TPTP Problem File: SLH0603^1.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 : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00085_003203__17012032_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1404 ( 496 unt; 128 typ; 0 def)
% Number of atoms : 3718 (1235 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 12878 ( 249 ~; 60 |; 188 &;10661 @)
% ( 0 <=>;1720 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 370 ( 370 >; 0 *; 0 +; 0 <<)
% Number of symbols : 114 ( 111 usr; 12 con; 0-4 aty)
% Number of variables : 3327 ( 69 ^;3173 !; 85 ?;3327 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:35:40.814
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2956882679547061052t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2670972154091845814t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_list_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
set_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
set_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (111)
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001tf__a_001tf__b,type,
bounde2262800523058855161ls_a_b: partia2175431115845679010xt_a_b > nat > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia2464479390973590831t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia5361259788508890537t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Divisibility_Oessentially__equal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
essent8953798148185448568xt_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid5798828371819920185xt_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd3793949463769647726t_unit: partia2670972154091845814t_unit > nat > set_list_a > set_list_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001tf__a_001tf__b,type,
embedd2795209813406577254on_a_b: partia2175431115845679010xt_a_b > nat > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd5150658419831591667t_unit: partia2670972154091845814t_unit > set_list_a > list_a > set_list_a > set_list_a ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Group_Om__inv_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
m_inv_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
mult_l7073676228092353617t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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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_I_062_It__Nat__Onat_Mtf__a_J_J,type,
minus_490503922182417452_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
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_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_Ocgenideal_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
genide3243992037924705879t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oprincipalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
princi8786919440553033881t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
bot_bot_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__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_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_less_set_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Oroots__on_001tf__a_001tf__b,type,
polyno5714441830345289050on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > multiset_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted__on_001tf__a_001tf__b,type,
polyno2453258491555121552on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomials_Oring_Ocoeff_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
coeff_6360649920519955023t_unit: partia2670972154091845814t_unit > list_list_a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Ocoeff_001tf__a_001tf__b,type,
coeff_a_b: partia2175431115845679010xt_a_b > list_a > nat > a ).
thf(sy_c_Polynomials_Oring_Omonom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
monom_7446464087056152608t_unit: partia2670972154091845814t_unit > list_a > nat > list_list_a ).
thf(sy_c_Polynomials_Oring_Omonom_001tf__a_001tf__b,type,
monom_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i7048835797181109658it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oadd__pow_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po2638046716968164713it_int: partia2670972154091845814t_unit > int > list_a > list_a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia2175431115845679010xt_a_b > int > a > a ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
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thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
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thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h7399960747407462284t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h405018892823518980t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x,type,
x: list_a ).
% Relevant facts (1268)
thf(fact_0_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_1_assms_I1_J,axiom,
member_list_a @ x @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% assms(1)
thf(fact_2_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_3_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_4_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_5_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_6_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_7_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_8_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_9_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_10_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_11_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_12_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_13_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_14_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_15_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_16_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_17_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_18_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_19_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_20_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_21_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_22_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_23_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_24_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_25_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_26_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_27_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_28_ring_Oring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_29_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_30_ring_Oring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_31_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_32_ring_Oonepideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_33_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_34_add_Oint__pow__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_35_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_36_ring__iso__memE_I3_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_37_ring__iso__memE_I3_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_38_ring__iso__memE_I3_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_39_ring__hom__zero,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_40_ring__hom__zero,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_a_b @ S )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_a_b @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_41_ring__hom__zero,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( ring_a_b @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_42_ring__hom__zero,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ring_a_b @ R )
=> ( ( ring_a_b @ S )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_a_b @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_43_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_44_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_45_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_46_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_47_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_48_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_49_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_50_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_51_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_52_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_53_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.int_pow_one
thf(fact_54_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_61_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_62_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_63_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_64_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_65_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_66_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_67_add_Oint__pow__1,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ one_one_int @ X )
= X ) ) ).
% add.int_pow_1
thf(fact_68_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_69_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_70_ring__hom__closed,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_71_ring__hom__closed,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_72_ring__hom__closed,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_73_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_74_ring__iso__memE_I1_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_75_ring__iso__memE_I1_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_76_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_77_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_78_ring_Oring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_79_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_80_ring_Oring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_81_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_82_ring_Oring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_83_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_84_ring_Oring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_85_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_86_ring_Oring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_87_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_88_ring__hom__add,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_89_ring__hom__add,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_90_ring__hom__add,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_91_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_92_ring_Ocgenideal__self,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_93_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_94_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_95_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_96_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_97_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_98_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_99_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_100_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_101_add_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% add.int_pow_mult
thf(fact_102_coeff__degree,axiom,
! [P2: list_a,I: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ I )
=> ( ( coeff_a_b @ r @ P2 @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_degree
thf(fact_103_coeff__in__carrier,axiom,
! [P2: list_a,I: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( coeff_a_b @ r @ P2 @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% coeff_in_carrier
thf(fact_104_add_Oint__pow__pow,axiom,
! [X: a,M: int,N: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ M @ ( add_pow_a_b_int @ r @ N @ X ) )
= ( add_pow_a_b_int @ r @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% add.int_pow_pow
thf(fact_105_add_Oint__pow__diff,axiom,
! [X: a,N: int,M: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_106_add__pow__ldistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_107_add__pow__rdistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_108_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_109_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_110_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_111_coeff__iff__length__cond,axiom,
! [P1: list_a,P22: list_a] :
( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P22 ) )
=> ( ( P1 = P22 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P22 ) ) ) ) ).
% coeff_iff_length_cond
thf(fact_112_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_113_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_114_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_115_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_116_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_117_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_118_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_119_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_120_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_121_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_122_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_123_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_124_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_125_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_126_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_127_add_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% add.int_pow_inv
thf(fact_128_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_129_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_130_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_131_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_132_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_133_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_134_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_135_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_136_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_137_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_138_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_148_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_149_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_150_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_151_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_152_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_153_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_154_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_155_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_165_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_166_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_167_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_168_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_ring_Ol__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_185_ring_Ol__minus,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_186_ring_Or__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_187_ring_Or__minus,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_197_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_216_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_217_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_218_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_219_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_220_ring_Oring__simprules_I20_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_221_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_222_ring_Oring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_223_ring_Ominus__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.minus_zero
thf(fact_224_ring_Ominus__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% ring.minus_zero
thf(fact_225_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_226_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_227_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_228_ring_Oring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_229_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_230_ring_Oring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_231_ring__hom__mult,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_232_ring__hom__mult,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_233_ring__hom__mult,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_234_ring__hom__mult,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_235_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_236_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_237_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_238_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_239_ring__iso__memE_I2_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_240_ring__iso__memE_I2_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_241_ring__iso__memE_I2_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_242_ring__iso__memE_I2_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_243_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_244_ring_Oring__simprules_I19_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_245_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_246_ring_Oring__simprules_I18_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_247_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_248_ring_Oring__simprules_I17_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_249_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_250_ring_Oring__simprules_I24_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_251_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_252_ring_Oring__simprules_I25_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_253_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_254_ring_Oring__simprules_I23_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_255_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_256_ring_Oring__simprules_I13_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_257_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_int @ R @ K @ A ) @ B )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_258_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,K: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_po2638046716968164713it_int @ R @ K @ A ) @ B )
= ( add_po2638046716968164713it_int @ R @ K @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_259_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_int @ R @ K @ B ) )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_260_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,K: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ A @ ( add_po2638046716968164713it_int @ R @ K @ B ) )
= ( add_po2638046716968164713it_int @ R @ K @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_261_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_262_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_263_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_264_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_265_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_266_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_267_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_268_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_269_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_270_ring_Oring__simprules_I16_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_271_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_272_ring_Oring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_273_ring_Ozeropideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_274_ring_Ozeropideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_275_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P2 @ I ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_276_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( member_a @ ( coeff_a_b @ R @ P2 @ I ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_277_assms_I2_J,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ n @ K )
=> ( ( coeff_a_b @ r @ x @ K )
= ( zero_a_b @ r ) ) ) ).
% assms(2)
thf(fact_278_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_279_ring_Ocoeff__degree,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ R @ P2 @ I )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_degree
thf(fact_280_ring_Ocoeff__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ I )
=> ( ( coeff_a_b @ R @ P2 @ I )
= ( zero_a_b @ R ) ) ) ) ).
% ring.coeff_degree
thf(fact_281_monoid__cancelI,axiom,
( ! [A4: a,B4: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B4 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: a,B4: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C2 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_282_bound_Ointro,axiom,
! [N: nat,F: nat > a,Z: a] :
( ! [M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( F @ M2 )
= Z ) )
=> ( bound_a @ Z @ N @ F ) ) ).
% bound.intro
thf(fact_283_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ! [Xa: a] :
( ( member_a @ Xa @ H2 )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_284_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I3: nat] : ( if_a @ ( I3 = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_285_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_286_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_287_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_288_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_289_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_290_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_291_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_292_subsetI,axiom,
! [A2: set_nat_a,B3: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B3 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_293_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_294_psubsetI,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_295_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_296_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_297_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_298_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_299_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_300_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_301_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_302_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_303_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_304_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_305_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_306_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_307_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_308_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_309_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_310_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_311_Diff__insert0,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_312_Diff__insert0,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_313_Diff__insert0,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_314_insert__Diff1,axiom,
! [X: list_a,B3: set_list_a,A2: set_list_a] :
( ( member_list_a @ X @ B3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_315_insert__Diff1,axiom,
! [X: nat > a,B3: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ X @ B3 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_316_insert__Diff1,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_317_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_318_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_319_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_320_insert__iff,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_321_insertCI,axiom,
! [A: list_a,B3: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B3 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_322_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_323_insertCI,axiom,
! [A: nat > a,B3: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_324_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_325_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_326_coeff__length,axiom,
! [P2: list_a,I: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P2 ) @ I )
=> ( ( coeff_a_b @ r @ P2 @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_length
thf(fact_327_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_328_non__empty__bounded__degree__polynomials,axiom,
! [K: nat] :
( ( bounde2262800523058855161ls_a_b @ r @ K )
!= bot_bot_set_list_a ) ).
% non_empty_bounded_degree_polynomials
thf(fact_329_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_330_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_331_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_332_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_333_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_334_Diff__eq__empty__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_335_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_336_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_337_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_338_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_339_insert__subset,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( ( member_list_a @ X @ B3 )
& ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_340_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( ( member_nat_a @ X @ B3 )
& ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_341_insert__subset,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_342_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_343_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_344_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_345_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_346_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_347_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_348_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_349_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_350_double__diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_351_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_352_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_353_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_354_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_355_subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_356_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_357_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_358_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A5 )
=> ( member_list_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_359_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A5 )
=> ( member_nat_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_360_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_361_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_362_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_363_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
=> ( member_list_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_364_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A5 )
=> ( member_nat_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_365_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_366_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_367_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_368_subsetD,axiom,
! [A2: set_nat_a,B3: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B3 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_369_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_370_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_371_in__mono,axiom,
! [A2: set_nat_a,B3: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B3 )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_372_in__mono,axiom,
! [A2: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_373_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_374_psubsetE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_375_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_376_psubset__imp__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_377_psubset__subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_378_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_379_subset__psubset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_380_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_381_subset__Diff__insert,axiom,
! [A2: set_list_a,B3: set_list_a,X: list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ ( insert_list_a @ X @ C3 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ C3 ) )
& ~ ( member_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_382_subset__Diff__insert,axiom,
! [A2: set_nat_a,B3: set_nat_a,X: nat > a,C3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B3 @ ( insert_nat_a @ X @ C3 ) ) )
= ( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B3 @ C3 ) )
& ~ ( member_nat_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_383_subset__Diff__insert,axiom,
! [A2: set_a,B3: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_384_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_385_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_386_subset__insert,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_387_subset__insert,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_388_subset__insert,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_389_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_390_bound__below,axiom,
! [Z: a,M: nat,F: nat > a,N: nat] :
( ( bound_a @ Z @ M @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_391_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H2 ) @ ( genideal_a_b @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_392_ring_Osubset__Idl__subset,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a,H2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ H2 ) @ ( genide3243992037924705879t_unit @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_393_ring_Ogenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_394_ring_Ogenideal__self,axiom,
! [R: partia2670972154091845814t_unit,S: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_395_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_396_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_397_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_398_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_399_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_400_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_401_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_402_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_403_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_404_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_405_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_406_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_407_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_408_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_409_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_410_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_411_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_412_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_413_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_414_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_415_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_416_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_417_subset__insert__iff,axiom,
! [A2: set_nat_a,X: nat > a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( ( ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B3 ) )
& ( ~ ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_418_subset__insert__iff,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( ( ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_419_subset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_420_subset__singletonD,axiom,
! [A2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_421_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_422_Diff__single__insert,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_423_Diff__single__insert,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_424_subset__singleton__iff,axiom,
! [X4: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X4 = bot_bot_set_list_a )
| ( X4
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_425_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_426_psubset__insert__iff,axiom,
! [A2: set_nat_a,X: nat > a,B3: set_nat_a] :
( ( ord_less_set_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( ( ( member_nat_a @ X @ B3 )
=> ( ord_less_set_nat_a @ A2 @ B3 ) )
& ( ~ ( member_nat_a @ X @ B3 )
=> ( ( ( member_nat_a @ X @ A2 )
=> ( ord_less_set_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B3 ) )
& ( ~ ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_427_psubset__insert__iff,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( ( ( member_list_a @ X @ B3 )
=> ( ord_less_set_list_a @ A2 @ B3 ) )
& ( ~ ( member_list_a @ X @ B3 )
=> ( ( ( member_list_a @ X @ A2 )
=> ( ord_less_set_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_428_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ B3 )
=> ( ord_less_set_a @ A2 @ B3 ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_429_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_430_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_431_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_432_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_433_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_434_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_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_450_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_451_ring_Omonom__coeff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) )
= ( ^ [I3: nat] : ( if_list_a @ ( I3 = N ) @ A @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_452_ring_Omonom__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( monom_a_b @ R @ A @ N ) )
= ( ^ [I3: nat] : ( if_a @ ( I3 = N ) @ A @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_453_ring_Ocoeff__length,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P2 ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ R @ P2 @ I )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_454_ring_Ocoeff__length,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ P2 ) @ I )
=> ( ( coeff_a_b @ R @ P2 @ I )
= ( zero_a_b @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_455_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_456_not__psubset__empty,axiom,
! [A2: set_list_a] :
~ ( ord_less_set_list_a @ A2 @ bot_bot_set_list_a ) ).
% not_psubset_empty
thf(fact_457_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_458_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_459_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_460_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y4: nat > a] :
~ ( member_nat_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_461_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_462_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_463_equals0D,axiom,
! [A2: set_nat_a,A: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A2 ) ) ).
% equals0D
thf(fact_464_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_465_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_466_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_467_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_468_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_469_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_470_ring_Oring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_471_insert__Diff__if,axiom,
! [X: list_a,B3: set_list_a,A2: set_list_a] :
( ( ( member_list_a @ X @ B3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( minus_646659088055828811list_a @ A2 @ B3 ) ) )
& ( ~ ( member_list_a @ X @ B3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( insert_list_a @ X @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_472_insert__Diff__if,axiom,
! [X: nat > a,B3: set_nat_a,A2: set_nat_a] :
( ( ( member_nat_a @ X @ B3 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) )
& ( ~ ( member_nat_a @ X @ B3 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( insert_nat_a @ X @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_473_insert__Diff__if,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_474_mk__disjoint__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ? [B6: set_list_a] :
( ( A2
= ( insert_list_a @ A @ B6 ) )
& ~ ( member_list_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_475_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_476_mk__disjoint__insert,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ? [B6: set_nat_a] :
( ( A2
= ( insert_nat_a @ A @ B6 ) )
& ~ ( member_nat_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_477_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_478_insert__eq__iff,axiom,
! [A: list_a,A2: set_list_a,B: list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ B @ B3 )
=> ( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C5: set_list_a] :
( ( A2
= ( insert_list_a @ B @ C5 ) )
& ~ ( member_list_a @ B @ C5 )
& ( B3
= ( insert_list_a @ A @ C5 ) )
& ~ ( member_list_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_479_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A2
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B3
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_480_insert__eq__iff,axiom,
! [A: nat > a,A2: set_nat_a,B: nat > a,B3: set_nat_a] :
( ~ ( member_nat_a @ A @ A2 )
=> ( ~ ( member_nat_a @ B @ B3 )
=> ( ( ( insert_nat_a @ A @ A2 )
= ( insert_nat_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C5: set_nat_a] :
( ( A2
= ( insert_nat_a @ B @ C5 ) )
& ~ ( member_nat_a @ B @ C5 )
& ( B3
= ( insert_nat_a @ A @ C5 ) )
& ~ ( member_nat_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_481_insert__absorb,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_482_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_483_insert__absorb,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_484_insert__ident,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ~ ( member_list_a @ X @ B3 )
=> ( ( ( insert_list_a @ X @ A2 )
= ( insert_list_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_485_insert__ident,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_486_insert__ident,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ~ ( member_nat_a @ X @ B3 )
=> ( ( ( insert_nat_a @ X @ A2 )
= ( insert_nat_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_487_Set_Oset__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ! [B6: set_list_a] :
( ( A2
= ( insert_list_a @ X @ B6 ) )
=> ( member_list_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_488_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_489_Set_Oset__insert,axiom,
! [X: nat > a,A2: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ~ ! [B6: set_nat_a] :
( ( A2
= ( insert_nat_a @ X @ B6 ) )
=> ( member_nat_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_490_insertI2,axiom,
! [A: list_a,B3: set_list_a,B: list_a] :
( ( member_list_a @ A @ B3 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_491_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_492_insertI2,axiom,
! [A: nat > a,B3: set_nat_a,B: nat > a] :
( ( member_nat_a @ A @ B3 )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_493_insertI1,axiom,
! [A: list_a,B3: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B3 ) ) ).
% insertI1
thf(fact_494_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_495_insertI1,axiom,
! [A: nat > a,B3: set_nat_a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ B3 ) ) ).
% insertI1
thf(fact_496_insertE,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_497_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_498_insertE,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_499_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y5: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_500_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X3: list_a,Y5: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_501_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_502_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X @ Y )
= ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_503_ring_Oring__simprules_I14_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_minu3984020753470702548t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_504_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_505_ring_Ogenideal__self_H,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_506_ring_Ogenideal__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.genideal_zero
thf(fact_507_ring_Ogenideal__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% ring.genideal_zero
thf(fact_508_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_509_principalideal_Ogenerate,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( I2
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_510_Diff__insert__absorb,axiom,
! [X: nat > a,A2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_511_Diff__insert__absorb,axiom,
! [X: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_512_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_513_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_514_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_515_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_516_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_517_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_518_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_519_singleton__iff,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_520_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_521_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_522_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_523_Diff__insert2,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_524_insert__Diff,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_525_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_526_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_527_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_528_Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_529_singletonD,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_530_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_531_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_532_bound_Obound,axiom,
! [Z: a,N: nat,F: nat > a,M: nat] :
( ( bound_a @ Z @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z ) ) ) ).
% bound.bound
thf(fact_533_bound__def,axiom,
( bound_a
= ( ^ [Z3: a,N2: nat,F2: nat > a] :
! [M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( ( F2 @ M3 )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_534_ring_Ocoeff__iff__length__cond,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P22 ) )
=> ( ( P1 = P22 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P22 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_535_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_536_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_537_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_538_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_539_subfield__long__division__theorem__shell,axiom,
! [K2: set_a,P2: list_a,B: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ? [Q2: list_a,R3: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ Q2 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_540_a__lcos__mult__one,axiom,
! [M4: set_a] :
( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M4 )
= M4 ) ) ).
% a_lcos_mult_one
thf(fact_541_a__lcos__m__assoc,axiom,
! [M4: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M4 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M4 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_542_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_543_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_544_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_545_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_546_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_547_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_548_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_549_subring__props_I5_J,axiom,
! [K2: set_a,H: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_550_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_551_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_552_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_553_Diff__idemp,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_554_Diff__iff,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( ( member_list_a @ C @ A2 )
& ~ ( member_list_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_555_Diff__iff,axiom,
! [C: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) )
= ( ( member_nat_a @ C @ A2 )
& ~ ( member_nat_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_556_Diff__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_557_DiffI,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ~ ( member_list_a @ C @ B3 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_558_DiffI,axiom,
! [C: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ A2 )
=> ( ~ ( member_nat_a @ C @ B3 )
=> ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_559_DiffI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_560_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_561_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_562_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_563_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_564_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_565_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_566_psubset__imp__ex__mem,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B3 )
=> ? [B4: list_a] : ( member_list_a @ B4 @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_567_psubset__imp__ex__mem,axiom,
! [A2: set_nat_a,B3: set_nat_a] :
( ( ord_less_set_nat_a @ A2 @ B3 )
=> ? [B4: nat > a] : ( member_nat_a @ B4 @ ( minus_490503922182417452_nat_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_568_psubset__imp__ex__mem,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_569_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_570_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_571_DiffD2,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
=> ~ ( member_list_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_572_DiffD2,axiom,
! [C: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) )
=> ~ ( member_nat_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_573_DiffD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( member_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_574_DiffD1,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_575_DiffD1,axiom,
! [C: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) )
=> ( member_nat_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_576_DiffD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_577_DiffE,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_578_DiffE,axiom,
! [C: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) )
=> ~ ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_579_DiffE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_580_psubsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_581_psubsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_582_psubsetD,axiom,
! [A2: set_nat_a,B3: set_nat_a,C: nat > a] :
( ( ord_less_set_nat_a @ A2 @ B3 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_583_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_584_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_585_size__neq__size__imp__neq,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X )
!= ( size_size_multiset_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_586_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_587_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_588_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_589_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_590_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_591_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_592_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_593_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_594_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_595_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_596_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_597_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_598_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_599_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_600_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_601_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_602_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_603_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_604_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_605_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_606_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_607_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_608_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_617_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_618_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_619_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_620_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_621_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_622_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_623_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_624_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_625_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_626_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_627_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_628_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_629_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_630_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_631_ring_Onon__empty__bounded__degree__polynomials,axiom,
! [R: partia2175431115845679010xt_a_b,K: nat] :
( ( ring_a_b @ R )
=> ( ( bounde2262800523058855161ls_a_b @ R @ K )
!= bot_bot_set_list_a ) ) ).
% ring.non_empty_bounded_degree_polynomials
thf(fact_632_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_633_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_634_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_635_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_636_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_637_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_638_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_639_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_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_651_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_652_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_653_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_654_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_655_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_656_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_657_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_658_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_659_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_660_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_661_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_662_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_663_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_ring_Osubfield__long__division__theorem__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a,B: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ? [Q2: list_a,R3: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K2 ) @ B @ Q2 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% ring.subfield_long_division_theorem_shell
thf(fact_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_683_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_684_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_685_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_686_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_687_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_688_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_689_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_690_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_691_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_692_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_693_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_694_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_695_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_696_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_697_pirreducible__degree,axiom,
! [K2: set_a,P2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_698_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_699_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,K: list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ A ) @ K2 )
=> ( member_list_a @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_700_line__extension__smult__closed,axiom,
! [K2: set_a,E2: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_701_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_702_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_703_space__subgroup__props_I6_J,axiom,
! [K2: set_a,N: nat,E2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ E2 )
=> ( member_a @ A @ E2 ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_704_dimension__is__inj,axiom,
! [K2: set_a,N: nat,E2: set_a,M: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ E2 )
=> ( N = M ) ) ) ) ).
% dimension_is_inj
thf(fact_705_space__subgroup__props_I3_J,axiom,
! [K2: set_a,N: nat,E2: set_a,V1: a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ V1 @ E2 )
=> ( ( member_a @ V2 @ E2 )
=> ( member_a @ ( add_a_b @ r @ V1 @ V2 ) @ E2 ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_706_space__subgroup__props_I2_J,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( member_a @ ( zero_a_b @ r ) @ E2 ) ) ) ).
% space_subgroup_props(2)
thf(fact_707_telescopic__base__aux,axiom,
! [K2: set_a,F3: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F3 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F3 )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F3 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_708_telescopic__base,axiom,
! [K2: set_a,F3: set_a,N: nat,M: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F3 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F3 )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ F3 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( times_times_nat @ N @ M ) @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base
thf(fact_709_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_710_space__subgroup__props_I5_J,axiom,
! [K2: set_a,N: nat,E2: set_a,K: a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V3 ) @ E2 ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_711_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E2: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E2 )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_712_space__subgroup__props_I4_J,axiom,
! [K2: set_a,N: nat,E2: set_a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ E2 ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_713_space__subgroup__props_I1_J,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_714_dimension__one,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K2 @ K2 ) ) ).
% dimension_one
thf(fact_715_subfieldE_I4_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K1 @ K2 )
=> ( ( member_list_a @ K22 @ K2 )
=> ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( mult_l7073676228092353617t_unit @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_716_subfieldE_I4_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_717_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_718_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_719_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_720_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_721_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_722_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_723_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_724_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_725_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_726_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_727_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_728_subfieldE_I3_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_729_subfieldE_I3_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_730_subfieldE_I5_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K1 @ K2 )
=> ( ( member_list_a @ K22 @ K2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( K22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_731_subfieldE_I5_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_732_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_733_ring_Opirreducible__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K2 ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ).
% ring.pirreducible_degree
thf(fact_734_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ E2 )
=> ( member_a @ A @ E2 ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_735_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a,K: list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ A ) @ E2 )
=> ( member_list_a @ A @ E2 ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_736_dimension__zero,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ E2 )
=> ( E2
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_737_ring_Oline__extension__smult__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,E2: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ! [K4: list_a,V: list_a] :
( ( member_list_a @ K4 @ K2 )
=> ( ( member_list_a @ V @ E2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ U ) @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_738_ring_Oline__extension__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a,A: a,K: a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_739_zero__dim,axiom,
! [K2: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_740_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_741_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_742_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_743_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_744_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_745_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_746_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_747_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_748_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_749_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_750_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_751_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_752_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_753_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_754_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_755_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_765_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_766_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_767_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_768_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_769_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_770_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_771_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_772_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_773_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_774_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_775_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_776_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_777_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_778_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_779_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_780_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_781_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_782_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_783_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_784_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_785_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_786_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_820_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_821_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_822_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_823_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_824_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_825_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_826_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_827_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_842_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_843_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_844_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_845_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_846_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_847_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_848_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_849_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_850_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_851_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_852_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_853_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_854_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_855_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_856_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_857_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_858_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_859_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_860_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_861_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_862_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_863_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_864_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_865_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_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_887_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_888_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_889_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_890_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_891_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_892_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_893_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_894_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_912_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_913_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_941_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_942_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_943_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_944_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_945_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_946_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_947_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_948_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_949_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_950_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_951_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_952_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_953_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_954_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_955_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_956_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_957_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_958_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_959_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_960_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_961_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_962_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_963_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_964_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_965_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_974_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_975_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_976_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_977_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_978_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_979_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_980_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_981_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_982_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_983_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_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_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_997_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_998_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_999_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_1000_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_1001_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1002_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1003_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1004_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1005_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1006_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_1007_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_1008_ring_Ozero__dim,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( embedd3793949463769647726t_unit @ R @ zero_zero_nat @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% ring.zero_dim
thf(fact_1009_ring_Ozero__dim,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.zero_dim
thf(fact_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V3: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U @ V3 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V3 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1019_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1020_ring_Odimension__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ zero_zero_nat @ K2 @ E2 )
=> ( E2
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_1021_ring_Odimension__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ E2 )
=> ( E2
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_1022_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V3: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U @ V3 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V3 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1023_ring_Osubring__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_1024_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_1025_ring_Osubring__props_I7_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,H1: list_a,H22: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_1026_ring_Osubring__props_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_1027_ring_Osubring__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( K2 != bot_bot_set_a ) ) ) ).
% ring.subring_props(4)
thf(fact_1028_ring_Osubring__props_I6_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,H1: list_a,H22: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_1029_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_1030_ring_Osubring__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_1031_ring_Odimension__is__inj,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,M: nat] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ E2 )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_1032_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_1033_ring_Osubring__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_1034_ring_Oline__extension__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,A: list_a,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_1035_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_1036_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ E2 ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_1037_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( member_a @ ( zero_a_b @ R ) @ E2 ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_1038_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a,V1: list_a,V2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ( member_list_a @ V1 @ E2 )
=> ( ( member_list_a @ V2 @ E2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ V1 @ V2 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_1039_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,V1: a,V2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( member_a @ V1 @ E2 )
=> ( ( member_a @ V2 @ E2 )
=> ( member_a @ ( add_a_b @ R @ V1 @ V2 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_1040_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a,K: list_a,V3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ V3 @ E2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ V3 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_1041_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,K: a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_1042_ring_Otelescopic__base__aux,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F3: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F3 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F3 )
=> ( ( embedd2795209813406577254on_a_b @ R @ one_one_nat @ F3 @ E2 )
=> ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 ) ) ) ) ) ) ).
% ring.telescopic_base_aux
thf(fact_1043_ring_Odimension__one,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( embedd2795209813406577254on_a_b @ R @ one_one_nat @ K2 @ K2 ) ) ) ).
% ring.dimension_one
thf(fact_1044_ring_Oline__extension__mem__iff,axiom,
! [R: partia2670972154091845814t_unit,U: list_a,K2: set_list_a,A: list_a,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K2 )
& ? [Y5: list_a] :
( ( member_list_a @ Y5 @ E2 )
& ( U
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ A ) @ Y5 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_1045_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,A: a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E2 )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ A ) @ Y5 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_1046_ring_Otelescopic__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F3: set_a,N: nat,M: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F3 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F3 )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ F3 @ E2 )
=> ( embedd2795209813406577254on_a_b @ R @ ( times_times_nat @ N @ M ) @ K2 @ E2 ) ) ) ) ) ) ).
% ring.telescopic_base
thf(fact_1047_ring_Ospace__subgroup__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( a_inv_a_b @ R @ V3 ) @ E2 ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_1048_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_1049_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_1050_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_1051_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_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_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1054_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1055_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1056_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1057_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1058_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1059_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1060_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1061_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1062_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1063_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_1064_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_1065_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1066_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1067_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_1068_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1069_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1070_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1071_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1072_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1073_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1074_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1075_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1076_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_1077_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1078_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1079_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_1080_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_1081_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1082_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_dimension_Osimps,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
= ( ? [K5: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V4: a,E3: set_a,N2: nat,K5: set_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K5 @ V4 @ E3 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V4 @ E3 )
& ( embedd2795209813406577254on_a_b @ r @ N2 @ K5 @ E3 ) ) ) ) ).
% dimension.simps
thf(fact_1093_dimension_Ocases,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ~ ! [V: a,E4: set_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A22 @ V @ E4 ) )
=> ( ( member_a @ V @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V @ E4 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N3 @ A22 @ E4 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_1094_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1095_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1096_Suc__dim,axiom,
! [V3: a,E2: set_a,N: nat,K2: set_a] :
( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V3 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ r @ K2 @ V3 @ E2 ) ) ) ) ) ).
% Suc_dim
thf(fact_1097_dimension__backwards,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ E2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E5: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E5 )
& ~ ( member_a @ X2 @ E5 )
& ( E2
= ( embedd971793762689825387on_a_b @ r @ K2 @ X2 @ E5 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_1098_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_1099_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1100_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1101_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_1102_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_1103_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_1104_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_1105_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1106_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1107_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_1108_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_1109_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_1117_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1118_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1119_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1120_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1121_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] :
( ( P @ X2 @ Y4 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1122_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1123_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1124_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1125_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1126_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1127_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1128_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1129_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1130_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1131_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1132_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1133_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1134_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1135_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1136_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1137_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1138_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1139_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1140_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1141_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1142_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1143_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1144_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_1145_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1146_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1147_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_1148_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_1149_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_1150_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1151_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_1152_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_1153_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_1154_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1155_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_1156_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1157_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_1158_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1159_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1160_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1161_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1162_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1163_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_1164_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_1165_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_1166_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1167_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K4 )
=> ( P @ I4 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1168_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_1169_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_1170_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1171_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1172_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1173_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1174_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_1175_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1176_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1177_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1178_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_1179_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1180_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1181_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1182_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1183_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_1184_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1185_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M2: nat] :
( M7
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1186_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_1187_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1188_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_1189_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1190_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1191_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y4: nat,Z4: nat] :
( ( R @ X2 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1192_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1193_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_1194_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1195_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_1196_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1197_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1198_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_1199_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_1200_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1201_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1202_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1203_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_1204_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1205_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_1206_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1207_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_1208_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_1209_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_1210_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_1211_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_1212_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1213_ring_OSuc__dim,axiom,
! [R: partia2670972154091845814t_unit,V3: list_a,E2: set_list_a,N: nat,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ V3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ~ ( member_list_a @ V3 @ E2 )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( embedd3793949463769647726t_unit @ R @ ( suc @ N ) @ K2 @ ( embedd5150658419831591667t_unit @ R @ K2 @ V3 @ E2 ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1214_ring_OSuc__dim,axiom,
! [R: partia2175431115845679010xt_a_b,V3: a,E2: set_a,N: nat,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V3 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ R @ K2 @ V3 @ E2 ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1215_ring_Odimension__backwards,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ ( suc @ N ) @ K2 @ E2 )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ? [E5: set_list_a] :
( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E5 )
& ~ ( member_list_a @ X2 @ E5 )
& ( E2
= ( embedd5150658419831591667t_unit @ R @ K2 @ X2 @ E5 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1216_ring_Odimension__backwards,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ E2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ? [E5: set_a] :
( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E5 )
& ~ ( member_a @ X2 @ E5 )
& ( E2
= ( embedd971793762689825387on_a_b @ R @ K2 @ X2 @ E5 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1217_ring_Odimension_Ocases,axiom,
! [R: partia2670972154091845814t_unit,A1: nat,A22: set_list_a,A32: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ~ ! [V: list_a,E4: set_list_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd5150658419831591667t_unit @ R @ A22 @ V @ E4 ) )
=> ( ( member_list_a @ V @ ( partia5361259788508890537t_unit @ R ) )
=> ( ~ ( member_list_a @ V @ E4 )
=> ~ ( embedd3793949463769647726t_unit @ R @ N3 @ A22 @ E4 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1218_ring_Odimension_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,A1: nat,A22: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ~ ! [V: a,E4: set_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ R @ A22 @ V @ E4 ) )
=> ( ( member_a @ V @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V @ E4 )
=> ~ ( embedd2795209813406577254on_a_b @ R @ N3 @ A22 @ E4 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1219_ring_Odimension_Osimps,axiom,
! [R: partia2670972154091845814t_unit,A1: nat,A22: set_list_a,A32: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ A1 @ A22 @ A32 )
= ( ? [K5: set_list_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
| ? [V4: list_a,E3: set_list_a,N2: nat,K5: set_list_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd5150658419831591667t_unit @ R @ K5 @ V4 @ E3 ) )
& ( member_list_a @ V4 @ ( partia5361259788508890537t_unit @ R ) )
& ~ ( member_list_a @ V4 @ E3 )
& ( embedd3793949463769647726t_unit @ R @ N2 @ K5 @ E3 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1220_ring_Odimension_Osimps,axiom,
! [R: partia2175431115845679010xt_a_b,A1: nat,A22: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A1 @ A22 @ A32 )
= ( ? [K5: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
| ? [V4: a,E3: set_a,N2: nat,K5: set_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ R @ K5 @ V4 @ E3 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ R ) )
& ~ ( member_a @ V4 @ E3 )
& ( embedd2795209813406577254on_a_b @ R @ N2 @ K5 @ E3 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1221_splitted__on__def,axiom,
! [K2: set_a,P2: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K2 @ P2 )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K2 @ P2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_1222_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1223_minf_I7_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T2 @ X5 ) ) ).
% minf(7)
thf(fact_1224_minf_I7_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_int @ T2 @ X5 ) ) ).
% minf(7)
thf(fact_1225_minf_I5_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T2 ) ) ).
% minf(5)
thf(fact_1226_minf_I5_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_int @ X5 @ T2 ) ) ).
% minf(5)
thf(fact_1227_minf_I4_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T2 ) ) ).
% minf(4)
thf(fact_1228_minf_I4_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T2 ) ) ).
% minf(4)
thf(fact_1229_minf_I3_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T2 ) ) ).
% minf(3)
thf(fact_1230_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P3: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1231_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P3: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1232_minusinfinity,axiom,
! [D: int,P12: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K4: int] :
( ( P12 @ X2 )
= ( P12 @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P12 @ X2 ) ) )
=> ( ? [X_1: int] : ( P12 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1233_plusinfinity,axiom,
! [D: int,P3: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K4: int] :
( ( P3 @ X2 )
= ( P3 @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [X_1: int] : ( P3 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1234_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1235_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_1236_coeff__nth,axiom,
! [I: nat,P2: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ P2 ) )
=> ( ( coeff_a_b @ r @ P2 @ I )
= ( nth_a @ P2 @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ I ) ) ) ) ).
% coeff_nth
thf(fact_1237_subfield__m__inv_I1_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_1238_subfield__m__inv_I2_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K @ ( m_inv_a_ring_ext_a_b @ r @ K ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_1239_subfield__m__inv_I3_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ K )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_1240_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_1241_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1242_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_1243_inv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_1244_inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_1245_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_1246_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_1247_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_1248_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_1249_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_1250_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_1251_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_1252_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_1253_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_1254_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_1255_subringI,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).
% subringI
thf(fact_1256_carrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_1257_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_1258_subalgebra__in__carrier,axiom,
! [K2: set_a,V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ord_less_eq_set_a @ V5 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_1259_carrier__polynomial__shell,axiom,
! [K2: set_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_1260_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E2: set_a,V5: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ( ord_less_eq_set_a @ V5 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V5 ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_1261_subcringI,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H2 @ r ) ) ) ).
% subcringI
thf(fact_1262_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F3: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F3 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F3 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_1263_finite__dimensionE_H,axiom,
! [K2: set_a,E2: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N3 @ K2 @ E2 ) ) ).
% finite_dimensionE'
thf(fact_1264_finite__dimensionI,axiom,
! [N: nat,K2: set_a,E2: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 ) ) ).
% finite_dimensionI
thf(fact_1265_finite__dimension__def,axiom,
! [K2: set_a,E2: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
= ( ? [N2: nat] : ( embedd2795209813406577254on_a_b @ r @ N2 @ K2 @ E2 ) ) ) ).
% finite_dimension_def
thf(fact_1266_unique__dimension,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X2 @ K2 @ E2 )
& ! [Y6: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y6 @ K2 @ E2 )
=> ( Y6 = X2 ) ) ) ) ) ).
% unique_dimension
thf(fact_1267_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ r ) ) ) ).
% finite_dimension_imp_subalgebra
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ x ) @ one_one_nat ) @ n )
| ( x
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
%------------------------------------------------------------------------------