TPTP Problem File: SLH0549^1.p
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%------------------------------------------------------------------------------
% 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_00215_008090__5500608_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1346 ( 677 unt; 73 typ; 0 def)
% Number of atoms : 3134 (1511 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10936 ( 295 ~; 89 |; 148 &;9415 @)
% ( 0 <=>; 989 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 247 ( 247 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 14 con; 0-3 aty)
% Number of variables : 3034 ( 106 ^;2821 !; 107 ?;3034 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:20:19.599
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (67)
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_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
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_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_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_Nat_OSuc,type,
suc: 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_Onum_Osize__num,type,
size_num: num > nat ).
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_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $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_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
set_or1222409239386451017an_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_v_a____,type,
a: nat ).
thf(sy_v_b____,type,
b: nat ).
thf(sy_v_c,type,
c: nat ).
thf(sy_v_d____,type,
d: nat ).
thf(sy_v_e____,type,
e: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: nat ).
% Relevant facts (1267)
thf(fact_0__092_060open_062n_A_092_060exclamdown_062_A0_A_060_ASuc_A1_092_060close_062,axiom,
ord_less_nat @ ( bits_nth_bit @ n @ zero_zero_nat ) @ ( suc @ one_one_nat ) ).
% \<open>n \<exclamdown> 0 < Suc 1\<close>
thf(fact_1__C0_Oprems_C_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ c ).
% "0.prems"(2)
thf(fact_2_assms_I1_J,axiom,
ord_less_nat @ n @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ c ) ).
% assms(1)
thf(fact_3_b__def,axiom,
( b
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ r ) ) ) ).
% b_def
thf(fact_4_digit__wise__equiv,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A: nat,B: nat] :
! [K: nat] :
( ( bits_nth_bit @ A @ K )
= ( bits_nth_bit @ B @ K ) ) ) ) ).
% digit_wise_equiv
thf(fact_5_aux1__digit__wise__equiv,axiom,
! [A2: nat] :
( ( ! [K: nat] :
( ( bits_nth_bit @ A2 @ K )
= zero_zero_nat ) )
= ( A2 = zero_zero_nat ) ) ).
% aux1_digit_wise_equiv
thf(fact_6_a__def,axiom,
( a
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ r ) ) ) ) ).
% a_def
thf(fact_7_zero__less__power2,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A2 != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_8_zero__eq__power2,axiom,
! [A2: nat] :
( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_9_zero__eq__power2,axiom,
! [A2: int] :
( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_10_power__eq__0__iff,axiom,
! [A2: nat,N: nat] :
( ( ( power_power_nat @ A2 @ N )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_11_power__eq__0__iff,axiom,
! [A2: int,N: nat] :
( ( ( power_power_int @ A2 @ N )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_12_odd__power__less__zero,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_13_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_14_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_15_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_16_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_17_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_18_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_19_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_20_assms_I2_J,axiom,
ord_less_nat @ r @ c ).
% assms(2)
thf(fact_21_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_22_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_23_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_24_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_25_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_26_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_27_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_28_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_29_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_30_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_31_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_32_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_33_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_34_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_35_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_36_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_37_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_38_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_39_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_40_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_41_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_42_power__one__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_43_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_44_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_45_power__inject__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ( power_power_nat @ A2 @ M )
= ( power_power_nat @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_46_power__inject__exp,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ( power_power_int @ A2 @ M )
= ( power_power_int @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_47_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_48_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_49_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_50_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_51_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_54_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_55_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_56_power__Suc0__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_57_power__Suc0__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_58_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_59_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_60_power__strict__increasing__iff,axiom,
! [B2: nat,X: nat,Y3: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y3 ) )
= ( ord_less_nat @ X @ Y3 ) ) ) ).
% power_strict_increasing_iff
thf(fact_61_power__strict__increasing__iff,axiom,
! [B2: int,X: nat,Y3: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y3 ) )
= ( ord_less_nat @ X @ Y3 ) ) ) ).
% power_strict_increasing_iff
thf(fact_62_d__def,axiom,
( d
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
@ ( set_or4665077453230672383an_nat @ ( suc @ r ) @ c ) ) ) ).
% d_def
thf(fact_63_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_64_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_65_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_66_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_67_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_68_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_69_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_70_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_71_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_72_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_73_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_74_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_75_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_76_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_77_left__right__inverse__power,axiom,
! [X: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X @ Y3 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_78_left__right__inverse__power,axiom,
! [X: int,Y3: int,N: nat] :
( ( ( times_times_int @ X @ Y3 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_79_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_80_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_81_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_82_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_83_power__less__power__Suc,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_84_power__less__power__Suc,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_85_power__gt1__lemma,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_86_power__gt1__lemma,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_87_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_88_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_89_power__gt1,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_90_power__gt1,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_91_power__strict__increasing,axiom,
! [N: nat,N3: nat,A2: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_92_power__strict__increasing,axiom,
! [N: nat,N3: nat,A2: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_93_power__less__imp__less__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_94_power__less__imp__less__exp,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_95_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_96_power__not__zero,axiom,
! [A2: nat,N: nat] :
( ( A2 != zero_zero_nat )
=> ( ( power_power_nat @ A2 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_97_power__not__zero,axiom,
! [A2: int,N: nat] :
( ( A2 != zero_zero_int )
=> ( ( power_power_int @ A2 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_98_power__commuting__commutes,axiom,
! [X: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X @ Y3 )
= ( times_times_nat @ Y3 @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y3 )
= ( times_times_nat @ Y3 @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_99_power__commuting__commutes,axiom,
! [X: int,Y3: int,N: nat] :
( ( ( times_times_int @ X @ Y3 )
= ( times_times_int @ Y3 @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y3 )
= ( times_times_int @ Y3 @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_100_power__mult__distrib,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B2 ) @ N )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% power_mult_distrib
thf(fact_101_power__mult__distrib,axiom,
! [A2: int,B2: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A2 @ B2 ) @ N )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).
% power_mult_distrib
thf(fact_102_power__commutes,axiom,
! [A2: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_103_power__commutes,axiom,
! [A2: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_104_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_105_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_106_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_107_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_108_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_109_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_110_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_111_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_112_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_113_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_114_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_115_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_116_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_117_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_118_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_119_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_120_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_121_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_122_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_123_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_124_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_125_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_126_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_127_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_128_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_129_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_130_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_131_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_132_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_133_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_134_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_135_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_136_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_137_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_138_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_139_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_140_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_141_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_142_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_143_power__mult,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_144_power__mult,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_145_power__Suc__less,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_146_power__Suc__less,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_147_power__Suc__less__one,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_148_power__Suc__less__one,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_149_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A2: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_150_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A2: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_151_one__less__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% one_less_power
thf(fact_152_one__less__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).
% one_less_power
thf(fact_153_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_154_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_155_zero__less__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_less_power
thf(fact_156_zero__less__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% zero_less_power
thf(fact_157_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_158_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_159_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_160_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_161_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_162_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_163_power__Suc2,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_164_power__Suc2,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_165_power__Suc,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_166_power__Suc,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ N ) )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_167_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_168_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_169_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_170_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_171_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_172_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_181_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_182_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_183_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_184_aux0__digit__wise__equiv,axiom,
! [A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ? [K3: nat] :
( ( bits_nth_bit @ A2 @ K3 )
= one_one_nat ) ) ).
% aux0_digit_wise_equiv
thf(fact_185_power__numeral__even,axiom,
! [Z2: nat,W: num] :
( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_186_power__numeral__even,axiom,
! [Z2: int,W: num] :
( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_187_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_188_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_189_power2__eq__square,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_190_power2__eq__square,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_191_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_192_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_193_power__even__eq,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_194_power__even__eq,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_195_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_196_power2__less__0,axiom,
! [A2: int] :
~ ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_197_power__odd__eq,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A2 @ ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_198_power__odd__eq,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A2 @ ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_199_e__def,axiom,
( e
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ K @ ( suc @ r ) ) ) )
@ ( set_or4665077453230672383an_nat @ ( suc @ r ) @ c ) ) ) ).
% e_def
thf(fact_200_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_201_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_202_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_203_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_204_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_205_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_206_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_207_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_208_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_209_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_210_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_211_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_212_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_213_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_214_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_215_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_216_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_217_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_218_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_219_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_220_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_221_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_222_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_223_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_224_mult__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_225_mult__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_226_mult__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_227_mult__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_228_mult__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_229_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_230_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_231_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_232_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_233_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_234_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_235_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_236_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_237_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_238_power__mult__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_239_power__mult__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_240_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_241_left__diff__distrib__numeral,axiom,
! [A2: int,B2: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_242_right__diff__distrib__numeral,axiom,
! [V: num,B2: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_243_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_244_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_245_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_246_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_247_d2r,axiom,
( d
= ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ r ) ) @ e ) ) ).
% d2r
thf(fact_248_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_249_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_250__092_060open_062d_A_061_A_I_092_060Sum_062k_A_061_ASuc_Ar_O_O_060c_O_A2_A_094_ASuc_Ar_A_K_A_In_A_092_060exclamdown_062_Ak_A_K_A2_A_094_A_Ik_A_N_ASuc_Ar_J_J_J_092_060close_062,axiom,
( d
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ r ) ) @ ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ K @ ( suc @ r ) ) ) ) )
@ ( set_or4665077453230672383an_nat @ ( suc @ r ) @ c ) ) ) ).
% \<open>d = (\<Sum>k = Suc r..<c. 2 ^ Suc r * (n \<exclamdown> k * 2 ^ (k - Suc r)))\<close>
thf(fact_251_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_252_left__diff__distrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_253_right__diff__distrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_254_left__diff__distrib_H,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_255_left__diff__distrib_H,axiom,
! [B2: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_256_right__diff__distrib_H,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_257_right__diff__distrib_H,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_258_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_259_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_260_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_261_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_262_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_263_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_264_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_265_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_266_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_267_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_268_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_269_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_270_linorder__neqE__linordered__idom,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_271_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_272_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_273_power2__commute,axiom,
! [X: int,Y3: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_274_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_275_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_276_power__eq__if,axiom,
( power_power_nat
= ( ^ [P2: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_277_power__eq__if,axiom,
( power_power_int
= ( ^ [P2: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P2 @ ( power_power_int @ P2 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_278_power__minus__mult,axiom,
! [N: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
= ( power_power_nat @ A2 @ N ) ) ) ).
% power_minus_mult
thf(fact_279_power__minus__mult,axiom,
! [N: nat,A2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
= ( power_power_int @ A2 @ N ) ) ) ).
% power_minus_mult
thf(fact_280_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_281_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_282_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_283_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_284_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_285_mult__not__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_286_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_287_divisors__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_288_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_289_no__zero__divisors,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_290_mult__left__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_291_mult__left__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_292_mult__right__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_293_mult__right__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_294_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_295_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_296_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_297_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_298_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_299_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_300_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_301_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_302_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_303_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_304_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_305_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_306_mult__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_307_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_308_mult__less__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_309_mult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_310_mult__neg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_311_mult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_312_mult__pos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_313_mult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_314_mult__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_315_mult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_316_mult__pos__neg2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_317_zero__less__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_318_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_319_zero__less__mult__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_320_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_321_zero__less__mult__pos2,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_322_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_323_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_324_mult__strict__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_325_mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_326_mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_327_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_328_mult__strict__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_329_mult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_330_mult__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_331_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_332_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_333_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_334_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_335_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_336_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_337_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_338_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_339_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_340_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_341_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_342_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_343_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_344_mult__numeral__1,axiom,
! [A2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_345_mult__numeral__1,axiom,
! [A2: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_346_mult__numeral__1__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_347_mult__numeral__1__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_348_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_349_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_350_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_351_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_352_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_353_ad,axiom,
( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ c ) )
= ( plus_plus_nat @ a @ d ) ) ).
% ad
thf(fact_354_sum__power2,axiom,
! [K2: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_355_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q2: nat] : ( ord_less_nat @ Q2 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_356_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q2: nat] : ( ord_less_nat @ Q2 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_357_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q2: nat] : ( ord_less_nat @ Q2 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_358_diff__gt__0__iff__gt,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_359_sum_Oneutral__const,axiom,
! [A3: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A3 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_360_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_361_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_362_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_363_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_364_add__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_365_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_366_add__right__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_367_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_368_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_369_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_370_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_371_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_372_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_373_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_374_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_375_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_376_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_377_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_378_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_379_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_380_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_381_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_382_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_383_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_384_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_385_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_386_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_387_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_388_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_389_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_390_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_391_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_392_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_393_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_394_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_395_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_396_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_397_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_398_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_399_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_400_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_401_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_402_add__diff__cancel,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel
thf(fact_403_diff__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% diff_add_cancel
thf(fact_404_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_405_add__diff__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_406_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_407_add__diff__cancel__left_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_408_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_409_add__diff__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_410_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_411_add__diff__cancel__right_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_412_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_413_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_414_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_415_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_416_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_417_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_418_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_419_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_420_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_421_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_422_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_423_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_424_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_425_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_426_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_427_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_428_distrib__right__numeral,axiom,
! [A2: nat,B2: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_429_distrib__right__numeral,axiom,
! [A2: int,B2: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_430_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_431_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_432_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_433_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_434_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_435_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_436_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_437_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_438_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_439_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_440_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_441_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_442_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_443_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_444_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_445_diff__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_446_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_447_add__implies__diff,axiom,
! [C: int,B2: int,A2: int] :
( ( ( plus_plus_int @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_448_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_449_diff__add__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_450_diff__diff__eq2,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_451_add__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_452_eq__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( A2
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_453_diff__eq__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_454_group__cancel_Osub1,axiom,
! [A3: int,K2: int,A2: int,B2: int] :
( ( A3
= ( plus_plus_int @ K2 @ A2 ) )
=> ( ( minus_minus_int @ A3 @ B2 )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_455_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_456_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_457_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_458_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_459_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_460_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_461_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_462_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_463_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_464_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_465_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_466_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_467_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_468_add__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_469_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_470_add__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_471_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_472_add__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_473_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_474_add__less__imp__less__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_475_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_476_add__less__imp__less__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_477_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_478_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_479_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_480_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_481_group__cancel_Oadd1,axiom,
! [A3: nat,K2: nat,A2: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A2 ) )
=> ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_482_group__cancel_Oadd1,axiom,
! [A3: int,K2: int,A2: int,B2: int] :
( ( A3
= ( plus_plus_int @ K2 @ A2 ) )
=> ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_483_group__cancel_Oadd2,axiom,
! [B3: nat,K2: nat,B2: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K2 @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_484_group__cancel_Oadd2,axiom,
! [B3: int,K2: int,B2: int,A2: int] :
( ( B3
= ( plus_plus_int @ K2 @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_485_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_486_add_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_487_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_488_add_Oright__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_489_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ) ).
% add.commute
thf(fact_490_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A: int,B: int] : ( plus_plus_int @ B @ A ) ) ) ).
% add.commute
thf(fact_491_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_492_add_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_493_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_494_add__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_495_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_496_add__right__imp__eq,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_497_sum_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A3: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A3 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A3 ) @ ( groups3542108847815614940at_nat @ H2 @ A3 ) ) ) ).
% sum.distrib
thf(fact_498_combine__common__factor,axiom,
! [A2: nat,E: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_499_combine__common__factor,axiom,
! [A2: int,E: int,B2: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_500_distrib__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_501_distrib__right,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_502_distrib__left,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_503_distrib__left,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_504_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_505_comm__semiring__class_Odistrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_506_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_507_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_508_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_509_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_510_nat__arith_Osuc1,axiom,
! [A3: nat,K2: nat,A2: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K2 @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_511_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_512_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_513_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_514_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_515_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_516_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_517_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_518_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_519_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_520_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_521_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_522_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_523_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_524_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_525_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_526_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_527_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_528_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_529_pos__add__strict,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_530_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_531_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_532_add__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_533_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_534_add__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_535_less__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_536_diff__less__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_537_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > nat,M: nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_538_add__less__zeroD,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y3 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_539_sum__squares__eq__zero__iff,axiom,
! [X: int,Y3: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_540_add__mono1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_541_add__mono1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_542_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_543_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_544_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_545_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: int,B2: int] :
( ~ ( ord_less_int @ A2 @ B2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_546_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_547_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_548_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_549_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_550_eq__add__iff1,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_551_eq__add__iff2,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_552_square__diff__square__factored,axiom,
! [X: int,Y3: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y3 ) @ ( minus_minus_int @ X @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_553_power__add,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_add
thf(fact_554_power__add,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).
% power_add
thf(fact_555_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_556_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_557_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_558_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
? [K: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M5 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_559_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_560_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_561_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_562_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_563_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_564_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_565_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_566_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_567_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_568_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_569_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_570_sum__power__add,axiom,
! [X: int,M: nat,I4: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I4 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_571_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_572_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_573_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_574_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_575_sum_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_576_sum_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_577_sum__squares__gt__zero__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) )
= ( ( X != zero_zero_int )
| ( Y3 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_578_not__sum__squares__lt__zero,axiom,
! [X: int,Y3: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_579_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_580_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_581_less__add__iff2,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_582_less__add__iff1,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_583_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_584_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_585_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_586_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_587_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_588_mult_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_589_mult_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_590_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A: nat,B: nat] : ( times_times_nat @ B @ A ) ) ) ).
% mult.commute
thf(fact_591_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A: int,B: int] : ( times_times_int @ B @ A ) ) ) ).
% mult.commute
thf(fact_592_mult_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_593_mult_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_594_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_595_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_596_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_597_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_598_diff__eq__diff__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_599_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_600_diff__right__commute,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_601_sum_OatLeastLessThan__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_602_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H2: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( member_nat @ ( I @ B4 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_603_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K2: nat > nat,A3: set_nat,H2: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B3 )
=> ( ( member_nat @ ( K2 @ Y4 ) @ A3 )
& ( ( H2 @ ( K2 @ Y4 ) )
= Y4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B3 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A3 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_604_sum_Oeq__general,axiom,
! [B3: set_nat,A3: set_nat,H2: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A3 )
& ( ( H2 @ X5 )
= Y4 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A3 )
& ( ( H2 @ Ya )
= Y4 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B3 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A3 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_605_sum_Ocong,axiom,
! [A3: set_nat,B3: set_nat,G: nat > nat,H2: nat > nat] :
( ( A3 = B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B3 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A3 )
= ( groups3542108847815614940at_nat @ H2 @ B3 ) ) ) ) ).
% sum.cong
thf(fact_606_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_607_left__add__twice,axiom,
! [A2: nat,B2: nat] :
( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_608_left__add__twice,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_609_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_610_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_611_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_612_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_613_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_614_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_615_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N5 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_616_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A3: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B3 )
@ A3 )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ J3 )
@ A3 )
@ B3 ) ) ).
% sum.swap
thf(fact_617_sum__power2__eq__zero__iff,axiom,
! [X: int,Y3: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_618_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_619_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_620_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_621_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_622_sum__power2__gt__zero__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y3 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_623_not__sum__power2__lt__zero,axiom,
! [X: int,Y3: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_624_power2__sum,axiom,
! [X: nat,Y3: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% power2_sum
thf(fact_625_power2__sum,axiom,
! [X: int,Y3: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% power2_sum
thf(fact_626_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_627_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_628_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_629_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_630_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A: int,B: int] :
( ( minus_minus_int @ A @ B )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_631_diff__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_632_diff__strict__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_633_diff__eq__diff__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_634_diff__strict__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_635_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_636_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_637_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_638_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_639_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A3: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A3 )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_640_sum_Oneutral,axiom,
! [A3: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A3 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_641_atLeastLessThan__eq__iff,axiom,
! [A2: num,B2: num,C: num,D: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_num @ C @ D )
=> ( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
= ( set_or1222409239386451017an_num @ C @ D ) )
= ( ( A2 = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_642_atLeastLessThan__eq__iff,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A2 = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_643_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A2 = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_644_Ico__eq__Ico,axiom,
! [L: num,H2: num,L2: num,H3: num] :
( ( ( set_or1222409239386451017an_num @ L @ H2 )
= ( set_or1222409239386451017an_num @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_num @ L @ H2 )
& ~ ( ord_less_num @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_645_Ico__eq__Ico,axiom,
! [L: int,H2: int,L2: int,H3: int] :
( ( ( set_or4662586982721622107an_int @ L @ H2 )
= ( set_or4662586982721622107an_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_int @ L @ H2 )
& ~ ( ord_less_int @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_646_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_647_atLeastLessThan__inj_I1_J,axiom,
! [A2: num,B2: num,C: num,D: num] :
( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
= ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_num @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_648_atLeastLessThan__inj_I1_J,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_649_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_650_atLeastLessThan__inj_I2_J,axiom,
! [A2: num,B2: num,C: num,D: num] :
( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
= ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_num @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_651_atLeastLessThan__inj_I2_J,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_652_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_653_power2__diff,axiom,
! [X: int,Y3: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% power2_diff
thf(fact_654_aux3__digit__sum__repr,axiom,
! [B2: nat,R: nat,A2: nat] :
( ( ord_less_nat @ B2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R ) )
=> ( ( bits_nth_bit @ ( plus_plus_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R ) ) @ B2 ) @ R )
= ( bits_nth_bit @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R ) ) @ R ) ) ) ).
% aux3_digit_sum_repr
thf(fact_655_aux2__digit__sum__repr,axiom,
! [N: nat,C: nat,R: nat,A2: nat] :
( ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ C ) )
=> ( ( ord_less_nat @ R @ C )
=> ( ( bits_nth_bit @ ( plus_plus_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ C ) ) @ N ) @ R )
= ( bits_nth_bit @ N @ R ) ) ) ) ).
% aux2_digit_sum_repr
thf(fact_656_sum__product,axiom,
! [F: nat > nat,A3: set_nat,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A3 ) ) ).
% sum_product
thf(fact_657_sum__distrib__right,axiom,
! [F: nat > nat,A3: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N5: nat] : ( times_times_nat @ ( F @ N5 ) @ R )
@ A3 ) ) ).
% sum_distrib_right
thf(fact_658_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A3: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A3 ) )
= ( groups3542108847815614940at_nat
@ ^ [N5: nat] : ( times_times_nat @ R @ ( F @ N5 ) )
@ A3 ) ) ).
% sum_distrib_left
thf(fact_659_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A: int,B: int] : ( ord_less_int @ ( minus_minus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_660_sum__cong__Suc,axiom,
! [A3: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A3 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A3 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A3 )
= ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ).
% sum_cong_Suc
thf(fact_661_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_662_sum__SucD,axiom,
! [F: nat > nat,A3: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A3 )
= ( suc @ N ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_663_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_664_calculation,axiom,
( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
@ ( set_ord_lessThan_nat @ c ) )
= ( plus_plus_nat @ a @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ r ) ) @ e ) ) ) ).
% calculation
thf(fact_665_digit__gen__pow2__reduct,axiom,
! [K2: nat,C: nat,A2: nat,T: nat] :
( ( ord_less_nat @ K2 @ C )
=> ( ( bits_nth_bit @ ( bits_nth_digit @ A2 @ T @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ C ) ) @ K2 )
= ( bits_nth_bit @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ T ) @ K2 ) ) ) ) ).
% digit_gen_pow2_reduct
thf(fact_666_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_667_general__digit__base,axiom,
! [T22: nat,T1: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ T22 @ T1 )
=> ( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( bits_nth_digit @ ( times_times_nat @ A2 @ ( power_power_nat @ B2 @ T1 ) ) @ T22 @ B2 )
= zero_zero_nat ) ) ) ).
% general_digit_base
thf(fact_668_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_669_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_670_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_671_lessThan__eq__iff,axiom,
! [X: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_672_lessThan__iff,axiom,
! [I: num,K2: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K2 ) )
= ( ord_less_num @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_673_lessThan__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_674_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_675_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_676_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_677_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_678_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_679_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_680_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_681_power__add__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_682_power__add__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_683_power__add__numeral2,axiom,
! [A2: nat,M: num,N: num,B2: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_684_power__add__numeral2,axiom,
! [A2: int,M: num,N: num,B2: int] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_685_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_686_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_687_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_688_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_689_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_690_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_691_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_692_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_693_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_694_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_695_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_696_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_697_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_698_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_699_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N5: nat] : ( minus_minus_int @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_700_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N5: nat] : ( minus_minus_int @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_701_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_702_sum_Onat__group,axiom,
! [G: nat > nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [M5: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M5 @ K2 ) @ ( plus_plus_nat @ ( times_times_nat @ M5 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K2 ) ) ) ) ).
% sum.nat_group
thf(fact_703_power__diff__1__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_704_one__diff__power__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_705_power__diff__sumr2,axiom,
! [X: int,N: nat,Y3: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y3 )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_706_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y3: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y3 )
@ ( groups3539618377306564664at_int
@ ^ [P2: nat] : ( times_times_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P2 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_707_one__diff__power__eq_H,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_708_nth__digit__base2__equiv,axiom,
( bits_nth_bit
= ( ^ [A: nat,K: nat] : ( bits_nth_digit @ A @ K @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% nth_digit_base2_equiv
thf(fact_709_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B2: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_710_add__scale__eq__noteq,axiom,
! [R: int,A2: int,B2: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus_int @ A2 @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B2 @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_711_mult__less__iff1,axiom,
! [Z2: int,X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
= ( ord_less_int @ X @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_712_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_713_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_714_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_715_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_716_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_717_add__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_718_add__0__iff,axiom,
! [B2: int,A2: int] :
( ( B2
= ( plus_plus_int @ B2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_719_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y3 ) ) )
= ( ( W = X )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_720_crossproduct__eq,axiom,
! [W: int,Y3: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y3 ) ) )
= ( ( W = X )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_721_crossproduct__noteq,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_722_crossproduct__noteq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_723_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_724_bits__div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_725_bits__div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_726_bits__div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_727_bits__div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_728_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_729_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_730_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_731_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_732_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_733_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_734_bits__div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% bits_div_by_1
thf(fact_735_bits__div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% bits_div_by_1
thf(fact_736_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_737_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_738_nonzero__mult__div__cancel__right,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_739_nonzero__mult__div__cancel__right,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_740_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_741_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_742_Suc__0__div__numeral_I2_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_743_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_744_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_745_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_746_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_747_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_748_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_749_div__mult2__numeral__eq,axiom,
! [A2: nat,K2: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ K2 @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_750_div__mult2__numeral__eq,axiom,
! [A2: int,K2: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( times_times_num @ K2 @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_751_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_752_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_753_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_754_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_755_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A5 )
& ~ ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_756_div__exp__eq,axiom,
! [A2: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_757_div__exp__eq,axiom,
! [A2: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_758_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_759_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_760_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_761_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_762_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_763_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_764_div__mult__mult1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_765_div__mult__mult1,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_766_div__mult__mult2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_767_div__mult__mult2,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_768_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_769_div__mult__mult1__if,axiom,
! [C: int,A2: int,B2: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_770_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_771_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_772_div__mult__self4,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A2 ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_773_div__mult__self4,axiom,
! [B2: int,C: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A2 ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_774_div__mult__self3,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A2 ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_775_div__mult__self3,axiom,
! [B2: int,C: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A2 ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_776_div__mult__self2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_777_div__mult__self2,axiom,
! [B2: int,A2: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_778_div__mult__self1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B2 ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_779_div__mult__self1,axiom,
! [B2: int,A2: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B2 ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_780_int__power__div__base,axiom,
! [M: nat,K2: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( divide_divide_int @ ( power_power_int @ K2 @ M ) @ K2 )
= ( power_power_int @ K2 @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_781_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_782_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_783_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_784_div__add__self2,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_785_div__add__self2,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_786_div__add__self1,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_787_div__add__self1,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_788_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_789_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_790_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_791_div__if,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N5: nat] :
( if_nat
@ ( ( ord_less_nat @ M5 @ N5 )
| ( N5 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N5 ) @ N5 ) ) ) ) ) ).
% div_if
thf(fact_792_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_793_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_794_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 )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_795_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_796_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_797_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_798_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_799_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_800_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_801_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_802_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_803_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_804_int__less__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_805_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_806_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_807_int__gr__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_808_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_809_set__bit__0,axiom,
! [A2: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_810_set__bit__0,axiom,
! [A2: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_811_unset__bit__0,axiom,
! [A2: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A2 )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_812_unset__bit__0,axiom,
! [A2: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A2 )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_813_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_814_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_815_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_816_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_817_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_818_plusinfinity,axiom,
! [D: int,P3: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P3 @ X4 )
= ( P3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [X_1: int] : ( P3 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_819_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_820_minf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ~ ( ord_less_num @ T @ X5 ) ) ).
% minf(7)
thf(fact_821_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_822_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_823_minf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ord_less_num @ X5 @ T ) ) ).
% minf(5)
thf(fact_824_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_825_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_826_minf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_827_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_828_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_829_minf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_830_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_831_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q5 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_832_minf_I2_J,axiom,
! [P: num > $o,P3: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q5 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_833_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q5 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_834_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q5 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_835_minf_I1_J,axiom,
! [P: num > $o,P3: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q5 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_836_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q5 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_837_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_838_pinf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ord_less_num @ T @ X5 ) ) ).
% pinf(7)
thf(fact_839_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_840_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_841_pinf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ~ ( ord_less_num @ X5 @ T ) ) ).
% pinf(5)
thf(fact_842_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_843_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_844_pinf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_845_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_846_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_847_pinf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_848_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_849_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q5 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_850_pinf_I2_J,axiom,
! [P: num > $o,P3: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q5 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_851_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q5 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_852_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q5 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_853_pinf_I1_J,axiom,
! [P: num > $o,P3: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q5 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_854_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q5 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q5 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_855_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( Q @ X4 )
= ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_856_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( Q @ X4 )
= ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_857_even__succ__div__exp,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_858_even__succ__div__exp,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_859_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_860_dvd__0__left__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_861_dvd__0__left__iff,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
= ( A2 = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_862_dvd__0__right,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_863_dvd__0__right,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ zero_zero_int ) ).
% dvd_0_right
thf(fact_864_dvd__add__triv__right__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_865_dvd__add__triv__right__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_866_dvd__add__triv__left__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_867_dvd__add__triv__left__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_868_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_869_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).
% dvd_1_left
thf(fact_870_div__dvd__div,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A2 ) @ ( divide_divide_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_871_div__dvd__div,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A2 ) @ ( divide_divide_int @ C @ A2 ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_872_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_873_dvd__times__right__cancel__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_874_dvd__times__right__cancel__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_875_dvd__times__left__cancel__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_876_dvd__times__left__cancel__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_877_dvd__mult__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_878_dvd__mult__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_879_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_880_dvd__add__times__triv__right__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A2 ) ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_881_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_882_dvd__add__times__triv__left__iff,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_883_unit__prod,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_884_unit__prod,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_885_dvd__div__mult__self,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_886_dvd__div__mult__self,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_887_dvd__mult__div__cancel,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ A2 ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_888_dvd__mult__div__cancel,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ A2 ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_889_div__add,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_890_div__add,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_891_unit__div,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_892_unit__div,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_893_unit__div__1__unit,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_894_unit__div__1__unit,axiom,
! [A2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A2 ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_895_unit__div__1__div__1,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) )
= A2 ) ) ).
% unit_div_1_div_1
thf(fact_896_unit__div__1__div__1,axiom,
! [A2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A2 ) )
= A2 ) ) ).
% unit_div_1_div_1
thf(fact_897_div__diff,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_diff
thf(fact_898_unit__div__mult__self,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_899_unit__div__mult__self,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_900_unit__mult__div__div,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A2 ) )
= ( divide_divide_nat @ B2 @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_901_unit__mult__div__div,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A2 ) )
= ( divide_divide_int @ B2 @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_902_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_903_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_904_even__mult__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_mult_iff
thf(fact_905_even__mult__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_mult_iff
thf(fact_906_even__add,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_add
thf(fact_907_even__add,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_add
thf(fact_908_odd__add,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% odd_add
thf(fact_909_odd__add,axiom,
! [A2: int,B2: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% odd_add
thf(fact_910_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_911_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_912_even__plus__one__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ).
% even_plus_one_iff
thf(fact_913_even__plus__one__iff,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ).
% even_plus_one_iff
thf(fact_914_even__diff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A2 @ B2 ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) ) ).
% even_diff
thf(fact_915_power__less__zero__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_916_power__less__zero__eq,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_917_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_918_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_919_even__succ__div__two,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_920_even__succ__div__two,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_921_odd__succ__div__two,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_922_odd__succ__div__two,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_923_even__succ__div__2,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_924_even__succ__div__2,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_925_zero__less__power__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A2 != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_926_even__power,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A2 @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_927_even__power,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A2 @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_928_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_929_odd__two__times__div__two__succ,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A2 ) ) ).
% odd_two_times_div_two_succ
thf(fact_930_odd__two__times__div__two__succ,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A2 ) ) ).
% odd_two_times_div_two_succ
thf(fact_931_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_932_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_933_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_934_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_935_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_936_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_937_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_938_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_939_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_940_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_941_div__power,axiom,
! [B2: nat,A2: nat,N: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( power_power_nat @ ( divide_divide_nat @ A2 @ B2 ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% div_power
thf(fact_942_div__power,axiom,
! [B2: int,A2: int,N: nat] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( power_power_int @ ( divide_divide_int @ A2 @ B2 ) @ N )
= ( divide_divide_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% div_power
thf(fact_943_dvd__div__eq__iff,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_944_dvd__div__eq__iff,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( ( divide_divide_int @ A2 @ C )
= ( divide_divide_int @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_945_dvd__div__eq__cancel,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B2 @ C ) )
=> ( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_946_dvd__div__eq__cancel,axiom,
! [A2: int,C: int,B2: int] :
( ( ( divide_divide_int @ A2 @ C )
= ( divide_divide_int @ B2 @ C ) )
=> ( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_947_div__div__div__same,axiom,
! [D: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ D @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ D ) @ ( divide_divide_nat @ B2 @ D ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_948_div__div__div__same,axiom,
! [D: int,B2: int,A2: int] :
( ( dvd_dvd_int @ D @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( divide_divide_int @ ( divide_divide_int @ A2 @ D ) @ ( divide_divide_int @ B2 @ D ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_949_strict__subset__divisors__dvd,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A2 ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
= ( ( dvd_dvd_nat @ A2 @ B2 )
& ~ ( dvd_dvd_nat @ B2 @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_950_strict__subset__divisors__dvd,axiom,
! [A2: int,B2: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A2 ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
= ( ( dvd_dvd_int @ A2 @ B2 )
& ~ ( dvd_dvd_int @ B2 @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_951_dvd__power__same,axiom,
! [X: nat,Y3: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y3 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% dvd_power_same
thf(fact_952_dvd__power__same,axiom,
! [X: int,Y3: int,N: nat] :
( ( dvd_dvd_int @ X @ Y3 )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% dvd_power_same
thf(fact_953_dvd__0__left,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_954_dvd__0__left,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
=> ( A2 = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_955_dvd__sum,axiom,
! [A3: set_nat,D: nat,F: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( dvd_dvd_nat @ D @ ( F @ A4 ) ) )
=> ( dvd_dvd_nat @ D @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% dvd_sum
thf(fact_956_dvd__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_957_dvd__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ B2 @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_958_dvd__refl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% dvd_refl
thf(fact_959_dvd__refl,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ A2 ) ).
% dvd_refl
thf(fact_960_dvd__div__unit__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_961_dvd__div__unit__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ C @ B2 ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_962_div__unit__dvd__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_963_div__unit__dvd__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_964_unit__div__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ B2 @ A2 )
= ( divide_divide_nat @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_965_unit__div__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( divide_divide_int @ B2 @ A2 )
= ( divide_divide_int @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_966_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_967_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_968_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_969_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_970_div__mult__div__if__dvd,axiom,
! [B2: nat,A2: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_971_div__mult__div__if__dvd,axiom,
! [B2: int,A2: int,D: int,C: int] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_972_dvd__mult__imp__div,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B2 )
=> ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_973_dvd__mult__imp__div,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B2 )
=> ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_974_dvd__div__mult2__eq,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_975_dvd__div__mult2__eq,axiom,
! [B2: int,C: int,A2: int] :
( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A2 )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_976_div__div__eq__right,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_977_div__div__eq__right,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( divide_divide_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_978_div__mult__swap,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_979_div__mult__swap,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_980_dvd__div__mult,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A2 )
= ( divide_divide_nat @ ( times_times_nat @ B2 @ A2 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_981_dvd__div__mult,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A2 )
= ( divide_divide_int @ ( times_times_int @ B2 @ A2 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_982_dvd__div__eq__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( ( divide_divide_nat @ A2 @ B2 )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_983_dvd__div__eq__0__iff,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( ( divide_divide_int @ A2 @ B2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_984_is__unit__mult__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_985_is__unit__mult__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
& ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_986_dvd__mult__unit__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_987_dvd__mult__unit__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B2 ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_988_mult__unit__dvd__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_989_mult__unit__dvd__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_990_dvd__mult__unit__iff_H,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_991_dvd__mult__unit__iff_H,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_992_mult__unit__dvd__iff_H,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_993_mult__unit__dvd__iff_H,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_994_unit__mult__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B2 )
= ( times_times_nat @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_995_unit__mult__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B2 )
= ( times_times_int @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_996_unit__mult__right__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ B2 @ A2 )
= ( times_times_nat @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_997_unit__mult__right__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ B2 @ A2 )
= ( times_times_int @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_998_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_999_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1000_dvd__diff__commute,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( dvd_dvd_int @ A2 @ ( minus_minus_int @ B2 @ C ) ) ) ).
% dvd_diff_commute
thf(fact_1001_dvd__diff__nat,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1002_dvd__diff,axiom,
! [X: int,Y3: int,Z2: int] :
( ( dvd_dvd_int @ X @ Y3 )
=> ( ( dvd_dvd_int @ X @ Z2 )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_1003_zdvd__zdiffD,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd_int @ K2 @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K2 @ N )
=> ( dvd_dvd_int @ K2 @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1004_one__dvd,axiom,
! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% one_dvd
thf(fact_1005_one__dvd,axiom,
! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).
% one_dvd
thf(fact_1006_unit__imp__dvd,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ B2 @ A2 ) ) ).
% unit_imp_dvd
thf(fact_1007_unit__imp__dvd,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ B2 @ A2 ) ) ).
% unit_imp_dvd
thf(fact_1008_dvd__unit__imp__unit,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1009_dvd__unit__imp__unit,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1010_dvd__add__right__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1011_dvd__add__right__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1012_dvd__add__left__iff,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_1013_dvd__add__left__iff,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_1014_dvd__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_1015_dvd__add,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_1016_dvd__triv__right,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_1017_dvd__triv__right,axiom,
! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_1018_dvd__mult__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
=> ( dvd_dvd_nat @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_1019_dvd__mult__right,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
=> ( dvd_dvd_int @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_1020_mult__dvd__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_1021_mult__dvd__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_1022_dvd__triv__left,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_1023_dvd__triv__left,axiom,
! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_1024_dvd__mult__left,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_1025_dvd__mult__left,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_1026_dvd__mult2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_1027_dvd__mult2,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_1028_dvd__mult,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_1029_dvd__mult,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_1030_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B: nat,A: nat] :
? [K: nat] :
( A
= ( times_times_nat @ B @ K ) ) ) ) ).
% dvd_def
thf(fact_1031_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B: int,A: int] :
? [K: int] :
( A
= ( times_times_int @ B @ K ) ) ) ) ).
% dvd_def
thf(fact_1032_dvdI,axiom,
! [A2: nat,B2: nat,K2: nat] :
( ( A2
= ( times_times_nat @ B2 @ K2 ) )
=> ( dvd_dvd_nat @ B2 @ A2 ) ) ).
% dvdI
thf(fact_1033_dvdI,axiom,
! [A2: int,B2: int,K2: int] :
( ( A2
= ( times_times_int @ B2 @ K2 ) )
=> ( dvd_dvd_int @ B2 @ A2 ) ) ).
% dvdI
thf(fact_1034_dvdE,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ~ ! [K3: nat] :
( A2
!= ( times_times_nat @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_1035_dvdE,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ~ ! [K3: int] :
( A2
!= ( times_times_int @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_1036_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_1037_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1038_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_1039_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_1040_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_1041_zdvd__period,axiom,
! [A2: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A2 @ D )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_1042_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
= ( ? [X3: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1043_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
= ( ? [X3: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1044_unit__dvdE,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [C2: nat] :
( B2
!= ( times_times_nat @ A2 @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_1045_unit__dvdE,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [C2: int] :
( B2
!= ( times_times_int @ A2 @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_1046_dvd__div__eq__mult,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( ( divide_divide_nat @ B2 @ A2 )
= C )
= ( B2
= ( times_times_nat @ C @ A2 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1047_dvd__div__eq__mult,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( ( divide_divide_int @ B2 @ A2 )
= C )
= ( B2
= ( times_times_int @ C @ A2 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1048_div__dvd__iff__mult,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1049_div__dvd__iff__mult,axiom,
! [B2: int,A2: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1050_dvd__div__iff__mult,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1051_dvd__div__iff__mult,axiom,
! [C: int,B2: int,A2: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1052_dvd__div__div__eq__mult,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( A2 != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B2 @ A2 )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B2 @ C )
= ( times_times_nat @ A2 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1053_dvd__div__div__eq__mult,axiom,
! [A2: int,C: int,B2: int,D: int] :
( ( A2 != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B2 @ A2 )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B2 @ C )
= ( times_times_int @ A2 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1054_inf__period_I4_J,axiom,
! [D: int,D3: int,T: int] :
( ( dvd_dvd_int @ D @ D3 )
=> ! [X5: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_1055_inf__period_I3_J,axiom,
! [D: int,D3: int,T: int] :
( ( dvd_dvd_int @ D @ D3 )
=> ! [X5: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_1056_unit__div__eq__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A2 @ B2 )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1057_unit__div__eq__0__iff,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A2 @ B2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1058_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_1059_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_1060_is__unit__div__mult2__eq,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1061_is__unit__div__mult2__eq,axiom,
! [B2: int,C: int,A2: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1062_unit__div__mult__swap,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_1063_unit__div__mult__swap,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_1064_unit__div__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_1065_unit__div__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
= ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_1066_div__mult__unit2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_1067_div__mult__unit2,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_1068_unit__eq__div2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( A2
= ( divide_divide_nat @ C @ B2 ) )
= ( ( times_times_nat @ A2 @ B2 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_1069_unit__eq__div2,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( A2
= ( divide_divide_int @ C @ B2 ) )
= ( ( times_times_int @ A2 @ B2 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_1070_unit__eq__div1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A2 @ B2 )
= C )
= ( A2
= ( times_times_nat @ C @ B2 ) ) ) ) ).
% unit_eq_div1
thf(fact_1071_unit__eq__div1,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A2 @ B2 )
= C )
= ( A2
= ( times_times_int @ C @ B2 ) ) ) ) ).
% unit_eq_div1
thf(fact_1072_is__unit__power__iff,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_1073_is__unit__power__iff,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_1074_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_1075_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_1076_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_1077_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_1078_is__unitE,axiom,
! [A2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [B4: nat] :
( ( B4 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B4 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A2 )
= B4 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B4 )
= A2 )
=> ( ( ( times_times_nat @ A2 @ B4 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A2 )
!= ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1079_is__unitE,axiom,
! [A2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [B4: int] :
( ( B4 != zero_zero_int )
=> ( ( dvd_dvd_int @ B4 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A2 )
= B4 )
=> ( ( ( divide_divide_int @ one_one_int @ B4 )
= A2 )
=> ( ( ( times_times_int @ A2 @ B4 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A2 )
!= ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1080_is__unit__div__mult__cancel__left,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) )
= ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1081_is__unit__div__mult__cancel__left,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ A2 @ B2 ) )
= ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1082_is__unit__div__mult__cancel__right,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) )
= ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1083_is__unit__div__mult__cancel__right,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ A2 ) )
= ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1084_evenE,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B4: nat] :
( A2
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% evenE
thf(fact_1085_evenE,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B4: int] :
( A2
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% evenE
thf(fact_1086_odd__even__add,axiom,
! [A2: nat,B2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% odd_even_add
thf(fact_1087_odd__even__add,axiom,
! [A2: int,B2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% odd_even_add
thf(fact_1088_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_1089_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_1090_bit__eq__rec,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
& ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_1091_bit__eq__rec,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A: int,B: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
& ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_1092_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_1093_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_1094_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_1095_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_1096_even__two__times__div__two,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= A2 ) ) ).
% even_two_times_div_two
thf(fact_1097_even__two__times__div__two,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= A2 ) ) ).
% even_two_times_div_two
thf(fact_1098_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_1099_even__unset__bit__iff,axiom,
! [M: nat,A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_1100_even__unset__bit__iff,axiom,
! [M: nat,A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_1101_even__set__bit__iff,axiom,
! [M: nat,A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_1102_even__set__bit__iff,axiom,
! [M: nat,A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_1103_even__diff__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K2 @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).
% even_diff_iff
thf(fact_1104_oddE,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B4: nat] :
( A2
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_1105_oddE,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B4: int] :
( A2
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_1106_zero__less__power__eq,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A2 != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).
% zero_less_power_eq
thf(fact_1107_aux__even__pow2__factor,axiom,
! [A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ? [K3: nat,B4: nat] :
( ( A2
= ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) @ B4 ) )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% aux_even_pow2_factor
thf(fact_1108_pow__divides__pow__iff,axiom,
! [N: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).
% pow_divides_pow_iff
thf(fact_1109_pow__divides__pow__iff,axiom,
! [N: nat,A2: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% pow_divides_pow_iff
thf(fact_1110_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ? [D4: nat,X4: nat,Y4: nat] :
( ( dvd_dvd_nat @ D4 @ A2 )
& ( dvd_dvd_nat @ D4 @ B2 )
& ( ( times_times_nat @ A2 @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1111_even__mult__exp__div__exp__iff,axiom,
! [A2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1112_even__mult__exp__div__exp__iff,axiom,
! [A2: int,M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1113_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1114_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_1115_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_1116_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1117_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1118_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1119_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1120_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_1121_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1122_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1123_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1124_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_1125_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1126_lessThan__subset__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_1127_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1128_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1129_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1130_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1131_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1132_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1133_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1134_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1135_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1136_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1137_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1138_le__add__diff__inverse,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1139_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1140_le__add__diff__inverse2,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1141_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1142_atLeastLessThan__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_num @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1143_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1144_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1145_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1146_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1147_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1148_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1149_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1150_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1151_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_1152_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_1153_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1154_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_1155_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_1156_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1157_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1158_power__increasing__iff,axiom,
! [B2: int,X: nat,Y3: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y3 ) )
= ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% power_increasing_iff
thf(fact_1159_power__increasing__iff,axiom,
! [B2: nat,X: nat,Y3: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y3 ) )
= ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% power_increasing_iff
thf(fact_1160_power__mono__iff,axiom,
! [A2: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1161_power__mono__iff,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1162_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_1163_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_1164_power2__less__eq__zero__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_1165_power2__eq__iff__nonneg,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y3 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1166_power2__eq__iff__nonneg,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y3 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1167_zero__le__power__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( 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 @ A2 ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1168_power__le__zero__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( 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 @ A2 @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A2 = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1169_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1170_dvd__diffD,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K2 @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1171_dvd__diffD1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1172_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1173_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1174_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1175_zero__le__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1176_mult__nonneg__nonpos2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1177_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1178_mult__nonpos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1179_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1180_mult__nonneg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1181_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1182_mult__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1183_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1184_split__mult__neg__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1185_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1186_mult__le__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_1187_mult__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1188_mult__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1189_mult__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1190_mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1191_mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1192_mult__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1193_mult__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1194_split__mult__pos__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1195_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1196_mult__mono_H,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1197_mult__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1198_mult__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ 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 @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1199_mult__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ 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 @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1200_add__nonpos__eq__0__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y3 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1201_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1202_add__nonneg__eq__0__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
=> ( ( ( plus_plus_int @ X @ Y3 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1203_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1204_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1205_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1206_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1207_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1208_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1209_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1210_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1211_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1212_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_1213_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_1214_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1215_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1216_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1217_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1218_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_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_add__le__less__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1227_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1228_add__less__le__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1229_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1230_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1231_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_1232_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1233_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1234_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1235_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1236_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1237_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1238_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1239_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1240_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1241_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1242_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1243_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1244_diff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% diff_add
thf(fact_1245_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1246_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1247_le__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_1248_diff__le__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_1249_zero__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_1250_zero__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_1251_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_1252_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1253_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1254_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_1255_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_1256_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1257_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_1258_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_1259_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_1260_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_1261_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
? [K: nat] :
( N5
= ( plus_plus_nat @ M5 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_1262_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1263_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1264_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_1265_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_1266_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
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y3: int] :
( ( if_int @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y3: int] :
( ( if_int @ $true @ X @ Y3 )
= 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,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_nat
@ ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_bit @ n @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ zero_zero_nat ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ zero_zero_nat ) ) ) ).
%------------------------------------------------------------------------------