TPTP Problem File: SLH0531^1.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 : Number_Theoretic_Transform/0008_Butterfly/prob_00876_044971__14183076_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1349 ( 731 unt; 77 typ; 0 def)
% Number of atoms : 3175 (1735 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10062 ( 328 ~; 74 |; 202 &;8589 @)
% ( 0 <=>; 869 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 224 ( 224 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 68 usr; 13 con; 0-3 aty)
% Number of variables : 2964 ( 102 ^;2766 !; 96 ?;2964 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:40:10.938
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $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__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (68)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Butterfly_Obutterfly_Oevens__odds_001t__Nat__Onat,type,
evens_odds_nat: $o > list_nat > list_nat ).
thf(sy_c_Butterfly_Obutterfly_Oevens__odds_001tf__b,type,
evens_odds_b: $o > list_b > list_b ).
thf(sy_c_Discrete_Olog,type,
log: nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001tf__b,type,
filter_b: ( b > $o ) > list_b > list_b ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__b,type,
cons_b: b > list_b > list_b ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__b,type,
map_nat_b: ( nat > b ) > list_nat > list_b ).
thf(sy_c_List_Olist_Omap_001tf__b_001t__Nat__Onat,type,
map_b_nat: ( b > nat ) > list_b > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__b,type,
map_b_b: ( b > b ) > list_b > list_b ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__b,type,
nth_b: list_b > nat > b ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
size_size_list_b: list_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_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__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > 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_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $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_v_k,type,
k: nat ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_x____,type,
x: b ).
thf(sy_v_xsa____,type,
xsa: list_b ).
% Relevant facts (1267)
thf(fact_0_evens__odds_Osimps_I3_J,axiom,
! [X: nat,Xs: list_nat] :
( ( evens_odds_nat @ $false @ ( cons_nat @ X @ Xs ) )
= ( evens_odds_nat @ $true @ Xs ) ) ).
% evens_odds.simps(3)
thf(fact_1_evens__odds_Osimps_I3_J,axiom,
! [X: b,Xs: list_b] :
( ( evens_odds_b @ $false @ ( cons_b @ X @ Xs ) )
= ( evens_odds_b @ $true @ Xs ) ) ).
% evens_odds.simps(3)
thf(fact_2_evens__odds_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( evens_odds_nat @ $true @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( evens_odds_nat @ $false @ Xs ) ) ) ).
% evens_odds.simps(2)
thf(fact_3_evens__odds_Osimps_I2_J,axiom,
! [X: b,Xs: list_b] :
( ( evens_odds_b @ $true @ ( cons_b @ X @ Xs ) )
= ( cons_b @ X @ ( evens_odds_b @ $false @ Xs ) ) ) ).
% evens_odds.simps(2)
thf(fact_4__092_060open_062map_A_I_I_B_J_Axs_J_A_Ifilter_Aeven_A_0910_O_O_060length_Axs_093_J_A_061_Aevens__odds_ATrue_Axs_A_092_060and_062_Amap_A_I_I_B_J_Axs_J_A_Ifilter_Aodd_A_0910_O_O_060length_Axs_093_J_A_061_Aevens__odds_AFalse_Axs_A_092_060Longrightarrow_062_Amap_A_I_I_B_J_A_Ix_A_D_Axs_J_J_A_Ifilter_Aeven_A_0910_O_O_060length_A_Ix_A_D_Axs_J_093_J_A_061_Aevens__odds_ATrue_A_Ix_A_D_Axs_J_092_060close_062,axiom,
( ( ( ( map_nat_b @ ( nth_b @ xsa ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_size_list_b @ xsa ) ) ) )
= ( evens_odds_b @ $true @ xsa ) )
& ( ( map_nat_b @ ( nth_b @ xsa )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_size_list_b @ xsa ) ) ) )
= ( evens_odds_b @ $false @ xsa ) ) )
=> ( ( map_nat_b @ ( nth_b @ ( cons_b @ x @ xsa ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_size_list_b @ ( cons_b @ x @ xsa ) ) ) ) )
= ( evens_odds_b @ $true @ ( cons_b @ x @ xsa ) ) ) ) ).
% \<open>map ((!) xs) (filter even [0..<length xs]) = evens_odds True xs \<and> map ((!) xs) (filter odd [0..<length xs]) = evens_odds False xs \<Longrightarrow> map ((!) (x # xs)) (filter even [0..<length (x # xs)]) = evens_odds True (x # xs)\<close>
thf(fact_5_nth__Cons__0,axiom,
! [X: b,Xs: list_b] :
( ( nth_b @ ( cons_b @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_6_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_7_length__odd__filter,axiom,
! [F: nat > b,L: nat] :
( ( size_size_list_b
@ ( map_nat_b @ F
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ L ) ) ) )
= ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% length_odd_filter
thf(fact_8_length__odd__filter,axiom,
! [F: nat > nat,L: nat] :
( ( size_size_list_nat
@ ( map_nat_nat @ F
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ L ) ) ) )
= ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% length_odd_filter
thf(fact_9_map__nth,axiom,
! [Xs: list_b] :
( ( map_nat_b @ ( nth_b @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_b @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_10_map__nth,axiom,
! [Xs: list_nat] :
( ( map_nat_nat @ ( nth_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_11_length__map,axiom,
! [F: b > b,Xs: list_b] :
( ( size_size_list_b @ ( map_b_b @ F @ Xs ) )
= ( size_size_list_b @ Xs ) ) ).
% length_map
thf(fact_12_length__map,axiom,
! [F: nat > b,Xs: list_nat] :
( ( size_size_list_b @ ( map_nat_b @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_13_length__map,axiom,
! [F: b > nat,Xs: list_b] :
( ( size_size_list_nat @ ( map_b_nat @ F @ Xs ) )
= ( size_size_list_b @ Xs ) ) ).
% length_map
thf(fact_14_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_15_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_16_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_17_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_18_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_19_dvd__0__right,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_20_dvd__0__right,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ zero_zero_int ) ).
% dvd_0_right
thf(fact_21_dvd__0__right,axiom,
! [A2: real] : ( dvd_dvd_real @ A2 @ zero_zero_real ) ).
% dvd_0_right
thf(fact_22_dvd__0__left__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_23_dvd__0__left__iff,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
= ( A2 = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_24_dvd__0__left__iff,axiom,
! [A2: real] :
( ( dvd_dvd_real @ zero_zero_real @ A2 )
= ( A2 = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_25_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_26_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_27_filter__filter,axiom,
! [P: nat > $o,Q: nat > $o,Xs: list_nat] :
( ( filter_nat @ P @ ( filter_nat @ Q @ Xs ) )
= ( filter_nat
@ ^ [X2: nat] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_28_map__ident,axiom,
( ( map_nat_nat
@ ^ [X2: nat] : X2 )
= ( ^ [Xs2: list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_29_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_30_list_Oinject,axiom,
! [X21: b,X22: list_b,Y21: b,Y22: list_b] :
( ( ( cons_b @ X21 @ X22 )
= ( cons_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_31_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_32_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_33_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_34_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_35_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_36_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_37_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_38_div__dvd__div,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A2 ) @ ( divide_divide_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_39_div__dvd__div,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A2 ) @ ( divide_divide_int @ C @ A2 ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_40_div__div__div__same,axiom,
! [D: nat,B: nat,A2: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A2 )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_div_div_same
thf(fact_41_div__div__div__same,axiom,
! [D: int,B: int,A2: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A2 )
=> ( ( divide_divide_int @ ( divide_divide_int @ A2 @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_div_div_same
thf(fact_42_dvd__div__eq__cancel,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( A2 = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_43_dvd__div__eq__cancel,axiom,
! [A2: int,C: int,B: int] :
( ( ( divide_divide_int @ A2 @ C )
= ( divide_divide_int @ B @ C ) )
=> ( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B )
=> ( A2 = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_44_dvd__div__eq__cancel,axiom,
! [A2: real,C: real,B: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B @ C ) )
=> ( ( dvd_dvd_real @ C @ A2 )
=> ( ( dvd_dvd_real @ C @ B )
=> ( A2 = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_45_dvd__div__eq__iff,axiom,
! [C: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B @ C ) )
= ( A2 = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_46_dvd__div__eq__iff,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( ( divide_divide_int @ A2 @ C )
= ( divide_divide_int @ B @ C ) )
= ( A2 = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_47_dvd__div__eq__iff,axiom,
! [C: real,A2: real,B: real] :
( ( dvd_dvd_real @ C @ A2 )
=> ( ( dvd_dvd_real @ C @ B )
=> ( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B @ C ) )
= ( A2 = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_48_dvd__div__eq__0__iff,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ A2 )
=> ( ( ( divide_divide_nat @ A2 @ B )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_49_dvd__div__eq__0__iff,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ A2 )
=> ( ( ( divide_divide_int @ A2 @ B )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_50_dvd__div__eq__0__iff,axiom,
! [B: real,A2: real] :
( ( dvd_dvd_real @ B @ A2 )
=> ( ( ( divide_divide_real @ A2 @ B )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_51_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_52_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_53_dvd__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_54_dvd__trans,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_55_dvd__refl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% dvd_refl
thf(fact_56_dvd__refl,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ A2 ) ).
% dvd_refl
thf(fact_57_not__Cons__self2,axiom,
! [X: b,Xs: list_b] :
( ( cons_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_58_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_59_neq__if__length__neq,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
!= ( size_size_list_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_60_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_61_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_b] :
( ( size_size_list_b @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_62_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_63_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X2: nat] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_64_dvd__0__left,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_65_dvd__0__left,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
=> ( A2 = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_66_dvd__0__left,axiom,
! [A2: real] :
( ( dvd_dvd_real @ zero_zero_real @ A2 )
=> ( A2 = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_67_map__eq__Cons__conv,axiom,
! [F: b > b,Xs: list_b,Y: b,Ys: list_b] :
( ( ( map_b_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
= ( ? [Z: b,Zs: list_b] :
( ( Xs
= ( cons_b @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_b_b @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_68_map__eq__Cons__conv,axiom,
! [F: nat > b,Xs: list_nat,Y: b,Ys: list_b] :
( ( ( map_nat_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
= ( ? [Z: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_b @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_69_map__eq__Cons__conv,axiom,
! [F: b > nat,Xs: list_b,Y: nat,Ys: list_nat] :
( ( ( map_b_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z: b,Zs: list_b] :
( ( Xs
= ( cons_b @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_b_nat @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_70_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_71_Cons__eq__map__conv,axiom,
! [X: b,Xs: list_b,F: b > b,Ys: list_b] :
( ( ( cons_b @ X @ Xs )
= ( map_b_b @ F @ Ys ) )
= ( ? [Z: b,Zs: list_b] :
( ( Ys
= ( cons_b @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_b_b @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_72_Cons__eq__map__conv,axiom,
! [X: b,Xs: list_b,F: nat > b,Ys: list_nat] :
( ( ( cons_b @ X @ Xs )
= ( map_nat_b @ F @ Ys ) )
= ( ? [Z: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_b @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_73_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: b > nat,Ys: list_b] :
( ( ( cons_nat @ X @ Xs )
= ( map_b_nat @ F @ Ys ) )
= ( ? [Z: b,Zs: list_b] :
( ( Ys
= ( cons_b @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_b_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_74_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_75_map__eq__Cons__D,axiom,
! [F: b > b,Xs: list_b,Y: b,Ys: list_b] :
( ( ( map_b_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
=> ? [Z2: b,Zs2: list_b] :
( ( Xs
= ( cons_b @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_b_b @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_76_map__eq__Cons__D,axiom,
! [F: nat > b,Xs: list_nat,Y: b,Ys: list_b] :
( ( ( map_nat_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_b @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_77_map__eq__Cons__D,axiom,
! [F: b > nat,Xs: list_b,Y: nat,Ys: list_nat] :
( ( ( map_b_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z2: b,Zs2: list_b] :
( ( Xs
= ( cons_b @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_b_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_78_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_79_Cons__eq__map__D,axiom,
! [X: b,Xs: list_b,F: b > b,Ys: list_b] :
( ( ( cons_b @ X @ Xs )
= ( map_b_b @ F @ Ys ) )
=> ? [Z2: b,Zs2: list_b] :
( ( Ys
= ( cons_b @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_b_b @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_80_Cons__eq__map__D,axiom,
! [X: b,Xs: list_b,F: nat > b,Ys: list_nat] :
( ( ( cons_b @ X @ Xs )
= ( map_nat_b @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_b @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_81_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: b > nat,Ys: list_b] :
( ( ( cons_nat @ X @ Xs )
= ( map_b_nat @ F @ Ys ) )
=> ? [Z2: b,Zs2: list_b] :
( ( Ys
= ( cons_b @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_b_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_82_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_83_list_Osimps_I9_J,axiom,
! [F: b > b,X21: b,X22: list_b] :
( ( map_b_b @ F @ ( cons_b @ X21 @ X22 ) )
= ( cons_b @ ( F @ X21 ) @ ( map_b_b @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_84_list_Osimps_I9_J,axiom,
! [F: b > nat,X21: b,X22: list_b] :
( ( map_b_nat @ F @ ( cons_b @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_b_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_85_list_Osimps_I9_J,axiom,
! [F: nat > b,X21: nat,X22: list_nat] :
( ( map_nat_b @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_b @ ( F @ X21 ) @ ( map_nat_b @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_86_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_87_map__eq__imp__length__eq,axiom,
! [F: b > b,Xs: list_b,G: nat > b,Ys: list_nat] :
( ( ( map_b_b @ F @ Xs )
= ( map_nat_b @ G @ Ys ) )
=> ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_88_map__eq__imp__length__eq,axiom,
! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
( ( ( map_b_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_89_map__eq__imp__length__eq,axiom,
! [F: nat > b,Xs: list_nat,G: b > b,Ys: list_b] :
( ( ( map_nat_b @ F @ Xs )
= ( map_b_b @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_90_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: b > nat,Ys: list_b] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_b_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_91_map__eq__imp__length__eq,axiom,
! [F: nat > b,Xs: list_nat,G: nat > b,Ys: list_nat] :
( ( ( map_nat_b @ F @ Xs )
= ( map_nat_b @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_92_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_93_filter_Osimps_I2_J,axiom,
! [P: b > $o,X: b,Xs: list_b] :
( ( ( P @ X )
=> ( ( filter_b @ P @ ( cons_b @ X @ Xs ) )
= ( cons_b @ X @ ( filter_b @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_b @ P @ ( cons_b @ X @ Xs ) )
= ( filter_b @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_94_filter_Osimps_I2_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( ( P @ X )
=> ( ( filter_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( filter_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( filter_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_95_length__even__filter,axiom,
! [F: nat > b,L: nat] :
( ( size_size_list_b @ ( map_nat_b @ F @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ L ) ) ) )
= ( minus_minus_nat @ L @ ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% length_even_filter
thf(fact_96_length__even__filter,axiom,
! [F: nat > nat,L: nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ L ) ) ) )
= ( minus_minus_nat @ L @ ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% length_even_filter
thf(fact_97_filter__even__map,axiom,
! [X: nat] :
( ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) )
= ( map_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ X ) ) ) ).
% filter_even_map
thf(fact_98_bit__eq__rec,axiom,
( ( ^ [Y2: nat,Z3: nat] : ( Y2 = Z3 ) )
= ( ^ [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_99_bit__eq__rec,axiom,
( ( ^ [Y2: int,Z3: int] : ( Y2 = Z3 ) )
= ( ^ [A: int,B2: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_100_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_101_bits__div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_102_bits__div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_103_bits__div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_104_bits__div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_105_divide__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_right
thf(fact_106_divide__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( divide_divide_real @ C @ A2 )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_left
thf(fact_107_divide__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_108_divide__numeral__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ ( numeral_numeral_real @ one ) )
= A2 ) ).
% divide_numeral_1
thf(fact_109_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y23: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y23 ) )
= ( X23 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_110_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_111_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_112_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_113_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_114_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_115_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_116_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_117_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_118_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_119_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_120_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_121_mult__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_122_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_123_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_124_mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_125_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_126_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_127_mult__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_128_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_129_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_130_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_131_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z4: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z4 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z4 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_132_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z4: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z4 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z4 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_133_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z4: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z4 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z4 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_134_times__divide__eq__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_135_divide__divide__eq__right,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_136_divide__divide__eq__left,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_137_times__divide__eq__left,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( divide_divide_real @ ( times_times_real @ B @ A2 ) @ C ) ) ).
% times_divide_eq_left
thf(fact_138_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_139_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_140_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_141_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_142_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_143_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A2: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_144_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_145_nonzero__mult__div__cancel__right,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_146_nonzero__mult__div__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_147_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_148_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_149_nonzero__mult__div__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_150_dvd__mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_151_dvd__mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_152_dvd__mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_153_dvd__mult__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_154_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_155_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_156_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_157_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_158_left__diff__distrib__numeral,axiom,
! [A2: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_159_left__diff__distrib__numeral,axiom,
! [A2: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_160_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_161_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_162_dvd__mult__div__cancel,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B @ A2 ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_163_dvd__mult__div__cancel,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B @ A2 ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_164_dvd__div__mult__self,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% dvd_div_mult_self
thf(fact_165_dvd__div__mult__self,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A2 ) @ A2 )
= B ) ) ).
% dvd_div_mult_self
thf(fact_166_div__diff,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_167_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_168_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A2: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A2 )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_169_eq__divide__eq__numeral1_I1_J,axiom,
! [A2: real,B: real,W: num] :
( ( A2
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_170_even__mult__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_171_even__mult__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_172_left__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_173_left__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_174_right__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_175_right__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_176_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_177_left__diff__distrib_H,axiom,
! [B: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_178_left__diff__distrib_H,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_179_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_180_right__diff__distrib_H,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_181_right__diff__distrib_H,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_182_diff__divide__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_183_divide__divide__eq__left_H,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_184_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z4: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z4 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z4 ) ) ) ).
% divide_divide_times_eq
thf(fact_185_times__divide__times__eq,axiom,
! [X: real,Y: real,Z4: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z4 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z4 ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_186_divide__diff__eq__iff,axiom,
! [Z4: real,X: real,Y: real] :
( ( Z4 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z4 ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z4 ) ) @ Z4 ) ) ) ).
% divide_diff_eq_iff
thf(fact_187_diff__divide__eq__iff,axiom,
! [Z4: real,X: real,Y: real] :
( ( Z4 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z4 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z4 ) @ Y ) @ Z4 ) ) ) ).
% diff_divide_eq_iff
thf(fact_188_diff__frac__eq,axiom,
! [Y: real,Z4: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z4 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z4 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z4 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z4 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_189_add__divide__eq__if__simps_I4_J,axiom,
! [Z4: real,A2: real,B: real] :
( ( ( Z4 = zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z4 ) )
= A2 ) )
& ( ( Z4 != zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z4 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z4 ) @ B ) @ Z4 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_190_dvd__diff,axiom,
! [X: int,Y: int,Z4: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z4 )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z4 ) ) ) ) ).
% dvd_diff
thf(fact_191_dvd__diff,axiom,
! [X: real,Y: real,Z4: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z4 )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z4 ) ) ) ) ).
% dvd_diff
thf(fact_192_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_193_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_194_mult__not__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_195_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_196_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_197_divisors__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_198_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_199_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_200_no__zero__divisors,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A2 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_201_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_202_mult__left__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_203_mult__left__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_204_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_205_mult__right__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_206_mult__right__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_207_dvdE,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ A2 )
=> ~ ! [K2: nat] :
( A2
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_208_dvdE,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ A2 )
=> ~ ! [K2: int] :
( A2
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_209_dvdE,axiom,
! [B: real,A2: real] :
( ( dvd_dvd_real @ B @ A2 )
=> ~ ! [K2: real] :
( A2
!= ( times_times_real @ B @ K2 ) ) ) ).
% dvdE
thf(fact_210_dvdI,axiom,
! [A2: nat,B: nat,K: nat] :
( ( A2
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A2 ) ) ).
% dvdI
thf(fact_211_dvdI,axiom,
! [A2: int,B: int,K: int] :
( ( A2
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A2 ) ) ).
% dvdI
thf(fact_212_dvdI,axiom,
! [A2: real,B: real,K: real] :
( ( A2
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A2 ) ) ).
% dvdI
thf(fact_213_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A: nat] :
? [K3: nat] :
( A
= ( times_times_nat @ B2 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_214_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A: int] :
? [K3: int] :
( A
= ( times_times_int @ B2 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_215_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B2: real,A: real] :
? [K3: real] :
( A
= ( times_times_real @ B2 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_216_dvd__mult,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_217_dvd__mult,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_218_dvd__mult,axiom,
! [A2: real,C: real,B: real] :
( ( dvd_dvd_real @ A2 @ C )
=> ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_219_dvd__mult2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_220_dvd__mult2,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_221_dvd__mult2,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_222_dvd__mult__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_223_dvd__mult__left,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_224_dvd__mult__left,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ C )
=> ( dvd_dvd_real @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_225_dvd__triv__left,axiom,
! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_226_dvd__triv__left,axiom,
! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_227_dvd__triv__left,axiom,
! [A2: real,B: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_228_mult__dvd__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_229_mult__dvd__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_230_mult__dvd__mono,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_231_dvd__mult__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_232_dvd__mult__right,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_233_dvd__mult__right,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_234_dvd__triv__right,axiom,
! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_235_dvd__triv__right,axiom,
! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_236_dvd__triv__right,axiom,
! [A2: real,B: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_237_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_238_nonzero__eq__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A2
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A2 @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_239_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A2 )
= ( B
= ( times_times_real @ A2 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_240_eq__divide__imp,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= B )
=> ( A2
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_241_divide__eq__imp,axiom,
! [C: real,B: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A2 @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A2 ) ) ) ).
% divide_eq_imp
thf(fact_242_eq__divide__eq,axiom,
! [A2: real,B: real,C: real] :
( ( A2
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A2 @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_243_divide__eq__eq,axiom,
! [B: real,C: real,A2: real] :
( ( ( divide_divide_real @ B @ C )
= A2 )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A2 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_244_frac__eq__eq,axiom,
! [Y: real,Z4: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z4 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z4 ) )
= ( ( times_times_real @ X @ Z4 )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_245_mult__numeral__1,axiom,
! [A2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_246_mult__numeral__1,axiom,
! [A2: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_247_mult__numeral__1,axiom,
! [A2: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_248_mult__numeral__1__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_249_mult__numeral__1__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_250_mult__numeral__1__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ ( numeral_numeral_real @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_251_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_252_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_253_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_254_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_255_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_256_div__mult__div__if__dvd,axiom,
! [B: nat,A2: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B @ A2 )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_257_div__mult__div__if__dvd,axiom,
! [B: int,A2: int,D: int,C: int] :
( ( dvd_dvd_int @ B @ A2 )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_258_dvd__mult__imp__div,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B )
=> ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_259_dvd__mult__imp__div,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B )
=> ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_260_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_261_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A2: int] :
( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A2 )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_262_div__div__eq__right,axiom,
! [C: nat,B: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ B @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( divide_divide_nat @ B @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A2 @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_263_div__div__eq__right,axiom,
! [C: int,B: int,A2: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ B @ A2 )
=> ( ( divide_divide_int @ A2 @ ( divide_divide_int @ B @ C ) )
= ( times_times_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_264_div__mult__swap,axiom,
! [C: nat,B: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_265_div__mult__swap,axiom,
! [C: int,B: int,A2: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_266_dvd__div__mult,axiom,
! [C: nat,B: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A2 )
= ( divide_divide_nat @ ( times_times_nat @ B @ A2 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_267_dvd__div__mult,axiom,
! [C: int,B: int,A2: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A2 )
= ( divide_divide_int @ ( times_times_int @ B @ A2 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_268_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_269_dvd__div__eq__mult,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A2 @ B )
=> ( ( ( divide_divide_nat @ B @ A2 )
= C )
= ( B
= ( times_times_nat @ C @ A2 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_270_dvd__div__eq__mult,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ A2 @ B )
=> ( ( ( divide_divide_int @ B @ A2 )
= C )
= ( B
= ( times_times_int @ C @ A2 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_271_div__dvd__iff__mult,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A2 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_272_div__dvd__iff__mult,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A2 )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_273_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A2: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_274_dvd__div__iff__mult,axiom,
! [C: int,B: int,A2: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_275_dvd__div__div__eq__mult,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( A2 != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B @ A2 )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B @ C )
= ( times_times_nat @ A2 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_276_dvd__div__div__eq__mult,axiom,
! [A2: int,C: int,B: int,D: int] :
( ( A2 != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B @ A2 )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B @ C )
= ( times_times_int @ A2 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_277_evenE,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B3: nat] :
( A2
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% evenE
thf(fact_278_evenE,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B3: int] :
( A2
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% evenE
thf(fact_279_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_280_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_281_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_282_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_283_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_284_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A: real,B2: real] :
( ( A = zero_zero_real )
=> ( B2 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_285_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_286_dvd__imp__mult__div__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B @ A2 ) )
= B ) ) ).
% dvd_imp_mult_div_cancel_left
thf(fact_287_dvd__imp__mult__div__cancel__left,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B @ A2 ) )
= B ) ) ).
% dvd_imp_mult_div_cancel_left
thf(fact_288_dvd__imp__mult__div__cancel__left,axiom,
! [A2: real,B: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( ( times_times_real @ A2 @ ( divide_divide_real @ B @ A2 ) )
= B ) ) ).
% dvd_imp_mult_div_cancel_left
thf(fact_289_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_290_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: real,B: real,C: real] :
( ( A2 != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_291_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_292_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: real,B: real,C: real] :
( ( A2 != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C @ A2 ) )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_293_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_294_div__mult__mult1__if,axiom,
! [C: int,A2: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_295_div__mult__mult2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_296_div__mult__mult2,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_297_div__mult__mult1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_298_div__mult__mult1,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_299_filter__even__nth,axiom,
! [J: nat,L: nat,X: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) @ L )
=> ( ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= L )
=> ( ( nth_nat @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ L ) ) @ J )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) ) ) ) ).
% filter_even_nth
thf(fact_300_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_301_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_302_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_303_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_304_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_305_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_306_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_307_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_308_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_309_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_310_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_311_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_312_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_313_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_314_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_315_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_316_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_317_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_318_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_319_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_320_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_321_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_322_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A2 )
= ( ord_less_real @ B @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_323_less__divide__eq__numeral1_I1_J,axiom,
! [A2: real,B: real,W: num] :
( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_324_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_325_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_326_nth__map,axiom,
! [N: nat,Xs: list_b,F: b > b] :
( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( map_b_b @ F @ Xs ) @ N )
= ( F @ ( nth_b @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_327_nth__map,axiom,
! [N: nat,Xs: list_b,F: b > nat] :
( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_nat @ ( map_b_nat @ F @ Xs ) @ N )
= ( F @ ( nth_b @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_328_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > b] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_b @ ( map_nat_b @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_329_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_330_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_331_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X3 ) ).
% linordered_field_no_lb
thf(fact_332_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_333_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_334_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_335_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_336_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_337_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_338_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_339_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_340_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_341_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_342_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_343_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_344_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_345_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_346_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_347_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_348_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_349_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_350_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_351_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_352_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_353_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_354_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_355_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_356_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_357_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_358_length__induct,axiom,
! [P: list_b > $o,Xs: list_b] :
( ! [Xs3: list_b] :
( ! [Ys2: list_b] :
( ( ord_less_nat @ ( size_size_list_b @ Ys2 ) @ ( size_size_list_b @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_359_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_360_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_361_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_362_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_363_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_364_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_365_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_366_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_367_mult__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_368_mult__neg__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_369_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_370_not__square__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_371_mult__less__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_372_mult__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_373_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_374_mult__neg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_375_mult__neg__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_376_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_377_mult__pos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_378_mult__pos__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_379_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_380_mult__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_381_mult__pos__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_382_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_383_mult__pos__neg2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_384_mult__pos__neg2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_385_zero__less__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_386_zero__less__mult__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_387_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_388_zero__less__mult__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_389_zero__less__mult__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_390_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_391_zero__less__mult__pos2,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_392_zero__less__mult__pos2,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_393_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_394_mult__less__cancel__left__neg,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_395_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_396_mult__less__cancel__left__pos,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_397_mult__strict__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_398_mult__strict__left__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_399_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_400_mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_401_mult__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_402_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_403_mult__less__cancel__left__disj,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_404_mult__strict__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_405_mult__strict__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_406_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_407_mult__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_408_mult__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_409_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_410_mult__less__cancel__right__disj,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_411_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_412_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_413_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_414_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_415_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_416_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_417_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_418_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_419_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_420_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_421_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_422_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_423_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_424_divide__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_425_divide__less__cancel,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A2 ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_426_zero__less__divide__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_427_divide__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_428_divide__strict__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_429_div__mult2__numeral__eq,axiom,
! [A2: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_430_div__mult2__numeral__eq,axiom,
! [A2: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_431_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_432_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_433_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_434_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_435_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_b,Z3: list_b] : ( Y2 = Z3 ) )
= ( ^ [Xs2: list_b,Ys3: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs2 ) )
=> ( ( nth_b @ Xs2 @ I2 )
= ( nth_b @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_436_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z3: list_nat] : ( Y2 = Z3 ) )
= ( ^ [Xs2: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_437_Skolem__list__nth,axiom,
! [K: nat,P: nat > b > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: b] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_b @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_438_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: nat] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_439_nth__equalityI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ Xs @ I3 )
= ( nth_b @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_440_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_441_divide__less__eq,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_442_less__divide__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_443_neg__divide__less__eq,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_444_neg__less__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_445_pos__divide__less__eq,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_446_pos__less__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_447_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z4: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z4 @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z4 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_448_mult__imp__less__div__pos,axiom,
! [Y: real,Z4: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z4 @ Y ) @ X )
=> ( ord_less_real @ Z4 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_449_divide__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_450_divide__strict__left__mono__neg,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_451_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_452_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_453_map__equality__iff,axiom,
! [F: b > b,Xs: list_b,G: nat > b,Ys: list_nat] :
( ( ( map_b_b @ F @ Xs )
= ( map_nat_b @ G @ Ys ) )
= ( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_b @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_454_map__equality__iff,axiom,
! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
( ( ( map_b_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_b @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_455_map__equality__iff,axiom,
! [F: nat > b,Xs: list_nat,G: b > b,Ys: list_b] :
( ( ( map_nat_b @ F @ Xs )
= ( map_b_b @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_b @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_456_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: b > nat,Ys: list_b] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_b_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_b @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_457_map__equality__iff,axiom,
! [F: nat > b,Xs: list_nat,G: nat > b,Ys: list_nat] :
( ( ( map_nat_b @ F @ Xs )
= ( map_nat_b @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_458_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_459_size__neq__size__imp__neq,axiom,
! [X: list_b,Y: list_b] :
( ( ( size_size_list_b @ X )
!= ( size_size_list_b @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_460_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_461_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_462_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_463_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_464_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_465_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_466_frac__less__eq,axiom,
! [Y: real,Z4: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z4 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z4 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z4 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z4 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_467_half__gt__zero__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% half_gt_zero_iff
thf(fact_468_half__gt__zero,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_469_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_470_dvd__diff__commute,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( dvd_dvd_int @ A2 @ ( minus_minus_int @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_471_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_472_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_473_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_474_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_475_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_476_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_477_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_478_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_479_diff__gt__0__iff__gt,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_480_diff__gt__0__iff__gt,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_real @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_481_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_482_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_483_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_484_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_485_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_486_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_487_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_488_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_489_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_490_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_491_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_492_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_493_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_494_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_495_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_496_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_497_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_498_div__neg__pos__less0,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_499_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_500_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_501_strict__subset__divisors__dvd,axiom,
! [A2: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A2 ) )
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ~ ( dvd_dvd_nat @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_502_strict__subset__divisors__dvd,axiom,
! [A2: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A2 ) )
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
= ( ( dvd_dvd_int @ A2 @ B )
& ~ ( dvd_dvd_int @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_503_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_504_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_505_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_506_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_507_mult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_508_mult_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_509_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A: nat,B2: nat] : ( times_times_nat @ B2 @ A ) ) ) ).
% mult.commute
thf(fact_510_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A: int,B2: int] : ( times_times_int @ B2 @ A ) ) ) ).
% mult.commute
thf(fact_511_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A: real,B2: real] : ( times_times_real @ B2 @ A ) ) ) ).
% mult.commute
thf(fact_512_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_513_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_514_mult_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_515_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_516_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_517_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_518_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_519_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_520_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_521_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_522_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_523_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_524_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_525_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_526_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_527_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_528_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z3: int] : ( Y2 = Z3 ) )
= ( ^ [A: int,B2: int] :
( ( minus_minus_int @ A @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_529_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z3: real] : ( Y2 = Z3 ) )
= ( ^ [A: real,B2: real] :
( ( minus_minus_real @ A @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_530_diff__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_531_diff__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_532_diff__strict__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_533_diff__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_534_diff__eq__diff__less,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_535_diff__eq__diff__less,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A2 @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_536_diff__strict__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_537_diff__strict__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_538_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_539_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_540_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_541_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_542_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_543_FNTT__termination__aux,axiom,
! [P: nat > $o,L: nat] : ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P @ ( upt @ zero_zero_nat @ L ) ) ) @ ( suc @ L ) ) ).
% FNTT_termination_aux
thf(fact_544_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_545_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_546_dvd__div__eq__2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( ( divide_divide_nat @ C @ A2 )
= ( divide_divide_nat @ C @ B ) )
=> ( A2 = B ) ) ) ) ) ).
% dvd_div_eq_2
thf(fact_547_bezout1__nat,axiom,
! [A2: nat,B: nat] :
? [D2: nat,X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X5 ) @ ( times_times_nat @ B @ Y3 ) )
= D2 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A2 @ Y3 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_548_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_549_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_550_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_551_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_552_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_553_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_554_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_555_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_556_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_557_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_558_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_559_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_560_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_561_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_562_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_563_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_564_nth__Cons__Suc,axiom,
! [X: b,Xs: list_b,N: nat] :
( ( nth_b @ ( cons_b @ X @ Xs ) @ ( suc @ N ) )
= ( nth_b @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_565_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_566_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_567_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_568_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_569_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_570_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_571_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_572_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_573_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_574_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_575_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_576_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_577_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_578_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_579_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_580_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_581_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X5: nat,Y3: nat] :
( ( P @ X5 @ Y3 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_582_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_583_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_584_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_585_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_586_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_587_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_588_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_589_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_590_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_591_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_592_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_593_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_594_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_595_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_596_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_597_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_598_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_599_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_600_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_601_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_602_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_603_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_604_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_605_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_606_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_607_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_608_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_609_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_610_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_611_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_612_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_613_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_614_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_615_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_616_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_617_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_618_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_619_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_620_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_621_length__Suc__conv,axiom,
! [Xs: list_b,N: nat] :
( ( ( size_size_list_b @ Xs )
= ( suc @ N ) )
= ( ? [Y4: b,Ys3: list_b] :
( ( Xs
= ( cons_b @ Y4 @ Ys3 ) )
& ( ( size_size_list_b @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_622_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_623_Suc__length__conv,axiom,
! [N: nat,Xs: list_b] :
( ( ( suc @ N )
= ( size_size_list_b @ Xs ) )
= ( ? [Y4: b,Ys3: list_b] :
( ( Xs
= ( cons_b @ Y4 @ Ys3 ) )
& ( ( size_size_list_b @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_624_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_625_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_626_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_627_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_628_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_629_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_630_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_631_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_632_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_633_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_634_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_635_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_636_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( A2 != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_637_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( dvd_dvd_nat @ A2 @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_638_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_639_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
| ( A = B2 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_640_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ~ ( dvd_dvd_nat @ B @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_641_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_642_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_643_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_644_gcd__nat_Oantisym,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% gcd_nat.antisym
thf(fact_645_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_646_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z3: nat] : ( Y2 = Z3 ) )
= ( ^ [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
& ( dvd_dvd_nat @ B2 @ A ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_647_gcd__nat_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% gcd_nat.trans
thf(fact_648_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_649_gcd__nat_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_650_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_651_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_652_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_653_div__if,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M5 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_654_map__upt__Suc,axiom,
! [F: nat > b,N: nat] :
( ( map_nat_b @ F @ ( upt @ zero_zero_nat @ ( suc @ N ) ) )
= ( cons_b @ ( F @ zero_zero_nat )
@ ( map_nat_b
@ ^ [I2: nat] : ( F @ ( suc @ I2 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_655_map__upt__Suc,axiom,
! [F: nat > nat,N: nat] :
( ( map_nat_nat @ F @ ( upt @ zero_zero_nat @ ( suc @ N ) ) )
= ( cons_nat @ ( F @ zero_zero_nat )
@ ( map_nat_nat
@ ^ [I2: nat] : ( F @ ( suc @ I2 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_656_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_657_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_658_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_659_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_660_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_661_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_662_division__decomp,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
=> ? [B4: nat,C3: nat] :
( ( A2
= ( times_times_nat @ B4 @ C3 ) )
& ( dvd_dvd_nat @ B4 @ B )
& ( dvd_dvd_nat @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_663_division__decomp,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
=> ? [B4: int,C3: int] :
( ( A2
= ( times_times_int @ B4 @ C3 ) )
& ( dvd_dvd_int @ B4 @ B )
& ( dvd_dvd_int @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_664_dvd__productE,axiom,
! [P2: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A2 @ B ) )
=> ~ ! [X5: nat,Y3: nat] :
( ( P2
= ( times_times_nat @ X5 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X5 @ A2 )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_665_dvd__productE,axiom,
! [P2: int,A2: int,B: int] :
( ( dvd_dvd_int @ P2 @ ( times_times_int @ A2 @ B ) )
=> ~ ! [X5: int,Y3: int] :
( ( P2
= ( times_times_int @ X5 @ Y3 ) )
=> ( ( dvd_dvd_int @ X5 @ A2 )
=> ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_666_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_667_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
& ( A2 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_668_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_669_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
& ( zero_zero_nat != A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_670_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_671_dvd__div__eq__1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( A2 = B ) ) ) ) ).
% dvd_div_eq_1
thf(fact_672_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_673_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_674_length__Cons,axiom,
! [X: b,Xs: list_b] :
( ( size_size_list_b @ ( cons_b @ X @ Xs ) )
= ( suc @ ( size_size_list_b @ Xs ) ) ) ).
% length_Cons
thf(fact_675_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_676_log__twice,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( log @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( suc @ ( log @ N ) ) ) ) ).
% log_twice
thf(fact_677_mult__less__iff1,axiom,
! [Z4: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z4 ) @ ( times_times_int @ Y @ Z4 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_678_mult__less__iff1,axiom,
! [Z4: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z4 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z4 ) @ ( times_times_real @ Y @ Z4 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_679_p__lst3,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).
% p_lst3
thf(fact_680_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_681_even__flip__bit__iff,axiom,
! [M: nat,A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_682_even__flip__bit__iff,axiom,
! [M: nat,A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_683_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_684_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_685_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_686_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_687_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_688_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_689_bits__div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% bits_div_by_1
thf(fact_690_bits__div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% bits_div_by_1
thf(fact_691_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_692_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_693_div__by__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ one_one_real )
= A2 ) ).
% div_by_1
thf(fact_694_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_695_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_696_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_697_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_698_log__zero,axiom,
( ( log @ zero_zero_nat )
= zero_zero_nat ) ).
% log_zero
thf(fact_699_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_700_mult__cancel__right2,axiom,
! [A2: real,C: real] :
( ( ( times_times_real @ A2 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_701_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_702_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_703_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_704_mult__cancel__left2,axiom,
! [C: real,A2: real] :
( ( ( times_times_real @ C @ A2 )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_705_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_706_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_707_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_708_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_709_zero__eq__1__divide__iff,axiom,
! [A2: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_710_one__divide__eq__0__iff,axiom,
! [A2: real] :
( ( ( divide_divide_real @ one_one_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_711_eq__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A2 ) )
= ( ( A2 != zero_zero_real )
& ( A2 = B ) ) ) ).
% eq_divide_eq_1
thf(fact_712_divide__eq__eq__1,axiom,
! [B: real,A2: real] :
( ( ( divide_divide_real @ B @ A2 )
= one_one_real )
= ( ( A2 != zero_zero_real )
& ( A2 = B ) ) ) ).
% divide_eq_eq_1
thf(fact_713_divide__self__if,axiom,
! [A2: real] :
( ( ( A2 = zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= zero_zero_real ) )
& ( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_714_divide__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% divide_self
thf(fact_715_one__eq__divide__iff,axiom,
! [A2: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A2 @ B ) )
= ( ( B != zero_zero_real )
& ( A2 = B ) ) ) ).
% one_eq_divide_iff
thf(fact_716_divide__eq__1__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A2 = B ) ) ) ).
% divide_eq_1_iff
thf(fact_717_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_718_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_719_div__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% div_self
thf(fact_720_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_721_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_722_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_723_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_724_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_725_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_726_algebraic__semidom__class_Ounit__prod,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_727_algebraic__semidom__class_Ounit__prod,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_728_comm__monoid__mult__class_Ounit__prod,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_729_comm__monoid__mult__class_Ounit__prod,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_730_comm__monoid__mult__class_Ounit__prod,axiom,
! [A2: real,B: real] :
( ( dvd_dvd_real @ A2 @ one_one_real )
=> ( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ one_one_real ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_731_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_732_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_733_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A2: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ one_one_real )
= ( ( dvd_dvd_real @ A2 @ one_one_real )
& ( dvd_dvd_real @ B @ one_one_real ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_734_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_735_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_736_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: real,A2: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ C )
= ( dvd_dvd_real @ A2 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_737_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_738_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_739_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ C )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_740_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_741_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_742_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_743_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_744_unit__div,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_745_unit__div,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_746_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_747_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_748_log__Suc__zero,axiom,
( ( log @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% log_Suc_zero
thf(fact_749_Discrete_Olog__one,axiom,
( ( log @ one_one_nat )
= zero_zero_nat ) ).
% Discrete.log_one
thf(fact_750_zero__less__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_divide_1_iff
thf(fact_751_less__divide__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_752_less__divide__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_real @ B @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_753_divide__less__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_real @ B @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_754_divide__less__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_real @ A2 @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_755_divide__less__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_756_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A2 @ B ) )
= ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_757_nonzero__divide__mult__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_758_unit__div__mult__self,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% unit_div_mult_self
thf(fact_759_unit__div__mult__self,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A2 ) @ A2 )
= B ) ) ).
% unit_div_mult_self
thf(fact_760_unit__mult__div__div,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A2 ) )
= ( divide_divide_nat @ B @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_761_unit__mult__div__div,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A2 ) )
= ( divide_divide_int @ B @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_762_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_763_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_764_nth__Cons__numeral,axiom,
! [X: b,Xs: list_b,V: num] :
( ( nth_b @ ( cons_b @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_b @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_765_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_766_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_767_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_768_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_769_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_770_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_771_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_772_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_773_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_774_nth__Cons__pos,axiom,
! [N: nat,X: b,Xs: list_b] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_b @ ( cons_b @ X @ Xs ) @ N )
= ( nth_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_775_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_776_log__half,axiom,
! [N: nat] :
( ( log @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ ( log @ N ) @ one_one_nat ) ) ).
% log_half
thf(fact_777_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_778_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_779_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_780_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_781_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_782_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_783_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_784_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_785_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_786_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_787_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_788_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_789_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_790_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_791_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_792_one__dvd,axiom,
! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% one_dvd
thf(fact_793_one__dvd,axiom,
! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).
% one_dvd
thf(fact_794_one__dvd,axiom,
! [A2: real] : ( dvd_dvd_real @ one_one_real @ A2 ) ).
% one_dvd
thf(fact_795_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A2 ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_796_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A2 ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_797_dvd__unit__imp__unit,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_798_dvd__unit__imp__unit,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_799_idom__class_Ounit__imp__dvd,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A2 ) ) ).
% idom_class.unit_imp_dvd
thf(fact_800_idom__class_Ounit__imp__dvd,axiom,
! [B: real,A2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ B @ A2 ) ) ).
% idom_class.unit_imp_dvd
thf(fact_801_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_802_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_803_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_804_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_805_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_806_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_807_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_808_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_809_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_810_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_811_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_812_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_813_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_814_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_815_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_816_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_817_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_818_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_819_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_820_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_821_right__inverse__eq,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A2 @ B )
= one_one_real )
= ( A2 = B ) ) ) ).
% right_inverse_eq
thf(fact_822_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_823_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_824_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_825_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_826_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_827_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A2 )
= ( times_times_nat @ C @ A2 ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_828_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ B @ A2 )
= ( times_times_int @ C @ A2 ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_829_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B )
= ( times_times_nat @ A2 @ C ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_830_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B )
= ( times_times_int @ A2 @ C ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_831_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_832_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_833_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_834_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_835_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_836_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_837_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_838_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_839_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_840_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_841_idom__class_Odvd__mult__unit__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_842_idom__class_Odvd__mult__unit__iff,axiom,
! [B: real,A2: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A2 @ ( times_times_real @ C @ B ) )
= ( dvd_dvd_real @ A2 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_843_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_844_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: real,A2: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ C ) )
= ( dvd_dvd_real @ A2 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_845_idom__class_Ounit__mult__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B )
= ( times_times_int @ A2 @ C ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_846_idom__class_Ounit__mult__left__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ one_one_real )
=> ( ( ( times_times_real @ A2 @ B )
= ( times_times_real @ A2 @ C ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_847_idom__class_Ounit__mult__right__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ B @ A2 )
= ( times_times_int @ C @ A2 ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_848_idom__class_Ounit__mult__right__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ one_one_real )
=> ( ( ( times_times_real @ B @ A2 )
= ( times_times_real @ C @ A2 ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_849_dvd__div__unit__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_850_dvd__div__unit__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_851_div__unit__dvd__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_852_div__unit__dvd__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_853_unit__div__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A2 )
= ( divide_divide_nat @ C @ A2 ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_854_unit__div__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A2 )
= ( divide_divide_int @ C @ A2 ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_855_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_856_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_857_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_858_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_859_less__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% less_divide_eq_1
thf(fact_860_divide__less__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ A2 @ B ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_861_unit__dvdE,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [C4: nat] :
( B
!= ( times_times_nat @ A2 @ C4 ) ) ) ) ).
% unit_dvdE
thf(fact_862_unit__dvdE,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [C4: int] :
( B
!= ( times_times_int @ A2 @ C4 ) ) ) ) ).
% unit_dvdE
thf(fact_863_unit__div__eq__0__iff,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A2 @ B )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_864_unit__div__eq__0__iff,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A2 @ B )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_865_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_866_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_867_unit__div__mult__swap,axiom,
! [C: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_868_unit__div__mult__swap,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_869_unit__div__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_870_unit__div__commute,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A2 @ B ) @ C )
= ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_871_div__mult__unit2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_872_div__mult__unit2,axiom,
! [C: int,B: int,A2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B @ A2 )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_873_unit__eq__div2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( A2
= ( divide_divide_nat @ C @ B ) )
= ( ( times_times_nat @ A2 @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_874_unit__eq__div2,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( A2
= ( divide_divide_int @ C @ B ) )
= ( ( times_times_int @ A2 @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_875_unit__eq__div1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A2 @ B )
= C )
= ( A2
= ( times_times_nat @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_876_unit__eq__div1,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A2 @ B )
= C )
= ( A2
= ( times_times_int @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_877_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_878_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_879_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_880_is__unitE,axiom,
! [A2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [B3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A2 )
= B3 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B3 )
= A2 )
=> ( ( ( times_times_nat @ A2 @ B3 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A2 )
!= ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_881_is__unitE,axiom,
! [A2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [B3: int] :
( ( B3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A2 )
= B3 )
=> ( ( ( divide_divide_int @ one_one_int @ B3 )
= A2 )
=> ( ( ( times_times_int @ A2 @ B3 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A2 )
!= ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_882_is__unit__div__mult__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ A2 @ B ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_883_is__unit__div__mult__cancel__left,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ A2 @ B ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_884_is__unit__div__mult__cancel__right,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B @ A2 ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_885_is__unit__div__mult__cancel__right,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ A2 ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_886_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_887_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_888_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_889_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_890_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_891_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_892_nth__Cons_H,axiom,
! [N: nat,X: b,Xs: list_b] :
( ( ( N = zero_zero_nat )
=> ( ( nth_b @ ( cons_b @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_b @ ( cons_b @ X @ Xs ) @ N )
= ( nth_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_893_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_894_nth__non__equal__first__eq,axiom,
! [X: b,Y: b,Xs: list_b,N: nat] :
( ( X != Y )
=> ( ( ( nth_b @ ( cons_b @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_895_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_896_Discrete_Olog_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( log @ X )
= Y )
=> ( ( ( ord_less_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( Y = zero_zero_nat ) )
& ( ~ ( ord_less_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( Y
= ( suc @ ( log @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% Discrete.log.elims
thf(fact_897_Discrete_Olog_Osimps,axiom,
( log
= ( ^ [N4: nat] : ( if_nat @ ( ord_less_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_nat @ ( suc @ ( log @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% Discrete.log.simps
thf(fact_898_filter__odd__nth,axiom,
! [J: nat,L: nat,X: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) @ L )
=> ( ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= L )
=> ( ( nth_nat
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ L ) )
@ J )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) @ one_one_nat ) ) ) ) ).
% filter_odd_nth
thf(fact_899_map__filter__shift,axiom,
! [F: nat > b,G: nat] :
( ( map_nat_b @ F @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( suc @ G ) ) ) )
= ( cons_b @ ( F @ zero_zero_nat )
@ ( map_nat_b
@ ^ [X2: nat] : ( F @ ( plus_plus_nat @ X2 @ one_one_nat ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ G ) ) ) ) ) ).
% map_filter_shift
thf(fact_900_map__filter__shift,axiom,
! [F: nat > nat,G: nat] :
( ( map_nat_nat @ F @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( suc @ G ) ) ) )
= ( cons_nat @ ( F @ zero_zero_nat )
@ ( map_nat_nat
@ ^ [X2: nat] : ( F @ ( plus_plus_nat @ X2 @ one_one_nat ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ G ) ) ) ) ) ).
% map_filter_shift
thf(fact_901_filter__odd__map,axiom,
! [X: nat] :
( ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) )
= ( map_nat_nat
@ ^ [Y4: nat] : ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y4 ) @ one_one_nat )
@ ( upt @ zero_zero_nat @ X ) ) ) ).
% filter_odd_map
thf(fact_902_map__filter__shift_H,axiom,
! [F: nat > b,G: nat] :
( ( map_nat_b @ F
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( suc @ G ) ) ) )
= ( map_nat_b
@ ^ [X2: nat] : ( F @ ( plus_plus_nat @ X2 @ one_one_nat ) )
@ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ G ) ) ) ) ).
% map_filter_shift'
thf(fact_903_map__filter__shift_H,axiom,
! [F: nat > nat,G: nat] :
( ( map_nat_nat @ F
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( suc @ G ) ) ) )
= ( map_nat_nat
@ ^ [X2: nat] : ( F @ ( plus_plus_nat @ X2 @ one_one_nat ) )
@ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ G ) ) ) ) ).
% map_filter_shift'
thf(fact_904_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_905_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_906_add__right__cancel,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_907_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_908_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_909_add__left__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_910_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_911_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_912_add__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add_0
thf(fact_913_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_914_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_915_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_916_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_917_add__cancel__right__right,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ A2 @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_918_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_919_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_920_add__cancel__right__left,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ B @ A2 ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_921_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_922_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_923_add__cancel__left__right,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_924_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_925_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_926_add__cancel__left__left,axiom,
! [B: real,A2: real] :
( ( ( plus_plus_real @ B @ A2 )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_927_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_928_double__zero__sym,axiom,
! [A2: real] :
( ( zero_zero_real
= ( plus_plus_real @ A2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_929_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_930_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_931_add_Oright__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.right_neutral
thf(fact_932_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_933_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_934_add__less__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_935_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_936_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_937_add__less__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_938_add__numeral__left,axiom,
! [V: num,W: num,Z4: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z4 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z4 ) ) ).
% add_numeral_left
thf(fact_939_add__numeral__left,axiom,
! [V: num,W: num,Z4: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z4 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z4 ) ) ).
% add_numeral_left
thf(fact_940_add__numeral__left,axiom,
! [V: num,W: num,Z4: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z4 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z4 ) ) ).
% add_numeral_left
thf(fact_941_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_942_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_943_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_944_add__diff__cancel,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_945_add__diff__cancel,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_946_diff__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_947_diff__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_948_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_949_add__diff__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_950_add__diff__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_951_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_952_add__diff__cancel__left_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_953_add__diff__cancel__left_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_954_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_955_add__diff__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_956_add__diff__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_957_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_958_add__diff__cancel__right_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_959_add__diff__cancel__right_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_960_dvd__add__triv__right__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_961_dvd__add__triv__right__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_962_dvd__add__triv__right__iff,axiom,
! [A2: real,B: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( dvd_dvd_real @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_963_dvd__add__triv__left__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_964_dvd__add__triv__left__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_965_dvd__add__triv__left__iff,axiom,
! [A2: real,B: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( dvd_dvd_real @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_966_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_967_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_968_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_969_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_970_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_971_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_972_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_973_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_974_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_975_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_976_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_977_less__add__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_978_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_979_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_980_less__add__same__cancel1,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_981_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_982_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_983_add__less__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ B )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_984_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_985_add__less__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_986_add__less__same__cancel1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A2 ) @ B )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_987_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_988_distrib__right__numeral,axiom,
! [A2: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_989_distrib__right__numeral,axiom,
! [A2: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_990_distrib__right__numeral,axiom,
! [A2: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_991_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_992_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_993_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_994_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_995_dvd__add__times__triv__left__iff,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_996_dvd__add__times__triv__left__iff,axiom,
! [A2: real,C: real,B: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ ( times_times_real @ C @ A2 ) @ B ) )
= ( dvd_dvd_real @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_997_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_998_dvd__add__times__triv__right__iff,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ ( times_times_int @ C @ A2 ) ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_999_dvd__add__times__triv__right__iff,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B @ ( times_times_real @ C @ A2 ) ) )
= ( dvd_dvd_real @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1000_div__add,axiom,
! [C: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_1001_div__add,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_1002_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_1003_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_1004_div__mult__self1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1005_div__mult__self1,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1006_div__mult__self2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1007_div__mult__self2,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1008_div__mult__self3,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1009_div__mult__self3,axiom,
! [B: int,C: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A2 ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1010_div__mult__self4,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1011_div__mult__self4,axiom,
! [B: int,C: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A2 ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1012_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_1013_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_1014_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1015_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_1016_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_1017_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1018_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1019_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1020_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1021_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1022_even__add,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_1023_even__add,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_1024_odd__add,axiom,
! [A2: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_1025_odd__add,axiom,
! [A2: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_1026_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_1027_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_1028_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_1029_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_1030_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_1031_even__diff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A2 @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) ) ).
% even_diff
thf(fact_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_1042_add_Ogroup__left__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_1043_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_1044_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_1045_add_Ocomm__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.comm_neutral
thf(fact_1046_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_1047_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_1048_comm__monoid__add__class_Oadd__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1049_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_1050_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_1051_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_1052_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1053_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1054_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1055_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1056_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1057_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1058_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1059_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1060_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1061_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1062_add__strict__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1063_add__strict__mono,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1064_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1065_add__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1066_add__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1067_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1068_add__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1069_add__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1070_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1071_add__less__imp__less__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1072_add__less__imp__less__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1073_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1074_add__less__imp__less__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1075_add__less__imp__less__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1076_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1077_ring__class_Oring__distribs_I2_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1078_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1079_ring__class_Oring__distribs_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1080_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1081_comm__semiring__class_Odistrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1082_comm__semiring__class_Odistrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1083_distrib__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_1084_distrib__left,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_1085_distrib__left,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_1086_distrib__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1087_distrib__right,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_1088_distrib__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_1089_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1090_combine__common__factor,axiom,
! [A2: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1091_combine__common__factor,axiom,
! [A2: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1092_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1093_mult__hom_Ohom__add,axiom,
! [C: int,X: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1094_mult__hom_Ohom__add,axiom,
! [C: real,X: real,Y: real] :
( ( times_times_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1095_group__cancel_Osub1,axiom,
! [A3: int,K: int,A2: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1096_group__cancel_Osub1,axiom,
! [A3: real,K: real,A2: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A2 ) )
=> ( ( minus_minus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1097_diff__eq__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= C )
= ( A2
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1098_diff__eq__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( minus_minus_real @ A2 @ B )
= C )
= ( A2
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1099_eq__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( A2
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1100_eq__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( A2
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1101_add__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1102_add__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1103_diff__diff__eq2,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1104_diff__diff__eq2,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1105_diff__add__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1106_diff__add__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1107_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1108_diff__add__eq__diff__diff__swap,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1109_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_1110_add__implies__diff,axiom,
! [C: int,B: int,A2: int] :
( ( ( plus_plus_int @ C @ B )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_1111_add__implies__diff,axiom,
! [C: real,B: real,A2: real] :
( ( ( plus_plus_real @ C @ B )
= A2 )
=> ( C
= ( minus_minus_real @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_1112_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1113_diff__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1114_diff__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1115_add__divide__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_1116_dvd__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1117_dvd__add,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1118_dvd__add,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( ( dvd_dvd_real @ A2 @ C )
=> ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1119_dvd__add__left__iff,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1120_dvd__add__left__iff,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1121_dvd__add__left__iff,axiom,
! [A2: real,C: real,B: real] :
( ( dvd_dvd_real @ A2 @ C )
=> ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A2 @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1122_dvd__add__right__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1123_dvd__add__right__iff,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1124_dvd__add__right__iff,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1125_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1126_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_1127_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1128_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A2: nat] :
( ( A3
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1129_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1130_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_1131_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1132_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_1133_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_1134_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1135_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1136_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1137_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1138_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1139_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1140_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1141_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1142_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1143_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1144_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1145_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1146_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1147_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1148_add__right__imp__eq,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1149_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1150_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1151_add__left__imp__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1152_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1153_add_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1154_add_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A2 @ C ) )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1155_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A: nat,B2: nat] : ( plus_plus_nat @ B2 @ A ) ) ) ).
% add.commute
thf(fact_1156_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A: int,B2: int] : ( plus_plus_int @ B2 @ A ) ) ) ).
% add.commute
thf(fact_1157_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A: real,B2: real] : ( plus_plus_real @ B2 @ A ) ) ) ).
% add.commute
thf(fact_1158_add_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_1159_add_Oright__cancel,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_1160_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_1161_add_Oleft__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_1162_add_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_1163_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_1164_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_1165_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1166_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_1167_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1168_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1169_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1170_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1171_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_1172_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1173_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1174_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1175_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_1176_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_1177_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1178_bezout__lemma__nat,axiom,
! [D: nat,A2: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A2 )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D ) ) )
=> ? [X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A2 @ B ) )
& ( ( ( times_times_nat @ A2 @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1179_bezout__add__nat,axiom,
! [A2: nat,B: nat] :
? [D2: nat,X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( times_times_nat @ A2 @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) )
| ( ( times_times_nat @ B @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1180_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A2
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1181_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A2
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1182_bezout__add__strong__nat,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ? [D2: nat,X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( times_times_nat @ A2 @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1183_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1184_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_1185_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_1186_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1187_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1188_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1189_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_1190_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_1191_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1192_num_Osize_I5_J,axiom,
! [X23: num] :
( ( size_size_num @ ( bit0 @ X23 ) )
= ( plus_plus_nat @ ( size_size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1193_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_1194_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1195_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_1196_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1197_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1198_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1199_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1200_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_1201_real__average__minus__first,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A2 )
= ( divide_divide_real @ ( minus_minus_real @ B @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_1202_real__average__minus__second,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A2 )
= ( divide_divide_real @ ( minus_minus_real @ B @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_1203_odd__less__0__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z4 ) @ Z4 ) @ zero_zero_int )
= ( ord_less_int @ Z4 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1204_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1205_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1206_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1207_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1208_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1209_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1210_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1211_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1212_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_1213_odd__nonzero,axiom,
! [Z4: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z4 ) @ Z4 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1214_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1215_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1216_zless__add1__eq,axiom,
! [W: int,Z4: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z4 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z4 )
| ( W = Z4 ) ) ) ).
% zless_add1_eq
thf(fact_1217_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_1218_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1219_plusinfinity,axiom,
! [D: int,P3: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int,K2: int] :
( ( P3 @ X5 )
= ( P3 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P3 @ X5 ) ) )
=> ( ? [X_12: int] : ( P3 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1220_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int,K2: int] :
( ( P1 @ X5 )
= ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P1 @ X5 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1221_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_1222_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1223_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_1224_log__exp,axiom,
! [N: nat] :
( ( log @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= N ) ).
% log_exp
thf(fact_1225_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_1226_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1227_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [X5: real] :
( ( ord_less_real @ zero_zero_real @ X5 )
& ( ( power_power_real @ X5 @ N )
= A2 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A2 ) )
=> ( Y5 = X5 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1228_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_1229_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1230_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_1231_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_1232_log__exp2__gt,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( log @ N ) ) ) ) ).
% log_exp2_gt
thf(fact_1233_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1234_two__powrs__div,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( times_times_nat @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ J ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) ) ) ) ).
% two_powrs_div
thf(fact_1235_two__powr__div,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ I @ J ) ) ) ) ).
% two_powr_div
thf(fact_1236_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1237_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1238_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1239_divides__rexp,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ Y @ ( suc @ N ) ) ) ) ).
% divides_rexp
thf(fact_1240_nat__exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P5 @ M5 ) ) ) ) ) ).
% nat_exists_least_iff
thf(fact_1241_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1242_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1243_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_1244_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1245_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_1246_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_1247_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_1248_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1249_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1250_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1251_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_1252_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1253_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1254_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1255_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1256_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1257_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_1258_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_1259_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1260_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1261_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1262_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1263_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1264_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1265_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1266_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
( ( ( map_nat_b @ ( nth_b @ xsa ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_size_list_b @ xsa ) ) ) )
= ( evens_odds_b @ $true @ xsa ) )
& ( ( map_nat_b @ ( nth_b @ xsa )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_size_list_b @ xsa ) ) ) )
= ( evens_odds_b @ $false @ xsa ) ) ) ).
thf(conj_1,conjecture,
( ( map_nat_b @ ( nth_b @ ( cons_b @ x @ xsa ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_size_list_b @ ( cons_b @ x @ xsa ) ) ) ) )
= ( evens_odds_b @ $false @ ( cons_b @ x @ xsa ) ) ) ).
%------------------------------------------------------------------------------