TPTP Problem File: SLH0550^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0017_Bipartite_Graphs/prob_00027_000968__13279614_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1404 ( 796 unt; 129 typ; 0 def)
% Number of atoms : 3078 (1834 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 9541 ( 406 ~; 57 |; 211 &;8098 @)
% ( 0 <=>; 769 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 247 ( 247 >; 0 *; 0 +; 0 <<)
% Number of symbols : 121 ( 118 usr; 20 con; 0-3 aty)
% Number of variables : 2700 ( 164 ^;2446 !; 90 ?;2700 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:39.453
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (118)
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
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__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
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__Extended____Nat__Oenat,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
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__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
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_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
inf_inf_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
neg_numeral_dbl_real: real > real ).
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__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
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_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).
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_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Nat__Onat,type,
undire7858122600432113898nt_nat: nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Set__Oset_Itf__a_J,type,
undire2320338297334612420_set_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
undire1521409233611534436dent_a: a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001t__Nat__Onat,type,
undire7290660292559394354ph_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001tf__a,type,
undire3507641187627840796raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph__axioms_001t__Nat__Onat,type,
undire6823751729292563413ms_nat: set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph__axioms_001tf__a,type,
undire3875311282895952441ioms_a: set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Nat__Onat,type,
undire3269267262472140706ph_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Set__Oset_Itf__a_J,type,
undire6886684016831807756_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001tf__a,type,
undire7251896706689453996raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Nat__Onat,type,
undire6814325412647357297en_nat: set_nat > set_nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Set__Oset_Itf__a_J,type,
undire2578756059399487229_set_a: set_set_a > set_set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001t__Nat__Onat,type,
undire8616788072062012598ge_nat: set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_X,type,
x: set_a ).
thf(sy_v_Y,type,
y: set_a ).
thf(sy_v_e_H____,type,
e: set_a ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_v1____,type,
v1: a ).
% Relevant facts (1270)
thf(fact_0_v1in,axiom,
member_a @ v1 @ e ).
% v1in
thf(fact_1_assms,axiom,
( ( inf_inf_set_a @ x @ y )
= bot_bot_set_a ) ).
% assms
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v1_O_A_092_060lbrakk_062v1_A_092_060in_062_Ae_H_059_Av1_A_092_060in_062_AX_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V1: a] :
( ( member_a @ V1 @ e )
=> ~ ( member_a @ V1 @ x ) ) ).
% \<open>\<And>thesis. (\<And>v1. \<lbrakk>v1 \<in> e'; v1 \<in> X\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_assm,axiom,
( member_set_a @ e
@ ( collect_set_a
@ ^ [E: set_a] :
( ( ( finite_card_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
& ( ( inf_inf_set_a @ E @ x )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ E @ y )
!= bot_bot_set_a ) ) ) ) ).
% assm
thf(fact_4_calculation_I2_J,axiom,
member_a @ v1 @ x ).
% calculation(2)
thf(fact_5_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_6_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_7_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_8_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_9_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_10_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_11_card__2__iff_H,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ S )
& ? [Y: a] :
( ( member_a @ Y @ S )
& ( X2 != Y )
& ! [Z: a] :
( ( member_a @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_12_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ? [Y: nat] :
( ( member_nat @ Y @ S )
& ( X2 != Y )
& ! [Z: nat] :
( ( member_nat @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_13_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_14_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_15_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_16_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_17_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_18_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_19_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_20_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_21_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_22_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_23_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_24_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_25_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_26_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_27_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_28_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_29_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_30_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_31_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_32_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_33_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_34_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_35_inf__right__idem,axiom,
! [X: set_a,Y2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_a @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_36_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_37_inf__left__idem,axiom,
! [X: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y2 ) )
= ( inf_inf_set_a @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_38_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_39_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_40_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_41_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A3 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_42_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_43_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_44_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_45_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_46_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_47_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_48_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_49_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_50_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_51_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_52_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_53_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_54_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_55_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_56_inf__left__commute,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_57_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_58_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_59_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_60_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_61_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_62_inf__assoc,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_63_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_64_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_65_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_66_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_67_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_74_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_75_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y2 ) )
= ( inf_inf_set_a @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_76_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_77_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_78_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_79_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_80_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_81_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_82_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_83_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_84_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_85_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_86_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_87_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_88_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_89_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_90_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X2: set_a] : $false ) ) ).
% empty_def
thf(fact_91_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_92_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_93_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_94_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_95_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_96_Int__Collect,axiom,
! [X: set_a,A: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_97_Int__Collect,axiom,
! [X: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_98_Int__Collect,axiom,
! [X: a,A: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_99_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( member_set_a @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_100_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_101_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( member_a @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_102_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X2 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_103_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_104_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_105_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_106_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_107_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_108_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_109_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_110_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_111_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_112_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_113_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_114_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_115_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_116_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_117_sgraph__axioms__def,axiom,
( undire3875311282895952441ioms_a
= ( ^ [Edges: set_set_a] :
! [E: set_a] :
( ( member_set_a @ E @ Edges )
=> ( ( finite_card_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sgraph_axioms_def
thf(fact_118_sgraph__axioms__def,axiom,
( undire6823751729292563413ms_nat
= ( ^ [Edges: set_set_nat] :
! [E: set_nat] :
( ( member_set_nat @ E @ Edges )
=> ( ( finite_card_nat @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sgraph_axioms_def
thf(fact_119_sgraph__axioms_Ointro,axiom,
! [Edges2: set_set_a] :
( ! [E2: set_a] :
( ( member_set_a @ E2 @ Edges2 )
=> ( ( finite_card_a @ E2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( undire3875311282895952441ioms_a @ Edges2 ) ) ).
% sgraph_axioms.intro
thf(fact_120_sgraph__axioms_Ointro,axiom,
! [Edges2: set_set_nat] :
( ! [E2: set_nat] :
( ( member_set_nat @ E2 @ Edges2 )
=> ( ( finite_card_nat @ E2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( undire6823751729292563413ms_nat @ Edges2 ) ) ).
% sgraph_axioms.intro
thf(fact_121_card__2__iff,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: nat,Y: nat] :
( ( S
= ( insert_nat @ X2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
& ( X2 != Y ) ) ) ) ).
% card_2_iff
thf(fact_122_card__2__iff,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a,Y: a] :
( ( S
= ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
& ( X2 != Y ) ) ) ) ).
% card_2_iff
thf(fact_123_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_124_inf__Int__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( inf_inf_set_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_125_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_126_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_127_comp__sgraph_Ois__sedge__def,axiom,
( undire4917966558017083288edge_a
= ( ^ [E: set_a] :
( ( finite_card_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% comp_sgraph.is_sedge_def
thf(fact_128_comp__sgraph_Ois__sedge__def,axiom,
( undire8616788072062012598ge_nat
= ( ^ [E: set_nat] :
( ( finite_card_nat @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% comp_sgraph.is_sedge_def
thf(fact_129_odd__card__imp__not__empty,axiom,
! [A: set_nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ A ) )
=> ( A != bot_bot_set_nat ) ) ).
% odd_card_imp_not_empty
thf(fact_130_odd__card__imp__not__empty,axiom,
! [A: set_a] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_a @ A ) )
=> ( A != bot_bot_set_a ) ) ).
% odd_card_imp_not_empty
thf(fact_131_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_132_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_133_insert__iff,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_134_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_135_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_136_insertCI,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_137_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_138_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_139_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_140_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_141_Int__insert__left__if0,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_142_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_143_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_144_Int__insert__left__if1,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_145_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_146_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_147_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_148_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_149_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_150_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_151_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_152_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_153_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_154_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_155_singleton__conv,axiom,
! [A2: set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( X2 = A2 ) )
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singleton_conv
thf(fact_156_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_157_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_158_singleton__conv2,axiom,
! [A2: set_a] :
( ( collect_set_a
@ ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 )
@ A2 ) )
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singleton_conv2
thf(fact_159_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_160_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 )
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_161_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_162_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_163_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_164_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_165_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_166_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_167_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_168_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_169_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_170_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_171_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_172_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_173_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_174_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B5: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B5 ) )
& ~ ( member_set_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_175_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_176_insert__commute,axiom,
! [X: nat,Y2: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y2 @ A ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_177_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_178_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_179_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_180_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_181_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_182_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_183_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_184_insert__ident,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A )
= ( insert_set_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_185_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_186_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_187_Set_Oset__insert,axiom,
! [X: set_a,A: set_set_a] :
( ( member_set_a @ X @ A )
=> ~ ! [B5: set_set_a] :
( ( A
= ( insert_set_a @ X @ B5 ) )
=> ( member_set_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_188_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_189_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_190_insertI2,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_191_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_192_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_193_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_194_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_195_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_196_insertE,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_197_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_198_insert__Collect,axiom,
! [A2: set_a,P: set_a > $o] :
( ( insert_set_a @ A2 @ ( collect_set_a @ P ) )
= ( collect_set_a
@ ^ [U: set_a] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_199_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_200_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( X2 = A4 )
| ( member_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_201_insert__compr,axiom,
( insert_set_a
= ( ^ [A4: set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( X2 = A4 )
| ( member_set_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_202_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A4 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_203_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_204_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_205_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_206_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_207_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_208_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_209_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_210_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_211_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_212_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_213_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_214_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_215_Int__insert__left,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_216_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_217_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_218_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_219_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_220_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_221_Collect__conv__if,axiom,
! [P: set_a > $o,A2: set_a] :
( ( ( P @ A2 )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if
thf(fact_222_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_223_Collect__conv__if,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_224_Collect__conv__if2,axiom,
! [P: set_a > $o,A2: set_a] :
( ( ( P @ A2 )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_225_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_226_Collect__conv__if2,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_227_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_228_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_229_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_230_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_231_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_232_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X4: set_set_a,Y5: set_set_a,E: set_set_a] :
? [X2: set_a,Y: set_a] :
( ( E
= ( insert_set_a @ X2 @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X2 @ X4 )
& ( member_set_a @ Y @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_233_comp__sgraph_Ois__edge__between__def,axiom,
( undire6814325412647357297en_nat
= ( ^ [X4: set_nat,Y5: set_nat,E: set_nat] :
? [X2: nat,Y: nat] :
( ( E
= ( insert_nat @ X2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
& ( member_nat @ X2 @ X4 )
& ( member_nat @ Y @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_234_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X4: set_a,Y5: set_a,E: set_a] :
? [X2: a,Y: a] :
( ( E
= ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X4 )
& ( member_a @ Y @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_235_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_236_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_237_Collect__empty__eq__bot,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( P = bot_bot_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_238_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_239_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_240_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_241_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_242_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_243_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_244_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_245_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_246_sgraph_Otwo__edges,axiom,
! [Vertices: set_a,Edges2: set_set_a,E3: set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges2 )
=> ( ( member_set_a @ E3 @ Edges2 )
=> ( ( finite_card_a @ E3 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sgraph.two_edges
thf(fact_247_sgraph_Otwo__edges,axiom,
! [Vertices: set_nat,Edges2: set_set_nat,E3: set_nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges2 )
=> ( ( member_set_nat @ E3 @ Edges2 )
=> ( ( finite_card_nat @ E3 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sgraph.two_edges
thf(fact_248_ulgraph_Ois__sedge__def,axiom,
! [Vertices: set_a,Edges2: set_set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges2 )
=> ( ( undire4917966558017083288edge_a @ E3 )
= ( ( finite_card_a @ E3 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% ulgraph.is_sedge_def
thf(fact_249_ulgraph_Ois__sedge__def,axiom,
! [Vertices: set_nat,Edges2: set_set_nat,E3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges2 )
=> ( ( undire8616788072062012598ge_nat @ E3 )
= ( ( finite_card_nat @ E3 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% ulgraph.is_sedge_def
thf(fact_250_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_251_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_252_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_253_of__nat__numeral,axiom,
! [N: num] :
( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N ) )
= ( numera1916890842035813515d_enat @ N ) ) ).
% of_nat_numeral
thf(fact_254_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_255_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_256_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges2: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges2 )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges2 ) ) ).
% ulgraph.empty_not_edge
thf(fact_257_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_258_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_set_a,Edges2: set_set_set_a,X5: set_set_a,Y6: set_set_a,E3: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges2 )
=> ( ( undire2578756059399487229_set_a @ X5 @ Y6 @ E3 )
= ( ? [X2: set_a,Y: set_a] :
( ( E3
= ( insert_set_a @ X2 @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X2 @ X5 )
& ( member_set_a @ Y @ Y6 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_259_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_nat,Edges2: set_set_nat,X5: set_nat,Y6: set_nat,E3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges2 )
=> ( ( undire6814325412647357297en_nat @ X5 @ Y6 @ E3 )
= ( ? [X2: nat,Y: nat] :
( ( E3
= ( insert_nat @ X2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
& ( member_nat @ X2 @ X5 )
& ( member_nat @ Y @ Y6 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_260_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_a,Edges2: set_set_a,X5: set_a,Y6: set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges2 )
=> ( ( undire8544646567961481629ween_a @ X5 @ Y6 @ E3 )
= ( ? [X2: a,Y: a] :
( ( E3
= ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X5 )
& ( member_a @ Y @ Y6 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_261_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_262_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_263_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_264_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_265_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_266_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_267_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_268_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_nat,Edges2: set_set_nat,X: nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges2 )
=> ~ ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ Edges2 ) ) ).
% sgraph.singleton_not_edge
thf(fact_269_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_a,Edges2: set_set_a,X: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges2 )
=> ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ Edges2 ) ) ).
% sgraph.singleton_not_edge
thf(fact_270_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_271_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_272_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X2: nat] :
( A3
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_273_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X2: a] :
( A3
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_274_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_275_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_276_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_277_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_278_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_279_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_280_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_281_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_282_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_283_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_284_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_285_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_286_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_287_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_288_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_289_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_290_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_291_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_292_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_293_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_294_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_295_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_296_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_297_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_298_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_299_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_300_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_301_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_302_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_303_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_304_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_305_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_306_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_307_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_308_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_309_divide__numeral__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ ( numeral_numeral_real @ one ) )
= A2 ) ).
% divide_numeral_1
thf(fact_310_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_311_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_312_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_313_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_314_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_315_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_316_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_317_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_318_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_319_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_320_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_321_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_322_ulgraph_Oalt__edge__size,axiom,
! [Vertices: set_a,Edges2: set_set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges2 )
=> ( ( member_set_a @ E3 @ Edges2 )
=> ( ( ( finite_card_a @ E3 )
= one_one_nat )
| ( ( finite_card_a @ E3 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% ulgraph.alt_edge_size
thf(fact_323_ulgraph_Oalt__edge__size,axiom,
! [Vertices: set_nat,Edges2: set_set_nat,E3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges2 )
=> ( ( member_set_nat @ E3 @ Edges2 )
=> ( ( ( finite_card_nat @ E3 )
= one_one_nat )
| ( ( finite_card_nat @ E3 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% ulgraph.alt_edge_size
thf(fact_324_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_325_unit__div,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_326_unit__div,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_327_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_328_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_329_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_330_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_331_div2__even__ext__nat,axiom,
! [X: nat,Y2: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% div2_even_ext_nat
thf(fact_332_div__dvd__div,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A2 ) @ ( divide_divide_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_333_div__dvd__div,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A2 ) @ ( divide_divide_int @ C @ A2 ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_334_bit__eq__rec,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
& ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_335_bit__eq__rec,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
& ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_336_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_337_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_338_div__by__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ one_one_real )
= A2 ) ).
% div_by_1
thf(fact_339_bits__div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% bits_div_by_1
thf(fact_340_bits__div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% bits_div_by_1
thf(fact_341_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_342_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_343_int__ops_I8_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(8)
thf(fact_344_dvd__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_345_dvd__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ B2 @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_trans
thf(fact_346_dvd__refl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% dvd_refl
thf(fact_347_dvd__refl,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ A2 ) ).
% dvd_refl
thf(fact_348_dvd__unit__imp__unit,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_349_dvd__unit__imp__unit,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_350_unit__imp__dvd,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ B2 @ A2 ) ) ).
% unit_imp_dvd
thf(fact_351_unit__imp__dvd,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ B2 @ A2 ) ) ).
% unit_imp_dvd
thf(fact_352_one__dvd,axiom,
! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% one_dvd
thf(fact_353_one__dvd,axiom,
! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).
% one_dvd
thf(fact_354_one__dvd,axiom,
! [A2: real] : ( dvd_dvd_real @ one_one_real @ A2 ) ).
% one_dvd
thf(fact_355_dvd__div__eq__iff,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_356_dvd__div__eq__iff,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( ( divide_divide_int @ A2 @ C )
= ( divide_divide_int @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_357_dvd__div__eq__iff,axiom,
! [C: real,A2: real,B2: real] :
( ( dvd_dvd_real @ C @ A2 )
=> ( ( dvd_dvd_real @ C @ B2 )
=> ( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_358_dvd__div__eq__cancel,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( divide_divide_nat @ A2 @ C )
= ( divide_divide_nat @ B2 @ C ) )
=> ( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_359_dvd__div__eq__cancel,axiom,
! [A2: int,C: int,B2: int] :
( ( ( divide_divide_int @ A2 @ C )
= ( divide_divide_int @ B2 @ C ) )
=> ( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_360_dvd__div__eq__cancel,axiom,
! [A2: real,C: real,B2: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B2 @ C ) )
=> ( ( dvd_dvd_real @ C @ A2 )
=> ( ( dvd_dvd_real @ C @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_361_div__div__div__same,axiom,
! [D: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ D @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ D ) @ ( divide_divide_nat @ B2 @ D ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_362_div__div__div__same,axiom,
! [D: int,B2: int,A2: int] :
( ( dvd_dvd_int @ D @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( divide_divide_int @ ( divide_divide_int @ A2 @ D ) @ ( divide_divide_int @ B2 @ D ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_363_dvd__div__unit__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_364_dvd__div__unit__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ C @ B2 ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_365_div__unit__dvd__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_366_div__unit__dvd__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_367_unit__div__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ B2 @ A2 )
= ( divide_divide_nat @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_368_unit__div__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( divide_divide_int @ B2 @ A2 )
= ( divide_divide_int @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_369_card__1__singletonE,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= one_one_nat )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_370_card__1__singletonE,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= one_one_nat )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_371_is__singleton__altdef,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( ( finite_card_a @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_372_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( ( finite_card_nat @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_373_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_374_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_384_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_385_add__numeral__left,axiom,
! [V: num,W: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_386_add__numeral__left,axiom,
! [V: num,W: num,Z2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_387_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_388_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_389_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_390_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_391_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_392_dvd__add__triv__left__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_393_dvd__add__triv__left__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_394_dvd__add__triv__left__iff,axiom,
! [A2: real,B2: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
= ( dvd_dvd_real @ A2 @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_395_dvd__add__triv__right__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_396_dvd__add__triv__right__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_397_dvd__add__triv__right__iff,axiom,
! [A2: real,B2: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
= ( dvd_dvd_real @ A2 @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_398_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_399_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_400_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_401_div__add,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_402_div__add,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_403_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_404_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_405_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_406_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_407_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_408_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_409_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_410_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_411_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_412_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_413_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_414_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_415_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_416_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_417_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_418_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_419_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_420_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_421_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_422_odd__add,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% odd_add
thf(fact_423_odd__add,axiom,
! [A2: int,B2: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% odd_add
thf(fact_424_even__add,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_add
thf(fact_425_even__add,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_add
thf(fact_426_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_427_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_428_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_429_is__num__normalize_I1_J,axiom,
! [A2: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_430_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_431_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_432_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_433_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_434_dvd__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_435_dvd__add,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_436_dvd__add,axiom,
! [A2: real,B2: real,C: real] :
( ( dvd_dvd_real @ A2 @ B2 )
=> ( ( dvd_dvd_real @ A2 @ C )
=> ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_437_dvd__add__left__iff,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_438_dvd__add__left__iff,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_439_dvd__add__left__iff,axiom,
! [A2: real,C: real,B2: real] :
( ( dvd_dvd_real @ A2 @ C )
=> ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
= ( dvd_dvd_real @ A2 @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_440_dvd__add__right__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_441_dvd__add__right__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_442_dvd__add__right__iff,axiom,
! [A2: real,B2: real,C: real] :
( ( dvd_dvd_real @ A2 @ B2 )
=> ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
= ( dvd_dvd_real @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_443_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_444_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_445_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_446_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_447_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_448_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_449_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_450_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_451_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_452_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_453_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_454_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_455_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_456_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_457_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_458_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_459_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_460_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_461_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_462_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_463_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_464_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_465_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_466_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_467_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_code(2)
thf(fact_468_field__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_469_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_470_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_471_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_472_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_473_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_474_odd__even__add,axiom,
! [A2: nat,B2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% odd_even_add
thf(fact_475_odd__even__add,axiom,
! [A2: int,B2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% odd_even_add
thf(fact_476_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_477_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_478_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_479_set__decode__0,axiom,
! [X: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_480_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices: set_nat,Edges2: set_set_nat,V: nat,E3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges2 )
=> ( ( ( undire7858122600432113898nt_nat @ V @ E3 )
& ( ( finite_card_nat @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_481_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices: set_a,Edges2: set_set_a,V: a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges2 )
=> ( ( ( undire1521409233611534436dent_a @ V @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_482_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_483_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect_nat
@ ^ [N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_484_add__divide__distrib,axiom,
! [A2: real,B2: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% add_divide_distrib
thf(fact_485_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_486_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_487_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_488_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_489_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_490_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_491_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_492_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_493_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_494_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_495_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_496_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_497_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_498_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_499_double__eq__0__iff,axiom,
! [A2: real] :
( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_500_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_501_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_502_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_503_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_504_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_505_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_506_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_507_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_508_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_509_divide__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% divide_cancel_right
thf(fact_510_divide__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ( divide_divide_real @ C @ A2 )
= ( divide_divide_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% divide_cancel_left
thf(fact_511_divide__eq__0__iff,axiom,
! [A2: real,B2: real] :
( ( ( divide_divide_real @ A2 @ B2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_512_bits__div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_513_bits__div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_514_bits__div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_515_bits__div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_516_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_517_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_518_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_519_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_520_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_521_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_522_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_523_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_524_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_525_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_526_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_527_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_528_dvd__0__left__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_529_dvd__0__left__iff,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
= ( A2 = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_530_dvd__0__left__iff,axiom,
! [A2: real] :
( ( dvd_dvd_real @ zero_zero_real @ A2 )
= ( A2 = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_531_dvd__0__right,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_532_dvd__0__right,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ zero_zero_int ) ).
% dvd_0_right
thf(fact_533_dvd__0__right,axiom,
! [A2: real] : ( dvd_dvd_real @ A2 @ zero_zero_real ) ).
% dvd_0_right
thf(fact_534_times__divide__eq__left,axiom,
! [B2: real,C: real,A2: real] :
( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
= ( divide_divide_real @ ( times_times_real @ B2 @ A2 ) @ C ) ) ).
% times_divide_eq_left
thf(fact_535_divide__divide__eq__left,axiom,
! [A2: real,B2: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B2 ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_536_divide__divide__eq__right,axiom,
! [A2: real,B2: real,C: real] :
( ( divide_divide_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ B2 ) ) ).
% divide_divide_eq_right
thf(fact_537_times__divide__eq__right,axiom,
! [A2: real,B2: real,C: real] :
( ( times_times_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ B2 ) @ C ) ) ).
% times_divide_eq_right
thf(fact_538_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_539_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_540_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_541_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_542_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_543_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_544_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_545_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_546_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_547_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_548_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_549_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_550_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_551_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_552_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_553_mult__cancel__right1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_554_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_555_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_556_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_557_mult__cancel__left1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_558_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A2: real,B2: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( divide_divide_real @ A2 @ B2 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_559_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( divide_divide_real @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_560_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_561_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_562_nonzero__mult__div__cancel__left,axiom,
! [A2: real,B2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_563_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B2 @ C ) )
= ( divide_divide_real @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_564_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
= ( divide_divide_real @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_565_nonzero__mult__div__cancel__right,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_566_nonzero__mult__div__cancel__right,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_567_nonzero__mult__div__cancel__right,axiom,
! [B2: real,A2: real] :
( ( B2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_568_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B2 ) )
= ( divide_divide_real @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_569_div__mult__mult1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_570_div__mult__mult1,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_571_div__mult__mult2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_572_div__mult__mult2,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_573_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_574_div__mult__mult1__if,axiom,
! [C: int,A2: int,B2: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_575_distrib__left__numeral,axiom,
! [V: num,B2: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_576_distrib__left__numeral,axiom,
! [V: num,B2: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_577_distrib__left__numeral,axiom,
! [V: num,B2: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_578_distrib__left__numeral,axiom,
! [V: num,B2: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B2 @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B2 ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_579_distrib__right__numeral,axiom,
! [A2: nat,B2: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_580_distrib__right__numeral,axiom,
! [A2: int,B2: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_581_distrib__right__numeral,axiom,
! [A2: real,B2: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_582_distrib__right__numeral,axiom,
! [A2: extended_enat,B2: extended_enat,V: num] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) @ ( numera1916890842035813515d_enat @ V ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A2 @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B2 @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_583_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_584_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_585_eq__divide__eq__1,axiom,
! [B2: real,A2: real] :
( ( one_one_real
= ( divide_divide_real @ B2 @ A2 ) )
= ( ( A2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% eq_divide_eq_1
thf(fact_586_divide__eq__eq__1,axiom,
! [B2: real,A2: real] :
( ( ( divide_divide_real @ B2 @ A2 )
= one_one_real )
= ( ( A2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% divide_eq_eq_1
thf(fact_587_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_588_divide__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% divide_self
thf(fact_589_one__eq__divide__iff,axiom,
! [A2: real,B2: real] :
( ( one_one_real
= ( divide_divide_real @ A2 @ B2 ) )
= ( ( B2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% one_eq_divide_iff
thf(fact_590_divide__eq__1__iff,axiom,
! [A2: real,B2: real] :
( ( ( divide_divide_real @ A2 @ B2 )
= one_one_real )
= ( ( B2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% divide_eq_1_iff
thf(fact_591_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_592_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_593_div__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% div_self
thf(fact_594_dvd__times__right__cancel__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_595_dvd__times__right__cancel__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_596_dvd__times__left__cancel__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_597_dvd__times__left__cancel__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_598_dvd__mult__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_599_dvd__mult__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_600_dvd__mult__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_601_dvd__mult__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A2 @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_602_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_603_dvd__add__times__triv__right__iff,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A2 ) ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_604_dvd__add__times__triv__right__iff,axiom,
! [A2: real,B2: real,C: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A2 ) ) )
= ( dvd_dvd_real @ A2 @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_605_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_606_dvd__add__times__triv__left__iff,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_607_dvd__add__times__triv__left__iff,axiom,
! [A2: real,C: real,B2: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ ( times_times_real @ C @ A2 ) @ B2 ) )
= ( dvd_dvd_real @ A2 @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_608_unit__prod,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_609_unit__prod,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_610_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_611_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_612_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_613_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_614_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_615_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_616_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_617_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_618_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_619_dvd__mult__div__cancel,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ A2 ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_620_dvd__mult__div__cancel,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ A2 ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_621_dvd__div__mult__self,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_622_dvd__div__mult__self,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_623_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_624_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_625_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_626_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_627_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_628_eq__divide__eq__numeral1_I1_J,axiom,
! [A2: real,B2: real,W: num] :
( ( A2
= ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) )
= B2 ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_629_divide__eq__eq__numeral1_I1_J,axiom,
! [B2: real,W: num,A2: real] :
( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
= A2 )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B2
= ( 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_630_div__mult__self4,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A2 ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_631_div__mult__self4,axiom,
! [B2: int,C: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A2 ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_632_div__mult__self3,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A2 ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_633_div__mult__self3,axiom,
! [B2: int,C: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A2 ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_634_div__mult__self2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_635_div__mult__self2,axiom,
! [B2: int,A2: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_636_div__mult__self1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B2 ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_637_div__mult__self1,axiom,
! [B2: int,A2: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B2 ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_638_nonzero__divide__mult__cancel__right,axiom,
! [B2: real,A2: real] :
( ( B2 != zero_zero_real )
=> ( ( divide_divide_real @ B2 @ ( times_times_real @ A2 @ B2 ) )
= ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_639_nonzero__divide__mult__cancel__left,axiom,
! [A2: real,B2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B2 ) )
= ( divide_divide_real @ one_one_real @ B2 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_640_unit__div__mult__self,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_641_unit__div__mult__self,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_642_unit__mult__div__div,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A2 ) )
= ( divide_divide_nat @ B2 @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_643_unit__mult__div__div,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A2 ) )
= ( divide_divide_int @ B2 @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_644_even__mult__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_mult_iff
thf(fact_645_even__mult__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% even_mult_iff
thf(fact_646_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_647_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_648_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: real,A2: real,B2: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z2 ) @ B2 )
= ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_649_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: real,A2: real,B2: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B2 @ Z2 ) )
= A2 ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B2 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_650_frac__eq__eq,axiom,
! [Y2: real,Z2: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y2 )
= ( divide_divide_real @ W @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_651_add__frac__eq,axiom,
! [Y2: real,Z2: real,X: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_652_add__frac__num,axiom,
! [Y2: real,X: real,Z2: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ Z2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_653_add__num__frac,axiom,
! [Y2: real,Z2: real,X: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_654_divide__eq__eq,axiom,
! [B2: real,C: real,A2: real] :
( ( ( divide_divide_real @ B2 @ C )
= A2 )
= ( ( ( C != zero_zero_real )
=> ( B2
= ( times_times_real @ A2 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_655_eq__divide__eq,axiom,
! [A2: real,B2: real,C: real] :
( ( A2
= ( divide_divide_real @ B2 @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A2 @ C )
= B2 ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_656_divide__eq__imp,axiom,
! [C: real,B2: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( B2
= ( times_times_real @ A2 @ C ) )
=> ( ( divide_divide_real @ B2 @ C )
= A2 ) ) ) ).
% divide_eq_imp
thf(fact_657_eq__divide__imp,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= B2 )
=> ( A2
= ( divide_divide_real @ B2 @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_658_times__divide__times__eq,axiom,
! [X: real,Y2: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_659_divide__divide__times__eq,axiom,
! [X: real,Y2: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y2 @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_660_add__divide__eq__iff,axiom,
! [Z2: real,X: real,Y2: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y2 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y2 ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_661_divide__add__eq__iff,axiom,
! [Z2: real,X: real,Y2: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y2 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_662_nonzero__divide__eq__eq,axiom,
! [C: real,B2: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B2 @ C )
= A2 )
= ( B2
= ( times_times_real @ A2 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_663_nonzero__eq__divide__eq,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( A2
= ( divide_divide_real @ B2 @ C ) )
= ( ( times_times_real @ A2 @ C )
= B2 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_664_divide__divide__eq__left_H,axiom,
! [A2: real,B2: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B2 ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ C @ B2 ) ) ) ).
% divide_divide_eq_left'
thf(fact_665_left__add__mult__distrib,axiom,
! [I: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_666_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_667_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_668_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_669_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_670_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_671_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_672_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_673_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_674_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_675_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_676_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_677_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_678_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_679_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_680_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_681_comp__sgraph_Oincident__def,axiom,
undire7858122600432113898nt_nat = member_nat ).
% comp_sgraph.incident_def
thf(fact_682_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_683_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A4: real,B4: real] :
( ( A4 = zero_zero_real )
=> ( B4 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_684_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_685_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_686_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_687_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_688_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B2: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B2 @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B2 ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_689_divide__eq__eq__numeral_I1_J,axiom,
! [B2: real,C: real,W: num] :
( ( ( divide_divide_real @ B2 @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B2
= ( 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_690_unit__dvdE,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [C4: nat] :
( B2
!= ( times_times_nat @ A2 @ C4 ) ) ) ) ).
% unit_dvdE
thf(fact_691_unit__dvdE,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [C4: int] :
( B2
!= ( times_times_int @ A2 @ C4 ) ) ) ) ).
% unit_dvdE
thf(fact_692_dvd__div__div__eq__mult,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( A2 != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B2 @ A2 )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B2 @ C )
= ( times_times_nat @ A2 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_693_dvd__div__div__eq__mult,axiom,
! [A2: int,C: int,B2: int,D: int] :
( ( A2 != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B2 @ A2 )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B2 @ C )
= ( times_times_int @ A2 @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_694_dvd__div__iff__mult,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_695_dvd__div__iff__mult,axiom,
! [C: int,B2: int,A2: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_696_div__dvd__iff__mult,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_697_div__dvd__iff__mult,axiom,
! [B2: int,A2: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_698_dvd__div__eq__mult,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( ( divide_divide_nat @ B2 @ A2 )
= C )
= ( B2
= ( times_times_nat @ C @ A2 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_699_dvd__div__eq__mult,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( ( divide_divide_int @ B2 @ A2 )
= C )
= ( B2
= ( times_times_int @ C @ A2 ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_700_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_701_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_702_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_703_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_704_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_705_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_706_dvdE,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ~ ! [K2: nat] :
( A2
!= ( times_times_nat @ B2 @ K2 ) ) ) ).
% dvdE
thf(fact_707_dvdE,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ~ ! [K2: int] :
( A2
!= ( times_times_int @ B2 @ K2 ) ) ) ).
% dvdE
thf(fact_708_dvdE,axiom,
! [B2: real,A2: real] :
( ( dvd_dvd_real @ B2 @ A2 )
=> ~ ! [K2: real] :
( A2
!= ( times_times_real @ B2 @ K2 ) ) ) ).
% dvdE
thf(fact_709_dvdI,axiom,
! [A2: nat,B2: nat,K: nat] :
( ( A2
= ( times_times_nat @ B2 @ K ) )
=> ( dvd_dvd_nat @ B2 @ A2 ) ) ).
% dvdI
thf(fact_710_dvdI,axiom,
! [A2: int,B2: int,K: int] :
( ( A2
= ( times_times_int @ B2 @ K ) )
=> ( dvd_dvd_int @ B2 @ A2 ) ) ).
% dvdI
thf(fact_711_dvdI,axiom,
! [A2: real,B2: real,K: real] :
( ( A2
= ( times_times_real @ B2 @ K ) )
=> ( dvd_dvd_real @ B2 @ A2 ) ) ).
% dvdI
thf(fact_712_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B4: nat,A4: nat] :
? [K3: nat] :
( A4
= ( times_times_nat @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_713_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B4: int,A4: int] :
? [K3: int] :
( A4
= ( times_times_int @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_714_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B4: real,A4: real] :
? [K3: real] :
( A4
= ( times_times_real @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_715_dvd__mult,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_716_dvd__mult,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_717_dvd__mult,axiom,
! [A2: real,C: real,B2: real] :
( ( dvd_dvd_real @ A2 @ C )
=> ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_718_dvd__mult2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_719_dvd__mult2,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_720_dvd__mult2,axiom,
! [A2: real,B2: real,C: real] :
( ( dvd_dvd_real @ A2 @ B2 )
=> ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_721_dvd__mult__left,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_722_dvd__mult__left,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_723_dvd__mult__left,axiom,
! [A2: real,B2: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B2 ) @ C )
=> ( dvd_dvd_real @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_724_dvd__triv__left,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_725_dvd__triv__left,axiom,
! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_726_dvd__triv__left,axiom,
! [A2: real,B2: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ A2 @ B2 ) ) ).
% dvd_triv_left
thf(fact_727_mult__dvd__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_728_mult__dvd__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( dvd_dvd_int @ A2 @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_729_mult__dvd__mono,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( dvd_dvd_real @ A2 @ B2 )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_730_dvd__mult__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
=> ( dvd_dvd_nat @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_731_dvd__mult__right,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
=> ( dvd_dvd_int @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_732_dvd__mult__right,axiom,
! [A2: real,B2: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B2 ) @ C )
=> ( dvd_dvd_real @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_733_dvd__triv__right,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_734_dvd__triv__right,axiom,
! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_735_dvd__triv__right,axiom,
! [A2: real,B2: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ A2 ) ) ).
% dvd_triv_right
thf(fact_736_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_737_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_738_mult__of__nat__commute,axiom,
! [X: nat,Y2: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y2 )
= ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_739_mult__of__nat__commute,axiom,
! [X: nat,Y2: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y2 )
= ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_740_mult__of__nat__commute,axiom,
! [X: nat,Y2: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y2 )
= ( times_times_real @ Y2 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_741_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_742_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_743_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_744_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_745_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_746_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_747_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_748_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_749_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_750_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_751_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_752_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_753_dvd__0__left,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_754_dvd__0__left,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
=> ( A2 = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_755_dvd__0__left,axiom,
! [A2: real] :
( ( dvd_dvd_real @ zero_zero_real @ A2 )
=> ( A2 = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_756_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_757_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_758_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_759_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_760_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_761_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_762_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_763_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_764_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_765_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_766_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_767_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_768_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_769_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_770_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_771_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_772_zdvd__period,axiom,
! [A2: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A2 @ D )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_773_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_774_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_775_right__inverse__eq,axiom,
! [B2: real,A2: real] :
( ( B2 != zero_zero_real )
=> ( ( ( divide_divide_real @ A2 @ B2 )
= one_one_real )
= ( A2 = B2 ) ) ) ).
% right_inverse_eq
thf(fact_776_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_777_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_778_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_779_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_780_is__unit__div__mult__cancel__right,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) )
= ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_781_is__unit__div__mult__cancel__right,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ A2 ) )
= ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_782_is__unit__div__mult__cancel__left,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) )
= ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_783_is__unit__div__mult__cancel__left,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ A2 @ B2 ) )
= ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_784_is__unitE,axiom,
! [A2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [B6: nat] :
( ( B6 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A2 )
= B6 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B6 )
= A2 )
=> ( ( ( times_times_nat @ A2 @ B6 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A2 )
!= ( times_times_nat @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_785_is__unitE,axiom,
! [A2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [B6: int] :
( ( B6 != zero_zero_int )
=> ( ( dvd_dvd_int @ B6 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A2 )
= B6 )
=> ( ( ( divide_divide_int @ one_one_int @ B6 )
= A2 )
=> ( ( ( times_times_int @ A2 @ B6 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A2 )
!= ( times_times_int @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_786_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_787_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_788_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_789_mult__numeral__1__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_790_mult__numeral__1__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_791_mult__numeral__1__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ ( numeral_numeral_real @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_792_mult__numeral__1__right,axiom,
! [A2: extended_enat] :
( ( times_7803423173614009249d_enat @ A2 @ ( numera1916890842035813515d_enat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_793_mult__numeral__1,axiom,
! [A2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_794_mult__numeral__1,axiom,
! [A2: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_795_mult__numeral__1,axiom,
! [A2: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_796_mult__numeral__1,axiom,
! [A2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_797_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_798_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_799_unit__mult__right__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ B2 @ A2 )
= ( times_times_nat @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_800_unit__mult__right__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ B2 @ A2 )
= ( times_times_int @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_801_unit__mult__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B2 )
= ( times_times_nat @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_802_unit__mult__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B2 )
= ( times_times_int @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_803_mult__unit__dvd__iff_H,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_804_mult__unit__dvd__iff_H,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_805_dvd__mult__unit__iff_H,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_806_dvd__mult__unit__iff_H,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_807_mult__unit__dvd__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_808_mult__unit__dvd__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_809_dvd__mult__unit__iff,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_810_dvd__mult__unit__iff,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B2 ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_811_is__unit__mult__iff,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_812_is__unit__mult__iff,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
& ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_813_div__mult__div__if__dvd,axiom,
! [B2: nat,A2: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_814_div__mult__div__if__dvd,axiom,
! [B2: int,A2: int,D: int,C: int] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_815_dvd__mult__imp__div,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B2 )
=> ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_816_dvd__mult__imp__div,axiom,
! [A2: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B2 )
=> ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_817_dvd__div__mult2__eq,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_818_dvd__div__mult2__eq,axiom,
! [B2: int,C: int,A2: int] :
( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A2 )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_819_div__div__eq__right,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_820_div__div__eq__right,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( divide_divide_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_821_div__mult__swap,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_822_div__mult__swap,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_823_dvd__div__mult,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A2 )
= ( divide_divide_nat @ ( times_times_nat @ B2 @ A2 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_824_dvd__div__mult,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A2 )
= ( divide_divide_int @ ( times_times_int @ B2 @ A2 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_825_div__mult2__eq_H,axiom,
! [A2: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_826_div__mult2__eq_H,axiom,
! [A2: int,M: nat,N: nat] :
( ( divide_divide_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_827_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_828_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_829_dvd__div__eq__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( ( divide_divide_nat @ A2 @ B2 )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_830_dvd__div__eq__0__iff,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( ( divide_divide_int @ A2 @ B2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_831_dvd__div__eq__0__iff,axiom,
! [B2: real,A2: real] :
( ( dvd_dvd_real @ B2 @ A2 )
=> ( ( ( divide_divide_real @ A2 @ B2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_832_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_833_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_834_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_835_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_836_is__unit__div__mult2__eq,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_837_is__unit__div__mult2__eq,axiom,
! [B2: int,C: int,A2: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_838_unit__div__mult__swap,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_839_unit__div__mult__swap,axiom,
! [C: int,A2: int,B2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_840_unit__div__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_841_unit__div__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
= ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_842_div__mult__unit2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_843_div__mult__unit2,axiom,
! [C: int,B2: int,A2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_844_unit__eq__div2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( A2
= ( divide_divide_nat @ C @ B2 ) )
= ( ( times_times_nat @ A2 @ B2 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_845_unit__eq__div2,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( A2
= ( divide_divide_int @ C @ B2 ) )
= ( ( times_times_int @ A2 @ B2 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_846_unit__eq__div1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A2 @ B2 )
= C )
= ( A2
= ( times_times_nat @ C @ B2 ) ) ) ) ).
% unit_eq_div1
thf(fact_847_unit__eq__div1,axiom,
! [B2: int,A2: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A2 @ B2 )
= C )
= ( A2
= ( times_times_int @ C @ B2 ) ) ) ) ).
% unit_eq_div1
thf(fact_848_div__add__self2,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_849_div__add__self2,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_850_div__add__self1,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_851_div__add__self1,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_852_unit__div__eq__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A2 @ B2 )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_853_unit__div__eq__0__iff,axiom,
! [B2: int,A2: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A2 @ B2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_854_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_855_div__exp__eq,axiom,
! [A2: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_856_div__exp__eq,axiom,
! [A2: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_857_left__add__twice,axiom,
! [A2: nat,B2: nat] :
( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_858_left__add__twice,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_859_left__add__twice,axiom,
! [A2: real,B2: real] :
( ( plus_plus_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_860_left__add__twice,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_861_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_862_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_863_mult__2__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_864_mult__2__right,axiom,
! [Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_865_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_866_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_867_mult__2,axiom,
! [Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_868_mult__2,axiom,
! [Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_869_evenE,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B6: nat] :
( A2
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% evenE
thf(fact_870_evenE,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B6: int] :
( A2
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% evenE
thf(fact_871_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_872_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_873_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_874_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_875_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
= ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_876_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_877_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_878_card__1__singleton__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: a] :
( A
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_879_card__eq__SucD,axiom,
! [A: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A )
= ( suc @ K ) )
=> ? [B6: set_a,B5: set_set_a] :
( ( A
= ( insert_set_a @ B6 @ B5 ) )
& ~ ( member_set_a @ B6 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_880_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B6: nat,B5: set_nat] :
( ( A
= ( insert_nat @ B6 @ B5 ) )
& ~ ( member_nat @ B6 @ B5 )
& ( ( finite_card_nat @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_881_card__eq__SucD,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
=> ? [B6: a,B5: set_a] :
( ( A
= ( insert_a @ B6 @ B5 ) )
& ~ ( member_a @ B6 @ B5 )
& ( ( finite_card_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_882_card__Suc__eq,axiom,
! [A: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A )
= ( suc @ K ) )
= ( ? [B4: set_a,B3: set_set_a] :
( ( A
= ( insert_set_a @ B4 @ B3 ) )
& ~ ( member_set_a @ B4 @ B3 )
& ( ( finite_card_set_a @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_883_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B4: nat,B3: set_nat] :
( ( A
= ( insert_nat @ B4 @ B3 ) )
& ~ ( member_nat @ B4 @ B3 )
& ( ( finite_card_nat @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_884_card__Suc__eq,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B4: a,B3: set_a] :
( ( A
= ( insert_a @ B4 @ B3 ) )
& ~ ( member_a @ B4 @ B3 )
& ( ( finite_card_a @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_885_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_886_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_887_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_888_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: nat,E3: set_nat] :
( ( ( undire7858122600432113898nt_nat @ V @ E3 )
& ( ( finite_card_nat @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_889_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: a,E3: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_890_oddE,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B6: nat] :
( A2
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_891_oddE,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B6: int] :
( A2
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_892_sum__power2__eq__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_893_sum__power2__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_894_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_895_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N )
= ( semiri4216267220026989637d_enat @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_896_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_897_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_898_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y2 )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_899_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri4216267220026989637d_enat @ Y2 )
= ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_900_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y2 )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_901_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y2 )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_902_zero__eq__power2,axiom,
! [A2: nat] :
( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_903_zero__eq__power2,axiom,
! [A2: real] :
( ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_904_zero__eq__power2,axiom,
! [A2: int] :
( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_905_power__add__numeral2,axiom,
! [A2: nat,M: num,N: num,B2: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_906_power__add__numeral2,axiom,
! [A2: int,M: num,N: num,B2: int] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_907_power__add__numeral2,axiom,
! [A2: real,M: num,N: num,B2: real] :
( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
= ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% power_add_numeral2
thf(fact_908_power__add__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_909_power__add__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_910_power__add__numeral,axiom,
! [A2: real,M: num,N: num] :
( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_911_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_912_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_913_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_914_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_915_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_916_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_917_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_918_power__one__right,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_919_power__one__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_920_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_921_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_922_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_923_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_924_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_925_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_926_power__Suc0__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_927_power__Suc0__right,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_928_power__Suc0__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_929_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_930_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_931_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
= ( X
= ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_932_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
= ( X
= ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_933_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
= ( X
= ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_934_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B2 @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_935_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B2 @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_936_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B2 @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_937_power__mult__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_938_power__mult__numeral,axiom,
! [A2: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_939_power__mult__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_940_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_941_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_942_power__divide,axiom,
! [A2: real,B2: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A2 @ B2 ) @ N )
= ( divide_divide_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% power_divide
thf(fact_943_dvd__power__same,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_944_dvd__power__same,axiom,
! [X: real,Y2: real,N: nat] :
( ( dvd_dvd_real @ X @ Y2 )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_945_dvd__power__same,axiom,
! [X: int,Y2: int,N: nat] :
( ( dvd_dvd_int @ X @ Y2 )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_946_left__right__inverse__power,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_947_left__right__inverse__power,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_948_left__right__inverse__power,axiom,
! [X: real,Y2: real,N: nat] :
( ( ( times_times_real @ X @ Y2 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_949_power__one__over,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A2 ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).
% power_one_over
thf(fact_950_power__Suc2,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_951_power__Suc2,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_952_power__Suc2,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ A2 @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_953_power__Suc,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_954_power__Suc,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ N ) )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_955_power__Suc,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ A2 @ ( suc @ N ) )
= ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_956_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_957_power__0,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_958_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_959_div__power,axiom,
! [B2: nat,A2: nat,N: nat] :
( ( dvd_dvd_nat @ B2 @ A2 )
=> ( ( power_power_nat @ ( divide_divide_nat @ A2 @ B2 ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% div_power
thf(fact_960_div__power,axiom,
! [B2: int,A2: int,N: nat] :
( ( dvd_dvd_int @ B2 @ A2 )
=> ( ( power_power_int @ ( divide_divide_int @ A2 @ B2 ) @ N )
= ( divide_divide_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% div_power
thf(fact_961_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_962_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_963_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_964_power__even__eq,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_965_power__even__eq,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_966_power__even__eq,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_967_is__unit__power__iff,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_968_is__unit__power__iff,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_969_power__numeral__even,axiom,
! [Z2: nat,W: num] :
( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_970_power__numeral__even,axiom,
! [Z2: int,W: num] :
( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_971_power__numeral__even,axiom,
! [Z2: real,W: num] :
( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_972_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_973_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_974_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_975_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_976_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_977_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_978_power2__eq__square,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_979_power2__eq__square,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_980_power2__eq__square,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_981_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_982_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_983_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_984_power__odd__eq,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A2 @ ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_985_power__odd__eq,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A2 @ ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_986_power__odd__eq,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A2 @ ( power_power_real @ ( power_power_real @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_987_power2__sum,axiom,
! [X: nat,Y2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_988_power2__sum,axiom,
! [X: int,Y2: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_989_power2__sum,axiom,
! [X: real,Y2: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_990_power2__sum,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_991_set__bit__0,axiom,
! [A2: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_992_set__bit__0,axiom,
! [A2: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_993_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_994_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_995_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B2 )
& ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_996_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
= ( ? [X2: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_997_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
= ( ? [X2: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_998_unity__coeff__ex,axiom,
! [P: real > $o,L: real] :
( ( ? [X2: real] : ( P @ ( times_times_real @ L @ X2 ) ) )
= ( ? [X2: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X2 @ zero_zero_real ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_999_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1000_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( A2 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1001_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( dvd_dvd_nat @ A2 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1002_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1003_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A4: nat,B4: nat] :
( ( ( dvd_dvd_nat @ A4 @ B4 )
& ( A4 != B4 ) )
| ( A4 = B4 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1004_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd_nat @ A2 @ B2 )
& ~ ( dvd_dvd_nat @ B2 @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1005_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1006_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1007_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1008_gcd__nat_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_1009_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_1010_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( dvd_dvd_nat @ A4 @ B4 )
& ( dvd_dvd_nat @ B4 @ A4 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1011_gcd__nat_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% gcd_nat.trans
thf(fact_1012_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_1013_gcd__nat_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_1014_dvd__productE,axiom,
! [P2: nat,A2: nat,B2: nat] :
( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A2 @ B2 ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P2
= ( times_times_nat @ X3 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X3 @ A2 )
=> ~ ( dvd_dvd_nat @ Y3 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_1015_dvd__productE,axiom,
! [P2: int,A2: int,B2: int] :
( ( dvd_dvd_int @ P2 @ ( times_times_int @ A2 @ B2 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P2
= ( times_times_int @ X3 @ Y3 ) )
=> ( ( dvd_dvd_int @ X3 @ A2 )
=> ~ ( dvd_dvd_int @ Y3 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_1016_division__decomp,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
=> ? [B7: nat,C5: nat] :
( ( A2
= ( times_times_nat @ B7 @ C5 ) )
& ( dvd_dvd_nat @ B7 @ B2 )
& ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% division_decomp
thf(fact_1017_division__decomp,axiom,
! [A2: int,B2: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) )
=> ? [B7: int,C5: int] :
( ( A2
= ( times_times_int @ B7 @ C5 ) )
& ( dvd_dvd_int @ B7 @ B2 )
& ( dvd_dvd_int @ C5 @ C ) ) ) ).
% division_decomp
thf(fact_1018_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1019_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
& ( zero_zero_nat != A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1020_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1021_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_1022_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1023_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_1024_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_1025_even__set__bit__iff,axiom,
! [M: nat,A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_1026_even__set__bit__iff,axiom,
! [M: nat,A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_1027_bezout__lemma__nat,axiom,
! [D: nat,A2: nat,B2: nat,X: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A2 )
=> ( ( dvd_dvd_nat @ D @ B2 )
=> ( ( ( ( times_times_nat @ A2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y2 ) @ D ) )
| ( ( times_times_nat @ B2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A2 @ B2 ) )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1028_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B2 )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D2 ) )
| ( ( times_times_nat @ B2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1029_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_1030_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_1031_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_1032_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_1033_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( semiri7260567687927622513x_real
@ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
@ N2
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_1034_even__succ__div__exp,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1035_even__succ__div__exp,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1036_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_1037_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1038_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_1039_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_1040_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_1041_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1042_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_1043_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1044_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1045_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_1046_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1047_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1048_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_1049_real__divide__square__eq,axiom,
! [R2: real,A2: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A2 ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A2 @ R2 ) ) ).
% real_divide_square_eq
thf(fact_1050_power__inject__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ( power_power_nat @ A2 @ M )
= ( power_power_nat @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1051_power__inject__exp,axiom,
! [A2: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ( power_power_real @ A2 @ M )
= ( power_power_real @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1052_power__inject__exp,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ( power_power_int @ A2 @ M )
= ( power_power_int @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1053_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1054_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1055_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1056_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1057_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1058_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1059_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_1060_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_1061_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_1062_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_1063_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1064_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1065_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1066_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_1067_divide__less__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1068_divide__less__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1069_less__divide__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1070_less__divide__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1071_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_1072_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_1073_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_1074_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_1075_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1076_divide__less__eq__numeral1_I1_J,axiom,
! [B2: real,W: num,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A2 )
= ( ord_less_real @ B2 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1077_less__divide__eq__numeral1_I1_J,axiom,
! [A2: real,B2: real,W: num] :
( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1078_power__strict__increasing__iff,axiom,
! [B2: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1079_power__strict__increasing__iff,axiom,
! [B2: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1080_power__strict__increasing__iff,axiom,
! [B2: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1081_pow__divides__pow__iff,axiom,
! [N: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).
% pow_divides_pow_iff
thf(fact_1082_pow__divides__pow__iff,axiom,
! [N: nat,A2: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
= ( dvd_dvd_int @ A2 @ B2 ) ) ) ).
% pow_divides_pow_iff
thf(fact_1083_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_1084_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_1085_power__strict__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1086_power__strict__decreasing__iff,axiom,
! [B2: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1087_power__strict__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1088_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1089_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1090_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1091_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1092_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1093_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1094_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1095_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1096_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1097_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1098_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1099_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1100_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1101_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1102_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1103_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1104_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1105_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_1106_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_1107_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_1108_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_1109_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_1110_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1111_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1120_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_1121_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_1122_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1123_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_1124_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_1125_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1126_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1127_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_1128_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_1129_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1130_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_1131_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1132_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1133_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1134_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1135_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1136_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1137_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1138_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_1139_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1140_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_1141_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1142_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_1143_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_1144_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_1145_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1146_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1147_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_card__less,axiom,
! [M6: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M6 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1159_card__less__Suc,axiom,
! [M6: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M6 )
& ( ord_less_nat @ K3 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M6 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1160_card__less__Suc2,axiom,
! [M6: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M6 )
& ( ord_less_nat @ K3 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M6 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1161_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1162_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_1163_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_1164_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_1165_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1166_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_1167_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_1168_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_1169_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_1170_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_1171_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_1172_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_1173_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_1174_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_1175_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_1176_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_1177_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_1178_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_1179_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_1180_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_1181_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_1182_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_1183_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_1184_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_1185_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1186_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_1187_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1188_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1189_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1190_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1191_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1192_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1193_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_1194_Suc__0__mod__numeral_I1_J,axiom,
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% Suc_0_mod_numeral(1)
thf(fact_1195_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_1196_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1197_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_1198_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_1199_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_1200_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_1201_Suc__0__mod__numeral_I2_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= one_one_nat ) ).
% Suc_0_mod_numeral(2)
thf(fact_1202_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_1203_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1204_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1205_zdiv__mono__strict,axiom,
! [A: int,B: int,N: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A @ N ) @ ( divide_divide_int @ B @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1206_real__arch__pow__inv,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1207_real__arch__pow,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1208_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1209_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1210_zmod__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zmod_int
thf(fact_1211_int__ops_I9_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A2 @ B2 ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(9)
thf(fact_1212_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1213_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P2 )
=> ( ( ord_less_nat @ M @ P2 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P2 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P2 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1214_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1215_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_1216_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_1217_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1218_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_1219_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1220_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1221_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1222_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_1223_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y7: real] :
? [N3: nat] : ( ord_less_real @ Y7 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1224_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1225_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_1226_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_1227_div__less__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A @ N ) @ ( divide_divide_nat @ B @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1228_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1229_div__mod__decomp,axiom,
! [A: nat,N: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N ) @ N ) @ ( modulo_modulo_nat @ A @ N ) ) ) ).
% div_mod_decomp
thf(fact_1230_div__mod__decomp__int,axiom,
! [A: int,N: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N ) @ N ) @ ( modulo_modulo_int @ A @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1231_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] :
( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_1232_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z: int] :
? [N2: nat] :
( Z
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1233_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_1234_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1235_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_1236_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_1237_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1238_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1239_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1240_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1241_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1242_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_1243_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_1244_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_1245_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1246_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1247_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1248_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1249_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1250_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_1251_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_1252_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1253_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_1254_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1255_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_1256_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_1257_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_1258_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
= ( zero_n2687167440665602831ol_nat
@ ( N
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_1259_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1260_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1261_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1262_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1263_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1264_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1265_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1266_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1267_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1268_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1269_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z: int] :
? [N2: nat] :
( Z
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
% 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,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [V2: a] :
( ( member_a @ V2 @ e )
=> ( ( member_a @ V2 @ y )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------