TPTP Problem File: SLH0329^1.p
View Solutions
- 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 : Digit_Expansions/0000_Bits_Digits/prob_00101_003959__5485044_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1314 ( 600 unt; 44 typ; 0 def)
% Number of atoms : 3318 (1343 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 11124 ( 303 ~; 88 |; 179 &;9391 @)
% ( 0 <=>;1163 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 140 ( 140 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 41 usr; 8 con; 0-3 aty)
% Number of variables : 2990 ( 38 ^;2836 !; 116 ?;2990 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:20:15.406
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
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 (41)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Bits__Digits_Onth__bit,type,
bits_nth_bit: nat > nat > nat ).
thf(sy_c_Bits__Digits_Onth__digit,type,
bits_nth_digit: nat > nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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_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_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_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_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_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
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__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_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__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_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_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_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_v_a,type,
a: nat ).
% Relevant facts (1264)
thf(fact_0_a__geq__0,axiom,
ord_less_nat @ zero_zero_nat @ a ).
% a_geq_0
thf(fact_1_even,axiom,
( ( modulo_modulo_nat @ a @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% even
thf(fact_2_aux__even__pow2__factor,axiom,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ? [K: nat,B: nat] :
( ( A
= ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ B ) )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% aux_even_pow2_factor
thf(fact_3_even__mult__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_mult_iff
thf(fact_4_even__mult__iff,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_mult_iff
thf(fact_5_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_6_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_7_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_8_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_9_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_10_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_11_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_12_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_13_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% evenE
thf(fact_14_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% evenE
thf(fact_15_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_16_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_17_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_18_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_19_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_20_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_21_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_22_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_23_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_24_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_25_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_26_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_27_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_28_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_29_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_30_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_31_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_32_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_33_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_34_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_35_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_36_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_37_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_38_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
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: 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_41_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_42_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_43_even__mod__2__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_44_even__mod__2__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_45_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_46_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_47_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_48_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_49_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_50_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_51_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_52_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_53_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_54_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_55_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_56_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_57_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_58_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_59_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_60_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_61_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_62_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_63_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_64_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_65_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_66_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_67_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_68_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_69_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_70_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_71_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_72_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_73_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_74_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_75_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_76_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_77_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_78_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_79_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_80_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_81_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_82_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_83_cong__exp__iff__simps_I6_J,axiom,
! [Q: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_84_cong__exp__iff__simps_I6_J,axiom,
! [Q: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_85_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_86_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_87_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_88_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_89_power__mult__distrib,axiom,
! [A: nat,B2: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% power_mult_distrib
thf(fact_90_power__mult__distrib,axiom,
! [A: int,B2: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).
% power_mult_distrib
thf(fact_91_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_92_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_93_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_94_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_95_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_96_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_97_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_98_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_99_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_100_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_101_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_102_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_103_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_104_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% pow_divides_pow_iff
thf(fact_105_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
= ( dvd_dvd_int @ A @ B2 ) ) ) ).
% pow_divides_pow_iff
thf(fact_106_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_107_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_108_dvd__imp__mod__0,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( modulo_modulo_nat @ B2 @ A )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_109_dvd__imp__mod__0,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( modulo_modulo_int @ B2 @ A )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_110_mod__mult__self2__is__0,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_111_mod__mult__self2__is__0,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_112_mod__mult__self1__is__0,axiom,
! [B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_113_mod__mult__self1__is__0,axiom,
! [B2: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_114_dvd__times__right__cancel__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_115_dvd__times__right__cancel__iff,axiom,
! [A: int,B2: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_116_dvd__times__left__cancel__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_117_dvd__times__left__cancel__iff,axiom,
! [A: int,B2: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_118_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_119_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_120__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062a_Amod_A2_A_061_A1_A_092_060Longrightarrow_062_Athesis_059_Aa_Amod_A2_A_061_A0_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ( ( modulo_modulo_nat @ a @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ a @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% \<open>\<And>thesis. \<lbrakk>a mod 2 = 1 \<Longrightarrow> thesis; a mod 2 = 0 \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_121_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_122_mod__mod__trivial,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mod_trivial
thf(fact_123_mod__mod__trivial,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mod_trivial
thf(fact_124_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_125_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_126_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_127_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_128_mult__eq__0__iff,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_129_mult__eq__0__iff,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_130_mult__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_131_mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_132_mult__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_133_mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_134_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_135_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_136_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_137_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_138_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_139_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_140_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_141_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_142_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_143_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_144_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_145_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_146_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_147_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_148_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_149_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_150_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_151_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_152_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_153_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_154_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_155_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_156_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_157_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_158_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_159_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_160_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_161_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_162_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_163_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_164_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_165_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_166_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_167_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_168_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_169_unit__prod,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_170_unit__prod,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_171_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_172_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_173_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_174_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_175_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_176_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_177_power__strict__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_178_power__strict__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_179_power__strict__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_180_power__strict__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_181_one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_mod_two_eq_one
thf(fact_182_one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_mod_two_eq_one
thf(fact_183_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_184_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_185_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_186_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_187_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_188_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_189_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_190_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_191_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_192_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_193_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_194_unit__imp__dvd,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ B2 @ A ) ) ).
% unit_imp_dvd
thf(fact_195_unit__imp__dvd,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ B2 @ A ) ) ).
% unit_imp_dvd
thf(fact_196_dvd__unit__imp__unit,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_197_dvd__unit__imp__unit,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_198_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_199_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_200_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_201_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_202_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_203_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_204_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_205_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_206_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_207_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_208_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_209_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_210_is__unit__mult__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_211_is__unit__mult__iff,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_212_dvd__mult__unit__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_213_dvd__mult__unit__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_214_mult__unit__dvd__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_215_mult__unit__dvd__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_216_dvd__mult__unit__iff_H,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_217_dvd__mult__unit__iff_H,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_218_mult__unit__dvd__iff_H,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_219_mult__unit__dvd__iff_H,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_220_unit__mult__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B2 )
= ( times_times_nat @ A @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_221_unit__mult__left__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B2 )
= ( times_times_int @ A @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_222_unit__mult__right__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B2 @ A )
= ( times_times_nat @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_223_unit__mult__right__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B2 @ A )
= ( times_times_int @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_224_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_225_unit__dvdE,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] :
( B2
!= ( times_times_nat @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_226_unit__dvdE,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] :
( B2
!= ( times_times_int @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_227_unit__imp__mod__eq__0,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B2 )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_228_unit__imp__mod__eq__0,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B2 )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_229_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_230_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_231_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_232_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_233_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_234_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_235_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_236_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_237_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_238_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_239_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_240_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_241_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_242_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_252_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_253_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_254_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_255_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_256_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_257_dvd__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_258_dvd__trans,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ B2 @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_259_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_260_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_261_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_262_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_263_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_264_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_265_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_266_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_267_gcd__nat_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ~ ( ( dvd_dvd_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% gcd_nat.asym
thf(fact_268_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_269_gcd__nat_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_270_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A2: nat,B3: nat] :
( ( dvd_dvd_nat @ A2 @ B3 )
& ( dvd_dvd_nat @ B3 @ A2 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_271_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_272_gcd__nat_Oantisym,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_273_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_274_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_275_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_276_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
= ( ( dvd_dvd_nat @ A @ B2 )
& ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_277_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A2: nat,B3: nat] :
( ( ( dvd_dvd_nat @ A2 @ B3 )
& ( A2 != B3 ) )
| ( A2 = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_278_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
= ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_279_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( dvd_dvd_nat @ A @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_280_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( A != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_281_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_282_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_283_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_284_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_285_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_286_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_287_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_288_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_289_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_290_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_291_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_292_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_293_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_294_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_295_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_296_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_297_mod2__eq__if,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_298_mod2__eq__if,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_299_parity__cases,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_300_parity__cases,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_301_mult__not__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_302_mult__not__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_303_divisors__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_304_divisors__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_305_no__zero__divisors,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_306_no__zero__divisors,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_307_mult__left__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_308_mult__left__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_309_mult__right__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_310_mult__right__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_311_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_312_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_313_dvdE,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ~ ! [K: nat] :
( A
!= ( times_times_nat @ B2 @ K ) ) ) ).
% dvdE
thf(fact_314_dvdE,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ~ ! [K: int] :
( A
!= ( times_times_int @ B2 @ K ) ) ) ).
% dvdE
thf(fact_315_dvdI,axiom,
! [A: nat,B2: nat,K2: nat] :
( ( A
= ( times_times_nat @ B2 @ K2 ) )
=> ( dvd_dvd_nat @ B2 @ A ) ) ).
% dvdI
thf(fact_316_dvdI,axiom,
! [A: int,B2: int,K2: int] :
( ( A
= ( times_times_int @ B2 @ K2 ) )
=> ( dvd_dvd_int @ B2 @ A ) ) ).
% dvdI
thf(fact_317_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A2: nat] :
? [K3: nat] :
( A2
= ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_318_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A2: int] :
? [K3: int] :
( A2
= ( times_times_int @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_319_dvd__productE,axiom,
! [P2: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B2 ) )
=> ~ ! [X2: nat,Y3: nat] :
( ( P2
= ( times_times_nat @ X2 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X2 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_320_dvd__productE,axiom,
! [P2: int,A: int,B2: int] :
( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B2 ) )
=> ~ ! [X2: int,Y3: int] :
( ( P2
= ( times_times_int @ X2 @ Y3 ) )
=> ( ( dvd_dvd_int @ X2 @ A )
=> ~ ( dvd_dvd_int @ Y3 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_321_dvd__mult,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_322_dvd__mult,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_323_division__decomp,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
=> ? [B4: nat,C3: nat] :
( ( A
= ( times_times_nat @ B4 @ C3 ) )
& ( dvd_dvd_nat @ B4 @ B2 )
& ( dvd_dvd_nat @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_324_division__decomp,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
=> ? [B4: int,C3: int] :
( ( A
= ( times_times_int @ B4 @ C3 ) )
& ( dvd_dvd_int @ B4 @ B2 )
& ( dvd_dvd_int @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_325_dvd__mult2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_326_dvd__mult2,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_327_dvd__mult__left,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_328_dvd__mult__left,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_329_dvd__triv__left,axiom,
! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).
% dvd_triv_left
thf(fact_330_dvd__triv__left,axiom,
! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).
% dvd_triv_left
thf(fact_331_mult__dvd__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_332_mult__dvd__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_333_dvd__mult__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
=> ( dvd_dvd_nat @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_334_dvd__mult__right,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
=> ( dvd_dvd_int @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_335_dvd__triv__right,axiom,
! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).
% dvd_triv_right
thf(fact_336_dvd__triv__right,axiom,
! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).
% dvd_triv_right
thf(fact_337_mod__mult__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_eq
thf(fact_338_mod__mult__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% mod_mult_eq
thf(fact_339_mod__mult__cong,axiom,
! [A: nat,C: nat,A3: nat,B2: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A3 @ C ) )
=> ( ( ( modulo_modulo_nat @ B2 @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_340_mod__mult__cong,axiom,
! [A: int,C: int,A3: int,B2: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A3 @ C ) )
=> ( ( ( modulo_modulo_int @ B2 @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_341_mod__mult__mult2,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_342_mod__mult__mult2,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_343_mult__mod__right,axiom,
! [C: nat,A: nat,B2: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).
% mult_mod_right
thf(fact_344_mult__mod__right,axiom,
! [C: int,A: int,B2: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).
% mult_mod_right
thf(fact_345_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_346_mod__mult__left__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_347_mod__mult__right__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_348_mod__mult__right__eq,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_349_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_350_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_351_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_352_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_353_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_354_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_355_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_356_dvd__mod__iff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
= ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_357_dvd__mod__iff,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
= ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_358_dvd__mod__imp__dvd,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_359_dvd__mod__imp__dvd,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_360_dvd__mod,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_361_dvd__mod,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd_int @ K2 @ M )
=> ( ( dvd_dvd_int @ K2 @ N )
=> ( dvd_dvd_int @ K2 @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_362_mod__mod__cancel,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
= ( modulo_modulo_nat @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_363_mod__mod__cancel,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_364_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_365_power__mod,axiom,
! [A: nat,B2: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
= ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).
% power_mod
thf(fact_366_power__mod,axiom,
! [A: int,B2: int,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
= ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).
% power_mod
thf(fact_367_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_368_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_369_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_370_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_371_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_372_mult__neg__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_373_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_374_mult__less__0__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_375_mult__neg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_376_mult__neg__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_377_mult__pos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_378_mult__pos__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_379_mult__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_380_mult__pos__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_381_mult__pos__neg2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_382_mult__pos__neg2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_383_zero__less__mult__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_384_zero__less__mult__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_385_zero__less__mult__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_386_zero__less__mult__pos2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_387_zero__less__mult__pos2,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_388_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_389_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_390_mult__strict__left__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_391_mult__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_392_mult__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_393_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_394_mult__strict__right__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_395_mult__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_396_mult__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_397_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_398_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_399_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_400_mod__0__imp__dvd,axiom,
! [A: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ B2 )
= zero_zero_nat )
=> ( dvd_dvd_nat @ B2 @ A ) ) ).
% mod_0_imp_dvd
thf(fact_401_mod__0__imp__dvd,axiom,
! [A: int,B2: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= zero_zero_int )
=> ( dvd_dvd_int @ B2 @ A ) ) ).
% mod_0_imp_dvd
thf(fact_402_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A2: nat,B3: nat] :
( ( modulo_modulo_nat @ B3 @ A2 )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_403_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A2: int,B3: int] :
( ( modulo_modulo_int @ B3 @ A2 )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_404_mod__eq__0__iff__dvd,axiom,
! [A: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ B2 )
= zero_zero_nat )
= ( dvd_dvd_nat @ B2 @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_405_mod__eq__0__iff__dvd,axiom,
! [A: int,B2: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= zero_zero_int )
= ( dvd_dvd_int @ B2 @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_406_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_407_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_408_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_409_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_410_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_411_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
=> ( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M3 @ N3 ) )
=> ( P @ M3 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_412_dvd__mult__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_413_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_414_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% bits_one_mod_two_eq_one
thf(fact_415_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_one_mod_two_eq_one
thf(fact_416_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_417_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_418_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_419_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_420_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_421_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_422_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_423_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_424_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_425_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_426_general__digit__base,axiom,
! [T2: nat,T1: nat,B2: nat,A: nat] :
( ( ord_less_nat @ T2 @ T1 )
=> ( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( bits_nth_digit @ ( times_times_nat @ A @ ( power_power_nat @ B2 @ T1 ) ) @ T2 @ B2 )
= zero_zero_nat ) ) ) ).
% general_digit_base
thf(fact_427_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_428_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_429_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_430_dbl__simps_I5_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) ).
% dbl_simps(5)
thf(fact_431_neg__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% neg_mod_bound
thf(fact_432_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K2 @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_433_zdvd__mono,axiom,
! [K2: int,M: int,T: int] :
( ( K2 != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K2 @ M ) @ ( times_times_int @ K2 @ T ) ) ) ) ).
% zdvd_mono
thf(fact_434_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_435_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_436_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_437_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ~ ( ord_less_num @ T @ X3 ) ) ).
% minf(7)
thf(fact_438_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_439_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_440_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ord_less_num @ X3 @ T ) ) ).
% minf(5)
thf(fact_441_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_442_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_443_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_444_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_445_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_446_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_447_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_448_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_449_minf_I2_J,axiom,
! [P: num > $o,P3: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_450_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_451_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_452_minf_I1_J,axiom,
! [P: num > $o,P3: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_453_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_454_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_455_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ord_less_num @ T @ X3 ) ) ).
% pinf(7)
thf(fact_456_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_457_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_458_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ~ ( ord_less_num @ X3 @ T ) ) ).
% pinf(5)
thf(fact_459_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_460_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_461_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_462_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_463_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_464_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_465_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_466_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_467_pinf_I2_J,axiom,
! [P: num > $o,P3: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_468_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_469_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_470_pinf_I1_J,axiom,
! [P: num > $o,P3: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: num] :
! [X2: num] :
( ( ord_less_num @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_471_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_472_mult_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_473_mult_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_474_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B3: nat] : ( times_times_nat @ B3 @ A2 ) ) ) ).
% mult.commute
thf(fact_475_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B3: int] : ( times_times_int @ B3 @ A2 ) ) ) ).
% mult.commute
thf(fact_476_mult_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_477_mult_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_478_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_479_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_480_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_481_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_482_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_483_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_484_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_485_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_486_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_487_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_488_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_489_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_490_nth__digit__base2__equiv,axiom,
( bits_nth_bit
= ( ^ [A2: nat,K3: nat] : ( bits_nth_digit @ A2 @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% nth_digit_base2_equiv
thf(fact_491_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_492_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_493_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_494_power__numeral,axiom,
! [K2: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K2 ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K2 @ L ) ) ) ).
% power_numeral
thf(fact_495_power__numeral,axiom,
! [K2: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K2 @ L ) ) ) ).
% power_numeral
thf(fact_496_even__unset__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_497_even__unset__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_498_even__flip__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_499_even__flip__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_500_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_501_add__left__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_502_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_503_add__right__cancel,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_504_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_505_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_506_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_507_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_508_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_509_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_510_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_511_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_512_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_513_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_514_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_515_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_516_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_517_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_518_add__le__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_519_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_520_add__le__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_521_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_522_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_523_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_524_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_525_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_526_add__cancel__right__right,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ A @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_527_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_528_add__cancel__right__left,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ B2 @ A ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_529_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_530_add__cancel__left__right,axiom,
! [A: int,B2: int] :
( ( ( plus_plus_int @ A @ B2 )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_531_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_532_add__cancel__left__left,axiom,
! [B2: int,A: int] :
( ( ( plus_plus_int @ B2 @ A )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_533_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_534_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_535_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_536_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_537_add__less__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_538_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_539_add__less__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_540_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_541_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_542_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_543_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_544_dvd__add__triv__left__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_545_dvd__add__triv__left__iff,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( dvd_dvd_int @ A @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_546_dvd__add__triv__right__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_547_dvd__add__triv__right__iff,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( dvd_dvd_int @ A @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_548_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_549_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_550_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_551_mod__add__self2,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_add_self2
thf(fact_552_mod__add__self2,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_add_self2
thf(fact_553_mod__add__self1,axiom,
! [B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_add_self1
thf(fact_554_mod__add__self1,axiom,
! [B2: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_add_self1
thf(fact_555_flip__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_556_unset__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_557_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_558_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_559_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_560_le__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_561_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_562_le__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_563_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_564_add__le__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_565_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_566_add__le__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_567_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_568_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_569_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_570_less__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_571_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_572_less__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_573_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_574_add__less__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_575_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_576_add__less__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_577_distrib__left__numeral,axiom,
! [V: num,B2: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_578_distrib__left__numeral,axiom,
! [V: num,B2: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_579_distrib__right__numeral,axiom,
! [A: nat,B2: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_580_distrib__right__numeral,axiom,
! [A: int,B2: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_581_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_582_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
= ( dvd_dvd_int @ A @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_583_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_584_dvd__add__times__triv__right__iff,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_585_mod__mult__self1,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self1
thf(fact_586_mod__mult__self1,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self1
thf(fact_587_mod__mult__self2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self2
thf(fact_588_mod__mult__self2,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self2
thf(fact_589_mod__mult__self3,axiom,
! [C: nat,B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self3
thf(fact_590_mod__mult__self3,axiom,
! [C: int,B2: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self3
thf(fact_591_mod__mult__self4,axiom,
! [B2: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self4
thf(fact_592_mod__mult__self4,axiom,
! [B2: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self4
thf(fact_593_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_594_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_595_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B2: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_596_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B2: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_597_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_598_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_599_mod__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_pos_pos_trivial
thf(fact_600_mod__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_neg_neg_trivial
thf(fact_601_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_602_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_603_power__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_604_power__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_605_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_606_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_607_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_608_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_609_power__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_610_power__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_611_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_612_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_613_odd__add,axiom,
! [A: nat,B2: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% odd_add
thf(fact_614_odd__add,axiom,
! [A: int,B2: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% odd_add
thf(fact_615_even__add,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_add
thf(fact_616_even__add,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_add
thf(fact_617_power__mono__iff,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_618_power__mono__iff,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_619_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_620_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_621_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_622_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_623_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_624_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_625_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_626_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_627_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_628_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_629_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_630_add__le__less__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_631_add__le__less__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_632_add__less__le__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_633_add__less__le__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_634_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_635_mod__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
= ( plus_plus_int @ K2 @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_636_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_637_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_638_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_639_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_640_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_641_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_642_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_643_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_644_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_645_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_646_group__cancel_Oadd1,axiom,
! [A4: nat,K2: nat,A: nat,B2: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A4 @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_647_group__cancel_Oadd1,axiom,
! [A4: int,K2: int,A: int,B2: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A4 @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_648_group__cancel_Oadd2,axiom,
! [B6: nat,K2: nat,B2: nat,A: nat] :
( ( B6
= ( plus_plus_nat @ K2 @ B2 ) )
=> ( ( plus_plus_nat @ A @ B6 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_649_group__cancel_Oadd2,axiom,
! [B6: int,K2: int,B2: int,A: int] :
( ( B6
= ( plus_plus_int @ K2 @ B2 ) )
=> ( ( plus_plus_int @ A @ B6 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_650_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_651_add_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_652_add_Oleft__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_653_add_Oright__cancel,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_654_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B3: nat] : ( plus_plus_nat @ B3 @ A2 ) ) ) ).
% add.commute
thf(fact_655_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B3: int] : ( plus_plus_int @ B3 @ A2 ) ) ) ).
% add.commute
thf(fact_656_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_657_add_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_658_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_659_add__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_660_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_661_add__left__imp__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_662_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_663_add__right__imp__eq,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_664_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_665_add__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_666_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_667_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_668_add__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_669_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_670_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_671_add__le__imp__le__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_672_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_673_add__le__imp__le__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_674_conj__le__cong,axiom,
! [X: int,X4: int,P: $o,P3: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_675_imp__le__cong,axiom,
! [X: int,X4: int,P: $o,P3: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_676_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_677_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_678_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_679_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_680_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_681_add__nonpos__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_682_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_683_add__nonneg__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_684_add__increasing2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_685_add__increasing2,axiom,
! [C: int,B2: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_686_add__decreasing2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_687_add__decreasing2,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_688_add__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_689_add__increasing,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_690_add__decreasing,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_691_add__decreasing,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_692_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_693_nth__bit__bounded,axiom,
! [A: nat,K2: nat] : ( ord_less_eq_nat @ ( bits_nth_bit @ A @ K2 ) @ one_one_nat ) ).
% nth_bit_bounded
thf(fact_694_is__num__normalize_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_695_add__strict__increasing2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_696_add__strict__increasing2,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_697_add__strict__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_698_add__strict__increasing,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_699_add__pos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_700_add__pos__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_701_add__nonpos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_702_add__nonpos__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_703_add__nonneg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_704_add__nonneg__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_705_add__neg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_706_add__neg__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_707_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_708_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_709_power__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_710_power__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_711_mod__eq__nat2E,axiom,
! [M: nat,Q: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q )
= ( modulo_modulo_nat @ N @ Q ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_712_mod__eq__nat1E,axiom,
! [M: nat,Q: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q )
= ( modulo_modulo_nat @ N @ Q ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_713_int__mod__pos__eq,axiom,
! [A: int,B2: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B2 )
=> ( ( modulo_modulo_int @ A @ B2 )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_714_int__mod__neg__eq,axiom,
! [A: int,B2: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ R )
=> ( ( modulo_modulo_int @ A @ B2 )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_715_split__zmod,axiom,
! [Q2: int > $o,N: int,K2: int] :
( ( Q2 @ ( modulo_modulo_int @ N @ K2 ) )
= ( ( ( K2 = zero_zero_int )
=> ( Q2 @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K2 )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I2 ) @ J2 ) ) )
=> ( Q2 @ J2 ) ) )
& ( ( ord_less_int @ K2 @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K2 @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I2 ) @ J2 ) ) )
=> ( Q2 @ J2 ) ) ) ) ) ).
% split_zmod
thf(fact_716_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_717_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_718_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_719_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_720_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_721_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_722_add__less__imp__less__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_723_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_724_add__less__imp__less__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_725_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_726_add__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_727_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_728_add__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_729_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_730_add__strict__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_731_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_732_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_733_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_734_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_735_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_736_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_737_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_738_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_739_comm__semiring__class_Odistrib,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_740_comm__semiring__class_Odistrib,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_741_distrib__left,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_742_distrib__left,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_743_distrib__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_744_distrib__right,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_745_combine__common__factor,axiom,
! [A: nat,E: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_746_combine__common__factor,axiom,
! [A: int,E: int,B2: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_747_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_748_dvd__add,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_749_dvd__add,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_750_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_751_dvd__add__left__iff,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( dvd_dvd_int @ A @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_752_dvd__add__right__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_753_dvd__add__right__iff,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_754_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_755_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_756_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_757_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ T ) ) ).
% pinf(6)
thf(fact_758_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_759_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_760_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ord_less_eq_num @ T @ X3 ) ) ).
% pinf(8)
thf(fact_761_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_762_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_763_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ord_less_eq_num @ X3 @ T ) ) ).
% minf(6)
thf(fact_764_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_765_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_766_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X3 ) ) ).
% minf(8)
thf(fact_767_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_768_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_769_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_770_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B: nat] :
( ( P @ A5 @ B )
= ( P @ B @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B: nat] :
( ( P @ A5 @ B )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_771_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_772_mod__add__right__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_773_mod__add__right__eq,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_774_mod__add__left__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_775_mod__add__left__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_776_mod__add__cong,axiom,
! [A: nat,C: nat,A3: nat,B2: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A3 @ C ) )
=> ( ( ( modulo_modulo_nat @ B2 @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_777_mod__add__cong,axiom,
! [A: int,C: int,A3: int,B2: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A3 @ C ) )
=> ( ( ( modulo_modulo_int @ B2 @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_778_mod__add__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% mod_add_eq
thf(fact_779_mod__add__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% mod_add_eq
thf(fact_780_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_781_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_782_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_783_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_784_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_785_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_786_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_787_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_788_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_789_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_790_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_791_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_792_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_793_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_794_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_795_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_796_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_797_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_798_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_799_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_800_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_801_obtain__smallest,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ( P @ K )
& ! [A6: nat] :
( ( ord_less_nat @ A6 @ K )
=> ~ ( P @ A6 ) ) ) ) ).
% obtain_smallest
thf(fact_802_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_803_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_804_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_805_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_806_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_807_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_808_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_809_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_810_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_811_zmod__le__nonneg__dividend,axiom,
! [M: int,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K2 ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_812_zmod__trivial__iff,axiom,
! [I: int,K2: int] :
( ( ( modulo_modulo_int @ I @ K2 )
= I )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K2 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K2 @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_813_mod__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K2 ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_814_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_815_neg__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K2 @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_816_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_817_power__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_818_power__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_819_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_820_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_821_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X5: int] : ( plus_plus_int @ X5 @ X5 ) ) ) ).
% dbl_def
thf(fact_822_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_823_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_824_pos__add__strict,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_825_pos__add__strict,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_826_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_827_add__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_828_add__pos__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_829_add__neg__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_830_add__neg__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_831_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_832_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_833_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_834_add__mono1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_835_add__mono1,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_836_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_837_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_838_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_839_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_840_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_841_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_842_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% minf(9)
thf(fact_843_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% minf(9)
thf(fact_844_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_845_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_846_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% pinf(9)
thf(fact_847_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% pinf(9)
thf(fact_848_mod__eqE,axiom,
! [A: int,C: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ~ ! [D2: int] :
( B2
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D2 ) ) ) ) ).
% mod_eqE
thf(fact_849_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_850_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_851_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_852_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_853_mult__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_854_mult__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_855_mult__mono_H,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_856_mult__mono_H,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_857_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_858_split__mult__pos__le,axiom,
! [A: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_859_mult__left__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_860_mult__nonpos__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_861_mult__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_862_mult__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_863_mult__right__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_864_mult__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_865_mult__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_866_mult__le__0__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_867_split__mult__neg__le,axiom,
! [A: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_868_split__mult__neg__le,axiom,
! [A: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_869_mult__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_870_mult__nonneg__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_871_mult__nonneg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_872_mult__nonneg__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_873_mult__nonpos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_874_mult__nonpos__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_875_mult__nonneg__nonpos2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_876_mult__nonneg__nonpos2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_877_zero__le__mult__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_878_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_879_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_880_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_881_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_882_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_883_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_884_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_885_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_886_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_887_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_888_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_889_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_890_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_891_power__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_892_power__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_893_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_894_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_895_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_896_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_897_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ~ ( P @ I4 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_898_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_899_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_900_power__le__dvd,axiom,
! [A: nat,N: nat,B2: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).
% power_le_dvd
thf(fact_901_power__le__dvd,axiom,
! [A: int,N: nat,B2: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).
% power_le_dvd
thf(fact_902_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_903_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_904_bezout__add__nat,axiom,
! [A: nat,B2: nat] :
? [D2: nat,X2: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B2 )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D2 ) )
| ( ( times_times_nat @ B2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_905_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B2 )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y ) @ D ) )
| ( ( times_times_nat @ B2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X2: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_906_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_907_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_908_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_909_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_910_pos__zmod__mult__2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_911_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_912_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_913_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_914_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_915_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X5: nat] : ( P @ ( times_times_nat @ L @ X5 ) ) )
= ( ? [X5: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X5 @ zero_zero_nat ) )
& ( P @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_916_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X5: int] : ( P @ ( times_times_int @ L @ X5 ) ) )
= ( ? [X5: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X5 @ zero_zero_int ) )
& ( P @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_917_ex__power__ivl1,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K2 )
& ( ord_less_nat @ K2 @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_918_ex__power__ivl2,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K2 )
& ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_919_mult__less__le__imp__less,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_920_mult__less__le__imp__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_921_mult__le__less__imp__less,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_922_mult__le__less__imp__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_923_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_924_mult__right__le__imp__le,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_925_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_926_mult__left__le__imp__le,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_927_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_928_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_929_mult__less__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_930_mult__strict__mono_H,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_931_mult__strict__mono_H,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_932_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_933_mult__right__less__imp__less,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_934_mult__less__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_935_mult__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_936_mult__strict__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_937_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_938_mult__left__less__imp__less,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_939_mult__le__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_940_mult__le__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_941_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_942_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_943_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_944_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_945_mult__le__one,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_946_mult__le__one,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_947_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_948_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_949_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_950_power__less__imp__less__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_951_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_952_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_953_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_954_bezout__add__strong__nat,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ? [D2: nat,X2: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B2 )
& ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_955_nat__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_956_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_957_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_958_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_959_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_960_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_961_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_962_left__add__twice,axiom,
! [A: nat,B2: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% left_add_twice
thf(fact_963_left__add__twice,axiom,
! [A: int,B2: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% left_add_twice
thf(fact_964_odd__even__add,axiom,
! [A: nat,B2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% odd_even_add
thf(fact_965_odd__even__add,axiom,
! [A: int,B2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% odd_even_add
thf(fact_966_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_967_mult__less__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_968_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_969_mult__less__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_970_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_971_mult__le__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_972_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_973_mult__le__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_974_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_975_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_976_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_977_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_978_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_979_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_980_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_981_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_982_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_983_split__mod,axiom,
! [Q2: nat > $o,M: nat,N: nat] :
( ( Q2 @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q2 @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( Q2 @ J2 ) ) ) ) ) ).
% split_mod
thf(fact_984_self__le__ge2__pow,axiom,
! [K2: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K2 @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_985_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_986_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_987_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_988_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_989_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_990_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_991_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_992_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_993_power__strict__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( 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 @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_994_power__strict__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( 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 @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_995_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_996_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_997_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_998_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_999_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_1000_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1001_power__mono__odd,axiom,
! [N: nat,A: int,B2: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono_odd
thf(fact_1002_dvd__power__iff__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M ) @ ( power_power_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1003_digit__gen__pow2__reduct,axiom,
! [K2: nat,C: nat,A: nat,T: nat] :
( ( ord_less_nat @ K2 @ C )
=> ( ( bits_nth_bit @ ( bits_nth_digit @ A @ T @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ C ) ) @ K2 )
= ( bits_nth_bit @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ T ) @ K2 ) ) ) ) ).
% digit_gen_pow2_reduct
thf(fact_1004_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_1005_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) ) ) ).
% oddE
thf(fact_1006_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_1007_sum__power2__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1008_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_1009_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_1010_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_1011_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_1012_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_1013_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_1014_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_1015_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_1016_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1017_mod__double__modulus,axiom,
! [M: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_nat @ X @ M ) )
| ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1018_mod__double__modulus,axiom,
! [M: int,X: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_int @ X @ M ) )
| ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1019_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1020_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
=> ? [Q4: int] :
( M
= ( times_times_int @ D @ Q4 ) ) ) ).
% zmod_eq_0D
thf(fact_1021_zdvd__mult__cancel,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K2 @ M ) @ ( times_times_int @ K2 @ N ) )
=> ( ( K2 != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1022_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1023_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1024_int__ge__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1025_unset__bit__less__eq,axiom,
! [N: nat,K2: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ K2 ) ).
% unset_bit_less_eq
thf(fact_1026_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1027_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1028_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1029_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1030_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1031_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1032_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1033_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1034_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1035_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_1036_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1037_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1038_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_1039_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_1040_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_1041_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1042_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1043_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1044_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1045_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1046_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1047_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1048_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1049_int__gr__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1050_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1051_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1052_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1053_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1054_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1055_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1056_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1057_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1058_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1059_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1060_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_1061_zdvd__reduce,axiom,
! [K2: int,N: int,M: int] :
( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N @ ( times_times_int @ K2 @ M ) ) )
= ( dvd_dvd_int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_1062_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1063_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1064_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1065_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1066_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1067_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1068_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1069_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1070_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1071_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1072_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1073_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1074_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1075_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1076_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1077_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1078_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1079_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_1080_div__dvd__div,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1081_div__dvd__div,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1082_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1083_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_1084_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1085_div__mult__mult1__if,axiom,
! [C: int,A: int,B2: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1086_div__mult__mult2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_1087_div__mult__mult2,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ A @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_1088_div__mult__mult1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_1089_div__mult__mult1,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_1090_nonzero__mult__div__cancel__right,axiom,
! [B2: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1091_nonzero__mult__div__cancel__right,axiom,
! [B2: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1092_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1093_nonzero__mult__div__cancel__left,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1094_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1095_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1096_dvd__mult__div__cancel,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_1097_dvd__mult__div__cancel,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_1098_dvd__div__mult__self,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_1099_dvd__div__mult__self,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_1100_div__add,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_1101_div__add,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_1102_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_1103_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_1104_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_1105_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_1106_unit__div,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_1107_unit__div,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_1108_bits__mod__div__trivial,axiom,
! [A: nat,B2: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_1109_bits__mod__div__trivial,axiom,
! [A: int,B2: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_1110_mod__div__trivial,axiom,
! [A: nat,B2: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_1111_mod__div__trivial,axiom,
! [A: int,B2: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_1112_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_1113_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_1114_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1115_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1116_div__mult__self1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_1117_div__mult__self1,axiom,
! [B2: int,A: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_1118_div__mult__self2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_1119_div__mult__self2,axiom,
! [B2: int,A: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_1120_div__mult__self3,axiom,
! [B2: nat,C: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_1121_div__mult__self3,axiom,
! [B2: int,C: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_1122_div__mult__self4,axiom,
! [B2: nat,C: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_1123_div__mult__self4,axiom,
! [B2: int,C: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_1124_unit__div__mult__self,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_1125_unit__div__mult__self,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_1126_unit__mult__div__div,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B2 @ A ) ) ) ).
% unit_mult_div_div
thf(fact_1127_unit__mult__div__div,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B2 @ A ) ) ) ).
% unit_mult_div_div
thf(fact_1128_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_1129_half__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% half_nonnegative_int_iff
thf(fact_1130_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1131_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_1132_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_1133_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_1134_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_1135_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1136_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1137_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1138_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1139_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_1140_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_1141_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_1142_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_1143_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_1144_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_1145_dvd__div__eq__0__iff,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ( ( ( divide_divide_nat @ A @ B2 )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1146_dvd__div__eq__0__iff,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( ( divide_divide_int @ A @ B2 )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1147_dvd__div__mult,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_1148_dvd__div__mult,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
= ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_1149_div__mult__swap,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_1150_div__mult__swap,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_1151_div__div__eq__right,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_1152_div__div__eq__right,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_1153_dvd__div__mult2__eq,axiom,
! [B2: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1154_dvd__div__mult2__eq,axiom,
! [B2: int,C: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1155_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_1156_dvd__mult__imp__div,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_1157_div__mult__div__if__dvd,axiom,
! [B2: nat,A: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1158_div__mult__div__if__dvd,axiom,
! [B2: int,A: int,D: int,C: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1159_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1160_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1161_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1162_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1163_unit__div__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B2 @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_1164_unit__div__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B2 @ A )
= ( divide_divide_int @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_1165_div__unit__dvd__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1166_div__unit__dvd__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1167_dvd__div__unit__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1168_dvd__div__unit__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1169_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ B2 )
= A )
= ( ( divide_divide_nat @ A @ B2 )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1170_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B2: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= A )
= ( ( divide_divide_int @ A @ B2 )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1171_div__add1__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1172_div__add1__eq,axiom,
! [A: int,B2: int,C: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1173_div__power,axiom,
! [B2: nat,A: nat,N: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% div_power
thf(fact_1174_div__power,axiom,
! [B2: int,A: int,N: nat] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
= ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% div_power
thf(fact_1175_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_1176_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_1177_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B2 @ C ) )
= ( A = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1178_dvd__div__eq__iff,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B2 @ C ) )
= ( A = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1179_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B2 @ C ) )
=> ( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( A = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1180_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B2: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B2 @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( A = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1181_div__div__div__same,axiom,
! [D: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_1182_div__div__div__same,axiom,
! [D: int,B2: int,A: int] :
( ( dvd_dvd_int @ D @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
= ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_1183_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1184_div__le__mono,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_1185_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1186_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_1187_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_1188_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_1189_div__neg__pos__less0,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1190_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1191_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1192_div__mult2__numeral__eq,axiom,
! [A: nat,K2: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K2 @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1193_div__mult2__numeral__eq,axiom,
! [A: int,K2: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K2 @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1194_div__add__self2,axiom,
! [B2: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1195_div__add__self2,axiom,
! [B2: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1196_div__add__self1,axiom,
! [B2: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1197_div__add__self1,axiom,
! [B2: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1198_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B2: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B2 @ A )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B2 @ C )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1199_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B2: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B2 @ A )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B2 @ C )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1200_dvd__div__iff__mult,axiom,
! [C: nat,B2: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1201_dvd__div__iff__mult,axiom,
! [C: int,B2: int,A: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1202_div__dvd__iff__mult,axiom,
! [B2: nat,A: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1203_div__dvd__iff__mult,axiom,
! [B2: int,A: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1204_dvd__div__eq__mult,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B2 )
=> ( ( ( divide_divide_nat @ B2 @ A )
= C )
= ( B2
= ( times_times_nat @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1205_dvd__div__eq__mult,axiom,
! [A: int,B2: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B2 )
=> ( ( ( divide_divide_int @ B2 @ A )
= C )
= ( B2
= ( times_times_int @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1206_unit__div__eq__0__iff,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B2 )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1207_unit__div__eq__0__iff,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B2 )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1208_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_1209_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_1210_is__unit__div__mult2__eq,axiom,
! [B2: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1211_is__unit__div__mult2__eq,axiom,
! [B2: int,C: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1212_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_1213_unit__div__mult__swap,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_1214_unit__div__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_1215_unit__div__commute,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_1216_div__mult__unit2,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_1217_div__le__mono2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1218_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_1219_nat__mult__div__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1220_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_1221_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_1222_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_1223_zdiv__mono1,axiom,
! [A: int,A3: int,B2: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1224_zdiv__mono2,axiom,
! [A: int,B5: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1225_zdiv__eq__0__iff,axiom,
! [I: int,K2: int] :
( ( ( divide_divide_int @ I @ K2 )
= zero_zero_int )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K2 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K2 @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1226_zdiv__mono1__neg,axiom,
! [A: int,A3: int,B2: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1227_zdiv__mono2__neg,axiom,
! [A: int,B5: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1228_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
= ( ( K2 = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1229_div__nonneg__neg__le0,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1230_div__nonpos__pos__le0,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1231_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K2 ) )
= ( ord_less_eq_int @ K2 @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1232_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1233_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1234_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1235_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K2 )
=> ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1236_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1237_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1238_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1239_nth__digit__def,axiom,
( bits_nth_digit
= ( ^ [Num: nat,K3: nat,Base: nat] : ( modulo_modulo_nat @ ( divide_divide_nat @ Num @ ( power_power_nat @ Base @ K3 ) ) @ Base ) ) ) ).
% nth_digit_def
thf(fact_1240_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_1241_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,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1242_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_1243_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_1244_int__div__pos__eq,axiom,
! [A: int,B2: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B2 )
=> ( ( divide_divide_int @ A @ B2 )
= Q ) ) ) ) ).
% int_div_pos_eq
thf(fact_1245_int__div__neg__eq,axiom,
! [A: int,B2: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ R )
=> ( ( divide_divide_int @ A @ B2 )
= Q ) ) ) ) ).
% int_div_neg_eq
thf(fact_1246_split__zdiv,axiom,
! [P: int > $o,N: int,K2: int] :
( ( P @ ( divide_divide_int @ N @ K2 ) )
= ( ( ( K2 = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K2 )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less_int @ K2 @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K2 @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1247_zmod__zmult2__eq,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1248_nth__bit__def,axiom,
( bits_nth_bit
= ( ^ [Num: nat,K3: nat] : ( modulo_modulo_nat @ ( divide_divide_nat @ Num @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% nth_bit_def
thf(fact_1249_neg__zdiv__mult__2,axiom,
! [A: int,B2: 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 ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1250_pos__zdiv__mult__2,axiom,
! [A: int,B2: 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 ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B2 @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1251_verit__le__mono__div__int,axiom,
! [A4: int,B6: int,N: int] :
( ( ord_less_int @ A4 @ B6 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B6 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B6 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1252_verit__le__mono__div,axiom,
! [A4: nat,B6: nat,N: nat] :
( ( ord_less_nat @ A4 @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B6 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B6 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1253_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1254_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1255_set__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1256_set__bit__greater__eq,axiom,
! [K2: int,N: nat] : ( ord_less_eq_int @ K2 @ ( bit_se7879613467334960850it_int @ N @ K2 ) ) ).
% set_bit_greater_eq
thf(fact_1257_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1258_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1259_div__less__mono,axiom,
! [A4: nat,B6: nat,N: nat] :
( ( ord_less_nat @ A4 @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A4 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B6 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B6 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1260_div__mod__decomp,axiom,
! [A4: nat,N: nat] :
( A4
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1261_zdiv__mono__strict,axiom,
! [A4: int,B6: int,N: int] :
( ( ord_less_int @ A4 @ B6 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A4 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B6 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B6 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1262_div__mod__decomp__int,axiom,
! [A4: int,N: int] :
( A4
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1263_neg__zmod__mult__2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% neg_zmod_mult_2
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [K4: nat,B7: nat] :
( ( a
= ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ B7 ) )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B7 ) ) ).
%------------------------------------------------------------------------------