TPTP Problem File: SLH0375^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Universal_Hash_Families/0028_Field/prob_00145_005194__18277198_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1280 ( 628 unt; 54 typ; 0 def)
% Number of atoms : 3068 (1337 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 13159 ( 185 ~; 55 |; 131 &;11649 @)
% ( 0 <=>;1139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 136 ( 136 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 12 con; 0-3 aty)
% Number of variables : 2811 ( 64 ^;2709 !; 38 ?;2811 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:41:20.825
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Int__Oint_Mt__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J_J,type,
partia2818514838349642498t_unit: $tType ).
thf(ty_n_t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
monoid8431999971278595628t_unit: $tType ).
thf(ty_n_t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
ring_e6626950497611839816t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (47)
thf(sy_c_AbelCoset_Oa__r__coset_001t__Int__Oint_001t__Product____Type__Ounit,type,
a_r_co6205493800230438172t_unit: partia2818514838349642498t_unit > set_int > int > set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Int__Oint_001t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
partia8426541738980984321t_unit: partia2818514838349642498t_unit > set_int ).
thf(sy_c_Congruence_Opartial__object_Opartial__object__ext_001t__Int__Oint_001t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
partia4118392927963588428t_unit: set_int > monoid8431999971278595628t_unit > partia2818514838349642498t_unit ).
thf(sy_c_Group_Omonoid_Omonoid__ext_001t__Int__Oint_001t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
monoid6080699973261426200t_unit: ( int > int > int ) > int > ring_e6626950497611839816t_unit > monoid8431999971278595628t_unit ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
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_Ideal_Ogenideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
genide1613390280493775889t_unit: partia2818514838349642498t_unit > set_int > set_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_IntRing_OZMod,type,
zMod: int > int > set_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
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__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $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__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Ring_Oa__inv_001t__Int__Oint_001t__Product____Type__Ounit,type,
a_inv_8811962894454695315t_unit: partia2818514838349642498t_unit > int > int ).
thf(sy_c_Ring_Oadd__pow_001t__Int__Oint_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po6254319245437977817it_int: partia2818514838349642498t_unit > int > int > int ).
thf(sy_c_Ring_Oadd__pow_001t__Int__Oint_001t__Product____Type__Ounit_001t__Nat__Onat,type,
add_po6256809715947028093it_nat: partia2818514838349642498t_unit > nat > int > int ).
thf(sy_c_Ring_Oring_Oadd_001t__Int__Oint_001t__Product____Type__Ounit,type,
add_int_Product_unit: partia2818514838349642498t_unit > int > int > int ).
thf(sy_c_Ring_Oring_Oring__ext_001t__Int__Oint_001t__Product____Type__Ounit,type,
ring_e5272872978682396362t_unit: int > ( int > int > int ) > product_unit > ring_e6626950497611839816t_unit ).
thf(sy_c_Ring_Oring_Ozero_001t__Int__Oint_001t__Product____Type__Ounit,type,
zero_i2266321264637750939t_unit: partia2818514838349642498t_unit > int ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
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_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x_H____,type,
x: nat ).
thf(sy_v_y_H____,type,
y: nat ).
% Relevant facts (1222)
thf(fact_0_int_Ogenideal__one,axiom,
( ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ one_one_int @ bot_bot_set_int ) )
= top_top_set_int ) ).
% int.genideal_one
thf(fact_1_int_Ogenideal__zero,axiom,
( ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ).
% int.genideal_zero
thf(fact_2_int_Ogenideal__self_H,axiom,
! [I: int] :
( ( member_int @ I @ top_top_set_int )
=> ( member_int @ I @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) ).
% int.genideal_self'
thf(fact_3_i_Oadd_Ocarrier__not__empty,axiom,
( ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
!= bot_bot_set_int ) ).
% i.add.carrier_not_empty
thf(fact_4_mod__mult__self1,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_5_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_6_mod__mult__self2,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_7_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_8_mod__mult__self3,axiom,
! [C: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_9_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_10_mod__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_11_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_12_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_13_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_14_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_15_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_16_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_17_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_18_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_19_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_20_UNIV_I3_J,axiom,
! [P: int > $o] :
( ( ! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( P @ X ) ) )
= ( ! [X2: int] : ( P @ X2 ) ) ) ).
% UNIV(3)
thf(fact_21_UNIV_I4_J,axiom,
! [P: int > $o] :
( ( ? [X: int] :
( ( member_int @ X @ top_top_set_int )
& ( P @ X ) ) )
= ( ? [X2: int] : ( P @ X2 ) ) ) ).
% UNIV(4)
thf(fact_22_mod__mod__trivial,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_23_mod__mod__trivial,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_24_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_25_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_26_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_27_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_28_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_29_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_30_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_31_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_32_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_33_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_34_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_35_int_Oadd_Oone__closed,axiom,
member_int @ zero_zero_int @ top_top_set_int ).
% int.add.one_closed
thf(fact_36_int_Oadd_Oright__cancel,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( ( plus_plus_int @ Y @ X3 )
= ( plus_plus_int @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% int.add.right_cancel
thf(fact_37_int_Oadd_Om__closed,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( plus_plus_int @ X3 @ Y ) @ top_top_set_int ) ) ) ).
% int.add.m_closed
thf(fact_38_int_Oone__closed,axiom,
member_int @ one_one_int @ top_top_set_int ).
% int.one_closed
thf(fact_39_int_Om__closed,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( times_times_int @ X3 @ Y ) @ top_top_set_int ) ) ) ).
% int.m_closed
thf(fact_40_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_41_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_42_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_43_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_44_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_45_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_46_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_47_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_48_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_49_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_50_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_51_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_52_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_53_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_54_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_55_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_56_int_Oadd_Or__cancel__one_H,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( X3
= ( plus_plus_int @ A @ X3 ) )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.r_cancel_one'
thf(fact_57_int_Oadd_Ol__cancel__one_H,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( X3
= ( plus_plus_int @ X3 @ A ) )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.l_cancel_one'
thf(fact_58_int_Oadd_Or__cancel__one,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( ( plus_plus_int @ A @ X3 )
= X3 )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.r_cancel_one
thf(fact_59_int_Oadd_Ol__cancel__one,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( ( plus_plus_int @ X3 @ A )
= X3 )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.l_cancel_one
thf(fact_60_int_Oadd_Or__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( plus_plus_int @ X3 @ zero_zero_int )
= X3 ) ) ).
% int.add.r_one
thf(fact_61_int_Oadd_Ol__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( plus_plus_int @ zero_zero_int @ X3 )
= X3 ) ) ).
% int.add.l_one
thf(fact_62_int_Or__null,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( times_times_int @ X3 @ zero_zero_int )
= zero_zero_int ) ) ).
% int.r_null
thf(fact_63_int_Ol__null,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( times_times_int @ zero_zero_int @ X3 )
= zero_zero_int ) ) ).
% int.l_null
thf(fact_64_int_Or__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( times_times_int @ X3 @ one_one_int )
= X3 ) ) ).
% int.r_one
thf(fact_65_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_68_int_Ol__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( times_times_int @ one_one_int @ X3 )
= X3 ) ) ).
% int.l_one
thf(fact_69_int__Zcarr,axiom,
! [K: int] : ( member_int @ K @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int_Zcarr
thf(fact_70_nat__mod__eq__iff,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X3 @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q2: nat] :
( ( plus_plus_nat @ X3 @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_71_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_72_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_73_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_74_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_75_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_76_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_77_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_78_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_79_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_80_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_81_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_82_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_83_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_84_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_85_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_86_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_87_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_88_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_89_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_90_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_91_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_92_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_93_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_94_int_Oadd_Om__lcomm,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( plus_plus_int @ X3 @ ( plus_plus_int @ Y @ Z ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X3 @ Z ) ) ) ) ) ) ).
% int.add.m_lcomm
thf(fact_95_int_Oadd_Om__assoc,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ X3 @ Y ) @ Z )
= ( plus_plus_int @ X3 @ ( plus_plus_int @ Y @ Z ) ) ) ) ) ) ).
% int.add.m_assoc
thf(fact_96_int_Oadd_Om__comm,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ X3 @ Y )
= ( plus_plus_int @ Y @ X3 ) ) ) ) ).
% int.add.m_comm
thf(fact_97_int_Om__lcomm,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ X3 @ ( times_times_int @ Y @ Z ) )
= ( times_times_int @ Y @ ( times_times_int @ X3 @ Z ) ) ) ) ) ) ).
% int.m_lcomm
thf(fact_98_int_Om__assoc,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ ( times_times_int @ X3 @ Y ) @ Z )
= ( times_times_int @ X3 @ ( times_times_int @ Y @ Z ) ) ) ) ) ) ).
% int.m_assoc
thf(fact_99_int_Om__comm,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X3 @ Y )
= ( times_times_int @ Y @ X3 ) ) ) ) ).
% int.m_comm
thf(fact_100_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_101_int__carrier__eq,axiom,
( ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
= top_top_set_int ) ).
% int_carrier_eq
thf(fact_102_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_103_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_104_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_105_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_106_mod__add__cong,axiom,
! [A: int,C: int,A3: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A3 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B2 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_107_mod__add__cong,axiom,
! [A: nat,C: nat,A3: nat,B: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A3 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B2 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_108_mod__add__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_109_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_110_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_111_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_112_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_113_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_114_mult__mod__right,axiom,
! [C: int,A: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_115_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_116_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_117_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_118_mod__mult__cong,axiom,
! [A: int,C: int,A3: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A3 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B2 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_119_mod__mult__cong,axiom,
! [A: nat,C: nat,A3: nat,B: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A3 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B2 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B2 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_120_mod__mult__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_121_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_122_int_Oadd_Ocarrier__not__empty,axiom,
top_top_set_int != bot_bot_set_int ).
% int.add.carrier_not_empty
thf(fact_123_int_Oadd_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
=> ( ( plus_plus_int @ U @ X4 )
= X4 ) )
=> ( U = zero_zero_int ) ) ) ).
% int.add.one_unique
thf(fact_124_int_Oadd_Oinv__unique,axiom,
! [Y: int,X3: int,Y2: int] :
( ( ( plus_plus_int @ Y @ X3 )
= zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y2 )
= zero_zero_int )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y2 @ top_top_set_int )
=> ( Y = Y2 ) ) ) ) ) ) ).
% int.add.inv_unique
thf(fact_125_int_Oadd_Or__inv__ex,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( ( plus_plus_int @ X3 @ X4 )
= zero_zero_int ) ) ) ).
% int.add.r_inv_ex
thf(fact_126_int_Oadd_Ol__inv__ex,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( ( plus_plus_int @ X4 @ X3 )
= zero_zero_int ) ) ) ).
% int.add.l_inv_ex
thf(fact_127_int_Oadd_Oinv__comm,axiom,
! [X3: int,Y: int] :
( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ Y @ X3 )
= zero_zero_int ) ) ) ) ).
% int.add.inv_comm
thf(fact_128_int_Ointegral__iff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral_iff
thf(fact_129_int_Om__rcancel,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% int.m_rcancel
thf(fact_130_int_Om__lcancel,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% int.m_lcancel
thf(fact_131_int_Ointegral,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral
thf(fact_132_int_Or__distr,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ Z @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ Z @ X3 ) @ ( times_times_int @ Z @ Y ) ) ) ) ) ) ).
% int.r_distr
thf(fact_133_int_Ol__distr,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ ( plus_plus_int @ X3 @ Y ) @ Z )
= ( plus_plus_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y @ Z ) ) ) ) ) ) ).
% int.l_distr
thf(fact_134_int_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
=> ( ( times_times_int @ U @ X4 )
= X4 ) )
=> ( U = one_one_int ) ) ) ).
% int.one_unique
thf(fact_135_int_Oinv__unique,axiom,
! [Y: int,X3: int,Y2: int] :
( ( ( times_times_int @ Y @ X3 )
= one_one_int )
=> ( ( ( times_times_int @ X3 @ Y2 )
= one_one_int )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y2 @ top_top_set_int )
=> ( Y = Y2 ) ) ) ) ) ) ).
% int.inv_unique
thf(fact_136_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D ) ) ) ) ).
% mod_eqE
thf(fact_137_int_Ocarrier__one__not__zero,axiom,
( ( top_top_set_int
!= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= ( one_one_int != zero_zero_int ) ) ).
% int.carrier_one_not_zero
thf(fact_138_int_Ocarrier__one__zero,axiom,
( ( top_top_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= ( one_one_int = zero_zero_int ) ) ).
% int.carrier_one_zero
thf(fact_139_int_Oone__zeroI,axiom,
( ( top_top_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
=> ( one_one_int = zero_zero_int ) ) ).
% int.one_zeroI
thf(fact_140_int_Oone__zeroD,axiom,
( ( one_one_int = zero_zero_int )
=> ( top_top_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% int.one_zeroD
thf(fact_141_i_Oadd_Or__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ A @ C )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B @ C ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ B @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ C @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% i.add.r_cancel
thf(fact_142_i_Oadd_Om__lcomm,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Z @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ Z ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Z ) ) ) ) ) ) ).
% i.add.m_lcomm
thf(fact_143_i_Oadd_Om__comm,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 ) ) ) ) ).
% i.add.m_comm
thf(fact_144_i_Oadd_Om__assoc,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Z @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) @ Z )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ Z ) ) ) ) ) ) ).
% i.add.m_assoc
thf(fact_145_i_Oadd_Ol__cancel,axiom,
! [C: int,A: int,B: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ C @ A )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ C @ B ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ B @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ C @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% i.add.l_cancel
thf(fact_146_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_147_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_148_sum__squares__eq__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_149_ZMod__def,axiom,
( zMod
= ( ^ [K2: int] : ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K2 @ bot_bot_set_int ) ) ) ) ) ).
% ZMod_def
thf(fact_150_i_Oadd_Onat__pow__pow,axiom,
! [X3: int,M: nat,N: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_nat @ N @ M ) @ X3 ) ) ) ).
% i.add.nat_pow_pow
thf(fact_151_i_Oadd_Oint__pow__pow,axiom,
! [X3: int,M: int,N: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) )
= ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ N @ M ) @ X3 ) ) ) ).
% i.add.int_pow_pow
thf(fact_152_zfact__iso__def,axiom,
( ring_zfact_iso
= ( ^ [P2: nat,K2: nat] : ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ P2 ) @ bot_bot_set_int ) ) @ ( semiri1314217659103216013at_int @ K2 ) ) ) ) ).
% zfact_iso_def
thf(fact_153_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_154_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_155_UNIV__I,axiom,
! [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% UNIV_I
thf(fact_156_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_157_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_158_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_159_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X: int] :
~ ( member_int @ X @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_160_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X: int] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_161_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X: int] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_162_insertCI,axiom,
! [A: int,B3: set_int,B: int] :
( ( ~ ( member_int @ A @ B3 )
=> ( A = B ) )
=> ( member_int @ A @ ( insert_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_163_insert__iff,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
= ( ( A = B )
| ( member_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_164_insert__absorb2,axiom,
! [X3: int,A2: set_int] :
( ( insert_int @ X3 @ ( insert_int @ X3 @ A2 ) )
= ( insert_int @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_165_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_166_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_167_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_168_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_169_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_170_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_171_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_172_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_173_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_174_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_175_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_176_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_177_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_178_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_179_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_180_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_181_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_182_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_183_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_184_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_185_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_186_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_187_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_188_i_Oadd_Ogroup__commutes__pow,axiom,
! [X3: int,Y: int,N: nat] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 ) )
=> ( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ Y )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ) ) ).
% i.add.group_commutes_pow
thf(fact_189_i_Oadd_Onat__pow__comm,axiom,
! [X3: int,N: nat,M: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ).
% i.add.nat_pow_comm
thf(fact_190_i_Oadd_Onat__pow__distrib,axiom,
! [X3: int,Y: int,N: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ).
% i.add.nat_pow_distrib
thf(fact_191_i_Oadd_Opow__mult__distrib,axiom,
! [X3: int,Y: int,N: nat] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 ) )
=> ( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ) ).
% i.add.pow_mult_distrib
thf(fact_192_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_193_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_194_i_Oadd_Oint__pow__distrib,axiom,
! [X3: int,Y: int,I: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ).
% i.add.int_pow_distrib
thf(fact_195_i_Oadd_Oint__pow__mult,axiom,
! [X3: int,I: int,J: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ I @ J ) @ X3 )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ J @ X3 ) ) ) ) ).
% i.add.int_pow_mult
thf(fact_196_i_Oadd_Oint__pow__mult__distrib,axiom,
! [X3: int,Y: int,I: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 ) )
=> ( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ) ).
% i.add.int_pow_mult_distrib
thf(fact_197_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_198_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_i_Oadd_Onat__pow__mult,axiom,
! [X3: int,N: nat,M: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_nat @ N @ M ) @ X3 ) ) ) ).
% i.add.nat_pow_mult
thf(fact_206_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_207_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_208_int_Oadd_Onat__pow__one,axiom,
! [N: nat] :
( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ zero_zero_int )
= zero_zero_int ) ).
% int.add.nat_pow_one
thf(fact_209_int_Oadd_Onat__pow__closed,axiom,
! [X3: int,N: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( member_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ top_top_set_int ) ) ).
% int.add.nat_pow_closed
thf(fact_210_int_Oadd_Onat__pow__0,axiom,
! [X3: int] :
( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ zero_zero_nat @ X3 )
= zero_zero_int ) ).
% int.add.nat_pow_0
thf(fact_211_int_Oadd_Oint__pow__1,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_int @ X3 )
= X3 ) ) ).
% int.add.int_pow_1
thf(fact_212_int_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Z @ zero_zero_int )
= zero_zero_int ) ).
% int.add.int_pow_one
thf(fact_213_int_Oadd_Oint__pow__closed,axiom,
! [X3: int,I: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( member_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ top_top_set_int ) ) ).
% int.add.int_pow_closed
thf(fact_214_i_Oadd_Om__closed,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( member_int @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% i.add.m_closed
thf(fact_215_i_Oadd_Oright__cancel,axiom,
! [X3: int,Y: int,Z: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Z @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% i.add.right_cancel
thf(fact_216_i_Oadd_Onat__pow__closed,axiom,
! [X3: int,N: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( member_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% i.add.nat_pow_closed
thf(fact_217_i_Oadd_Oint__pow__1,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_int @ X3 )
= X3 ) ) ).
% i.add.int_pow_1
thf(fact_218_i_Oadd_Oint__pow__closed,axiom,
! [X3: int,I: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( member_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% i.add.int_pow_closed
thf(fact_219_i_Oadd_Onat__pow__eone,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_nat @ X3 )
= X3 ) ) ).
% i.add.nat_pow_eone
thf(fact_220_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_221_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_222_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_223_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_224_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_225_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_226_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_227_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_228_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_229_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_230_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_231_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_232_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_233_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_234_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_235_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_236_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_237_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_238_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_239_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_240_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_241_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_242_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_243_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_244_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_245_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_246_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_247_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_248_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_249_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_250_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_251_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_252_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_253_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_254_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_255_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_256_split__mod,axiom,
! [Q: nat > $o,M: nat,N: nat] :
( ( Q @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( Q @ J2 ) ) ) ) ) ).
% split_mod
thf(fact_257_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_258_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_259_sum__squares__gt__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X3 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_260_add__less__zeroD,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X3 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X3 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_261_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_262_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_263_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_264_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_265_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_266_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_267_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_268_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_269_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_270_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_271_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_272_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_273_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_274_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_275_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_276_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_277_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_278_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_279_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_280_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_281_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_282_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_283_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_284_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_285_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_286_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_287_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_288_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_289_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_290_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_291_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_292_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_293_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_294_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_295_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_296_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_297_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_298_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_299_ZMod__mod,axiom,
( zMod
= ( ^ [M3: int,A4: int] : ( zMod @ M3 @ ( modulo_modulo_int @ A4 @ M3 ) ) ) ) ).
% ZMod_mod
thf(fact_300_ZMod__eq__mod,axiom,
! [M: int,A: int,B: int] :
( ( ( zMod @ M @ A )
= ( zMod @ M @ B ) )
= ( ( modulo_modulo_int @ A @ M )
= ( modulo_modulo_int @ B @ M ) ) ) ).
% ZMod_eq_mod
thf(fact_301_ZMod__imp__zmod,axiom,
! [M: int,A: int,B: int] :
( ( ( zMod @ M @ A )
= ( zMod @ M @ B ) )
=> ( ( modulo_modulo_int @ A @ M )
= ( modulo_modulo_int @ B @ M ) ) ) ).
% ZMod_imp_zmod
thf(fact_302_zmod__imp__ZMod,axiom,
! [A: int,M: int,B: int] :
( ( ( modulo_modulo_int @ A @ M )
= ( modulo_modulo_int @ B @ M ) )
=> ( ( zMod @ M @ A )
= ( zMod @ M @ B ) ) ) ).
% zmod_imp_ZMod
thf(fact_303_not__sum__squares__lt__zero,axiom,
! [X3: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_304_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_305_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_306_UNIV__eq__I,axiom,
! [A2: set_int] :
( ! [X4: int] : ( member_int @ X4 @ A2 )
=> ( top_top_set_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_307_UNIV__witness,axiom,
? [X4: int] : ( member_int @ X4 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_308_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_309_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_310_equals0I,axiom,
! [A2: set_int] :
( ! [Y3: int] :
~ ( member_int @ Y3 @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_311_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X: int] : ( member_int @ X @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_312_insertE,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_313_insertI1,axiom,
! [A: int,B3: set_int] : ( member_int @ A @ ( insert_int @ A @ B3 ) ) ).
% insertI1
thf(fact_314_insertI2,axiom,
! [A: int,B3: set_int,B: int] :
( ( member_int @ A @ B3 )
=> ( member_int @ A @ ( insert_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_315_Set_Oset__insert,axiom,
! [X3: int,A2: set_int] :
( ( member_int @ X3 @ A2 )
=> ~ ! [B4: set_int] :
( ( A2
= ( insert_int @ X3 @ B4 ) )
=> ( member_int @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_316_insert__ident,axiom,
! [X3: int,A2: set_int,B3: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ~ ( member_int @ X3 @ B3 )
=> ( ( ( insert_int @ X3 @ A2 )
= ( insert_int @ X3 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_317_insert__absorb,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_318_insert__eq__iff,axiom,
! [A: int,A2: set_int,B: int,B3: set_int] :
( ~ ( member_int @ A @ A2 )
=> ( ~ ( member_int @ B @ B3 )
=> ( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C2: set_int] :
( ( A2
= ( insert_int @ B @ C2 ) )
& ~ ( member_int @ B @ C2 )
& ( B3
= ( insert_int @ A @ C2 ) )
& ~ ( member_int @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_319_insert__commute,axiom,
! [X3: int,Y: int,A2: set_int] :
( ( insert_int @ X3 @ ( insert_int @ Y @ A2 ) )
= ( insert_int @ Y @ ( insert_int @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_320_mk__disjoint__insert,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ? [B4: set_int] :
( ( A2
= ( insert_int @ A @ B4 ) )
& ~ ( member_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_321_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_322_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_323_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_324_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_325_rcos__zfact,axiom,
! [K: int,L: int,R: int] :
( ( member_int @ K @ ( zMod @ L @ R ) )
=> ? [X4: int] :
( K
= ( plus_plus_int @ ( times_times_int @ X4 @ L ) @ R ) ) ) ).
% rcos_zfact
thf(fact_326_mult__of__nat__commute,axiom,
! [X3: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_327_mult__of__nat__commute,axiom,
! [X3: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_328_empty__not__UNIV,axiom,
bot_bot_set_int != top_top_set_int ).
% empty_not_UNIV
thf(fact_329_insert__UNIV,axiom,
! [X3: int] :
( ( insert_int @ X3 @ top_top_set_int )
= top_top_set_int ) ).
% insert_UNIV
thf(fact_330_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_331_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_332_doubleton__eq__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_333_insert__not__empty,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ A2 )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_334_singleton__inject,axiom,
! [A: int,B: int] :
( ( ( insert_int @ A @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_335_int_Oadd_Oint__pow__pow,axiom,
! [X3: int,M: int,N: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) )
= ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ N @ M ) @ X3 ) ) ) ).
% int.add.int_pow_pow
thf(fact_336_int_Oadd_Oint__pow__mult,axiom,
! [X3: int,I: int,J: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ I @ J ) @ X3 )
= ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ J @ X3 ) ) ) ) ).
% int.add.int_pow_mult
thf(fact_337_int_Oadd__pow__ldistr__int,axiom,
! [A: int,B: int,K: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( times_times_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ A ) @ B )
= ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).
% int.add_pow_ldistr_int
thf(fact_338_int_Oadd__pow__rdistr__int,axiom,
! [A: int,B: int,K: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( times_times_int @ A @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ B ) )
= ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).
% int.add_pow_rdistr_int
thf(fact_339_int_Oadd_Oint__pow__distrib,axiom,
! [X3: int,Y: int,I: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ).
% int.add.int_pow_distrib
thf(fact_340_int_Oadd_Oint__pow__mult__distrib,axiom,
! [X3: int,Y: int,I: int] :
( ( ( plus_plus_int @ X3 @ Y )
= ( plus_plus_int @ Y @ X3 ) )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ) ).
% int.add.int_pow_mult_distrib
thf(fact_341_int_Oadd__pow__ldistr,axiom,
! [A: int,B: int,K: nat] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( times_times_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ A ) @ B )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).
% int.add_pow_ldistr
thf(fact_342_int_Oadd__pow__rdistr,axiom,
! [A: int,B: int,K: nat] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( times_times_int @ A @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ B ) )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).
% int.add_pow_rdistr
thf(fact_343_int_Oadd_Onat__pow__comm,axiom,
! [X3: int,N: nat,M: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) )
= ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ).
% int.add.nat_pow_comm
thf(fact_344_int_Oadd_Onat__pow__distrib,axiom,
! [X3: int,Y: int,N: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ).
% int.add.nat_pow_distrib
thf(fact_345_int_Oadd_Opow__mult__distrib,axiom,
! [X3: int,Y: int,N: nat] :
( ( ( plus_plus_int @ X3 @ Y )
= ( plus_plus_int @ Y @ X3 ) )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ) ).
% int.add.pow_mult_distrib
thf(fact_346_int_Oadd_Ogroup__commutes__pow,axiom,
! [X3: int,Y: int,N: nat] :
( ( ( plus_plus_int @ X3 @ Y )
= ( plus_plus_int @ Y @ X3 ) )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ Y )
= ( plus_plus_int @ Y @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ) ) ).
% int.add.group_commutes_pow
thf(fact_347_int_Oadd_Onat__pow__pow,axiom,
! [X3: int,M: nat,N: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_nat @ N @ M ) @ X3 ) ) ) ).
% int.add.nat_pow_pow
thf(fact_348_int__add__eq,axiom,
! [X3: int,Y: int] :
( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y )
= ( plus_plus_int @ X3 @ Y ) ) ).
% int_add_eq
thf(fact_349_int_Oadd_Onat__pow__mult,axiom,
! [X3: int,N: nat,M: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_nat @ N @ M ) @ X3 ) ) ) ).
% int.add.nat_pow_mult
thf(fact_350_int__cosetI,axiom,
! [N: nat,U: int,V: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_int @ U @ ( semiri1314217659103216013at_int @ N ) )
= ( modulo_modulo_int @ V @ ( semiri1314217659103216013at_int @ N ) ) )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ N ) @ bot_bot_set_int ) ) @ U )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ N ) @ bot_bot_set_int ) ) @ V ) ) ) ) ).
% int_cosetI
thf(fact_351_i_Oadd_Onat__pow__Suc2,axiom,
! [X3: int,N: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ X3 )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ).
% i.add.nat_pow_Suc2
thf(fact_352_i_Oadd_Oinv__comm,axiom,
! [X3: int,Y: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ) ).
% i.add.inv_comm
thf(fact_353_i_Oadd_Oinv__unique,axiom,
! [Y: int,X3: int,Y2: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y2 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y2 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% i.add.inv_unique
thf(fact_354_i_Oadd_Ol__inv__ex,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
& ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X4 @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% i.add.l_inv_ex
thf(fact_355_i_Oadd_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ U @ X4 )
= X4 ) )
=> ( U
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% i.add.one_unique
thf(fact_356_i_Oadd_Or__inv__ex,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
& ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ X4 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% i.add.r_inv_ex
thf(fact_357_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_358_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_359_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_360_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_361_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_362_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_363_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_364_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_365_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_366_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_367_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_368_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_369_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_370_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_371_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_372_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_373_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_374_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_375_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_376_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_377_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_378_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_379_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_380_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_381_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_382_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_383_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_384_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_385_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_386_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_387_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_388_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_389_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_390_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_391_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_392_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_393_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_394_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_395_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_396_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_397_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_398_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_399_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_400_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_401_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_402_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_403_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_404_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_405_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_406_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_407_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_408_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_409_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_410_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_411_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_412_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_413_i_Oadd_Oone__closed,axiom,
member_int @ ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ).
% i.add.one_closed
thf(fact_414_i_Oadd_Onat__pow__one,axiom,
! [N: nat] :
( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% i.add.nat_pow_one
thf(fact_415_int_Oadd_Onat__pow__Suc,axiom,
! [N: nat,X3: int] :
( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ X3 )
= ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ X3 ) ) ).
% int.add.nat_pow_Suc
thf(fact_416_i_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Z @ ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% i.add.int_pow_one
thf(fact_417_i_Oadd_Or__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
= X3 ) ) ).
% i.add.r_one
thf(fact_418_i_Oadd_Or__cancel__one_H,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( X3
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ A @ X3 ) )
= ( A
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ) ).
% i.add.r_cancel_one'
thf(fact_419_i_Oadd_Or__cancel__one,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ A @ X3 )
= X3 )
= ( A
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ) ).
% i.add.r_cancel_one
thf(fact_420_i_Oadd_Ol__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ X3 )
= X3 ) ) ).
% i.add.l_one
thf(fact_421_i_Oadd_Ol__cancel__one_H,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( X3
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ A ) )
= ( A
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ) ).
% i.add.l_cancel_one'
thf(fact_422_i_Oadd_Ol__cancel__one,axiom,
! [X3: int,A: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ A )
= X3 )
= ( A
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ) ).
% i.add.l_cancel_one
thf(fact_423_i_Oadd_Onat__pow__0,axiom,
! [X3: int] :
( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ zero_zero_nat @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% i.add.nat_pow_0
thf(fact_424_i_Oadd_Onat__pow__Suc,axiom,
! [N: nat,X3: int] :
( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ X3 )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ X3 ) ) ).
% i.add.nat_pow_Suc
thf(fact_425_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_426_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_427_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_428_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_429_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_430_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_431_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_432_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_433_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_434_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_435_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_436_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_437_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_438_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_439_int_Olless__antisym,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ) ) ).
% int.lless_antisym
thf(fact_440_int_Olless__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ord_less_int @ A @ C ) ) ) ) ) ) ).
% int.lless_trans
thf(fact_441_not__psubset__empty,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_442_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_443_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_444_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_445_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_446_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_447_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_448_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_449_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_450_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_451_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_452_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_453_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_454_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_455_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_456_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,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_457_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_458_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_459_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_460_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_461_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_462_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_463_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_464_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_465_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_466_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_467_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_468_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_469_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_470_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_471_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_472_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_473_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_474_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_475_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_476_lift__Suc__mono__less,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_477_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_478_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_479_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_set_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_480_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_481_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_482_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_483_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_484_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_485_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_486_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_487_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_488_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_489_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_490_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_491_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_492_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_493_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_494_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_495_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_496_mod__induct,axiom,
! [P: nat > $o,N: nat,P3: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P3 )
=> ( ( ord_less_nat @ M @ P3 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P3 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P3 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_497_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_498_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_499_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_500_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_501_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_502_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_503_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_504_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_505_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_506_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_507_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B5: int] : ( plus_plus_int @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_508_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( plus_plus_nat @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_509_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_510_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_511_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_512_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_513_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_514_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_515_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_516_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_517_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_518_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_519_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_520_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_521_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_522_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_523_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B5: int] : ( times_times_int @ B5 @ A4 ) ) ) ).
% mult.commute
thf(fact_524_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B5: nat] : ( times_times_nat @ B5 @ A4 ) ) ) ).
% mult.commute
thf(fact_525_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_526_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_527_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_528_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_529_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_530_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_531_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_532_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_533_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_534_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_535_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_536_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_537_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_538_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_539_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_540_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_541_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_542_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_543_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_544_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_545_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_546_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_547_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_548_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_549_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_550_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_551_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_552_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_553_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_554_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_555_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_556_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_557_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_558_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_559_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_560_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_561_int__zero__eq,axiom,
( ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
= zero_zero_int ) ).
% int_zero_eq
thf(fact_562_int_Oadd_Onat__pow__Suc2,axiom,
! [X3: int,N: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ X3 )
= ( plus_plus_int @ X3 @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ).
% int.add.nat_pow_Suc2
thf(fact_563_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_564_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_565_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_566_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_567_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_568_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_569_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_570_i_Oadd_Opow__eq__div2,axiom,
! [X3: int,M: nat,N: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_nat @ M @ N ) @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% i.add.pow_eq_div2
thf(fact_571_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_572_i_Oadd_Oinv__equality,axiom,
! [Y: int,X3: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 )
= Y ) ) ) ) ).
% i.add.inv_equality
thf(fact_573_i_Oadd_Oint__pow__inv,axiom,
! [X3: int,I: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) )
= ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) ) ) ) ).
% i.add.int_pow_inv
thf(fact_574_i_Oadd_Onat__pow__inv,axiom,
! [X3: int,I: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) )
= ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) ) ) ) ).
% i.add.nat_pow_inv
thf(fact_575_i_Oadd_Oinv__solve__right_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ B @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ C @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ C ) )
= A )
= ( B
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ A @ C ) ) ) ) ) ) ).
% i.add.inv_solve_right'
thf(fact_576_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_577_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_578_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_579_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_580_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_581_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_582_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_583_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_584_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_585_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_586_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_587_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_588_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_589_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_590_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_591_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_592_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_593_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_594_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_595_minus__mod__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mod_self2
thf(fact_596_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_597_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_598_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_599_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_600_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_601_i_Oadd_Oinv__mult,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ) ) ).
% i.add.inv_mult
thf(fact_602_i_Oadd_Oinv__mult__group,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ Y ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) ) ) ) ) ).
% i.add.inv_mult_group
thf(fact_603_i_Oadd_Oinv__solve__left,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ B @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ C @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( A
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B ) @ C ) )
= ( C
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B @ A ) ) ) ) ) ) ).
% i.add.inv_solve_left
thf(fact_604_i_Oadd_Oinv__solve__left_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ B @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ C @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B ) @ C )
= A )
= ( C
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B @ A ) ) ) ) ) ) ).
% i.add.inv_solve_left'
thf(fact_605_i_Oadd_Oinv__solve__right,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ B @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( member_int @ C @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( A
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ B @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ C ) ) )
= ( B
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ A @ C ) ) ) ) ) ) ).
% i.add.inv_solve_right
thf(fact_606_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_607_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_608_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_609_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_610_i_Oadd_Oinv__closed,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( member_int @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% i.add.inv_closed
thf(fact_611_i_Oadd_Oinv__inv,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) )
= X3 ) ) ).
% i.add.inv_inv
thf(fact_612_i_Oadd_Oinv__eq__1__iff,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
= ( X3
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% i.add.inv_eq_1_iff
thf(fact_613_i_Oadd_Ol__inv,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) @ X3 )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% i.add.l_inv
thf(fact_614_i_Oadd_Or__inv,axiom,
! [X3: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 ) )
= ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% i.add.r_inv
thf(fact_615_psubset__trans,axiom,
! [A2: set_int,B3: set_int,C4: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ( ( ord_less_set_int @ B3 @ C4 )
=> ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% psubset_trans
thf(fact_616_psubsetD,axiom,
! [A2: set_int,B3: set_int,C: int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_617_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_618_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_619_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_620_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_621_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A4: int,B5: int] :
( ( minus_minus_int @ A4 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_622_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_623_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_624_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_625_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_626_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_627_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_628_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_629_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_630_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_631_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_632_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_633_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_634_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_635_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_636_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_637_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_638_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_639_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_640_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_641_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_642_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_643_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_644_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_645_mod__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_646_mod__diff__cong,axiom,
! [A: int,C: int,A3: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A3 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B2 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_647_mod__diff__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_648_mod__diff__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_649_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_650_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_651_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_652_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_653_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_654_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_655_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_656_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_657_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_658_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_659_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_660_square__diff__square__factored,axiom,
! [X3: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X3 @ Y ) @ ( minus_minus_int @ X3 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_661_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_662_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_663_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_664_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_665_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_666_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_667_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_668_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_669_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_670_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_671_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_672_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_673_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N4 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_674_square__diff__one__factored,axiom,
! [X3: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_675_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_676_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_677_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_678_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_679_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_680_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_681_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_682_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_683_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_684_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_685_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_686_int_Oadd_Opow__eq__div2,axiom,
! [X3: int,M: nat,N: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 )
= ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_nat @ M @ N ) @ X3 )
= zero_zero_int ) ) ) ).
% int.add.pow_eq_div2
thf(fact_687_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_688_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_689_i_Oadd_Oone__in__subset,axiom,
! [H: set_int] :
( ( ord_less_eq_set_int @ H @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( H != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H )
=> ( member_int @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X4 ) @ H ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H )
=> ! [Xa: int] :
( ( member_int @ Xa @ H )
=> ( member_int @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X4 @ Xa ) @ H ) ) )
=> ( member_int @ ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ H ) ) ) ) ) ).
% i.add.one_in_subset
thf(fact_690_i_Oadd_Oint__pow__neg__int,axiom,
! [X3: int,N: nat] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ X3 )
= ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ).
% i.add.int_pow_neg_int
thf(fact_691_i_Oadd_Oint__pow__diff,axiom,
! [X3: int,N: int,M: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_int @ N @ M ) @ X3 )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) ) ) ) ) ).
% i.add.int_pow_diff
thf(fact_692_i_Oadd_Oint__pow__neg,axiom,
! [X3: int,I: int] :
( ( member_int @ X3 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ I ) @ X3 )
= ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) ) ) ) ).
% i.add.int_pow_neg
thf(fact_693_subset__antisym,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_694_subsetI,axiom,
! [A2: set_int,B3: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( member_int @ X4 @ B3 ) )
=> ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_695_Diff__cancel,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ A2 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_696_empty__Diff,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_697_Diff__empty,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% Diff_empty
thf(fact_698_insert__Diff1,axiom,
! [X3: int,B3: set_int,A2: set_int] :
( ( member_int @ X3 @ B3 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B3 )
= ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_699_Diff__insert0,axiom,
! [X3: int,A2: set_int,B3: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ B3 ) )
= ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_700_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_701_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_702_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_703_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_704_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_705_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_706_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_707_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_708_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_709_Diff__UNIV,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ top_top_set_int )
= bot_bot_set_int ) ).
% Diff_UNIV
thf(fact_710_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_711_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_712_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_713_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_714_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_715_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_716_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_717_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_718_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_719_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_720_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_721_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_722_Diff__eq__empty__iff,axiom,
! [A2: set_int,B3: set_int] :
( ( ( minus_minus_set_int @ A2 @ B3 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_723_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_724_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_725_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_726_insert__subset,axiom,
! [X3: int,A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X3 @ A2 ) @ B3 )
= ( ( member_int @ X3 @ B3 )
& ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_727_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_728_insert__Diff__single,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( insert_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_729_int_Oadd_Oinv__closed,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( member_int @ ( uminus_uminus_int @ X3 ) @ top_top_set_int ) ) ).
% int.add.inv_closed
thf(fact_730_int_Ominus__minus,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( uminus_uminus_int @ X3 ) )
= X3 ) ) ).
% int.minus_minus
thf(fact_731_int_Ominus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% int.minus_zero
thf(fact_732_mod__minus__minus,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_733_int_Ominus__closed,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( minus_minus_int @ X3 @ Y ) @ top_top_set_int ) ) ) ).
% int.minus_closed
thf(fact_734_psubsetI,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_735_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_736_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_737_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_738_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_739_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_740_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_741_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_742_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_743_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_744_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_745_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_746_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_747_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_748_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_749_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_750_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_751_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_752_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_753_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_754_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_755_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_756_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_757_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_758_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_759_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_760_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_761_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_762_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_763_singleton__insert__inj__eq_H,axiom,
! [A: int,A2: set_int,B: int] :
( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_764_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A2: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_765_int_Oadd_Oinv__eq__1__iff,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( ( uminus_uminus_int @ X3 )
= zero_zero_int )
= ( X3 = zero_zero_int ) ) ) ).
% int.add.inv_eq_1_iff
thf(fact_766_minus__mod__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_767_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_768_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_769_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_770_int_Oa__coset__add__zero,axiom,
! [M6: set_int] :
( ( ord_less_eq_set_int @ M6 @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ zero_zero_int )
= M6 ) ) ).
% int.a_coset_add_zero
thf(fact_771_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B6: set_int] :
( ( ord_less_set_int @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_772_subset__psubset__trans,axiom,
! [A2: set_int,B3: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( ord_less_set_int @ B3 @ C4 )
=> ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_773_subset__not__subset__eq,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A5 @ B6 )
& ~ ( ord_less_eq_set_int @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_774_psubset__subset__trans,axiom,
! [A2: set_int,B3: set_int,C4: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ C4 )
=> ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_775_psubset__imp__subset,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_776_psubset__imp__ex__mem,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ? [B7: int] : ( member_int @ B7 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_777_psubset__eq,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_778_psubsetE,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_779_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_780_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_781_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_782_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_783_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_784_int_Ominus__eq,axiom,
( minus_minus_int
= ( ^ [X: int,Y5: int] : ( plus_plus_int @ X @ ( uminus_uminus_int @ Y5 ) ) ) ) ).
% int.minus_eq
thf(fact_785_Diff__mono,axiom,
! [A2: set_int,C4: set_int,D4: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ C4 )
=> ( ( ord_less_eq_set_int @ D4 @ B3 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_786_Diff__subset,axiom,
! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_787_double__diff,axiom,
! [A2: set_int,B3: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ C4 )
=> ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_788_subset__Diff__insert,axiom,
! [A2: set_int,B3: set_int,X3: int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B3 @ ( insert_int @ X3 @ C4 ) ) )
= ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B3 @ C4 ) )
& ~ ( member_int @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_789_subset__insert__iff,axiom,
! [A2: set_int,X3: int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B3 ) )
= ( ( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B3 ) )
& ( ~ ( member_int @ X3 @ A2 )
=> ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_790_Diff__single__insert,axiom,
! [A2: set_int,X3: int,B3: set_int] :
( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B3 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_791_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_792_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_793_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_794_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_795_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_796_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_797_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_798_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_799_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_800_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_801_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_802_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_803_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_804_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_805_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_806_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_807_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_808_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_809_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_810_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_811_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_812_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_813_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
? [C5: nat] :
( B5
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_814_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_815_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_816_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_817_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_818_insert__Diff__if,axiom,
! [X3: int,B3: set_int,A2: set_int] :
( ( ( member_int @ X3 @ B3 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B3 )
= ( minus_minus_set_int @ A2 @ B3 ) ) )
& ( ~ ( member_int @ X3 @ B3 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B3 )
= ( insert_int @ X3 @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_819_subset__UNIV,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% subset_UNIV
thf(fact_820_UNIV_I2_J,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% UNIV(2)
thf(fact_821_mod__minus__eq,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_822_mod__minus__cong,axiom,
! [A: int,B: int,A3: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A3 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_823_mod__minus__right,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_824_insert__mono,axiom,
! [C4: set_int,D4: set_int,A: int] :
( ( ord_less_eq_set_int @ C4 @ D4 )
=> ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D4 ) ) ) ).
% insert_mono
thf(fact_825_subset__insert,axiom,
! [X3: int,A2: set_int,B3: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B3 ) )
= ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_826_subset__insertI,axiom,
! [B3: set_int,A: int] : ( ord_less_eq_set_int @ B3 @ ( insert_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_827_subset__insertI2,axiom,
! [A2: set_int,B3: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_828_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X: int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_829_set__eq__subset,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
= ( ^ [A5: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A5 @ B6 )
& ( ord_less_eq_set_int @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_830_subset__trans,axiom,
! [A2: set_int,B3: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ C4 )
=> ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_831_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_832_subset__refl,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_833_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A5 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_834_equalityD2,axiom,
! [A2: set_int,B3: set_int] :
( ( A2 = B3 )
=> ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_835_equalityD1,axiom,
! [A2: set_int,B3: set_int] :
( ( A2 = B3 )
=> ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_836_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B6: set_int] :
! [X: int] :
( ( member_int @ X @ A5 )
=> ( member_int @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_837_equalityE,axiom,
! [A2: set_int,B3: set_int] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ~ ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_838_subsetD,axiom,
! [A2: set_int,B3: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_839_in__mono,axiom,
! [A2: set_int,B3: set_int,X3: int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_840_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_841_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_842_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_843_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_844_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_845_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_846_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_847_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_848_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_849_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_850_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_851_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_852_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_853_psubset__insert__iff,axiom,
! [A2: set_int,X3: int,B3: set_int] :
( ( ord_less_set_int @ A2 @ ( insert_int @ X3 @ B3 ) )
= ( ( ( member_int @ X3 @ B3 )
=> ( ord_less_set_int @ A2 @ B3 ) )
& ( ~ ( member_int @ X3 @ B3 )
=> ( ( ( member_int @ X3 @ A2 )
=> ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B3 ) )
& ( ~ ( member_int @ X3 @ A2 )
=> ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_854_int_Oadd_Oone__in__subset,axiom,
! [H: set_int] :
( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( H != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H )
=> ( member_int @ ( uminus_uminus_int @ X4 ) @ H ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H )
=> ! [Xa: int] :
( ( member_int @ Xa @ H )
=> ( member_int @ ( plus_plus_int @ X4 @ Xa ) @ H ) ) )
=> ( member_int @ zero_zero_int @ H ) ) ) ) ) ).
% int.add.one_in_subset
thf(fact_855_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_856_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_857_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_858_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_859_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_860_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_861_square__eq__1__iff,axiom,
! [X3: int] :
( ( ( times_times_int @ X3 @ X3 )
= one_one_int )
= ( ( X3 = one_one_int )
| ( X3
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_862_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_863_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B5: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_864_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B5: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_865_int_Oadd_Oinv__solve__right_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( plus_plus_int @ B @ ( uminus_uminus_int @ C ) )
= A )
= ( B
= ( plus_plus_int @ A @ C ) ) ) ) ) ) ).
% int.add.inv_solve_right'
thf(fact_866_int_Oadd_Oinv__solve__right,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( A
= ( plus_plus_int @ B @ ( uminus_uminus_int @ C ) ) )
= ( B
= ( plus_plus_int @ A @ C ) ) ) ) ) ) ).
% int.add.inv_solve_right
thf(fact_867_int_Oadd_Oinv__solve__left_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( plus_plus_int @ ( uminus_uminus_int @ B ) @ C )
= A )
= ( C
= ( plus_plus_int @ B @ A ) ) ) ) ) ) ).
% int.add.inv_solve_left'
thf(fact_868_int_Oadd_Oinv__solve__left,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( A
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ C ) )
= ( C
= ( plus_plus_int @ B @ A ) ) ) ) ) ) ).
% int.add.inv_solve_left
thf(fact_869_int_Oadd_Oinv__mult__group,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X3 ) ) ) ) ) ).
% int.add.inv_mult_group
thf(fact_870_int_Oa__transpose__inv,axiom,
! [X3: int,Y: int,Z: int] :
( ( ( plus_plus_int @ X3 @ Y )
= Z )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( plus_plus_int @ ( uminus_uminus_int @ X3 ) @ Z )
= Y ) ) ) ) ) ).
% int.a_transpose_inv
thf(fact_871_int_Oadd_Oinv__mult,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( plus_plus_int @ X3 @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y ) ) ) ) ) ).
% int.add.inv_mult
thf(fact_872_int_Or__neg2,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ X3 @ ( plus_plus_int @ ( uminus_uminus_int @ X3 ) @ Y ) )
= Y ) ) ) ).
% int.r_neg2
thf(fact_873_int_Or__neg1,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ ( uminus_uminus_int @ X3 ) @ ( plus_plus_int @ X3 @ Y ) )
= Y ) ) ) ).
% int.r_neg1
thf(fact_874_int_Or__minus,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X3 @ ( uminus_uminus_int @ Y ) )
= ( uminus_uminus_int @ ( times_times_int @ X3 @ Y ) ) ) ) ) ).
% int.r_minus
thf(fact_875_int_Ol__minus,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ ( uminus_uminus_int @ X3 ) @ Y )
= ( uminus_uminus_int @ ( times_times_int @ X3 @ Y ) ) ) ) ) ).
% int.l_minus
thf(fact_876_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_877_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_878_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_879_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_880_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_881_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_882_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_883_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_884_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_885_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_886_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_887_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_888_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_889_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_890_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_891_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_892_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_893_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_894_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_895_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_896_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_897_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_898_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_899_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_900_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_901_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_902_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_903_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_904_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_905_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_906_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_907_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_908_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_909_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_910_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_911_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_912_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_913_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_914_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_915_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_916_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_917_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_918_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_919_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_920_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_921_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_922_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_923_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_924_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_925_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_926_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_927_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_928_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_929_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_930_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_931_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_932_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_933_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_934_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_935_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_936_Diff__insert__absorb,axiom,
! [X3: int,A2: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_937_Diff__insert2,axiom,
! [A2: set_int,A: int,B3: set_int] :
( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_938_insert__Diff,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_939_Diff__insert,axiom,
! [A2: set_int,A: int,B3: set_int] :
( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% Diff_insert
thf(fact_940_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_941_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_942_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_943_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_944_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_945_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_946_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_947_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_948_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_949_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_950_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_951_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_952_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_953_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_954_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_955_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_956_subset__singleton__iff,axiom,
! [X5: set_int,A: int] :
( ( ord_less_eq_set_int @ X5 @ ( insert_int @ A @ bot_bot_set_int ) )
= ( ( X5 = bot_bot_set_int )
| ( X5
= ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_957_subset__singletonD,axiom,
! [A2: set_int,X3: int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) )
=> ( ( A2 = bot_bot_set_int )
| ( A2
= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_958_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus2_not_zero
thf(fact_959_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus1_not_zero
thf(fact_960_int_Oadd_Oinv__equality,axiom,
! [Y: int,X3: int] :
( ( ( plus_plus_int @ Y @ X3 )
= zero_zero_int )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( uminus_uminus_int @ X3 )
= Y ) ) ) ) ).
% int.add.inv_equality
thf(fact_961_int_Osum__zero__eq__neg,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
=> ( X3
= ( uminus_uminus_int @ Y ) ) ) ) ) ).
% int.sum_zero_eq_neg
thf(fact_962_int_Oadd_Or__inv,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( plus_plus_int @ X3 @ ( uminus_uminus_int @ X3 ) )
= zero_zero_int ) ) ).
% int.add.r_inv
thf(fact_963_int_Oadd_Ol__inv,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( plus_plus_int @ ( uminus_uminus_int @ X3 ) @ X3 )
= zero_zero_int ) ) ).
% int.add.l_inv
thf(fact_964_int_Osquare__eq__one,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( ( times_times_int @ X3 @ X3 )
= one_one_int )
=> ( ( X3 = one_one_int )
| ( X3
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% int.square_eq_one
thf(fact_965_sum__squares__ge__zero,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_966_sum__squares__le__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_967_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_968_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_969_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_970_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_971_mult__right__le__one__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X3 @ Y ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_972_mult__left__le__one__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_973_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_974_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_975_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_976_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_977_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_978_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_979_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_980_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_981_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_982_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_983_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_984_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_985_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_986_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_987_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_988_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_989_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_990_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_991_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_992_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_993_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_994_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_995_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_996_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_997_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_998_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_999_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1000_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1001_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1002_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1003_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1004_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1005_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1006_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1007_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1008_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1009_convex__bound__le,axiom,
! [X3: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X3 @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1010_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1011_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1012_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1013_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1014_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1015_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1016_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1017_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1018_int_Oadd_Oint__pow__diff,axiom,
! [X3: int,N: int,M: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_int @ N @ M ) @ X3 )
= ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) @ ( uminus_uminus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X3 ) ) ) ) ) ).
% int.add.int_pow_diff
thf(fact_1019_int_Oa__coset__add__inv2,axiom,
! [M6: set_int,X3: int,Y: int] :
( ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ X3 )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ Y ) )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ M6 @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ ( plus_plus_int @ X3 @ ( uminus_uminus_int @ Y ) ) )
= M6 ) ) ) ) ) ).
% int.a_coset_add_inv2
thf(fact_1020_int_Oa__coset__add__inv1,axiom,
! [M6: set_int,X3: int,Y: int] :
( ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ ( plus_plus_int @ X3 @ ( uminus_uminus_int @ Y ) ) )
= M6 )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ M6 @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ X3 )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ Y ) ) ) ) ) ) ).
% int.a_coset_add_inv1
thf(fact_1021_convex__bound__lt,axiom,
! [X3: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X3 @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1022_int_Oadd_Onat__pow__inv,axiom,
! [X3: int,I: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( uminus_uminus_int @ X3 ) )
= ( uminus_uminus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) ) ) ) ).
% int.add.nat_pow_inv
thf(fact_1023_int__a__inv__eq,axiom,
! [X3: int] :
( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X3 )
= ( uminus_uminus_int @ X3 ) ) ).
% int_a_inv_eq
thf(fact_1024_int_Oadd_Oint__pow__neg,axiom,
! [X3: int,I: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ I ) @ X3 )
= ( uminus_uminus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) ) ) ) ).
% int.add.int_pow_neg
thf(fact_1025_int_Oadd_Oint__pow__inv,axiom,
! [X3: int,I: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( uminus_uminus_int @ X3 ) )
= ( uminus_uminus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X3 ) ) ) ) ).
% int.add.int_pow_inv
thf(fact_1026_int_Oa__r__coset__subset__G,axiom,
! [H: set_int,X3: int] :
( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X3 ) @ top_top_set_int ) ) ) ).
% int.a_r_coset_subset_G
thf(fact_1027_int_Oa__coset__add__assoc,axiom,
! [M6: set_int,G: int,H2: int] :
( ( ord_less_eq_set_int @ M6 @ top_top_set_int )
=> ( ( member_int @ G @ top_top_set_int )
=> ( ( member_int @ H2 @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ G ) @ H2 )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M6 @ ( plus_plus_int @ G @ H2 ) ) ) ) ) ) ).
% int.a_coset_add_assoc
thf(fact_1028_int_Oa__rcosI,axiom,
! [H2: int,H: set_int,X3: int] :
( ( member_int @ H2 @ H )
=> ( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( member_int @ ( plus_plus_int @ H2 @ X3 ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X3 ) ) ) ) ) ).
% int.a_rcosI
thf(fact_1029_int_Osubset__Idl__subset,axiom,
! [I4: set_int,H: set_int] :
( ( ord_less_eq_set_int @ I4 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ H @ I4 )
=> ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I4 ) ) ) ) ).
% int.subset_Idl_subset
thf(fact_1030_int_Ogenideal__self,axiom,
! [S2: set_int] :
( ( ord_less_eq_set_int @ S2 @ top_top_set_int )
=> ( ord_less_eq_set_int @ S2 @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ S2 ) ) ) ).
% int.genideal_self
thf(fact_1031_int_Oadd_Oint__pow__neg__int,axiom,
! [X3: int,N: nat] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ X3 )
= ( uminus_uminus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X3 ) ) ) ) ).
% int.add.int_pow_neg_int
thf(fact_1032_int__Idl__subset__ideal,axiom,
! [K: int,L: int] :
( ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K @ bot_bot_set_int ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) )
= ( member_int @ K @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) ) ) ).
% int_Idl_subset_ideal
thf(fact_1033_int_OIdl__subset__ideal_H,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ A @ bot_bot_set_int ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( member_int @ A @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ).
% int.Idl_subset_ideal'
thf(fact_1034_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1035_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1036_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1037_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1038_int_Ole__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( X3 = Y ) ) ) ) ) ).
% int.le_antisym
thf(fact_1039_int_Ole__refl,axiom,
! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ord_less_eq_int @ X3 @ X3 ) ) ).
% int.le_refl
thf(fact_1040_Compl__subset__Compl__iff,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B3 ) )
= ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1041_Compl__anti__mono,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1042_DiffI,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ A2 )
=> ( ~ ( member_int @ C @ B3 )
=> ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1043_Diff__iff,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
= ( ( member_int @ C @ A2 )
& ~ ( member_int @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_1044_Diff__idemp,axiom,
! [A2: set_int,B3: set_int] :
( ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_int @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_1045_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1046_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1047_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1048_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1049_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_1050_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1051_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_1052_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1053_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_1054_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_1055_subset__Compl__singleton,axiom,
! [A2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1056_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1057_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1058_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1059_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1060_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1061_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1062_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1063_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1064_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1065_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1066_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_1067_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_1068_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_1069_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1070_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1071_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% int.lless_eq
thf(fact_1072_DiffE,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
=> ~ ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B3 ) ) ) ).
% DiffE
thf(fact_1073_DiffD1,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
=> ( member_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_1074_DiffD2,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
=> ~ ( member_int @ C @ B3 ) ) ).
% DiffD2
thf(fact_1075_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1076_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1077_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1078_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1079_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1080_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1081_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1082_Compl__eq__Diff__UNIV,axiom,
( uminus1532241313380277803et_int
= ( minus_minus_set_int @ top_top_set_int ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1083_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1084_int_Ole__trans,axiom,
! [X3: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ord_less_eq_int @ X3 @ Z ) ) ) ) ) ) ).
% int.le_trans
thf(fact_1085_int_Ototal__order__total,axiom,
! [X3: int,Y: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ) ) ).
% int.total_order_total
thf(fact_1086_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1087_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1088_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1089_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1090_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1091_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1092_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1093_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1094_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1095_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1096_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1097_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1098_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1099_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1100_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1101_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1102_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1103_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1104_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1105_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1106_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1107_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1108_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1109_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1110_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1111_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1112_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1113_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1114_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1115_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1116_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1117_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1118_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1119_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1120_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1121_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1122_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1123_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1124_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1125_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1126_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_1127_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1128_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1129_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1130_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1131_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1132_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1133_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1134_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1135_Compl__UNIV__eq,axiom,
( ( uminus1532241313380277803et_int @ top_top_set_int )
= bot_bot_set_int ) ).
% Compl_UNIV_eq
thf(fact_1136_Compl__empty__eq,axiom,
( ( uminus1532241313380277803et_int @ bot_bot_set_int )
= top_top_set_int ) ).
% Compl_empty_eq
thf(fact_1137_subset__Compl__self__eq,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_1138_Compl__insert,axiom,
! [X3: int,A2: set_int] :
( ( uminus1532241313380277803et_int @ ( insert_int @ X3 @ A2 ) )
= ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% Compl_insert
thf(fact_1139_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1140_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1141_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1142_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1143_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1144_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1145_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1146_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1147_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1148_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1149_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1150_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1151_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1152_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1153_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1154_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1155_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1156_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1157_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1158_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1159_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1160_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1161_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1162_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1163_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1164_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1165_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1166_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1167_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_1168_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_1169_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1170_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1171_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1172_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1173_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1174_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1175_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1176_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1177_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1178_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1179_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1180_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1181_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1182_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1183_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1184_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1185_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_1186_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1187_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo_int @ I @ K )
= I )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1188_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1189_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S3: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1190_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1191_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1192_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
= ( plus_plus_int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1193_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1194_split__zmod,axiom,
! [Q: int > $o,N: int,K: int] :
( ( Q @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( Q @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( Q @ J2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( Q @ J2 ) ) ) ) ) ).
% split_zmod
thf(fact_1195_int__mod__neg__eq,axiom,
! [A: int,B: int,Q4: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( modulo_modulo_int @ A @ B )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1196_int__mod__pos__eq,axiom,
! [A: int,B: int,Q4: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1197_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
= ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_1198_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1199_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1200_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1201_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1202_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1203_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1204_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1205_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1206_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1207_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1208_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1209_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1210_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_1211_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_1212_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_1213_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1214_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1215_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1216_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1217_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1218_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1219_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1220_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1221_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
% 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,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ n ) @ bot_bot_set_int ) ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ x ) @ ( semiri1314217659103216013at_int @ y ) ) )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ n ) @ bot_bot_set_int ) ) @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ x @ y ) ) @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
%------------------------------------------------------------------------------