TPTP Problem File: SLH0639^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Number_Theoretic_Transform/0008_Butterfly/prob_00386_019173__14139696_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1370 ( 777 unt; 90 typ; 0 def)
% Number of atoms : 2962 (1586 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 11075 ( 338 ~; 54 |; 171 &;9571 @)
% ( 0 <=>; 941 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 169 ( 169 >; 0 *; 0 +; 0 <<)
% Number of symbols : 87 ( 84 usr; 30 con; 0-6 aty)
% Number of variables : 2626 ( 32 ^;2565 !; 29 ?;2626 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:39:13.465
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
list_F4626807571770296779ring_a: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (84)
thf(sy_c_Butterfly_Obutterfly_001tf__a,type,
butterfly_a: nat > nat > nat > finite_mod_ring_a > finite_mod_ring_a > nat > $o ).
thf(sy_c_Butterfly_Obutterfly_OFNTT_001tf__a,type,
fNTT_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_ONTT__gen_001tf__a,type,
nTT_gen_a: nat > finite_mod_ring_a > nat > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_Ontt__gen_001tf__a,type,
ntt_gen_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > nat > nat > finite_mod_ring_a ).
thf(sy_c_Butterfly_Obutterfly__axioms,type,
butterfly_axioms: nat > nat > $o ).
thf(sy_c_Discrete_Osqrt,type,
sqrt: nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
minus_3609261664126569004ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Field__Omod____ring_Itf__a_J,type,
one_on2109788427901206336ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
plus_p6165643967897163644ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Finite____Field__Omod____ring_Itf__a_J,type,
times_5121417576591743744ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
uminus3100561713750211260ring_a: finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_z7902377541816115708ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Finite____Field__Omod____ring_Itf__a_J,type,
if_Finite_mod_ring_a: $o > finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_List_Olist_ONil_001t__Finite____Field__Omod____ring_Itf__a_J,type,
nil_Fi5353433074977123787ring_a: list_F4626807571770296779ring_a ).
thf(sy_c_List_Onth_001t__Finite____Field__Omod____ring_Itf__a_J,type,
nth_Fi694352073394265932ring_a: list_F4626807571770296779ring_a > nat > finite_mod_ring_a ).
thf(sy_c_NTT_Ontt_001tf__a,type,
ntt_a: nat > nat > nat > finite_mod_ring_a > finite_mod_ring_a > $o ).
thf(sy_c_NTT_Ontt_OINTT_001tf__a,type,
iNTT_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_NTT_Ontt_ONTT_001tf__a,type,
nTT_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
size_s7115545719440041015ring_a: list_F4626807571770296779ring_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Finite____Field__Omod____ring_Itf__a_J,type,
numera7938180240421336042ring_a: num > finite_mod_ring_a ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Finite____Field__Omod____ring_Itf__a_J,type,
power_6826135765519566523ring_a: finite_mod_ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Finite____Field__Omod____ring_Itf__a_J,type,
divide972148758386938611ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_v_N,type,
n: nat ).
thf(sy_v__092_060mu_062,type,
mu: finite_mod_ring_a ).
thf(sy_v__092_060omega_062,type,
omega: finite_mod_ring_a ).
thf(sy_v_fntt1____,type,
fntt1: list_F4626807571770296779ring_a ).
thf(sy_v_fntt2____,type,
fntt2: list_F4626807571770296779ring_a ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l1____,type,
l1: nat ).
thf(sy_v_l2____,type,
l2: nat ).
thf(sy_v_la____,type,
la: nat ).
thf(sy_v_llen____,type,
llen: nat ).
thf(sy_v_n,type,
n2: nat ).
thf(sy_v_numbers1____,type,
numbers1: list_F4626807571770296779ring_a ).
thf(sy_v_numbers2____,type,
numbers2: list_F4626807571770296779ring_a ).
thf(sy_v_numbersa____,type,
numbersa: list_F4626807571770296779ring_a ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_sum2____,type,
sum2: list_F4626807571770296779ring_a ).
% Relevant facts (1270)
thf(fact_0_llen__def,axiom,
( llen
= ( size_s7115545719440041015ring_a @ numbersa ) ) ).
% llen_def
thf(fact_1_n__two__pot,axiom,
( n2
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n ) ) ).
% n_two_pot
thf(fact_2_two__powr__div,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ I @ J ) ) ) ) ).
% two_powr_div
thf(fact_3_ass,axiom,
ord_less_nat @ j @ l1 ).
% ass
thf(fact_4_Suc_Oprems_I2_J,axiom,
ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ la ) ) @ n2 ).
% Suc.prems(2)
thf(fact_5_Suc_Oprems_I1_J,axiom,
( ( size_s7115545719440041015ring_a @ numbersa )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ la ) ) ) ).
% Suc.prems(1)
thf(fact_6__092_060open_062l_A_060_AN_A_092_060Longrightarrow_062_A2_A_094_AN_Adiv_A2_A_094_ASuc_Al_A_K_A2_A_061_A2_A_094_AN_Adiv_A2_A_094_Al_092_060close_062,axiom,
( ( ord_less_nat @ la @ n )
=> ( ( times_times_nat @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ la ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ) ) ).
% \<open>l < N \<Longrightarrow> 2 ^ N div 2 ^ Suc l * 2 = 2 ^ N div 2 ^ l\<close>
thf(fact_7_n__lst2,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n2 ).
% n_lst2
thf(fact_8_div__exp__sub,axiom,
! [L: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) @ n2 )
=> ( ( divide_divide_nat @ n2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ n @ L ) ) ) ) ).
% div_exp_sub
thf(fact_9_that,axiom,
ord_less_nat @ i @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ).
% that
thf(fact_10__C01_C,axiom,
( ( size_s7115545719440041015ring_a @ sum2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% "01"
thf(fact_11_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_12_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_13_power__even__eq,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_14_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_15_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_16_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_17_power4__eq__xxxx,axiom,
! [X: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_18_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_19_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_20_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_21_power2__eq__square,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_5121417576591743744ring_a @ A @ A ) ) ).
% power2_eq_square
thf(fact_22_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_23_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_24_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_25_numbers2__even,axiom,
( ( size_s7115545719440041015ring_a @ numbers2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% numbers2_even
thf(fact_26_times__divide__eq__left,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( divide972148758386938611ring_a @ B @ C ) @ A )
= ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_27_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_28_divide__divide__eq__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ ( divide972148758386938611ring_a @ A @ B ) @ C )
= ( divide972148758386938611ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_29_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_30_divide__divide__eq__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ A @ ( divide972148758386938611ring_a @ B @ C ) )
= ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_31_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_32_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_33_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_34_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_35_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_36_l2__def,axiom,
( l2
= ( size_s7115545719440041015ring_a @ numbers2 ) ) ).
% l2_def
thf(fact_37_times__divide__eq__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( divide972148758386938611ring_a @ B @ C ) )
= ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_38_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_39_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_40_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_41_power__mult__numeral,axiom,
! [A: finite_mod_ring_a,M: num,N: num] :
( ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_6826135765519566523ring_a @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_42_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_43_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y: real] : ( ord_less_real @ Y @ X2 ) ).
% linordered_field_no_lb
thf(fact_44_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_45_diff__divide__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ C )
= ( minus_3609261664126569004ring_a @ ( divide972148758386938611ring_a @ A @ C ) @ ( divide972148758386938611ring_a @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_46_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_47_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_48_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_49_power__Suc2,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ N ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_50_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_51_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_52_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_53_power__Suc,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ N ) )
= ( times_5121417576591743744ring_a @ A @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_Suc
thf(fact_54_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_55_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_56_divide__divide__eq__left_H,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ ( divide972148758386938611ring_a @ A @ B ) @ C )
= ( divide972148758386938611ring_a @ A @ ( times_5121417576591743744ring_a @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_57_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_58_divide__divide__times__eq,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,Z: finite_mod_ring_a,W: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ ( divide972148758386938611ring_a @ X @ Y2 ) @ ( divide972148758386938611ring_a @ Z @ W ) )
= ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ X @ W ) @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_59_divide__divide__times__eq,axiom,
! [X: real,Y2: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_60_times__divide__times__eq,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,Z: finite_mod_ring_a,W: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( divide972148758386938611ring_a @ X @ Y2 ) @ ( divide972148758386938611ring_a @ Z @ W ) )
= ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ X @ Z ) @ ( times_5121417576591743744ring_a @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_61_times__divide__times__eq,axiom,
! [X: real,Y2: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_62_power__commuting__commutes,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= ( times_times_nat @ Y2 @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y2 )
= ( times_times_nat @ Y2 @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_63_power__commuting__commutes,axiom,
! [X: real,Y2: real,N: nat] :
( ( ( times_times_real @ X @ Y2 )
= ( times_times_real @ Y2 @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y2 )
= ( times_times_real @ Y2 @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_64_power__commuting__commutes,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ X ) )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X @ N ) @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ ( power_6826135765519566523ring_a @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_65_power__commuting__commutes,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= ( times_times_int @ Y2 @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y2 )
= ( times_times_int @ Y2 @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_66_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_67_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_68_power__mult__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ N )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_69_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_70_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_71_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_72_power__commutes,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ A )
= ( times_5121417576591743744ring_a @ A @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_commutes
thf(fact_73_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_74_power__divide,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( divide972148758386938611ring_a @ A @ B ) @ N )
= ( divide972148758386938611ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) ) ) ).
% power_divide
thf(fact_75_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_76_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_77_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_78_power__mult,axiom,
! [A: finite_mod_ring_a,M: nat,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( times_times_nat @ M @ N ) )
= ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_79_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_80_power2__commute,axiom,
! [X: real,Y2: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_81_power2__commute,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ ( minus_3609261664126569004ring_a @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_6826135765519566523ring_a @ ( minus_3609261664126569004ring_a @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_82_power2__commute,axiom,
! [X: int,Y2: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y2 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_83_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_84_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_85_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_86_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_87_two__powrs__div,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( times_times_nat @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ J ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) ) ) ) ).
% two_powrs_div
thf(fact_88_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_89_power__odd__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_90_power__odd__eq,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_5121417576591743744ring_a @ A @ ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_91_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_92_numbers1__even,axiom,
( ( size_s7115545719440041015ring_a @ numbers1 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% numbers1_even
thf(fact_93_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_94_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_95_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_96_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_97_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_98_fntt1__length,axiom,
( ( size_s7115545719440041015ring_a @ fntt1 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% fntt1_length
thf(fact_99_l1__def,axiom,
( l1
= ( size_s7115545719440041015ring_a @ numbers1 ) ) ).
% l1_def
thf(fact_100_right__diff__distrib__numeral,axiom,
! [V: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ B ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_101_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_102_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_103_left__diff__distrib__numeral,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,V: num] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ ( numera7938180240421336042ring_a @ V ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ ( numera7938180240421336042ring_a @ V ) ) @ ( times_5121417576591743744ring_a @ B @ ( numera7938180240421336042ring_a @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_104_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_105_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_106_fntt2__length,axiom,
( ( size_s7115545719440041015ring_a @ fntt2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% fntt2_length
thf(fact_107_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_108_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_109_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_110_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_111_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_112_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_113_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_114_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_115_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_116_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_117_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_118_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_119_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_120_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_121_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_122_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_123_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ M ) @ ( numera7938180240421336042ring_a @ N ) )
= ( numera7938180240421336042ring_a @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_124_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_125_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_126_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_127_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ Z ) )
= ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_128_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_129_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_130_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_131_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_132_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_133_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_134_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_135_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_136_mult__numeral__1,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_137_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_138_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_139_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_140_mult__numeral__1__right,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( numera7938180240421336042ring_a @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_141_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_142_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_143_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_144_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_145_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_146_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_147_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_148_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_149_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_150_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_151_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_152_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_153_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_154_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_155_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_156_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_157_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_158_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_159_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_160_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_161_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_162_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_163_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_164_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_165_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_166_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_167_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_168_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_169_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_170_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_171_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_172_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_173_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_174_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_175_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_176_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_177_size__neq__size__imp__neq,axiom,
! [X: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ X )
!= ( size_s7115545719440041015ring_a @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_178_size__neq__size__imp__neq,axiom,
! [X: num,Y2: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_179_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_180_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_181_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_182_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_183_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_184_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_185_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_186_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_187_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_188_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_189_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_190_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_191_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_192_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_193_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_194_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_195_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_196_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_197_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_198_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_199_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_200_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_201_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_202_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_203_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_204_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_205_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_206_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z2: nat] :
( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_207_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_208_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_209_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_210_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_211_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_212_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_213_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_214_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_215_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_216_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_217_eq__diff__iff,axiom,
! [K: nat,M: nat,N: 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 ) ) ) ) ).
% eq_diff_iff
thf(fact_218_le__diff__iff,axiom,
! [K: nat,M: nat,N: 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 ) ) ) ) ).
% le_diff_iff
thf(fact_219_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: 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 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_220_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_221_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_222_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_223_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_224_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_225_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_226_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_227_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_228_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_229_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 ) ) ) ).
% diff_mult_distrib2
thf(fact_230_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 ) ) ) ).
% diff_mult_distrib
thf(fact_231_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_232_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_233_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_234_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_235_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_236_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_237_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_238_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_239_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_240_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_241_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_242_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_243_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_244_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_245_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_246_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_247_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_248_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_249_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_250_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_251_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_252_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_253_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_254_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_255_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_256_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_257_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_258_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_259_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_260_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_261_less__diff__iff,axiom,
! [K: nat,M: nat,N: 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 ) ) ) ) ).
% less_diff_iff
thf(fact_262_diff__less__mono,axiom,
! [A: nat,B: nat,C: 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 ) ) ) ) ).
% diff_less_mono
thf(fact_263_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_264_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_265_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_266_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_267_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_268_butterfly__axioms__def,axiom,
( butterfly_axioms
= ( ^ [N3: nat,N5: nat] :
( N3
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ).
% butterfly_axioms_def
thf(fact_269_butterfly__axioms_Ointro,axiom,
! [N: nat,N6: nat] :
( ( N
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) )
=> ( butterfly_axioms @ N @ N6 ) ) ).
% butterfly_axioms.intro
thf(fact_270_sqrt__unique,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ( ord_less_nat @ N @ ( power_power_nat @ ( suc @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( sqrt @ N )
= M ) ) ) ).
% sqrt_unique
thf(fact_271_power__numeral,axiom,
! [K: num,L: num] :
( ( power_6826135765519566523ring_a @ ( numera7938180240421336042ring_a @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera7938180240421336042ring_a @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_272_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_273_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_274_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_275_p__lst3,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).
% p_lst3
thf(fact_276_butterfly_Odiv__exp__sub,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat,L: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) @ N )
=> ( ( divide_divide_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N6 @ L ) ) ) ) ) ).
% butterfly.div_exp_sub
thf(fact_277_numbers1__fntt,axiom,
( fntt1
= ( nTT_gen_a @ n2 @ omega @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ numbers1 ) ) ).
% numbers1_fntt
thf(fact_278_numbers2__fntt,axiom,
( fntt2
= ( nTT_gen_a @ n2 @ omega @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ numbers2 ) ) ).
% numbers2_fntt
thf(fact_279_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_280_sqrt__inverse__power2,axiom,
! [N: nat] :
( ( sqrt @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= N ) ).
% sqrt_inverse_power2
thf(fact_281_butterfly_Oaxioms_I2_J,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( butterfly_axioms @ N @ N6 ) ) ).
% butterfly.axioms(2)
thf(fact_282_butterfly_ONTT__gen_Ocong,axiom,
nTT_gen_a = nTT_gen_a ).
% butterfly.NTT_gen.cong
thf(fact_283_mono__sqrt_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( sqrt @ M ) @ ( sqrt @ N ) ) ) ).
% mono_sqrt'
thf(fact_284_sqrt__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( sqrt @ N ) @ N ) ).
% sqrt_le
thf(fact_285_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_286_butterfly_On__two__pot,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( N
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ).
% butterfly.n_two_pot
thf(fact_287_le__sqrtI,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ord_less_eq_nat @ X @ ( sqrt @ Y2 ) ) ) ).
% le_sqrtI
thf(fact_288_sqrt__leI,axiom,
! [Y2: nat,X: nat] :
( ! [Z2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ord_less_eq_nat @ Z2 @ X ) )
=> ( ord_less_eq_nat @ ( sqrt @ Y2 ) @ X ) ) ).
% sqrt_leI
thf(fact_289_le__sqrt__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ ( sqrt @ Y2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) ).
% le_sqrt_iff
thf(fact_290_sqrt__le__iff,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sqrt @ Y2 ) @ X )
= ( ! [Z3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
=> ( ord_less_eq_nat @ Z3 @ X ) ) ) ) ).
% sqrt_le_iff
thf(fact_291_sqrt__power2__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( power_power_nat @ ( sqrt @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ).
% sqrt_power2_le
thf(fact_292_Suc__sqrt__power2__gt,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( suc @ ( sqrt @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_sqrt_power2_gt
thf(fact_293_NTT__gen__NTT__full__length,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( nTT_gen_a @ n2 @ omega @ n2 @ Numbers )
= ( nTT_a @ n2 @ omega @ Numbers ) ) ) ).
% NTT_gen_NTT_full_length
thf(fact_294_length__NTT,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( size_s7115545719440041015ring_a @ ( nTT_a @ n2 @ omega @ Numbers ) )
= n2 ) ) ).
% length_NTT
thf(fact_295_Suc_OIH,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ n2 )
=> ( ( fNTT_a @ n2 @ omega @ Numbers )
= ( nTT_gen_a @ n2 @ omega @ ( size_s7115545719440041015ring_a @ Numbers ) @ Numbers ) ) ) ) ).
% Suc.IH
thf(fact_296_fntt1__def,axiom,
( fntt1
= ( fNTT_a @ n2 @ omega @ numbers1 ) ) ).
% fntt1_def
thf(fact_297_fntt2__def,axiom,
( fntt2
= ( fNTT_a @ n2 @ omega @ numbers2 ) ) ).
% fntt2_def
thf(fact_298__C004_C,axiom,
( ( nth_Fi694352073394265932ring_a @ sum2 @ i )
= ( minus_3609261664126569004ring_a @ ( nth_Fi694352073394265932ring_a @ fntt1 @ i ) @ ( times_5121417576591743744ring_a @ ( nth_Fi694352073394265932ring_a @ fntt2 @ i ) @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ llen ) @ i ) ) ) ) ) ).
% "004"
thf(fact_299_butterfly__axioms,axiom,
butterfly_a @ p @ n2 @ k @ omega @ mu @ n ).
% butterfly_axioms
thf(fact_300_fntt1__by__index,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) )
=> ( ( nth_Fi694352073394265932ring_a @ fntt1 @ I )
= ( ntt_gen_a @ n2 @ omega @ numbers1 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ I ) ) ) ).
% fntt1_by_index
thf(fact_301_exp__rule,axiom,
! [C: finite_mod_ring_a,D: finite_mod_ring_a,E: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ C @ D ) @ E )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ C @ E ) @ ( power_6826135765519566523ring_a @ D @ E ) ) ) ).
% exp_rule
thf(fact_302_fntt2__by__index,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) )
=> ( ( nth_Fi694352073394265932ring_a @ fntt2 @ I )
= ( ntt_gen_a @ n2 @ omega @ numbers2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ I ) ) ) ).
% fntt2_by_index
thf(fact_303_length__INTT,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( size_s7115545719440041015ring_a @ ( iNTT_a @ n2 @ mu @ Numbers ) )
= n2 ) ) ).
% length_INTT
thf(fact_304_butterfly_OFNTT_Ocong,axiom,
fNTT_a = fNTT_a ).
% butterfly.FNTT.cong
thf(fact_305_butterfly_Ontt__gen_Ocong,axiom,
ntt_gen_a = ntt_gen_a ).
% butterfly.ntt_gen.cong
thf(fact_306_butterfly_ONTT__gen__NTT__full__length,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat,Numbers: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( ( size_s7115545719440041015ring_a @ Numbers )
= N )
=> ( ( nTT_gen_a @ N @ Omega @ N @ Numbers )
= ( nTT_a @ N @ Omega @ Numbers ) ) ) ) ).
% butterfly.NTT_gen_NTT_full_length
thf(fact_307_ntt__axioms,axiom,
ntt_a @ p @ n2 @ k @ omega @ mu ).
% ntt_axioms
thf(fact_308_mu__properties,axiom,
( ( times_5121417576591743744ring_a @ mu @ omega )
= one_on2109788427901206336ring_a ) ).
% mu_properties
thf(fact_309_omega__properties_I1_J,axiom,
( ( power_6826135765519566523ring_a @ omega @ n2 )
= one_on2109788427901206336ring_a ) ).
% omega_properties(1)
thf(fact_310_FNTT_Osimps_I1_J,axiom,
( ( fNTT_a @ n2 @ omega @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% FNTT.simps(1)
thf(fact_311__C003_C,axiom,
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( nth_Fi694352073394265932ring_a @ fntt2 @ i ) ) @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ llen ) @ i ) ) )
= ( times_5121417576591743744ring_a @ ( nth_Fi694352073394265932ring_a @ fntt2 @ i ) @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ llen ) @ ( plus_plus_nat @ i @ ( divide_divide_nat @ llen @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% "003"
thf(fact_312_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_313_nth__equalityI,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7115545719440041015ring_a @ Xs ) )
=> ( ( nth_Fi694352073394265932ring_a @ Xs @ I3 )
= ( nth_Fi694352073394265932ring_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_314_neg__cong,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ X )
= ( uminus3100561713750211260ring_a @ Y2 ) )
=> ( X = Y2 ) ) ).
% neg_cong
thf(fact_315_omega__properties_I2_J,axiom,
omega != one_on2109788427901206336ring_a ).
% omega_properties(2)
thf(fact_316_mu__properties_H,axiom,
mu != one_on2109788427901206336ring_a ).
% mu_properties'
thf(fact_317_omega__exists,axiom,
? [Omega2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega2 @ n2 )
= one_on2109788427901206336ring_a )
& ( Omega2 != one_on2109788427901206336ring_a )
& ! [M2: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega2 @ M2 )
= one_on2109788427901206336ring_a )
& ( M2 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M2 ) ) ) ).
% omega_exists
thf(fact_318_omega__properties__ex,axiom,
~ ! [Omega2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega2 @ n2 )
= one_on2109788427901206336ring_a )
=> ( ( Omega2 != one_on2109788427901206336ring_a )
=> ~ ! [M2: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega2 @ M2 )
= one_on2109788427901206336ring_a )
& ( M2 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M2 ) ) ) ) ).
% omega_properties_ex
thf(fact_319_omega__properties_I3_J,axiom,
! [M2: nat] :
( ( ( ( power_6826135765519566523ring_a @ omega @ M2 )
= one_on2109788427901206336ring_a )
& ( M2 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M2 ) ) ).
% omega_properties(3)
thf(fact_320_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_321_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_322_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_323_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_324_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_325_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_326_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_327_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_328_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_329_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_330_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_331_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_332_power__one,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ one_on2109788427901206336ring_a @ N )
= one_on2109788427901206336ring_a ) ).
% power_one
thf(fact_333_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_334_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_335_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_336_div__minus__minus,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( uminus3100561713750211260ring_a @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ).
% div_minus_minus
thf(fact_337_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_338_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_339_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_340_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_341_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_342_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_343_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_344_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_345_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_346_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_347_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_348_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_349_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 ) ) ) ).
% mult_cancel1
thf(fact_350_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 ) ) ) ).
% mult_cancel2
thf(fact_351_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_352_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 ) ) ).
% nat_add_left_cancel_less
thf(fact_353_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 ) ) ).
% nat_add_left_cancel_le
thf(fact_354_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_355_n__min1__2,axiom,
( ( n2
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( omega
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ).
% n_min1_2
thf(fact_356_sqrt__zero,axiom,
( ( sqrt @ zero_zero_nat )
= zero_zero_nat ) ).
% sqrt_zero
thf(fact_357_omg__n__2__min1,axiom,
( ( power_6826135765519566523ring_a @ omega @ ( divide_divide_nat @ n2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ).
% omg_n_2_min1
thf(fact_358_sum__squares__eq__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_359_sum__squares__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_360_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_361_diff__numeral__special_I9_J,axiom,
( ( minus_3609261664126569004ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% diff_numeral_special(9)
thf(fact_362_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_363_div__mult__mult1,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_364_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_365_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_366_div__mult__mult2,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_367_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_368_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_369_div__mult__mult1__if,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( C = zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= zero_z7902377541816115708ring_a ) )
& ( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_370_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_371_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_372_mult__divide__mult__cancel__left__if,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( C = zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= zero_z7902377541816115708ring_a ) )
& ( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_373_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_374_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_375_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_376_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_377_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_378_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_379_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_380_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_381_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_382_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_383_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_384_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_385_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_386_divide__self__if,axiom,
! [A: finite_mod_ring_a] :
( ( ( A = zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ A @ A )
= zero_z7902377541816115708ring_a ) )
& ( ( A != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ A @ A )
= one_on2109788427901206336ring_a ) ) ) ).
% divide_self_if
thf(fact_387_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_388_divide__self,axiom,
! [A: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ A @ A )
= one_on2109788427901206336ring_a ) ) ).
% divide_self
thf(fact_389_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_390_one__eq__divide__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( one_on2109788427901206336ring_a
= ( divide972148758386938611ring_a @ A @ B ) )
= ( ( B != zero_z7902377541816115708ring_a )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_391_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_392_divide__eq__1__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( divide972148758386938611ring_a @ A @ B )
= one_on2109788427901206336ring_a )
= ( ( B != zero_z7902377541816115708ring_a )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_393_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_394_distrib__left__numeral,axiom,
! [V: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ B ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_395_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_396_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_397_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_398_distrib__right__numeral,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,V: num] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ ( numera7938180240421336042ring_a @ V ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ ( numera7938180240421336042ring_a @ V ) ) @ ( times_5121417576591743744ring_a @ B @ ( numera7938180240421336042ring_a @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_399_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_400_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_401_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_402_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_403_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_404_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_405_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_406_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_407_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_408_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_409_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_410_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_411_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_412_mult__minus1,axiom,
! [Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ Z )
= ( uminus3100561713750211260ring_a @ Z ) ) ).
% mult_minus1
thf(fact_413_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_414_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_415_mult__minus1__right,axiom,
! [Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ Z @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( uminus3100561713750211260ring_a @ Z ) ) ).
% mult_minus1_right
thf(fact_416_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_417_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_418_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ M ) ) @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ N ) ) )
= ( uminus3100561713750211260ring_a @ ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ M ) @ ( numera7938180240421336042ring_a @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_419_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_420_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(167)
thf(fact_421_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y2: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ V ) ) @ ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) @ Y2 ) )
= ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(167)
thf(fact_422_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(167)
thf(fact_423_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_424_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_425_power__0__Suc,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( suc @ N ) )
= zero_z7902377541816115708ring_a ) ).
% power_0_Suc
thf(fact_426_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_427_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_428_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_429_power__zero__numeral,axiom,
! [K: num] :
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( numeral_numeral_nat @ K ) )
= zero_z7902377541816115708ring_a ) ).
% power_zero_numeral
thf(fact_430_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_431_div__minus1__right,axiom,
! [A: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ A @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( uminus3100561713750211260ring_a @ A ) ) ).
% div_minus1_right
thf(fact_432_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_433_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_434_divide__minus1,axiom,
! [X: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ X @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( uminus3100561713750211260ring_a @ X ) ) ).
% divide_minus1
thf(fact_435_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_436_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_437_power__Suc0__right,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_438_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_439_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_440_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_441_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 ) ) ) ).
% add_gr_0
thf(fact_442_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_443_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_444_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_445_nat__mult__less__cancel__disj,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 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_446_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 ) ) ) ).
% mult_less_cancel2
thf(fact_447_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 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_448_length__0__conv,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_Fi5353433074977123787ring_a ) ) ).
% length_0_conv
thf(fact_449_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_450_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_451_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_452_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_453_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_454_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_455_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_456_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_457_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_458_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_459_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_460_n__min1__gr2,axiom,
( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n2 )
=> ( ( power_6826135765519566523ring_a @ omega @ ( divide_divide_nat @ n2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ).
% n_min1_gr2
thf(fact_461_sqrt__greater__zero__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sqrt @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% sqrt_greater_zero_iff
thf(fact_462_omega__div__exp__min1,axiom,
! [L: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ L ) ) @ n2 )
=> ( ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ omega @ ( divide_divide_nat @ n2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ L ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ).
% omega_div_exp_min1
thf(fact_463_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_464_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_465_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_466_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_467_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_468_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_469_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_470_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_471_nonzero__divide__mult__cancel__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ A @ ( times_5121417576591743744ring_a @ A @ B ) )
= ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_472_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_473_nonzero__divide__mult__cancel__right,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ B @ ( times_5121417576591743744ring_a @ A @ B ) )
= ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_474_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_475_divide__eq__eq__numeral1_I1_J,axiom,
! [B: finite_mod_ring_a,W: num,A: finite_mod_ring_a] :
( ( ( divide972148758386938611ring_a @ B @ ( numera7938180240421336042ring_a @ W ) )
= A )
= ( ( ( ( numera7938180240421336042ring_a @ W )
!= zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ A @ ( numera7938180240421336042ring_a @ W ) ) ) )
& ( ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_476_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_477_eq__divide__eq__numeral1_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,W: num] :
( ( A
= ( divide972148758386938611ring_a @ B @ ( numera7938180240421336042ring_a @ W ) ) )
= ( ( ( ( numera7938180240421336042ring_a @ W )
!= zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ ( numera7938180240421336042ring_a @ W ) )
= B ) )
& ( ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_478_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_479_div__mult__self1,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A @ ( times_5121417576591743744ring_a @ C @ B ) ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_480_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_481_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_482_div__mult__self2,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_483_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_484_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_485_div__mult__self3,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ B ) @ A ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_486_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_487_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_488_div__mult__self4,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ C ) @ A ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_489_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_490_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_491_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_492_add__neg__numeral__special_I8_J,axiom,
( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% add_neg_numeral_special(8)
thf(fact_493_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_494_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_495_add__neg__numeral__special_I7_J,axiom,
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= zero_z7902377541816115708ring_a ) ).
% add_neg_numeral_special(7)
thf(fact_496_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_497_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_498_diff__numeral__special_I12_J,axiom,
( ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= zero_z7902377541816115708ring_a ) ).
% diff_numeral_special(12)
thf(fact_499_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_500_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( numera7938180240421336042ring_a @ N ) )
= ( numera7938180240421336042ring_a @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_501_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_502_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_503_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_504_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ N ) @ one_on2109788427901206336ring_a )
= ( numera7938180240421336042ring_a @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_505_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_506_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_507_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_508_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_509_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_510_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_511_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_512_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_513_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ N ) @ ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ N ) )
= one_on2109788427901206336ring_a ) ).
% minus_one_mult_self
thf(fact_514_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_515_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_516_left__minus__one__mult__self,axiom,
! [N: nat,A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ N ) @ ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_517_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_518_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_519_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_520_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_521_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_522_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_523_power__eq__0__iff,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( ( power_6826135765519566523ring_a @ A @ N )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_524_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_525_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_526_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_527_length__greater__0__conv,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ Xs ) )
= ( Xs != nil_Fi5353433074977123787ring_a ) ) ).
% length_greater_0_conv
thf(fact_528_nat__mult__le__cancel__disj,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 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_529_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 ) ) ) ).
% mult_le_cancel2
thf(fact_530_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_531_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_532_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_533_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_534_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_535_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ M ) @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ N ) ) )
= ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_536_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_537_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_538_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ M ) ) @ ( numera7938180240421336042ring_a @ N ) )
= ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_539_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_540_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_541_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ M ) ) @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ N ) ) )
= ( numera7938180240421336042ring_a @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_542_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_543_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_544_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ V ) ) @ ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) @ Y2 ) )
= ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_545_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_546_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_547_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) @ Y2 ) )
= ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_548_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_549_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y2 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(169)
thf(fact_550_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ V ) ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ Y2 ) )
= ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(169)
thf(fact_551_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y2 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(169)
thf(fact_552_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_553_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_554_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_555_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_556_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_557_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_558_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_559_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_560_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_561_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_562_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_563_divide__eq__eq__numeral1_I2_J,axiom,
! [B: finite_mod_ring_a,W: num,A: finite_mod_ring_a] :
( ( ( divide972148758386938611ring_a @ B @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) )
= A )
= ( ( ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
!= zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ A @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) ) ) )
& ( ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
= zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_564_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_565_eq__divide__eq__numeral1_I2_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,W: num] :
( ( A
= ( divide972148758386938611ring_a @ B @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) ) )
= ( ( ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
!= zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) )
= B ) )
& ( ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
= zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_566_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_567_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_568_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_569_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_570_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_571_one__add__one,axiom,
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_572_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_573_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_574_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_575_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_576_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_577_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_578_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_579_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_580_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_581_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_582_power__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_583_power__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_584_power__increasing__iff,axiom,
! [B: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_585_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_586_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_587_zero__eq__power2,axiom,
! [A: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7902377541816115708ring_a )
= ( A = zero_z7902377541816115708ring_a ) ) ).
% zero_eq_power2
thf(fact_588_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_589_power2__minus,axiom,
! [A: real] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_590_power2__minus,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_6826135765519566523ring_a @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_591_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_592_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_593_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_594_Suc__0__div__numeral_I2_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_595_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_596_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_597_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_598_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_599_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_600_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_601_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_602_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_603_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_604_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_605_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_606_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_607_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_608_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_609_power2__eq__iff__nonneg,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_610_power2__eq__iff__nonneg,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_611_power2__eq__iff__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_612_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_613_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_614_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_615_add__neg__numeral__special_I9_J,axiom,
( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_616_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_617_diff__numeral__special_I11_J,axiom,
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_618_diff__numeral__special_I11_J,axiom,
( ( minus_3609261664126569004ring_a @ one_on2109788427901206336ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_619_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_620_diff__numeral__special_I10_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_621_diff__numeral__special_I10_J,axiom,
( ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ one_on2109788427901206336ring_a )
= ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_622_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_623_sum__power2__eq__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_624_sum__power2__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_625_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_626_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_627_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_6826135765519566523ring_a @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_628_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_629_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_real ) ).
% power_minus1_even
thf(fact_630_power__minus1__even,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_on2109788427901206336ring_a ) ).
% power_minus1_even
thf(fact_631_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_632_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_633_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_634_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= one_on2109788427901206336ring_a ) )
& ( ( N != zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ) ).
% power_0_left
thf(fact_635_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_636_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_637_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_638_power__0,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ).
% power_0
thf(fact_639_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_640_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_641_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_642_add__divide__eq__if__simps_I3_J,axiom,
! [Z: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( Z = zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ Z ) ) @ B )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( times_5121417576591743744ring_a @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_643_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_644_minus__divide__add__eq__iff,axiom,
! [Z: finite_mod_ring_a,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ X @ Z ) ) @ Y2 )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ X ) @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_645_right__inverse__eq,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( ( divide972148758386938611ring_a @ A @ B )
= one_on2109788427901206336ring_a )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_646_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_647_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_648_nonzero__minus__divide__divide,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( uminus3100561713750211260ring_a @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_649_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_650_nonzero__minus__divide__right,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ B ) )
= ( divide972148758386938611ring_a @ A @ ( uminus3100561713750211260ring_a @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_651_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_652_divide__eq__minus__1__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( divide972148758386938611ring_a @ A @ B )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( ( B != zero_z7902377541816115708ring_a )
& ( A
= ( uminus3100561713750211260ring_a @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_653_div__add__self2,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ B )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% div_add_self2
thf(fact_654_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_655_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_656_div__add__self1,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ B @ A ) @ B )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% div_add_self1
thf(fact_657_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_658_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_659_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_660_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_661_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_662_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_663_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_664_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_665_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_666_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_667_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_668_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_669_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_670_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_671_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_672_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_673_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_674_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_675_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_676_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_677_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_678_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_679_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_680_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_681_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_682_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_683_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_684_is__num__normalize_I8_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) )
= ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ B ) @ ( uminus3100561713750211260ring_a @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_685_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_686_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_687_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_688_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_689_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_690_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_691_list_Osize_I3_J,axiom,
( ( size_s7115545719440041015ring_a @ nil_Fi5353433074977123787ring_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_692_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_693_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_694_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( numera7938180240421336042ring_a @ X ) )
= ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ X ) @ one_on2109788427901206336ring_a ) ) ).
% one_plus_numeral_commute
thf(fact_695_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_696_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_697_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_698_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_699_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_700_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_701_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_702_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_703_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ ( plus_plus_nat @ N @ K ) )
= ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_704_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_705_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_706_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_707_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_708_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_709_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_710_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_711_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_712_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_le_numeral
thf(fact_713_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_714_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_715_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_716_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_717_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_718_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_719_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_720_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_721_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_less_numeral
thf(fact_722_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_723_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_724_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_725_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_726_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_727_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_728_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_729_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_730_uminus__numeral__One,axiom,
( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ one ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ).
% uminus_numeral_One
thf(fact_731_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_732_field__le__epsilon,axiom,
! [X: real,Y2: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y2 @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_epsilon
thf(fact_733_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_734_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_735_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_736_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_737_sum__squares__le__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_738_sum__squares__le__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_739_power__minus,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% power_minus
thf(fact_740_power__minus,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ A ) @ N )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ N ) @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_minus
thf(fact_741_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_742_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_743_power__eq__if,axiom,
( power_power_real
= ( ^ [P3: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_744_power__eq__if,axiom,
( power_6826135765519566523ring_a
= ( ^ [P3: finite_mod_ring_a,M6: nat] : ( if_Finite_mod_ring_a @ ( M6 = zero_zero_nat ) @ one_on2109788427901206336ring_a @ ( times_5121417576591743744ring_a @ P3 @ ( power_6826135765519566523ring_a @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_745_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_746_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_747_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_748_sum__squares__gt__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) )
= ( ( X != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_749_sum__squares__gt__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) )
= ( ( X != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_750_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_751_eq__minus__divide__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A
= ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ B @ C ) ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ C )
= ( uminus3100561713750211260ring_a @ B ) ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_752_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_753_minus__divide__eq__eq,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ B @ C ) )
= A )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( ( uminus3100561713750211260ring_a @ B )
= ( times_5121417576591743744ring_a @ A @ C ) ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_754_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_755_nonzero__neg__divide__eq__eq,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ B ) )
= C )
= ( ( uminus3100561713750211260ring_a @ A )
= ( times_5121417576591743744ring_a @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_756_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_757_nonzero__neg__divide__eq__eq2,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( C
= ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ B ) ) )
= ( ( times_5121417576591743744ring_a @ C @ B )
= ( uminus3100561713750211260ring_a @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_758_add__divide__eq__if__simps_I2_J,axiom,
! [Z: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( Z = zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A @ Z ) @ B )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A @ ( times_5121417576591743744ring_a @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_759_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_760_add__divide__eq__if__simps_I1_J,axiom,
! [Z: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( Z = zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ A @ ( divide972148758386938611ring_a @ B @ Z ) )
= A ) )
& ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ A @ ( divide972148758386938611ring_a @ B @ Z ) )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_761_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_762_add__frac__eq,axiom,
! [Y2: finite_mod_ring_a,Z: finite_mod_ring_a,X: finite_mod_ring_a,W: finite_mod_ring_a] :
( ( Y2 != zero_z7902377541816115708ring_a )
=> ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ X @ Y2 ) @ ( divide972148758386938611ring_a @ W @ Z ) )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ X @ Z ) @ ( times_5121417576591743744ring_a @ W @ Y2 ) ) @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_763_add__frac__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_764_add__frac__num,axiom,
! [Y2: finite_mod_ring_a,X: finite_mod_ring_a,Z: finite_mod_ring_a] :
( ( Y2 != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ X @ Y2 ) @ Z )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ X @ ( times_5121417576591743744ring_a @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_765_add__frac__num,axiom,
! [Y2: real,X: real,Z: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_766_add__num__frac,axiom,
! [Y2: finite_mod_ring_a,Z: finite_mod_ring_a,X: finite_mod_ring_a] :
( ( Y2 != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ Z @ ( divide972148758386938611ring_a @ X @ Y2 ) )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ X @ ( times_5121417576591743744ring_a @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_767_add__num__frac,axiom,
! [Y2: real,Z: real,X: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_768_add__divide__eq__iff,axiom,
! [Z: finite_mod_ring_a,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ X @ ( divide972148758386938611ring_a @ Y2 @ Z ) )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_769_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y2 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_770_divide__add__eq__iff,axiom,
! [Z: finite_mod_ring_a,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( Z != zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ X @ Z ) @ Y2 )
= ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ X @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_771_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_772_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_773_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_774_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_775_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_776_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_777_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_778_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_779_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_780_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_781_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_782_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ).
% zero_power
thf(fact_783_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_784_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_785_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_786_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_787_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_788_div__minus__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ A @ ( uminus3100561713750211260ring_a @ B ) )
= ( divide972148758386938611ring_a @ ( uminus3100561713750211260ring_a @ A ) @ B ) ) ).
% div_minus_right
thf(fact_789_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_790_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_791_minus__divide__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ B ) )
= ( divide972148758386938611ring_a @ A @ ( uminus3100561713750211260ring_a @ B ) ) ) ).
% minus_divide_right
thf(fact_792_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_793_minus__divide__divide,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( uminus3100561713750211260ring_a @ B ) )
= ( divide972148758386938611ring_a @ A @ B ) ) ).
% minus_divide_divide
thf(fact_794_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_795_minus__divide__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ B ) )
= ( divide972148758386938611ring_a @ ( uminus3100561713750211260ring_a @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_796_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_797_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_798_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_799_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_800_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_801_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_802_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_803_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_804_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_805_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_806_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_807_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_808_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_809_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_810_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_811_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_812_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_813_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_814_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 ) ) ).
% add_leD2
thf(fact_815_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 ) ) ).
% add_leD1
thf(fact_816_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_817_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_818_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 ) ) ) ).
% add_leE
thf(fact_819_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_820_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_821_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_822_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_823_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_824_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 ) ) ).
% diff_cancel2
thf(fact_825_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 ) ) ).
% Nat.diff_cancel
thf(fact_826_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_827_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_828_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_829_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_830_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 ) ) ) ).
% add_mult_distrib
thf(fact_831_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 ) ) ) ).
% add_mult_distrib2
thf(fact_832_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_833_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_834_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_835_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_836_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_837_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_838_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_839_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_840_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_841_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_842_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_843_power__not__zero,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( power_6826135765519566523ring_a @ A @ N )
!= zero_z7902377541816115708ring_a ) ) ).
% power_not_zero
thf(fact_844_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_845_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_846_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_847_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_848_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_849_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_850_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_851_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X3: nat,Y: nat] :
( ( P @ X3 @ Y )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_852_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_853_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_854_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_855_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_856_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_857_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_858_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_859_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_860_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_861_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_862_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_863_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_864_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_865_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_866_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_867_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_868_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_869_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 ) ) ) ).
% diffs0_imp_equal
thf(fact_870_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_871_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 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_872_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_873_butterfly_Oneg__cong,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( ( uminus3100561713750211260ring_a @ X )
= ( uminus3100561713750211260ring_a @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% butterfly.neg_cong
thf(fact_874_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_875_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_876_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_877_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_878_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_879_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_880_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_881_divide__eq__eq__numeral_I2_J,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,W: num] :
( ( ( divide972148758386938611ring_a @ B @ C )
= ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) @ C ) ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_882_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_883_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
= ( divide972148758386938611ring_a @ B @ C ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) @ C )
= B ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_884_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_885_add__divide__eq__if__simps_I6_J,axiom,
! [Z: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( Z = zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ Z ) ) @ B )
= ( uminus3100561713750211260ring_a @ B ) ) )
& ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ A @ Z ) ) @ B )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( times_5121417576591743744ring_a @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_886_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_887_add__divide__eq__if__simps_I5_J,axiom,
! [Z: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( Z = zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( divide972148758386938611ring_a @ A @ Z ) @ B )
= ( uminus3100561713750211260ring_a @ B ) ) )
& ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( divide972148758386938611ring_a @ A @ Z ) @ B )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ A @ ( times_5121417576591743744ring_a @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_888_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y2 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_889_minus__divide__diff__eq__iff,axiom,
! [Z: finite_mod_ring_a,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ ( divide972148758386938611ring_a @ X @ Z ) ) @ Y2 )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ ( uminus3100561713750211260ring_a @ X ) @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_890_field__le__mult__one__interval,axiom,
! [X: real,Y2: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_891_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_892_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_893_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_894_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_895_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_896_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_897_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_898_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_899_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_900_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_901_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_902_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_903_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_904_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_905_power__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_906_power__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_907_power__decreasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_908_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_909_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_910_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_911_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_912_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_913_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_914_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_915_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_916_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_917_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_918_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_919_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_920_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_921_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_922_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_923_butterfly_Oaxioms_I1_J,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ntt_a @ P2 @ N @ K @ Omega @ Mu ) ) ).
% butterfly.axioms(1)
thf(fact_924_le__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_925_minus__divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_926_neg__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_927_neg__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_928_pos__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_929_pos__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_930_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_931_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_932_power2__eq__1__iff,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( A = one_one_real )
| ( A
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% power2_eq_1_iff
thf(fact_933_power2__eq__1__iff,axiom,
! [A: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on2109788427901206336ring_a )
= ( ( A = one_on2109788427901206336ring_a )
| ( A
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ) ).
% power2_eq_1_iff
thf(fact_934_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_935_scaling__mono,axiom,
! [U: real,V: real,R2: real,S: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_936_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_937_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_938_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_939_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_940_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_941_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_942_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_943_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_944_numeral__times__minus__swap,axiom,
! [W: num,X: real] :
( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
= ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_945_numeral__times__minus__swap,axiom,
! [W: num,X: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ ( uminus3100561713750211260ring_a @ X ) )
= ( times_5121417576591743744ring_a @ X @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_946_numeral__times__minus__swap,axiom,
! [W: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_947_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_948_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_949_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_950_power__diff__power__eq,axiom,
! [A: finite_mod_ring_a,N: nat,M: nat] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide972148758386938611ring_a @ ( power_6826135765519566523ring_a @ A @ M ) @ ( power_6826135765519566523ring_a @ A @ N ) )
= ( power_6826135765519566523ring_a @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide972148758386938611ring_a @ ( power_6826135765519566523ring_a @ A @ M ) @ ( power_6826135765519566523ring_a @ A @ N ) )
= ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ ( power_6826135765519566523ring_a @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_951_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_952_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_953_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_954_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_955_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_956_butterfly_On__min1__2,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( N
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( Omega
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ) ).
% butterfly.n_min1_2
thf(fact_957_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_958_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_959_power__add,axiom,
! [A: finite_mod_ring_a,M: nat,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ M ) @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_add
thf(fact_960_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_961_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_962_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_963_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_964_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_965_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_966_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_967_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_968_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_969_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_970_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_971_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_972_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_973_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_974_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_975_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_976_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_977_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_978_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_979_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_980_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_981_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_982_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_983_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_984_numeral__One,axiom,
( ( numera7938180240421336042ring_a @ one )
= one_on2109788427901206336ring_a ) ).
% numeral_One
thf(fact_985_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_986_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_987_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_988_left__right__inverse__power,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_989_left__right__inverse__power,axiom,
! [X: real,Y2: real,N: nat] :
( ( ( times_times_real @ X @ Y2 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_990_left__right__inverse__power,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X @ Y2 )
= one_on2109788427901206336ring_a )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X @ N ) @ ( power_6826135765519566523ring_a @ Y2 @ N ) )
= one_on2109788427901206336ring_a ) ) ).
% left_right_inverse_power
thf(fact_991_left__right__inverse__power,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_992_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_993_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_994_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_995_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_996_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_997_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_998_power__one__over,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ A ) @ N )
= ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_one_over
thf(fact_999_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_1000_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1001_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1002_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_1003_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1004_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1005_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_1006_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_1007_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_1008_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1009_divide__nonneg__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1010_divide__nonneg__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1011_divide__nonpos__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1012_divide__nonpos__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1013_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1014_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1015_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1016_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1017_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1018_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1019_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1020_divide__neg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_1021_divide__neg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_1022_divide__pos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_1023_divide__pos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_1024_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_1025_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_1026_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1027_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1028_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1029_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1030_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1031_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1032_frac__eq__eq,axiom,
! [Y2: finite_mod_ring_a,Z: finite_mod_ring_a,X: finite_mod_ring_a,W: finite_mod_ring_a] :
( ( Y2 != zero_z7902377541816115708ring_a )
=> ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( ( divide972148758386938611ring_a @ X @ Y2 )
= ( divide972148758386938611ring_a @ W @ Z ) )
= ( ( times_5121417576591743744ring_a @ X @ Z )
= ( times_5121417576591743744ring_a @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1033_frac__eq__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y2 )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1034_divide__eq__eq,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( divide972148758386938611ring_a @ B @ C )
= A )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ A @ C ) ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq
thf(fact_1035_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_1036_eq__divide__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A
= ( divide972148758386938611ring_a @ B @ C ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ C )
= B ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( A = zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq
thf(fact_1037_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_1038_divide__eq__imp,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( B
= ( times_5121417576591743744ring_a @ A @ C ) )
=> ( ( divide972148758386938611ring_a @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1039_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1040_eq__divide__imp,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ A @ C )
= B )
=> ( A
= ( divide972148758386938611ring_a @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1041_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1042_nonzero__divide__eq__eq,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( divide972148758386938611ring_a @ B @ C )
= A )
= ( B
= ( times_5121417576591743744ring_a @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1043_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1044_nonzero__eq__divide__eq,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( A
= ( divide972148758386938611ring_a @ B @ C ) )
= ( ( times_5121417576591743744ring_a @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1045_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1046_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1047_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1048_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1049_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1050_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1051_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1052_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1053_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1054_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: 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 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1055_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1056_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1057_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1058_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1059_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1060_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1061_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1062_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_1063_square__le__1,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% square_le_1
thf(fact_1064_sum__power2__ge__zero,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1065_sum__power2__ge__zero,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1066_sum__power2__le__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1067_sum__power2__le__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1068_not__sum__power2__lt__zero,axiom,
! [X: real,Y2: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_1069_not__sum__power2__lt__zero,axiom,
! [X: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_1070_sum__power2__gt__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1071_sum__power2__gt__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1072_butterfly_OFNTT_Osimps_I1_J,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( fNTT_a @ N @ Omega @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ) ).
% butterfly.FNTT.simps(1)
thf(fact_1073_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% power_minus1_odd
thf(fact_1074_power__minus1__odd,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ).
% power_minus1_odd
thf(fact_1075_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_1076_mult__1s__ring__1_I2_J,axiom,
! [B: real] :
( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_1077_mult__1s__ring__1_I2_J,axiom,
! [B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ B @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ one ) ) )
= ( uminus3100561713750211260ring_a @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_1078_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_1079_mult__1s__ring__1_I1_J,axiom,
! [B: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_1080_mult__1s__ring__1_I1_J,axiom,
! [B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ ( numera7938180240421336042ring_a @ one ) ) @ B )
= ( uminus3100561713750211260ring_a @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_1081_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_1082_power__minus__Bit0,axiom,
! [X: real,K: num] :
( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_1083_power__minus__Bit0,axiom,
! [X: finite_mod_ring_a,K: num] :
( ( power_6826135765519566523ring_a @ ( uminus3100561713750211260ring_a @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_6826135765519566523ring_a @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_1084_power__minus__Bit0,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_1085_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1086_butterfly_Oomg__n__2__min1,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( power_6826135765519566523ring_a @ Omega @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ).
% butterfly.omg_n_2_min1
thf(fact_1087_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1088_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1089_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1090_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1091_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1092_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1093_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1094_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1095_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1096_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1097_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1098_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1099_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1100_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1101_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1102_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1103_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1104_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1105_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1106_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1107_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1108_power__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1109_power__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1110_power__increasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1111_frac__le,axiom,
! [Y2: real,X: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_1112_frac__less,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_1113_frac__less2,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_1114_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_1115_divide__nonneg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_1116_divide__nonneg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1117_divide__nonpos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1118_divide__nonpos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_1119_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1120_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1121_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1122_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1123_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1124_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_1125_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1126_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1127_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_1128_mult__imp__div__pos__less,axiom,
! [Y2: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1129_mult__imp__less__div__pos,axiom,
! [Y2: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y2 ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1130_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1131_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1132_divide__eq__eq__numeral_I1_J,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,W: num] :
( ( ( divide972148758386938611ring_a @ B @ C )
= ( numera7938180240421336042ring_a @ W ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ C ) ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1133_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1134_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( numera7938180240421336042ring_a @ W )
= ( divide972148758386938611ring_a @ B @ C ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ C )
= B ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1135_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1136_add__divide__eq__if__simps_I4_J,axiom,
! [Z: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( Z = zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ A @ ( divide972148758386938611ring_a @ B @ Z ) )
= A ) )
& ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ A @ ( divide972148758386938611ring_a @ B @ Z ) )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1137_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1138_diff__frac__eq,axiom,
! [Y2: finite_mod_ring_a,Z: finite_mod_ring_a,X: finite_mod_ring_a,W: finite_mod_ring_a] :
( ( Y2 != zero_z7902377541816115708ring_a )
=> ( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( divide972148758386938611ring_a @ X @ Y2 ) @ ( divide972148758386938611ring_a @ W @ Z ) )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ X @ Z ) @ ( times_5121417576591743744ring_a @ W @ Y2 ) ) @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1139_diff__frac__eq,axiom,
! [Y2: real,Z: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1140_diff__divide__eq__iff,axiom,
! [Z: finite_mod_ring_a,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ X @ ( divide972148758386938611ring_a @ Y2 @ Z ) )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_1141_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y2 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y2 ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_1142_divide__diff__eq__iff,axiom,
! [Z: finite_mod_ring_a,X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( Z != zero_z7902377541816115708ring_a )
=> ( ( minus_3609261664126569004ring_a @ ( divide972148758386938611ring_a @ X @ Z ) @ Y2 )
= ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ X @ ( times_5121417576591743744ring_a @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_1143_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y2 )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_1144_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1145_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1146_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1147_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1148_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1149_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1150_neq__if__length__neq,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
!= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1151_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1152_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1153_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1154_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1155_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1156_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1157_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1158_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: 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 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1159_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1160_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1161_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1162_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1163_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1164_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1165_butterfly_On__min1__gr2,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 )
=> ( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_6826135765519566523ring_a @ Omega @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ) ) ).
% butterfly.n_min1_gr2
thf(fact_1166_arith__geo__mean,axiom,
! [U: real,X: real,Y2: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1167_mult__2,axiom,
! [Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) @ Z )
= ( plus_p6165643967897163644ring_a @ Z @ Z ) ) ).
% mult_2
thf(fact_1168_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_1169_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_1170_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_1171_mult__2__right,axiom,
! [Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ Z @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) )
= ( plus_p6165643967897163644ring_a @ Z @ Z ) ) ).
% mult_2_right
thf(fact_1172_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_1173_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_1174_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_1175_left__add__twice,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ A @ B ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1176_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1177_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1178_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1179_power2__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus_real @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_1180_power2__eq__iff,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_6826135765519566523ring_a @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus3100561713750211260ring_a @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_1181_power2__eq__iff,axiom,
! [X: int,Y2: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus_int @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_1182_butterfly__def,axiom,
( butterfly_a
= ( ^ [P3: nat,N3: nat,K3: nat,Omega3: finite_mod_ring_a,Mu2: finite_mod_ring_a,N5: nat] :
( ( ntt_a @ P3 @ N3 @ K3 @ Omega3 @ Mu2 )
& ( butterfly_axioms @ N3 @ N5 ) ) ) ) ).
% butterfly_def
thf(fact_1183_butterfly_Ointro,axiom,
! [P2: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N6: nat] :
( ( ntt_a @ P2 @ N @ K @ Omega @ Mu )
=> ( ( butterfly_axioms @ N @ N6 )
=> ( butterfly_a @ P2 @ N @ K @ Omega @ Mu @ N6 ) ) ) ).
% butterfly.intro
thf(fact_1184_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1185_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1186_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1187_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1188_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1189_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1190_div__if,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M6 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_1191_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1192_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1193_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1194_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1195_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_1196_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_1197_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_1198_p__fact,axiom,
( p
= ( plus_plus_nat @ ( times_times_nat @ k @ n2 ) @ one_one_nat ) ) ).
% p_fact
thf(fact_1199_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1200_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1201_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1202_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1203_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 ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1204_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 ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1205_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1206_sqrt__one,axiom,
( ( sqrt @ one_one_nat )
= one_one_nat ) ).
% sqrt_one
thf(fact_1207_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1208_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1209_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1210_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1211_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1212_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1213_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1214_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1215_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_1216_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1217_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1218_int__div__pos__eq,axiom,
! [A: int,B: int,Q: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_pos_eq
thf(fact_1219_int__div__neg__eq,axiom,
! [A: int,B: int,Q: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_neg_eq
thf(fact_1220_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1221_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1222_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1223_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1224_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1225_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1226_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1227_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1228_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1229_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1230_zdiv__mono1,axiom,
! [A: int,A3: int,B: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1231_zdiv__mono2,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1232_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1233_zdiv__mono1__neg,axiom,
! [A: int,A3: int,B: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1234_zdiv__mono2__neg,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1235_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1236_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1237_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1238_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1239_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1240_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1241_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1242_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1243_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1244_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1245_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1246_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_1247_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_1248_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1249_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1250_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1251_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1252_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1253_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1254_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1255_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1256_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1257_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1258_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1259_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N )
= A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1260_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1261_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1262_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1263_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1264_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1265_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1266_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1267_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1268_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1269_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_nat @ ( divide_divide_nat @ n2 @ llen ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ j ) )
= ( times_times_nat @ ( divide_divide_nat @ n2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) @ j ) ) ).
%------------------------------------------------------------------------------