B_18: real,A_19: real] :
( ( ord_less_real @ B_18 @ A_19 )
=> ( ( ord_less_real @ C_9 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_19 @ C_9 ) @ ( times_times_real @ B_18 @ C_9 ) ) ) ) ).
thf(fact_817_mult__strict__right__mono__neg,axiom,
! [C_9: int,B_18: int,A_19: int] :
( ( ord_less_int @ B_18 @ A_19 )
=> ( ( ord_less_int @ C_9 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_19 @ C_9 ) @ ( times_times_int @ B_18 @ C_9 ) ) ) ) ).
thf(fact_818_comm__mult__strict__left__mono,axiom,
! [C_8: real,A_18: real,B_17: real] :
( ( ord_less_real @ A_18 @ B_17 )
=> ( ( ord_less_real @ zero_zero_real @ C_8 )
=> ( ord_less_real @ ( times_times_real @ C_8 @ A_18 ) @ ( times_times_real @ C_8 @ B_17 ) ) ) ) ).
thf(fact_819_comm__mult__strict__left__mono,axiom,
! [C_8: nat,A_18: nat,B_17: nat] :
( ( ord_less_nat @ A_18 @ B_17 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_8 )
=> ( ord_less_nat @ ( times_times_nat @ C_8 @ A_18 ) @ ( times_times_nat @ C_8 @ B_17 ) ) ) ) ).
thf(fact_820_comm__mult__strict__left__mono,axiom,
! [C_8: int,A_18: int,B_17: int] :
( ( ord_less_int @ A_18 @ B_17 )
=> ( ( ord_less_int @ zero_zero_int @ C_8 )
=> ( ord_less_int @ ( times_times_int @ C_8 @ A_18 ) @ ( times_times_int @ C_8 @ B_17 ) ) ) ) ).
thf(fact_821_mult__strict__left__mono,axiom,
! [C_7: real,A_17: real,B_16: real] :
( ( ord_less_real @ A_17 @ B_16 )
=> ( ( ord_less_real @ zero_zero_real @ C_7 )
=> ( ord_less_real @ ( times_times_real @ C_7 @ A_17 ) @ ( times_times_real @ C_7 @ B_16 ) ) ) ) ).
thf(fact_822_mult__strict__left__mono,axiom,
! [C_7: nat,A_17: nat,B_16: nat] :
( ( ord_less_nat @ A_17 @ B_16 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_7 )
=> ( ord_less_nat @ ( times_times_nat @ C_7 @ A_17 ) @ ( times_times_nat @ C_7 @ B_16 ) ) ) ) ).
thf(fact_823_mult__strict__left__mono,axiom,
! [C_7: int,A_17: int,B_16: int] :
( ( ord_less_int @ A_17 @ B_16 )
=> ( ( ord_less_int @ zero_zero_int @ C_7 )
=> ( ord_less_int @ ( times_times_int @ C_7 @ A_17 ) @ ( times_times_int @ C_7 @ B_16 ) ) ) ) ).
thf(fact_824_mult__strict__right__mono,axiom,
! [C_6: real,A_16: real,B_15: real] :
( ( ord_less_real @ A_16 @ B_15 )
=> ( ( ord_less_real @ zero_zero_real @ C_6 )
=> ( ord_less_real @ ( times_times_real @ A_16 @ C_6 ) @ ( times_times_real @ B_15 @ C_6 ) ) ) ) ).
thf(fact_825_mult__strict__right__mono,axiom,
! [C_6: nat,A_16: nat,B_15: nat] :
( ( ord_less_nat @ A_16 @ B_15 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_6 )
=> ( ord_less_nat @ ( times_times_nat @ A_16 @ C_6 ) @ ( times_times_nat @ B_15 @ C_6 ) ) ) ) ).
thf(fact_826_mult__strict__right__mono,axiom,
! [C_6: int,A_16: int,B_15: int] :
( ( ord_less_int @ A_16 @ B_15 )
=> ( ( ord_less_int @ zero_zero_int @ C_6 )
=> ( ord_less_int @ ( times_times_int @ A_16 @ C_6 ) @ ( times_times_int @ B_15 @ C_6 ) ) ) ) ).
thf(fact_827_mult__neg__neg,axiom,
! [B_14: real,A_15: real] :
( ( ord_less_real @ A_15 @ zero_zero_real )
=> ( ( ord_less_real @ B_14 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_15 @ B_14 ) ) ) ) ).
thf(fact_828_mult__neg__neg,axiom,
! [B_14: int,A_15: int] :
( ( ord_less_int @ A_15 @ zero_zero_int )
=> ( ( ord_less_int @ B_14 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_15 @ B_14 ) ) ) ) ).
thf(fact_829_mult__neg__pos,axiom,
! [B_13: real,A_14: real] :
( ( ord_less_real @ A_14 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B_13 )
=> ( ord_less_real @ ( times_times_real @ A_14 @ B_13 ) @ zero_zero_real ) ) ) ).
thf(fact_830_mult__neg__pos,axiom,
! [B_13: nat,A_14: nat] :
( ( ord_less_nat @ A_14 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_13 )
=> ( ord_less_nat @ ( times_times_nat @ A_14 @ B_13 ) @ zero_zero_nat ) ) ) ).
thf(fact_831_mult__neg__pos,axiom,
! [B_13: int,A_14: int] :
( ( ord_less_int @ A_14 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_13 )
=> ( ord_less_int @ ( times_times_int @ A_14 @ B_13 ) @ zero_zero_int ) ) ) ).
thf(fact_832_mult__less__cancel__left__neg,axiom,
! [A_13: real,B_12: real,C_5: real] :
( ( ord_less_real @ C_5 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C_5 @ A_13 ) @ ( times_times_real @ C_5 @ B_12 ) )
<=> ( ord_less_real @ B_12 @ A_13 ) ) ) ).
thf(fact_833_mult__less__cancel__left__neg,axiom,
! [A_13: int,B_12: int,C_5: int] :
( ( ord_less_int @ C_5 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_5 @ A_13 ) @ ( times_times_int @ C_5 @ B_12 ) )
<=> ( ord_less_int @ B_12 @ A_13 ) ) ) ).
thf(fact_834_zero__less__mult__pos2,axiom,
! [B_11: real,A_12: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B_11 @ A_12 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_12 )
=> ( ord_less_real @ zero_zero_real @ B_11 ) ) ) ).
thf(fact_835_zero__less__mult__pos2,axiom,
! [B_11: nat,A_12: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_11 @ A_12 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_12 )
=> ( ord_less_nat @ zero_zero_nat @ B_11 ) ) ) ).
thf(fact_836_zero__less__mult__pos2,axiom,
! [B_11: int,A_12: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_11 @ A_12 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_12 )
=> ( ord_less_int @ zero_zero_int @ B_11 ) ) ) ).
thf(fact_837_zero__less__mult__pos,axiom,
! [A_11: real,B_10: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_11 @ B_10 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_11 )
=> ( ord_less_real @ zero_zero_real @ B_10 ) ) ) ).
thf(fact_838_zero__less__mult__pos,axiom,
! [A_11: nat,B_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_11 @ B_10 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_11 )
=> ( ord_less_nat @ zero_zero_nat @ B_10 ) ) ) ).
thf(fact_839_zero__less__mult__pos,axiom,
! [A_11: int,B_10: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_11 @ B_10 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_11 )
=> ( ord_less_int @ zero_zero_int @ B_10 ) ) ) ).
thf(fact_840_mult__pos__neg2,axiom,
! [B_9: real,A_10: real] :
( ( ord_less_real @ zero_zero_real @ A_10 )
=> ( ( ord_less_real @ B_9 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B_9 @ A_10 ) @ zero_zero_real ) ) ) ).
thf(fact_841_mult__pos__neg2,axiom,
! [B_9: nat,A_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_10 )
=> ( ( ord_less_nat @ B_9 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_9 @ A_10 ) @ zero_zero_nat ) ) ) ).
thf(fact_842_mult__pos__neg2,axiom,
! [B_9: int,A_10: int] :
( ( ord_less_int @ zero_zero_int @ A_10 )
=> ( ( ord_less_int @ B_9 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_9 @ A_10 ) @ zero_zero_int ) ) ) ).
thf(fact_843_mult__pos__neg,axiom,
! [B_8: real,A_9: real] :
( ( ord_less_real @ zero_zero_real @ A_9 )
=> ( ( ord_less_real @ B_8 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_9 @ B_8 ) @ zero_zero_real ) ) ) ).
thf(fact_844_mult__pos__neg,axiom,
! [B_8: nat,A_9: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_9 )
=> ( ( ord_less_nat @ B_8 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_9 @ B_8 ) @ zero_zero_nat ) ) ) ).
thf(fact_845_mult__pos__neg,axiom,
! [B_8: int,A_9: int] :
( ( ord_less_int @ zero_zero_int @ A_9 )
=> ( ( ord_less_int @ B_8 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_9 @ B_8 ) @ zero_zero_int ) ) ) ).
thf(fact_846_mult__pos__pos,axiom,
! [B_7: real,A_8: real] :
( ( ord_less_real @ zero_zero_real @ A_8 )
=> ( ( ord_less_real @ zero_zero_real @ B_7 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_8 @ B_7 ) ) ) ) ).
thf(fact_847_mult__pos__pos,axiom,
! [B_7: nat,A_8: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_8 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_7 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_8 @ B_7 ) ) ) ) ).
thf(fact_848_mult__pos__pos,axiom,
! [B_7: int,A_8: int] :
( ( ord_less_int @ zero_zero_int @ A_8 )
=> ( ( ord_less_int @ zero_zero_int @ B_7 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_8 @ B_7 ) ) ) ) ).
thf(fact_849_mult__less__cancel__left__pos,axiom,
! [A_7: real,B_6: real,C_4: real] :
( ( ord_less_real @ zero_zero_real @ C_4 )
=> ( ( ord_less_real @ ( times_times_real @ C_4 @ A_7 ) @ ( times_times_real @ C_4 @ B_6 ) )
<=> ( ord_less_real @ A_7 @ B_6 ) ) ) ).
thf(fact_850_mult__less__cancel__left__pos,axiom,
! [A_7: int,B_6: int,C_4: int] :
( ( ord_less_int @ zero_zero_int @ C_4 )
=> ( ( ord_less_int @ ( times_times_int @ C_4 @ A_7 ) @ ( times_times_int @ C_4 @ B_6 ) )
<=> ( ord_less_int @ A_7 @ B_6 ) ) ) ).
thf(fact_851_mult__less__cancel__left__disj,axiom,
! [C_3: real,A_6: real,B_5: real] :
( ( ord_less_real @ ( times_times_real @ C_3 @ A_6 ) @ ( times_times_real @ C_3 @ B_5 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_3 )
& ( ord_less_real @ A_6 @ B_5 ) )
| ( ( ord_less_real @ C_3 @ zero_zero_real )
& ( ord_less_real @ B_5 @ A_6 ) ) ) ) ).
thf(fact_852_mult__less__cancel__left__disj,axiom,
! [C_3: int,A_6: int,B_5: int] :
( ( ord_less_int @ ( times_times_int @ C_3 @ A_6 ) @ ( times_times_int @ C_3 @ B_5 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_3 )
& ( ord_less_int @ A_6 @ B_5 ) )
| ( ( ord_less_int @ C_3 @ zero_zero_int )
& ( ord_less_int @ B_5 @ A_6 ) ) ) ) ).
thf(fact_853_mult__less__cancel__right__disj,axiom,
! [A_5: real,C_2: real,B_4: real] :
( ( ord_less_real @ ( times_times_real @ A_5 @ C_2 ) @ ( times_times_real @ B_4 @ C_2 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_2 )
& ( ord_less_real @ A_5 @ B_4 ) )
| ( ( ord_less_real @ C_2 @ zero_zero_real )
& ( ord_less_real @ B_4 @ A_5 ) ) ) ) ).
thf(fact_854_mult__less__cancel__right__disj,axiom,
! [A_5: int,C_2: int,B_4: int] :
( ( ord_less_int @ ( times_times_int @ A_5 @ C_2 ) @ ( times_times_int @ B_4 @ C_2 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_2 )
& ( ord_less_int @ A_5 @ B_4 ) )
| ( ( ord_less_int @ C_2 @ zero_zero_int )
& ( ord_less_int @ B_4 @ A_5 ) ) ) ) ).
thf(fact_855_not__square__less__zero,axiom,
! [A_4: real] :
~ ( ord_less_real @ ( times_times_real @ A_4 @ A_4 ) @ zero_zero_real ) ).
thf(fact_856_not__square__less__zero,axiom,
! [A_4: int] :
~ ( ord_less_int @ ( times_times_int @ A_4 @ A_4 ) @ zero_zero_int ) ).
thf(fact_857_pos__add__strict,axiom,
! [B_3: nat,C_1: nat,A_3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_3 )
=> ( ( ord_less_nat @ B_3 @ C_1 )
=> ( ord_less_nat @ B_3 @ ( plus_plus_nat @ A_3 @ C_1 ) ) ) ) ).
thf(fact_858_pos__add__strict,axiom,
! [B_3: int,C_1: int,A_3: int] :
( ( ord_less_int @ zero_zero_int @ A_3 )
=> ( ( ord_less_int @ B_3 @ C_1 )
=> ( ord_less_int @ B_3 @ ( plus_plus_int @ A_3 @ C_1 ) ) ) ) ).
thf(fact_859_divides__ge,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = zero_zero_nat )
| ( ord_less_eq_nat @ A @ B ) ) ) ).
thf(fact_860_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_861_real__mult__less__iff1,axiom,
! [X_1: real,Y_1: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X_1 @ Z ) @ ( times_times_real @ Y_1 @ Z ) )
<=> ( ord_less_real @ X_1 @ Y_1 ) ) ) ).
thf(fact_862_real__mult__le__cancel__iff1,axiom,
! [X_1: real,Y_1: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X_1 @ Z ) @ ( times_times_real @ Y_1 @ Z ) )
<=> ( ord_less_eq_real @ X_1 @ Y_1 ) ) ) ).
thf(fact_863_real__mult__le__cancel__iff2,axiom,
! [X_1: real,Y_1: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X_1 ) @ ( times_times_real @ Z @ Y_1 ) )
<=> ( ord_less_eq_real @ X_1 @ Y_1 ) ) ) ).
thf(fact_864_real__mult__order,axiom,
! [Y_1: real,X_1: real] :
( ( ord_less_real @ zero_zero_real @ X_1 )
=> ( ( ord_less_real @ zero_zero_real @ Y_1 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ X_1 @ Y_1 ) ) ) ) ).
thf(fact_865_real__mult__less__mono2,axiom,
! [X_1: real,Y_1: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ X_1 @ Y_1 )
=> ( ord_less_real @ ( times_times_real @ Z @ X_1 ) @ ( times_times_real @ Z @ Y_1 ) ) ) ) ).
thf(fact_866_real__two__squares__add__zero__iff,axiom,
! [X_1: real,Y_1: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_1 @ X_1 ) @ ( times_times_real @ Y_1 @ Y_1 ) )
= zero_zero_real )
<=> ( ( X_1 = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
thf(fact_867_divides__exp2,axiom,
! [X_1: nat,Y_1: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X_1 @ N ) @ Y_1 )
=> ( dvd_dvd_nat @ X_1 @ Y_1 ) ) ) ).
thf(fact_868_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_869_exp__eq__1,axiom,
! [X_1: nat,N: nat] :
( ( ( power_power_nat @ X_1 @ N )
= one_one_nat )
<=> ( ( X_1 = one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_870_divides__div__not,axiom,
! [X_1: nat,Q: nat,N: nat,R: nat] :
( ( X_1
= ( 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_1 ) ) ) ) ).
thf(fact_871_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_872_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_873_realpow__pos__nth__unique,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
& ( ( power_power_real @ X @ N )
= A )
& ! [Y: real] :
( ( ( ord_less_real @ zero_zero_real @ Y )
& ( ( power_power_real @ Y @ N )
= A ) )
=> ( Y = X ) ) ) ) ) ).
thf(fact_874_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_875_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_876_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_877_dvd_Oorder__refl,axiom,
! [X_1: nat] : ( dvd_dvd_nat @ X_1 @ X_1 ) ).
thf(fact_878_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_879_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_880_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_881_real__le__linear,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
| ( ord_less_eq_real @ W @ Z ) ) ).
thf(fact_882_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_883_real__le__antisym,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_884_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_885_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_886_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_887_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_888_nat__less__cases,axiom,
! [P_1: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P_1 @ N @ M ) )
=> ( ( ( M = N )
=> ( P_1 @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P_1 @ N @ M ) )
=> ( P_1 @ N @ M ) ) ) ) ).
thf(fact_889_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_890_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_891_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_892_linorder__neqE__nat,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 != Y_1 )
=> ( ~ ( ord_less_nat @ X_1 @ Y_1 )
=> ( ord_less_nat @ Y_1 @ X_1 ) ) ) ).
thf(fact_893_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_894_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_895_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_896_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_897_dvd_Oless__asym,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ~ ( ( dvd_dvd_nat @ Y_1 @ X_1 )
& ~ ( dvd_dvd_nat @ X_1 @ Y_1 ) ) ) ).
thf(fact_898_dvd_Oless__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( ( ( dvd_dvd_nat @ Y_1 @ Z )
& ~ ( dvd_dvd_nat @ Z @ Y_1 ) )
=> ( ( dvd_dvd_nat @ X_1 @ Z )
& ~ ( dvd_dvd_nat @ Z @ X_1 ) ) ) ) ).
thf(fact_899_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_900_dvd_Oless__le__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( ( dvd_dvd_nat @ Y_1 @ Z )
=> ( ( dvd_dvd_nat @ X_1 @ Z )
& ~ ( dvd_dvd_nat @ Z @ X_1 ) ) ) ) ).
thf(fact_901_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_902_dvd_Oless__imp__triv,axiom,
! [P_1: $o,X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( ( ( dvd_dvd_nat @ Y_1 @ X_1 )
& ~ ( dvd_dvd_nat @ X_1 @ Y_1 ) )
=> P_1 ) ) ).
thf(fact_903_dvd_Oless__imp__not__eq2,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( Y_1 != X_1 ) ) ).
thf(fact_904_dvd_Oless__imp__not__eq,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( X_1 != Y_1 ) ) ).
thf(fact_905_dvd_Oless__imp__not__less,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ~ ( ( dvd_dvd_nat @ Y_1 @ X_1 )
& ~ ( dvd_dvd_nat @ X_1 @ Y_1 ) ) ) ).
thf(fact_906_dvd_Oless__imp__le,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( dvd_dvd_nat @ X_1 @ Y_1 ) ) ).
thf(fact_907_dvd_Oless__not__sym,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ~ ( ( dvd_dvd_nat @ Y_1 @ X_1 )
& ~ ( dvd_dvd_nat @ X_1 @ Y_1 ) ) ) ).
thf(fact_908_dvd_Oless__imp__neq,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
=> ( X_1 != Y_1 ) ) ).
thf(fact_909_dvd_Ole__less__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( ( dvd_dvd_nat @ X_1 @ Y_1 )
=> ( ( ( dvd_dvd_nat @ Y_1 @ Z )
& ~ ( dvd_dvd_nat @ Z @ Y_1 ) )
=> ( ( dvd_dvd_nat @ X_1 @ Z )
& ~ ( dvd_dvd_nat @ Z @ X_1 ) ) ) ) ).
thf(fact_910_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_911_dvd_Oorder__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( ( dvd_dvd_nat @ X_1 @ Y_1 )
=> ( ( dvd_dvd_nat @ Y_1 @ Z )
=> ( dvd_dvd_nat @ X_1 @ Z ) ) ) ).
thf(fact_912_dvd_Oantisym,axiom,
! [X_1: nat,Y_1: nat] :
( ( dvd_dvd_nat @ X_1 @ Y_1 )
=> ( ( dvd_dvd_nat @ Y_1 @ X_1 )
=> ( X_1 = Y_1 ) ) ) ).
thf(fact_913_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_914_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_915_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_916_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_917_dvd_Ole__imp__less__or__eq,axiom,
! [X_1: nat,Y_1: nat] :
( ( dvd_dvd_nat @ X_1 @ Y_1 )
=> ( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
| ( X_1 = Y_1 ) ) ) ).
thf(fact_918_dvd_Oantisym__conv,axiom,
! [Y_1: nat,X_1: nat] :
( ( dvd_dvd_nat @ Y_1 @ X_1 )
=> ( ( dvd_dvd_nat @ X_1 @ Y_1 )
<=> ( X_1 = Y_1 ) ) ) ).
thf(fact_919_dvd_Oeq__refl,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 = Y_1 )
=> ( dvd_dvd_nat @ X_1 @ Y_1 ) ) ).
thf(fact_920_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_921_dvd_Oless__le__not__le,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
<=> ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) ) ) ).
thf(fact_922_dvd_Oless__le,axiom,
! [X_1: nat,Y_1: nat] :
( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
<=> ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ( X_1 != Y_1 ) ) ) ).
thf(fact_923_dvd_Ole__less,axiom,
! [X_1: nat,Y_1: nat] :
( ( dvd_dvd_nat @ X_1 @ Y_1 )
<=> ( ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ~ ( dvd_dvd_nat @ Y_1 @ X_1 ) )
| ( X_1 = Y_1 ) ) ) ).
thf(fact_924_dvd_Oeq__iff,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 = Y_1 )
<=> ( ( dvd_dvd_nat @ X_1 @ Y_1 )
& ( dvd_dvd_nat @ Y_1 @ X_1 ) ) ) ).
thf(fact_925_dvd_Oless__irrefl,axiom,
! [X_1: nat] :
~ ( ( dvd_dvd_nat @ X_1 @ X_1 )
& ~ ( dvd_dvd_nat @ X_1 @ X_1 ) ) ).
thf(fact_926_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_927_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_928_nat__add__left__commute,axiom,
! [X_1: nat,Y_1: nat,Z: nat] :
( ( plus_plus_nat @ X_1 @ ( plus_plus_nat @ Y_1 @ Z ) )
= ( plus_plus_nat @ Y_1 @ ( plus_plus_nat @ X_1 @ Z ) ) ) ).
thf(fact_929_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_930_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_931_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_932_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_933_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_934_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_935_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_936_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_937_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_938_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_939_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_940_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_941_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_942_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_951_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_952_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_953_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_954_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_955_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_956_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_957_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_958_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_959_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_960_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_961_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_962_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_963_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_964_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_965_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_966_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_967_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_968_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_969_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_970_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_971_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_972_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_973_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_974_not__add__less1,axiom,
! [I_1: nat,J_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ I_1 ) ).
thf(fact_975_not__add__less2,axiom,
! [J_1: nat,I_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ I_1 ) ).
thf(fact_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_986_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_987_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_988_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_989_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_990_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_991_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_992_dvd__reduce,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( plus_plus_nat @ N @ K ) )
<=> ( dvd_dvd_nat @ K @ N ) ) ).
thf(fact_993_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_994_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_995_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_996_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
<=> ( M = one_one_nat ) ) ).
thf(fact_1020_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_1021_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_1022_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_1023_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_1024_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_1025_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_1026_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_1027_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_1028_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_1029_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_1030_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_1031_nat__diff__split,axiom,
! [P_1: nat > $o,A: nat,B: nat] :
( ( P_1 @ ( minus_minus_nat @ A @ B ) )
<=> ( ( ( ord_less_nat @ A @ B )
=> ( P_1 @ zero_zero_nat ) )
& ! [D_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
=> ( P_1 @ D_2 ) ) ) ) ).
thf(fact_1032_nat__diff__split__asm,axiom,
! [P_1: nat > $o,A: nat,B: nat] :
( ( P_1 @ ( minus_minus_nat @ A @ B ) )
<=> ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P_1 @ zero_zero_nat ) )
| ? [D_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
& ~ ( P_1 @ D_2 ) ) ) ) ).
thf(fact_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_ex__least__nat__less,axiom,
! [N: nat,P_1: nat > $o] :
( ~ ( P_1 @ zero_zero_nat )
=> ( ( P_1 @ N )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K_1 )
=> ~ ( P_1 @ I ) )
& ( P_1 @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_1044_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_1045_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_1046_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_1047_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_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__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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_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_is__sum2sq__def,axiom,
! [X_1: int] :
( ( twoSqu919416604sum2sq @ X_1 )
<=> ? [A_2: int,B_2: int] :
( ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ A_2 @ B_2 ) )
= X_1 ) ) ).
thf(fact_1061_imp__le__cong,axiom,
! [P_3: $o,P_1: $o,X_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P_1
<=> P_3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> P_1 )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> P_3 ) ) ) ).
thf(fact_1062_conj__le__cong,axiom,
! [P_3: $o,P_1: $o,X_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P_1
<=> P_3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& P_1 )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& P_3 ) ) ) ).
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_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P_1 @ X )
=> ( P_1 @ ( minus_minus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X: int] :
( ( P_1 @ X )
=> ( P_1 @ ( minus_minus_int @ X @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1066_incr__mult__lemma,axiom,
! [K: int,P_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P_1 @ X )
=> ( P_1 @ ( plus_plus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X: int] :
( ( P_1 @ X )
=> ( P_1 @ ( plus_plus_int @ X @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1067_zprime__factor__exists,axiom,
! [A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ? [P_2: int] :
( ( zprime @ P_2 )
& ( dvd_dvd_int @ P_2 @ A ) ) ) ).
thf(fact_1068_zcong__zless__unique,axiom,
! [A: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ? [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
& ( ord_less_int @ X @ M )
& ( zcong @ A @ X @ M )
& ! [Y: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y )
& ( ord_less_int @ Y @ M )
& ( zcong @ A @ Y @ M ) )
=> ( Y = X ) ) ) ) ).
thf(fact_1069_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_1070_Wilson__Russ,axiom,
! [P: int] :
( ( zprime @ P )
=> ( zcong @ ( zfact @ ( minus_minus_int @ P @ one_one_int ) ) @ ( number_number_of_int @ min ) @ P ) ) ).
thf(fact_1071_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_1072_inv__inv,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P )
=> ( ( inv @ P @ ( inv @ P @ A ) )
= A ) ) ) ) ) ).
thf(fact_1073_MultInvPair__distinct,axiom,
! [J_1: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ~ ( zcong @ A @ zero_zero_int @ P )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P )
=> ( ~ ( quadRes @ P @ A )
=> ~ ( zcong @ J_1 @ ( times_times_int @ A @ ( multInv @ P @ J_1 ) ) @ P ) ) ) ) ) ) ).
thf(fact_1074_aux______3,axiom,
! [J_1: int,K: int,A: int,P: int] :
( ( zcong @ ( times_times_int @ J_1 @ K ) @ A @ P )
=> ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P @ J_1 ) @ J_1 ) @ K ) @ ( times_times_int @ ( multInv @ P @ J_1 ) @ A ) @ P ) ) ).
thf(fact_1075_aux______1,axiom,
! [J_1: int,A: int,P: int,K: int] :
( ( zcong @ J_1 @ ( times_times_int @ A @ ( multInv @ P @ K ) ) @ P )
=> ( zcong @ ( times_times_int @ J_1 @ K ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P @ K ) ) @ K ) @ P ) ) ).
thf(fact_1076_inv__distinct,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( A
!= ( inv @ P @ A ) ) ) ) ) ).
thf(fact_1077_inv__not__1,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( inv @ P @ A )
!= one_one_int ) ) ) ) ).
thf(fact_1078_inv__not__p__minus__1,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( inv @ P @ A )
!= ( minus_minus_int @ P @ one_one_int ) ) ) ) ) ).
thf(fact_1079_inv__g__1,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ord_less_int @ one_one_int @ ( inv @ P @ A ) ) ) ) ) ).
thf(fact_1080_inv__less__p__minus__1,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ord_less_int @ ( inv @ P @ A ) @ ( minus_minus_int @ P @ one_one_int ) ) ) ) ) ).
thf(fact_1081_MultInv__prop1,axiom,
! [X_1: int,Y_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zcong @ X_1 @ Y_1 @ P )
=> ( zcong @ ( multInv @ P @ X_1 ) @ ( multInv @ P @ Y_1 ) @ P ) ) ) ).
thf(fact_1082_inv__not__0,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( inv @ P @ A )
!= zero_zero_int ) ) ) ) ).
thf(fact_1083_MultInv__zcong__prop1,axiom,
! [A: int,J_1: int,K: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zcong @ J_1 @ K @ P )
=> ( zcong @ ( times_times_int @ A @ ( multInv @ P @ J_1 ) ) @ ( times_times_int @ A @ ( multInv @ P @ K ) ) @ P ) ) ) ).
thf(fact_1084_MultInv__prop5,axiom,
! [Y_1: int,X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( ~ ( zcong @ Y_1 @ zero_zero_int @ P )
=> ( ( zcong @ ( multInv @ P @ X_1 ) @ ( multInv @ P @ Y_1 ) @ P )
=> ( zcong @ X_1 @ Y_1 @ P ) ) ) ) ) ) ).
thf(fact_1085_MultInv__prop4,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( zcong @ ( multInv @ P @ ( multInv @ P @ X_1 ) ) @ X_1 @ P ) ) ) ) ).
thf(fact_1086_MultInv__prop3,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ~ ( zcong @ ( multInv @ P @ X_1 ) @ zero_zero_int @ P ) ) ) ) ).
thf(fact_1087_inv__is__inv,axiom,
! [A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P )
=> ( zcong @ ( times_times_int @ A @ ( inv @ P @ A ) ) @ one_one_int @ P ) ) ) ) ).
thf(fact_1088_aux______4,axiom,
! [K: int,A: int,J_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P )
=> ( ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P @ J_1 ) @ J_1 ) @ K ) @ ( times_times_int @ ( multInv @ P @ J_1 ) @ A ) @ P )
=> ( zcong @ K @ ( times_times_int @ A @ ( multInv @ P @ J_1 ) ) @ P ) ) ) ) ) ).
thf(fact_1089_aux______2,axiom,
! [J_1: int,A: int,K: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ K @ zero_zero_int @ P )
=> ( ( zcong @ ( times_times_int @ J_1 @ K ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P @ K ) ) @ K ) @ P )
=> ( zcong @ ( times_times_int @ J_1 @ K ) @ A @ P ) ) ) ) ) ).
thf(fact_1090_MultInv__zcong__prop2,axiom,
! [A: int,J_1: int,K: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ K @ zero_zero_int @ P )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P )
=> ( ( zcong @ J_1 @ ( times_times_int @ A @ ( multInv @ P @ K ) ) @ P )
=> ( zcong @ K @ ( times_times_int @ A @ ( multInv @ P @ J_1 ) ) @ P ) ) ) ) ) ) ).
thf(fact_1091_MultInv__zcong__prop3,axiom,
! [J_1: int,K: int,A: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ A @ zero_zero_int @ P )
=> ( ~ ( zcong @ K @ zero_zero_int @ P )
=> ( ~ ( zcong @ J_1 @ zero_zero_int @ P )
=> ( ( zcong @ ( times_times_int @ A @ ( multInv @ P @ J_1 ) ) @ ( times_times_int @ A @ ( multInv @ P @ K ) ) @ P )
=> ( zcong @ J_1 @ K @ P ) ) ) ) ) ) ) ).
thf(fact_1092_Int2_Oaux____2,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( zcong @ ( times_times_int @ ( times_times_int @ X_1 @ ( multInv @ P @ X_1 ) ) @ ( multInv @ P @ ( multInv @ P @ X_1 ) ) ) @ X_1 @ P ) ) ) ) ).
thf(fact_1093_Int2_Oaux____1,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( zcong @ ( multInv @ P @ ( multInv @ P @ X_1 ) ) @ ( times_times_int @ ( times_times_int @ X_1 @ ( multInv @ P @ X_1 ) ) @ ( multInv @ P @ ( multInv @ P @ X_1 ) ) ) @ P ) ) ) ) ).
thf(fact_1094_MultInv__prop2a,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( zcong @ ( times_times_int @ ( multInv @ P @ X_1 ) @ X_1 ) @ one_one_int @ P ) ) ) ) ).
thf(fact_1095_MultInv__prop2,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P )
=> ( ( zprime @ P )
=> ( ~ ( zcong @ X_1 @ zero_zero_int @ P )
=> ( zcong @ ( times_times_int @ X_1 @ ( multInv @ P @ X_1 ) ) @ one_one_int @ P ) ) ) ) ).
thf(fact_1096_wset__mem__inv__mem,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P ) )
=> ( member_int @ ( inv @ P @ B ) @ ( wset @ A @ P ) ) ) ) ) ) ).
thf(fact_1097_wset__inv__mem__mem,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ B @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( member_int @ ( inv @ P @ B ) @ ( wset @ A @ P ) )
=> ( member_int @ B @ ( wset @ A @ P ) ) ) ) ) ) ) ) ).
thf(fact_1098_wset__mem__mem,axiom,
! [P: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( member_int @ A @ ( wset @ A @ P ) ) ) ).
thf(fact_1099_wset__subset,axiom,
! [B: int,P: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P ) )
=> ( member_int @ B @ ( wset @ A @ P ) ) ) ) ).
thf(fact_1100_wset__less,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P ) )
=> ( ord_less_int @ B @ ( minus_minus_int @ P @ one_one_int ) ) ) ) ) ).
thf(fact_1101_wset__g__1,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P ) )
=> ( ord_less_int @ one_one_int @ B ) ) ) ) ).
thf(fact_1102_wset__mem__imp__or,axiom,
! [B: int,P: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ~ ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P ) )
=> ( ( member_int @ B @ ( wset @ A @ P ) )
=> ( ( B = A )
| ( B
= ( inv @ P @ A ) ) ) ) ) ) ).
thf(fact_1103_wset__mem,axiom,
! [B: int,A: int,P: int] :
( ( zprime @ P )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( wset @ A @ P ) ) ) ) ) ) ).
thf(fact_1104_int__le__induct,axiom,
! [P_1: int > $o,I_1: int,K: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P_1 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P_1 @ I )
=> ( P_1 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P_1 @ I_1 ) ) ) ) ).
thf(fact_1105_int__less__induct,axiom,
! [P_1: int > $o,I_1: int,K: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P_1 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P_1 @ I )
=> ( P_1 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P_1 @ I_1 ) ) ) ) ).
thf(fact_1106_d22set__induct__old,axiom,
! [X_1: int,P_1: int > $o] :
( ! [A_2: int] :
( ( ( ord_less_int @ one_one_int @ A_2 )
=> ( P_1 @ ( minus_minus_int @ A_2 @ one_one_int ) ) )
=> ( P_1 @ A_2 ) )
=> ( P_1 @ X_1 ) ) ).
thf(fact_1107_int__ge__induct,axiom,
! [P_1: int > $o,K: int,I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P_1 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P_1 @ I )
=> ( P_1 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P_1 @ I_1 ) ) ) ) ).
thf(fact_1108_int__gr__induct,axiom,
! [P_1: int > $o,K: int,I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P_1 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P_1 @ I )
=> ( P_1 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P_1 @ I_1 ) ) ) ) ).
thf(fact_1109_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_1110_d22set__eq__wset,axiom,
! [P: int] :
( ( zprime @ P )
=> ( ( d22set @ ( minus_minus_int @ P @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( wset @ ( minus_minus_int @ P @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P ) ) ) ).
thf(fact_1111_ex__least__nat__le,axiom,
! [N: nat,P_1: nat > $o] :
( ~ ( P_1 @ zero_zero_nat )
=> ( ( P_1 @ N )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K_1 )
=> ~ ( P_1 @ I ) )
& ( P_1 @ K_1 ) ) ) ) ).
thf(fact_1112_d22set__le,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
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__g__1,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_int @ one_one_int @ B ) ) ).
thf(fact_1115_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_1116_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_1117_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_1118_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
thf(fact_1119_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
<=> ( A = zero_zero_nat ) ) ).
thf(fact_1120_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_1121_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
thf(fact_1122_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_1123_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_1124_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
thf(fact_1125_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_1126_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
<=> ( A = one_one_nat ) ) ).
thf(fact_1127_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( A = one_one_nat ) ) ).
thf(fact_1128_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_1129_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_1130_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_1131_divides__le,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( ord_less_eq_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1132_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_1133_SR__def,axiom,
! [P: int] :
( ( sr @ P )
= ( collect_int
@ ^ [X: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X ) @ ( ord_less_int @ X @ P ) ) ) ) ).
thf(fact_1134_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_1135_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_1136_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X_1: int,Y_1: int] :
( ( dvd_dvd_int @ ( number_number_of_int @ X_1 ) @ ( number_number_of_int @ Y_1 ) )
<=> ( ( div_mod_int @ ( number_number_of_int @ Y_1 ) @ ( number_number_of_int @ X_1 ) )
= zero_zero_int ) ) ).
thf(fact_1137_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_1138_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_1139_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1: int,X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ X_1 @ Y_1 ) ) ) ) ).
thf(fact_1140_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_1141_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_1142_zmod__eq__dvd__iff,axiom,
! [X_1: int,N: int,Y_1: int] :
( ( ( div_mod_int @ X_1 @ N )
= ( div_mod_int @ Y_1 @ N ) )
<=> ( dvd_dvd_int @ N @ ( minus_minus_int @ X_1 @ Y_1 ) ) ) ).
thf(fact_1143_Residues_Oaux,axiom,
! [X_1: int,M: int,Y_1: int] :
( ( ( div_mod_int @ X_1 @ M )
= ( div_mod_int @ Y_1 @ M ) )
=> ( zcong @ X_1 @ Y_1 @ M ) ) ).
thf(fact_1144_mod__mod__is__mod,axiom,
! [X_1: int,M: int] : ( zcong @ X_1 @ ( div_mod_int @ X_1 @ M ) @ M ) ).
thf(fact_1145_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_1146_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_1147_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_1148_zpower__zmod,axiom,
! [X_1: int,M: int,Y_1: nat] :
( ( div_mod_int @ ( power_power_int @ ( div_mod_int @ X_1 @ M ) @ Y_1 ) @ M )
= ( div_mod_int @ ( power_power_int @ X_1 @ Y_1 ) @ M ) ) ).
thf(fact_1149_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_1150_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_1151_zmod__self,axiom,
! [A: int] :
( ( div_mod_int @ A @ A )
= zero_zero_int ) ).
thf(fact_1152_zmod__zero,axiom,
! [B: int] :
( ( div_mod_int @ zero_zero_int @ B )
= zero_zero_int ) ).
thf(fact_1153_zdiff__zmod__right,axiom,
! [X_1: int,Y_1: int,M: int] :
( ( div_mod_int @ ( minus_minus_int @ X_1 @ ( div_mod_int @ Y_1 @ M ) ) @ M )
= ( div_mod_int @ ( minus_minus_int @ X_1 @ Y_1 ) @ M ) ) ).
thf(fact_1154_zdiff__zmod__left,axiom,
! [X_1: int,M: int,Y_1: int] :
( ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ X_1 @ M ) @ Y_1 ) @ M )
= ( div_mod_int @ ( minus_minus_int @ X_1 @ Y_1 ) @ M ) ) ).
thf(fact_1155_zmod__minus1__right,axiom,
! [A: int] :
( ( div_mod_int @ A @ ( number_number_of_int @ min ) )
= zero_zero_int ) ).
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_split__zmod,axiom,
! [P_1: int > $o,N: int,K: int] :
( ( P_1 @ ( div_mod_int @ N @ K ) )
<=> ( ( ( K = zero_zero_int )
=> ( P_1 @ 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_1 @ 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_1 @ J ) ) ) ) ) ).
thf(fact_1167_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_1168_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_1169_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_1170_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_1171_divmod__int__rel__mod__eq,axiom,
! [A_1: int,B_1: int,Q_2: int,Y_1: int] :
( ( A_1
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_2 ) @ Y_1 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ B_1 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_int @ B_1 @ Y_1 )
& ( ord_less_eq_int @ Y_1 @ zero_zero_int ) ) ) )
=> ( ( B_1 != zero_zero_int )
=> ( ( div_mod_int @ A_1 @ B_1 )
= Y_1 ) ) ) ) ).
thf(fact_1172_neq__one__mod__two,axiom,
! [X_1: int] :
( ( ( div_mod_int @ X_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_int )
<=> ( ( div_mod_int @ X_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ) ).
thf(fact_1173_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_1174_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_1175_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_1176_StandardRes__prop4,axiom,
! [X_1: int,Y_1: int,M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ( zcong @ ( times_times_int @ ( standardRes @ M @ X_1 ) @ ( standardRes @ M @ Y_1 ) ) @ ( times_times_int @ X_1 @ Y_1 ) @ M ) ) ).
thf(fact_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_StandardRes__def,axiom,
! [M: int,X_1: int] :
( ( standardRes @ M @ X_1 )
= ( div_mod_int @ X_1 @ M ) ) ).
thf(fact_1185_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_1186_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_1187_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) ) ).
thf(fact_1188_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ M ) ).
thf(fact_1189_StandardRes__eq__zcong,axiom,
! [M: int,X_1: int] :
( ( ( standardRes @ M @ X_1 )
= zero_zero_int )
<=> ( zcong @ X_1 @ zero_zero_int @ M ) ) ).
thf(fact_1190_StandardRes__prop3,axiom,
! [X_1: int,P: int] :
( ~ ( zcong @ X_1 @ zero_zero_int @ P )
<=> ( ( standardRes @ P @ X_1 )
!= zero_zero_int ) ) ).
thf(fact_1191_StandardRes__prop1,axiom,
! [X_1: int,M: int] : ( zcong @ X_1 @ ( standardRes @ M @ X_1 ) @ M ) ).
thf(fact_1192_StandardRes__ubound,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ zero_zero_int @ P )
=> ( ord_less_int @ ( standardRes @ P @ X_1 ) @ P ) ) ).
thf(fact_1193_StandardRes__SR__prop,axiom,
! [X_1: int,P: int] :
( ( member_int @ X_1 @ ( sr @ P ) )
=> ( ( standardRes @ P @ X_1 )
= X_1 ) ) ).
thf(fact_1194_split__mod,axiom,
! [P_1: nat > $o,N: nat,K: nat] :
( ( P_1 @ ( div_mod_nat @ N @ K ) )
<=> ( ( ( K = zero_zero_nat )
=> ( P_1 @ 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_1 @ J ) ) ) ) ) ) ).
thf(fact_1195_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 ) ) ) ) ).
thf(fact_1196_StandardRes__lbound,axiom,
! [X_1: int,P: int] :
( ( ord_less_int @ zero_zero_int @ P )
=> ( ord_less_eq_int @ zero_zero_int @ ( standardRes @ P @ X_1 ) ) ) ).
thf(fact_1197_StandardRes__prop2,axiom,
! [X1: int,X2: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( standardRes @ M @ X1 )
= ( standardRes @ M @ X2 ) )
<=> ( zcong @ X1 @ X2 @ M ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
? [X: int,Y: int] :
( ( plus_plus_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ).
%------------------------------------------------------------------------------