TPTP Problem File: SLH0416^1.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_00129_004897__17061610_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1557 ( 361 unt; 285 typ; 0 def)
% Number of atoms : 4069 (1367 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 14511 ( 418 ~; 48 |; 278 &;11699 @)
% ( 0 <=>;2068 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 1048 (1048 >; 0 *; 0 +; 0 <<)
% Number of symbols : 256 ( 253 usr; 19 con; 0-4 aty)
% Number of variables : 3615 ( 241 ^;3134 !; 240 ?;3615 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:36:02.922
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia5333488208502193986t_unit: $tType ).
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,
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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,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_li5608457238520824219list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
set_list_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
set_nat_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_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__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
list_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
set_nat_int: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mtf__a_J_J,type,
list_nat_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__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $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 (253)
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_AbelCoset_Oset__add_001tf__a_001tf__b,type,
set_add_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001tf__a_001tf__b,type,
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thf(sy_c_Bounded__Degree__Polynomials_Oring_Obuild__poly_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Bounded__Degree__Polynomials_Oring_Obuild__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
bounde968622122064583553t_unit: partia2670972154091845814t_unit > ( nat > list_a ) > nat > list_list_a ).
thf(sy_c_Bounded__Degree__Polynomials_Oring_Obuild__poly_001tf__a_001tf__b,type,
bounde1002222742488328185ly_a_b: partia2175431115845679010xt_a_b > ( nat > a ) > nat > list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia5038748322285217333t_unit: partia5333488208502193986t_unit > set_list_list_list_a ).
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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_Oassociated_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
associ5860276527279195403xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
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_Ofactors_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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thf(sy_c_Divisibility_Oirreducible_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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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_Divisibility_Owfactors_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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thf(sy_c_Embedded__Algebras_Oring_OSpan_001tf__a_001tf__b,type,
embedded_Span_a_b: partia2175431115845679010xt_a_b > set_a > list_a > set_a ).
thf(sy_c_Embedded__Algebras_Oring_Ocombine_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Ocombine_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Oring_Oindependent_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
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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_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
embedd1097489486847499020t_unit: set_list_list_a > set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd1768981623711841426t_unit: set_list_a > set_list_a > partia2670972154091845814t_unit > $o ).
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_OUnits_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
units_2932844235741507942t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_a_ring_ext_a_b: partia2175431115845679010xt_a_b > set_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_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
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one_li8328186300101108157t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Group_Opow_001tf__a_001t__Product____Type__Ounit_001t__Nat__Onat,type,
pow_a_1875594501834816709it_nat: partia8223610829204095565t_unit > a > nat > a ).
thf(sy_c_Group_Opow_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
pow_a_1026414303147256608_b_nat: partia2175431115845679010xt_a_b > a > nat > a ).
thf(sy_c_Group_Ounits__of_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_8174867845824275201xt_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
cgenid24865672677839267t_unit: partia2956882679547061052t_unit > list_list_a > set_list_list_a ).
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,
cgenid9131348535277946915t_unit: partia2670972154091845814t_unit > list_a > set_list_a ).
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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
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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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
princi2534607884127416211t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
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_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
append_nat_int: list_nat_int > list_nat_int > list_nat_int ).
thf(sy_c_List_Oappend_001_062_It__Nat__Onat_Mtf__a_J,type,
append_nat_a: list_nat_a > list_nat_a > list_nat_a ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
append_list_list_a: list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
cons_nat_int: ( nat > int ) > list_nat_int > list_nat_int ).
thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mtf__a_J,type,
cons_nat_a: ( nat > a ) > list_nat_a > list_nat_a ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
nil_nat_int: list_nat_int ).
thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mtf__a_J,type,
nil_nat_a: list_nat_a ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
hd_nat_int: list_nat_int > nat > int ).
thf(sy_c_List_Olist_Ohd_001_062_It__Nat__Onat_Mtf__a_J,type,
hd_nat_a: list_nat_a > nat > a ).
thf(sy_c_List_Olist_Ohd_001t__Int__Oint,type,
hd_int: list_int > int ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
hd_list_list_a: list_list_list_a > list_list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Olist__all2_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
list_a9087535860334789575_nat_a: ( ( nat > a ) > ( nat > a ) > $o ) > list_nat_a > list_nat_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
list_all2_nat_a_a: ( ( nat > a ) > a > $o ) > list_nat_a > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_a6511221006964818183list_a: ( list_list_a > list_list_a > $o ) > list_list_list_a > list_list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
list_a2339687664058784123st_a_a: ( list_list_a > a > $o ) > list_list_list_a > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_Itf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
list_a1636186360766657384_nat_a: ( list_a > ( nat > a ) > $o ) > list_list_a > list_nat_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_a1787013445579514253list_a: ( list_a > list_list_a > $o ) > list_list_a > list_list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
list_a3802133873445908231list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_Itf__a_J_001tf__a,type,
list_all2_list_a_a: ( list_a > a > $o ) > list_list_a > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
list_all2_a_nat_a: ( a > ( nat > a ) > $o ) > list_a > list_nat_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_a1736694032391712403list_a: ( a > list_list_a > $o ) > list_a > list_list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001t__List__Olist_Itf__a_J,type,
list_all2_a_list_a: ( a > list_a > $o ) > list_a > list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001tf__a,type,
list_all2_a_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
set_nat_int2: list_nat_int > set_nat_int ).
thf(sy_c_List_Olist_Oset_001_062_It__Nat__Onat_Mtf__a_J,type,
set_nat_a2: list_nat_a > set_nat_a ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
set_list_list_a2: list_list_list_a > set_list_list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Oreplicate_001_062_It__Nat__Onat_Mtf__a_J,type,
replicate_nat_a: nat > ( nat > a ) > list_nat_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
replic3997036819131463498list_a: nat > list_list_a > list_list_list_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_Itf__a_J,type,
replicate_list_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oreplicate_001tf__a,type,
replicate_a: nat > a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
size_s5718426915756887103at_int: list_nat_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_Mtf__a_J_J,type,
size_size_list_nat_a: list_nat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
size_s2403821588304063868list_a: list_list_list_a > 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_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_M_Eo_J,type,
bot_bo4370453251804172807st_a_o: list_list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
bot_bot_set_nat_int: set_nat_int ).
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__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
bot_bo1875519244922727510list_a: set_list_list_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_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_M_Eo_J,type,
ord_le3894349897559202247st_a_o: ( list_list_a > $o ) > ( list_list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_list_a_o: ( list_a > $o ) > ( list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $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_It__List__Olist_Itf__a_J_J_J,type,
ord_le5338140678153942166list_a: set_list_list_a > set_list_list_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_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_M_Eo_J,type,
ord_le1801313680655002067st_a_o: ( list_list_a > $o ) > ( list_list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_eq_list_a_o: ( list_a > $o ) > ( list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $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_Mt__Int__Oint_J_J,type,
ord_le6569500216720880561at_int: set_nat_int > set_nat_int > $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__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_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_Oexp__base_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
polyno6819740552565085946t_unit: partia2956882679547061052t_unit > list_list_a > nat > list_list_list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno3522816881121920896t_unit: partia2670972154091845814t_unit > list_a > nat > list_list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001tf__a_001tf__b,type,
polyno2922411391617481336se_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
polyno5142720416380192742t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno6951661231331188332t_unit: partia2670972154091845814t_unit > list_list_a > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001tf__a_001tf__b,type,
polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Polynomials_Oring_Ocoeff_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
coeff_1681977662540381769t_unit: partia2956882679547061052t_unit > list_list_list_a > nat > list_list_a ).
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_Oconst__term_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
const_6738166269504826821t_unit: partia2670972154091845814t_unit > list_list_a > list_a ).
thf(sy_c_Polynomials_Oring_Oconst__term_001tf__a_001tf__b,type,
const_term_a_b: partia2175431115845679010xt_a_b > list_a > a ).
thf(sy_c_Polynomials_Oring_Oeval_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
eval_l1088911609197519410t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Oeval_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
eval_l34571156754992824t_unit: partia2670972154091845814t_unit > list_list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Oeval_001tf__a_001tf__b,type,
eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).
thf(sy_c_Polynomials_Oring_Omonom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
monom_4043874212805408666t_unit: partia2956882679547061052t_unit > list_list_a > nat > list_list_list_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_Oring_Onormalize_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
normal1297324897130370429t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_list_a ).
thf(sy_c_Polynomials_Oring_Onormalize_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
normal637505603836502915t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Onormalize_001tf__a_001tf__b,type,
normalize_a_b: partia2175431115845679010xt_a_b > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
poly_a7341706734723628101t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_a7601779127272115787t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001tf__a_001tf__b,type,
poly_add_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__mult_001tf__a_001tf__b,type,
poly_mult_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
univ_p2250591967980070728t_unit: partia2956882679547061052t_unit > set_list_list_a > partia5333488208502193986t_unit ).
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_Polynomials_Ovar_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
var_li3532061862469730199t_unit: partia2956882679547061052t_unit > list_list_list_a ).
thf(sy_c_Polynomials_Ovar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
var_li8453953174693405341t_unit: partia2670972154091845814t_unit > list_list_a ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i6186174840089424918t_unit: partia2956882679547061052t_unit > partia2956882679547061052t_unit > set_li5608457238520824219list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i4611353245267337884t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i5684343068699926420it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i7582117978422105628t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > set_li6773872926390105121list_a ).
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,
ring_i7414513579304222626t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i4464730343205239444t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > set_a_list_list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i4557880751517319194t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_inv_7033018035630854991t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_inv_8944721093294617173t_unit: partia2670972154091845814t_unit > list_a > list_a ).
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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_minu2241224857956505934t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_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_Oring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_l1939023646219158831t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring_Ozero__update_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_u1196785550890449590t_unit: ( list_a > list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).
thf(sy_c_Ring_Oring_Ozero__update_001tf__a_001tf__b,type,
zero_update_a_b: ( a > a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h8129544334414776832t_unit: partia2956882679547061052t_unit > partia2956882679547061052t_unit > set_li5608457238520824219list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h5031276006722532742t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h8078271382950527358it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h8002040739877300486t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > set_li6773872926390105121list_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h6858658657455840382t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > set_a_list_list_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__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
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_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
collec1292721268053437947list_a: ( list_list_list_a > $o ) > set_list_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_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_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
insert_nat_int: ( nat > int ) > set_nat_int > set_nat_int ).
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__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
insert_list_list_a: list_list_a > set_list_list_a > set_list_list_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_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_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
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_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8231385768148312316list_a: ( list_list_a > list_list_a ) > set_li5608457238520824219list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
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_Mt__Int__Oint_J,type,
member_nat_int: ( nat > int ) > set_nat_int > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_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_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_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__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_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_f,type,
f: nat > a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1266)
thf(fact_0_length__build__poly,axiom,
! [F: nat > a,N: nat] : ( ord_less_eq_nat @ ( size_size_list_a @ ( bounde1002222742488328185ly_a_b @ r @ F @ N ) ) @ N ) ).
% length_build_poly
thf(fact_1_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_2_ring_Obuild__poly_Ocong,axiom,
bounde1002222742488328185ly_a_b = bounde1002222742488328185ly_a_b ).
% ring.build_poly.cong
thf(fact_3_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_4_assms,axiom,
! [K: nat] :
( ( ord_less_nat @ K @ n )
=> ( member_a @ ( f @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% assms
thf(fact_5_build__poly__poly,axiom,
! [N: nat,F: nat > a] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( member_a @ ( F @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( bounde1002222742488328185ly_a_b @ r @ F @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% build_poly_poly
thf(fact_6_bounded__degree__polynomials__length,axiom,
( bounde1385458654359096187t_unit
= ( ^ [F2: partia2956882679547061052t_unit,N2: nat] :
( collec1292721268053437947list_a
@ ^ [X: list_list_list_a] :
( ( member5342144027231129785list_a @ X @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ F2 @ ( partia2464479390973590831t_unit @ F2 ) ) ) )
& ( ord_less_eq_nat @ ( size_s2403821588304063868list_a @ X ) @ N2 ) ) ) ) ) ).
% bounded_degree_polynomials_length
thf(fact_7_bounded__degree__polynomials__length,axiom,
( bounde872414301498847361t_unit
= ( ^ [F2: partia2670972154091845814t_unit,N2: nat] :
( collect_list_list_a
@ ^ [X: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ F2 @ ( partia5361259788508890537t_unit @ F2 ) ) ) )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ X ) @ N2 ) ) ) ) ) ).
% bounded_degree_polynomials_length
thf(fact_8_bounded__degree__polynomials__length,axiom,
( bounde2262800523058855161ls_a_b
= ( ^ [F2: partia2175431115845679010xt_a_b,N2: nat] :
( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ F2 @ ( partia707051561876973205xt_a_b @ F2 ) ) ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ X ) @ N2 ) ) ) ) ) ).
% bounded_degree_polynomials_length
thf(fact_9_cgenideal__self,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) ) ) ).
% cgenideal_self
thf(fact_10_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_11_coeff__in__carrier,axiom,
! [P: list_a,I2: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( coeff_a_b @ r @ P @ I2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% coeff_in_carrier
thf(fact_12_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_13_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_15_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_16_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_17_coeff__iff__length__cond,axiom,
! [P1: list_a,P2: list_a] :
( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P2 ) )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P2 ) ) ) ) ).
% coeff_iff_length_cond
thf(fact_18_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_19_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_20_less__imp__neq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_21_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_22_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_23_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_24_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_25_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_26_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_27_less__induct,axiom,
! [P3: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X3 )
=> ( P3 @ Y2 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A ) ) ).
% less_induct
thf(fact_28_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_29_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_30_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_31_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_32_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_33_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_34_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X4: nat] : ( P4 @ X4 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P5 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_35_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P3 @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_36_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_37_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_38_bot_Oextremum__strict,axiom,
! [A: set_list_a] :
~ ( ord_less_set_list_a @ A @ bot_bot_set_list_a ) ).
% bot.extremum_strict
thf(fact_39_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_40_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_41_bot_Onot__eq__extremum,axiom,
! [A: set_list_a] :
( ( A != bot_bot_set_list_a )
= ( ord_less_set_list_a @ bot_bot_set_list_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_42_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_43_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_44_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_45_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_46_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_47_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_48_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_49_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_50_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_51_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_52_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_53_order__less__asym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_54_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_55_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_56_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_57_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_58_order__less__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_59_mem__Collect__eq,axiom,
! [A: nat > a,P3: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: list_a,P3: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: a,P3: a > $o] :
( ( member_a @ A @ ( collect_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: list_list_a,P3: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A3: set_nat_a] :
( ( collect_nat_a
@ ^ [X: nat > a] : ( member_nat_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A3: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_67_Collect__cong,axiom,
! [P3: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_a @ P3 )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_68_Collect__cong,axiom,
! [P3: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P3 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_69_Collect__cong,axiom,
! [P3: list_list_a > $o,Q: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_list_a @ P3 )
= ( collect_list_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_70_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_71_ord__eq__less__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_72_ord__eq__less__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_73_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_74_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_75_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_76_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_77_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_78_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_79_order__less__irrefl,axiom,
! [X2: set_a] :
~ ( ord_less_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_80_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_81_order__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_82_order__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_83_order__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_84_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_85_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_86_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_87_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_88_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_89_order__less__not__sym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_90_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_91_order__less__imp__triv,axiom,
! [X2: set_a,Y: set_a,P3: $o] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ X2 )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_92_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_93_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_94_order__less__imp__not__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_95_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_96_order__less__imp__not__eq2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_97_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_98_order__less__imp__not__less,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_99_bot_Oextremum__uniqueI,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
=> ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_100_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_101_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_102_bot_Oextremum__unique,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_103_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_104_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_105_bot_Oextremum,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% bot.extremum
thf(fact_106_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_107_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_108_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_109_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_110_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_111_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_112_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_113_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_114_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_115_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_116_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_117_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_118_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_119_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_120_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_121_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_122_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_123_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_124_order__le__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_125_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_126_order__le__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_127_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_128_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_129_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_130_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_131_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_132_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_133_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_134_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_135_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_136_order__less__imp__le,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_137_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_138_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_139_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_140_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_141_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_142_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_143_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_144_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_145_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_146_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_147_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_148_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_149_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_150_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_151_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_152_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_153_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_154_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_155_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_156_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_157_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_158_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_159_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_160_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_161_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_162_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_163_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_164_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_165_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_166_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_167_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_168_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_169_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_170_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_171_antisym__conv2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_172_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_173_antisym__conv1,axiom,
! [X2: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_174_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_175_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_176_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_177_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_178_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_179_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,I2: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( member_list_list_a @ ( coeff_1681977662540381769t_unit @ R @ P @ I2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_180_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I2: nat] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( member_a @ ( coeff_a_b @ R @ P @ I2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_181_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I2: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P @ I2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_182_ring_Obuild__poly__poly,axiom,
! [R: partia2670972154091845814t_unit,N: nat,F: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( member_list_a @ ( F @ I ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( member_list_list_a @ ( bounde968622122064583553t_unit @ R @ F @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.build_poly_poly
thf(fact_183_ring_Obuild__poly__poly,axiom,
! [R: partia2956882679547061052t_unit,N: nat,F: nat > list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( member_list_list_a @ ( F @ I ) @ ( partia2464479390973590831t_unit @ R ) ) )
=> ( member5342144027231129785list_a @ ( bounde7005217870566028923t_unit @ R @ F @ N ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% ring.build_poly_poly
thf(fact_184_ring_Obuild__poly__poly,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,F: nat > a] :
( ( ring_a_b @ R )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( member_a @ ( F @ I ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( member_list_a @ ( bounde1002222742488328185ly_a_b @ R @ F @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.build_poly_poly
thf(fact_185_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_186_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_187_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_188_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_189_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_190_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_191_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_192_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_193_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_194_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_195_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_196_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_197_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_198_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_199_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_200_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_201_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_202_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_203_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_204_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_205_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_206_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_207_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_208_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_209_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_210_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_211_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_212_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_213_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_214_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_215_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_216_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_217_linorder__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_218_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_219_order__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_220_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_221_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_222_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_223_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_224_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_225_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_226_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_227_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_228_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_229_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_230_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_231_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_232_ring_Olength__build__poly,axiom,
! [R: partia2175431115845679010xt_a_b,F: nat > a,N: nat] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( bounde1002222742488328185ly_a_b @ R @ F @ N ) ) @ N ) ) ).
% ring.length_build_poly
thf(fact_233_poly__degree__bound__from__coeff__1,axiom,
! [X2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ N @ K2 )
=> ( ( coeff_a_b @ r @ X2 @ K2 )
= ( zero_a_b @ r ) ) )
=> ( member_list_a @ X2 @ ( bounde2262800523058855161ls_a_b @ r @ N ) ) ) ) ).
% poly_degree_bound_from_coeff_1
thf(fact_234_build__poly__coeff,axiom,
! [I2: nat,N: nat,F: nat > a] :
( ( ( ord_less_nat @ I2 @ N )
=> ( ( coeff_a_b @ r @ ( bounde1002222742488328185ly_a_b @ r @ F @ N ) @ I2 )
= ( F @ I2 ) ) )
& ( ~ ( ord_less_nat @ I2 @ N )
=> ( ( coeff_a_b @ r @ ( bounde1002222742488328185ly_a_b @ r @ F @ N ) @ I2 )
= ( zero_a_b @ r ) ) ) ) ).
% build_poly_coeff
thf(fact_235_coeff__length,axiom,
! [P: list_a,I2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I2 )
=> ( ( coeff_a_b @ r @ P @ I2 )
= ( zero_a_b @ r ) ) ) ).
% coeff_length
thf(fact_236_ring_Oonepideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_237_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_238_ring_Oonepideal,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( princi2534607884127416211t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_239_subalgebra__in__carrier,axiom,
! [K3: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_240_carrier__is__subalgebra,axiom,
! [K3: set_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_241_ring_Ocgenideal__self,axiom,
! [R: partia2670972154091845814t_unit,I2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I2 @ ( cgenid9131348535277946915t_unit @ R @ I2 ) ) ) ) ).
% ring.cgenideal_self
thf(fact_242_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ R @ I2 ) ) ) ) ).
% ring.cgenideal_self
thf(fact_243_ring_Ocgenideal__self,axiom,
! [R: partia2956882679547061052t_unit,I2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ I2 @ ( cgenid24865672677839267t_unit @ R @ I2 ) ) ) ) ).
% ring.cgenideal_self
thf(fact_244_ring_Ocoeff__iff__length__cond,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P2 ) )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P2 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_245_ee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( essent8953798148185448568xt_a_b @ r @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ Cs ) ) ) ) ) ) ).
% ee_trans
thf(fact_246_ee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ Bs @ As ) ) ) ) ).
% ee_sym
thf(fact_247_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_248_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_249_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).
% ee_refl
thf(fact_250_Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I2: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P @ I2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% Polynomials.ring.coeff_in_carrier
thf(fact_251_Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I2: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( coeff_a_b @ R @ P @ I2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% Polynomials.ring.coeff_in_carrier
thf(fact_252_Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,I2: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( coeff_1681977662540381769t_unit @ R @ P @ I2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% Polynomials.ring.coeff_in_carrier
thf(fact_253_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_254_ring_Onon__empty__bounded__degree__polynomials,axiom,
! [R: partia2670972154091845814t_unit,K: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( bounde872414301498847361t_unit @ R @ K )
!= bot_bo1875519244922727510list_a ) ) ).
% ring.non_empty_bounded_degree_polynomials
thf(fact_255_ring_Ocoeff__length,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I2: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P ) @ I2 )
=> ( ( coeff_6360649920519955023t_unit @ R @ P @ I2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_256_ring_Ocoeff__length,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I2: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I2 )
=> ( ( coeff_a_b @ R @ P @ I2 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_257_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_258_principalideal_Ois__principalideal,axiom,
! [I3: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I3 @ R )
=> ( principalideal_a_b @ I3 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_259_ring_Obuild__poly__coeff,axiom,
! [R: partia2670972154091845814t_unit,I2: nat,N: nat,F: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( ord_less_nat @ I2 @ N )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( bounde968622122064583553t_unit @ R @ F @ N ) @ I2 )
= ( F @ I2 ) ) )
& ( ~ ( ord_less_nat @ I2 @ N )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( bounde968622122064583553t_unit @ R @ F @ N ) @ I2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.build_poly_coeff
thf(fact_260_ring_Obuild__poly__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,I2: nat,N: nat,F: nat > a] :
( ( ring_a_b @ R )
=> ( ( ( ord_less_nat @ I2 @ N )
=> ( ( coeff_a_b @ R @ ( bounde1002222742488328185ly_a_b @ R @ F @ N ) @ I2 )
= ( F @ I2 ) ) )
& ( ~ ( ord_less_nat @ I2 @ N )
=> ( ( coeff_a_b @ R @ ( bounde1002222742488328185ly_a_b @ R @ F @ N ) @ I2 )
= ( zero_a_b @ R ) ) ) ) ) ).
% ring.build_poly_coeff
thf(fact_261_ring_Opoly__degree__bound__from__coeff__1,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member5342144027231129785list_a @ X2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ N @ K2 )
=> ( ( coeff_1681977662540381769t_unit @ R @ X2 @ K2 )
= ( zero_l347298301471573063t_unit @ R ) ) )
=> ( member5342144027231129785list_a @ X2 @ ( bounde1385458654359096187t_unit @ R @ N ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff_1
thf(fact_262_ring_Opoly__degree__bound__from__coeff__1,axiom,
! [R: partia2175431115845679010xt_a_b,X2: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ N @ K2 )
=> ( ( coeff_a_b @ R @ X2 @ K2 )
= ( zero_a_b @ R ) ) )
=> ( member_list_a @ X2 @ ( bounde2262800523058855161ls_a_b @ R @ N ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff_1
thf(fact_263_ring_Opoly__degree__bound__from__coeff__1,axiom,
! [R: partia2670972154091845814t_unit,X2: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ N @ K2 )
=> ( ( coeff_6360649920519955023t_unit @ R @ X2 @ K2 )
= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ( member_list_list_a @ X2 @ ( bounde872414301498847361t_unit @ R @ N ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff_1
thf(fact_264_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_265_ring__iso__memE_I1_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_266_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X2: list_a] :
( ( member6714375691612171394list_a @ H @ ( ring_i7582117978422105628t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_267_ring__iso__memE_I1_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_268_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_269_ring__iso__memE_I1_J,axiom,
! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X2: a] :
( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_270_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_271_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_272_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X2: list_list_a] :
( ( member8231385768148312316list_a @ H @ ( ring_i6186174840089424918t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_273_exp__base__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X2 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_274_ring__iso__imp__img__ring,axiom,
! [H: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_ring
thf(fact_275_boundD__carrier,axiom,
! [N: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_276_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I4: nat] : ( if_a @ ( I4 = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_277_a__lcos__mult__one,axiom,
! [M3: set_a] :
( ( ord_less_eq_set_a @ M3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M3 )
= M3 ) ) ).
% a_lcos_mult_one
thf(fact_278_empty__subsetI,axiom,
! [A3: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A3 ) ).
% empty_subsetI
thf(fact_279_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_280_subset__empty,axiom,
! [A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ bot_bot_set_list_a )
= ( A3 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_281_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_282_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_283_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_284_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_285_psubsetI,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_less_set_a @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_286_subsetI,axiom,
! [A3: set_list_a,B4: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ( member_list_a @ X3 @ B4 ) )
=> ( ord_le8861187494160871172list_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_287_subsetI,axiom,
! [A3: set_nat_a,B4: set_nat_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A3 )
=> ( member_nat_a @ X3 @ B4 ) )
=> ( ord_le871467723717165285_nat_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_288_subsetI,axiom,
! [A3: set_list_list_a,B4: set_list_list_a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A3 )
=> ( member_list_list_a @ X3 @ B4 ) )
=> ( ord_le8488217952732425610list_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_289_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_290_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_291_empty__iff,axiom,
! [C: list_list_a] :
~ ( member_list_list_a @ C @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_292_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_293_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_294_all__not__in__conv,axiom,
! [A3: set_nat_a] :
( ( ! [X: nat > a] :
~ ( member_nat_a @ X @ A3 ) )
= ( A3 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_295_all__not__in__conv,axiom,
! [A3: set_list_list_a] :
( ( ! [X: list_list_a] :
~ ( member_list_list_a @ X @ A3 ) )
= ( A3 = bot_bo1875519244922727510list_a ) ) ).
% all_not_in_conv
thf(fact_296_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_297_all__not__in__conv,axiom,
! [A3: set_list_a] :
( ( ! [X: list_a] :
~ ( member_list_a @ X @ A3 ) )
= ( A3 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_298_Collect__empty__eq,axiom,
! [P3: list_list_a > $o] :
( ( ( collect_list_list_a @ P3 )
= bot_bo1875519244922727510list_a )
= ( ! [X: list_list_a] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_299_Collect__empty__eq,axiom,
! [P3: a > $o] :
( ( ( collect_a @ P3 )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_300_Collect__empty__eq,axiom,
! [P3: list_a > $o] :
( ( ( collect_list_a @ P3 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_301_empty__Collect__eq,axiom,
! [P3: list_list_a > $o] :
( ( bot_bo1875519244922727510list_a
= ( collect_list_list_a @ P3 ) )
= ( ! [X: list_list_a] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_302_empty__Collect__eq,axiom,
! [P3: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P3 ) )
= ( ! [X: a] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_303_empty__Collect__eq,axiom,
! [P3: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P3 ) )
= ( ! [X: list_a] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_304_insertCI,axiom,
! [A: list_a,B4: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B4 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_305_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_306_insertCI,axiom,
! [A: nat > a,B4: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B4 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_307_insertCI,axiom,
! [A: list_list_a,B4: set_list_list_a,B: list_list_a] :
( ( ~ ( member_list_list_a @ A @ B4 )
=> ( A = B ) )
=> ( member_list_list_a @ A @ ( insert_list_list_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_308_insert__iff,axiom,
! [A: list_a,B: list_a,A3: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A3 ) )
= ( ( A = B )
| ( member_list_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_309_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_310_insert__iff,axiom,
! [A: nat > a,B: nat > a,A3: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A3 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_311_insert__iff,axiom,
! [A: list_list_a,B: list_list_a,A3: set_list_list_a] :
( ( member_list_list_a @ A @ ( insert_list_list_a @ B @ A3 ) )
= ( ( A = B )
| ( member_list_list_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_312_insert__absorb2,axiom,
! [X2: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A3 ) )
= ( insert_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_313_a__l__coset__subset__G,axiom,
! [H2: set_a,X2: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X2 @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_314_insert__subset,axiom,
! [X2: list_a,A3: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X2 @ A3 ) @ B4 )
= ( ( member_list_a @ X2 @ B4 )
& ( ord_le8861187494160871172list_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_315_insert__subset,axiom,
! [X2: nat > a,A3: set_nat_a,B4: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X2 @ A3 ) @ B4 )
= ( ( member_nat_a @ X2 @ B4 )
& ( ord_le871467723717165285_nat_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_316_insert__subset,axiom,
! [X2: list_list_a,A3: set_list_list_a,B4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( insert_list_list_a @ X2 @ A3 ) @ B4 )
= ( ( member_list_list_a @ X2 @ B4 )
& ( ord_le8488217952732425610list_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_317_insert__subset,axiom,
! [X2: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B4 )
= ( ( member_a @ X2 @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_318_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_319_singletonI,axiom,
! [A: list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singletonI
thf(fact_320_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_321_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_322_singleton__conv,axiom,
! [A: list_list_a] :
( ( collect_list_list_a
@ ^ [X: list_list_a] : ( X = A ) )
= ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singleton_conv
thf(fact_323_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X: a] : ( X = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_324_singleton__conv,axiom,
! [A: list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( X = A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_325_singleton__conv2,axiom,
! [A: list_list_a] :
( ( collect_list_list_a
@ ( ^ [Y5: list_list_a,Z2: list_list_a] : ( Y5 = Z2 )
@ A ) )
= ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singleton_conv2
thf(fact_326_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_327_singleton__conv2,axiom,
! [A: list_a] :
( ( collect_list_a
@ ( ^ [Y5: list_a,Z2: list_a] : ( Y5 = Z2 )
@ A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_328_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A3: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_329_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_330_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A3: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A3 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_331_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_332_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_333_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_334_subset__insert,axiom,
! [X2: list_a,A3: set_list_a,B4: set_list_a] :
( ~ ( member_list_a @ X2 @ A3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X2 @ B4 ) )
= ( ord_le8861187494160871172list_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_335_subset__insert,axiom,
! [X2: nat > a,A3: set_nat_a,B4: set_nat_a] :
( ~ ( member_nat_a @ X2 @ A3 )
=> ( ( ord_le871467723717165285_nat_a @ A3 @ ( insert_nat_a @ X2 @ B4 ) )
= ( ord_le871467723717165285_nat_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_336_subset__insert,axiom,
! [X2: list_list_a,A3: set_list_list_a,B4: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A3 )
=> ( ( ord_le8488217952732425610list_a @ A3 @ ( insert_list_list_a @ X2 @ B4 ) )
= ( ord_le8488217952732425610list_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_337_subset__insert,axiom,
! [X2: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B4 ) )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_338_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_339_subset__insertI2,axiom,
! [A3: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_340_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_341_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_342_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_343_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_344_insert__not__empty,axiom,
! [A: list_a,A3: set_list_a] :
( ( insert_list_a @ A @ A3 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_345_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_346_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_347_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_348_singleton__iff,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_349_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_350_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_351_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_352_singletonD,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_353_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_354_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_355_insertE,axiom,
! [A: list_a,B: list_a,A3: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_356_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_357_insertE,axiom,
! [A: nat > a,B: nat > a,A3: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_nat_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_358_insertE,axiom,
! [A: list_list_a,B: list_list_a,A3: set_list_list_a] :
( ( member_list_list_a @ A @ ( insert_list_list_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_list_list_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_359_insertI1,axiom,
! [A: list_a,B4: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B4 ) ) ).
% insertI1
thf(fact_360_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_361_insertI1,axiom,
! [A: nat > a,B4: set_nat_a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ B4 ) ) ).
% insertI1
thf(fact_362_insertI1,axiom,
! [A: list_list_a,B4: set_list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ B4 ) ) ).
% insertI1
thf(fact_363_insertI2,axiom,
! [A: list_a,B4: set_list_a,B: list_a] :
( ( member_list_a @ A @ B4 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_364_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_365_insertI2,axiom,
! [A: nat > a,B4: set_nat_a,B: nat > a] :
( ( member_nat_a @ A @ B4 )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_366_insertI2,axiom,
! [A: list_list_a,B4: set_list_list_a,B: list_list_a] :
( ( member_list_list_a @ A @ B4 )
=> ( member_list_list_a @ A @ ( insert_list_list_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_367_Set_Oset__insert,axiom,
! [X2: list_a,A3: set_list_a] :
( ( member_list_a @ X2 @ A3 )
=> ~ ! [B5: set_list_a] :
( ( A3
= ( insert_list_a @ X2 @ B5 ) )
=> ( member_list_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_368_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B5: set_a] :
( ( A3
= ( insert_a @ X2 @ B5 ) )
=> ( member_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_369_Set_Oset__insert,axiom,
! [X2: nat > a,A3: set_nat_a] :
( ( member_nat_a @ X2 @ A3 )
=> ~ ! [B5: set_nat_a] :
( ( A3
= ( insert_nat_a @ X2 @ B5 ) )
=> ( member_nat_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_370_Set_Oset__insert,axiom,
! [X2: list_list_a,A3: set_list_list_a] :
( ( member_list_list_a @ X2 @ A3 )
=> ~ ! [B5: set_list_list_a] :
( ( A3
= ( insert_list_list_a @ X2 @ B5 ) )
=> ( member_list_list_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_371_insert__compr,axiom,
( insert_nat_a
= ( ^ [A4: nat > a,B6: set_nat_a] :
( collect_nat_a
@ ^ [X: nat > a] :
( ( X = A4 )
| ( member_nat_a @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_372_insert__compr,axiom,
( insert_list_a
= ( ^ [A4: list_a,B6: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( X = A4 )
| ( member_list_a @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_373_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B6: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A4 )
| ( member_a @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_374_insert__compr,axiom,
( insert_list_list_a
= ( ^ [A4: list_list_a,B6: set_list_list_a] :
( collect_list_list_a
@ ^ [X: list_list_a] :
( ( X = A4 )
| ( member_list_list_a @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_375_insert__ident,axiom,
! [X2: list_a,A3: set_list_a,B4: set_list_a] :
( ~ ( member_list_a @ X2 @ A3 )
=> ( ~ ( member_list_a @ X2 @ B4 )
=> ( ( ( insert_list_a @ X2 @ A3 )
= ( insert_list_a @ X2 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_376_insert__ident,axiom,
! [X2: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ~ ( member_a @ X2 @ B4 )
=> ( ( ( insert_a @ X2 @ A3 )
= ( insert_a @ X2 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_377_insert__ident,axiom,
! [X2: nat > a,A3: set_nat_a,B4: set_nat_a] :
( ~ ( member_nat_a @ X2 @ A3 )
=> ( ~ ( member_nat_a @ X2 @ B4 )
=> ( ( ( insert_nat_a @ X2 @ A3 )
= ( insert_nat_a @ X2 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_378_insert__ident,axiom,
! [X2: list_list_a,A3: set_list_list_a,B4: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A3 )
=> ( ~ ( member_list_list_a @ X2 @ B4 )
=> ( ( ( insert_list_list_a @ X2 @ A3 )
= ( insert_list_list_a @ X2 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_379_insert__absorb,axiom,
! [A: list_a,A3: set_list_a] :
( ( member_list_a @ A @ A3 )
=> ( ( insert_list_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_380_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_381_insert__absorb,axiom,
! [A: nat > a,A3: set_nat_a] :
( ( member_nat_a @ A @ A3 )
=> ( ( insert_nat_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_382_insert__absorb,axiom,
! [A: list_list_a,A3: set_list_list_a] :
( ( member_list_list_a @ A @ A3 )
=> ( ( insert_list_list_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_383_insert__eq__iff,axiom,
! [A: list_a,A3: set_list_a,B: list_a,B4: set_list_a] :
( ~ ( member_list_a @ A @ A3 )
=> ( ~ ( member_list_a @ B @ B4 )
=> ( ( ( insert_list_a @ A @ A3 )
= ( insert_list_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_list_a] :
( ( A3
= ( insert_list_a @ B @ C3 ) )
& ~ ( member_list_a @ B @ C3 )
& ( B4
= ( insert_list_a @ A @ C3 ) )
& ~ ( member_list_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_384_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B4
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_385_insert__eq__iff,axiom,
! [A: nat > a,A3: set_nat_a,B: nat > a,B4: set_nat_a] :
( ~ ( member_nat_a @ A @ A3 )
=> ( ~ ( member_nat_a @ B @ B4 )
=> ( ( ( insert_nat_a @ A @ A3 )
= ( insert_nat_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_nat_a] :
( ( A3
= ( insert_nat_a @ B @ C3 ) )
& ~ ( member_nat_a @ B @ C3 )
& ( B4
= ( insert_nat_a @ A @ C3 ) )
& ~ ( member_nat_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_386_insert__eq__iff,axiom,
! [A: list_list_a,A3: set_list_list_a,B: list_list_a,B4: set_list_list_a] :
( ~ ( member_list_list_a @ A @ A3 )
=> ( ~ ( member_list_list_a @ B @ B4 )
=> ( ( ( insert_list_list_a @ A @ A3 )
= ( insert_list_list_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_list_list_a] :
( ( A3
= ( insert_list_list_a @ B @ C3 ) )
& ~ ( member_list_list_a @ B @ C3 )
& ( B4
= ( insert_list_list_a @ A @ C3 ) )
& ~ ( member_list_list_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_387_insert__Collect,axiom,
! [A: list_a,P3: list_a > $o] :
( ( insert_list_a @ A @ ( collect_list_a @ P3 ) )
= ( collect_list_a
@ ^ [U: list_a] :
( ( U != A )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_388_insert__Collect,axiom,
! [A: a,P3: a > $o] :
( ( insert_a @ A @ ( collect_a @ P3 ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_389_insert__Collect,axiom,
! [A: list_list_a,P3: list_list_a > $o] :
( ( insert_list_list_a @ A @ ( collect_list_list_a @ P3 ) )
= ( collect_list_list_a
@ ^ [U: list_list_a] :
( ( U != A )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_390_insert__commute,axiom,
! [X2: a,Y: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_391_mk__disjoint__insert,axiom,
! [A: list_a,A3: set_list_a] :
( ( member_list_a @ A @ A3 )
=> ? [B5: set_list_a] :
( ( A3
= ( insert_list_a @ A @ B5 ) )
& ~ ( member_list_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_392_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B5: set_a] :
( ( A3
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_393_mk__disjoint__insert,axiom,
! [A: nat > a,A3: set_nat_a] :
( ( member_nat_a @ A @ A3 )
=> ? [B5: set_nat_a] :
( ( A3
= ( insert_nat_a @ A @ B5 ) )
& ~ ( member_nat_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_394_mk__disjoint__insert,axiom,
! [A: list_list_a,A3: set_list_list_a] :
( ( member_list_list_a @ A @ A3 )
=> ? [B5: set_list_list_a] :
( ( A3
= ( insert_list_list_a @ A @ B5 ) )
& ~ ( member_list_list_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_395_Collect__conv__if,axiom,
! [P3: list_list_a > $o,A: list_list_a] :
( ( ( P3 @ A )
=> ( ( collect_list_list_a
@ ^ [X: list_list_a] :
( ( X = A )
& ( P3 @ X ) ) )
= ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) )
& ( ~ ( P3 @ A )
=> ( ( collect_list_list_a
@ ^ [X: list_list_a] :
( ( X = A )
& ( P3 @ X ) ) )
= bot_bo1875519244922727510list_a ) ) ) ).
% Collect_conv_if
thf(fact_396_Collect__conv__if,axiom,
! [P3: a > $o,A: a] :
( ( ( P3 @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P3 @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P3 @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P3 @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_397_Collect__conv__if,axiom,
! [P3: list_a > $o,A: list_a] :
( ( ( P3 @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P3 @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P3 @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P3 @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_398_Collect__conv__if2,axiom,
! [P3: list_list_a > $o,A: list_list_a] :
( ( ( P3 @ A )
=> ( ( collect_list_list_a
@ ^ [X: list_list_a] :
( ( A = X )
& ( P3 @ X ) ) )
= ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) )
& ( ~ ( P3 @ A )
=> ( ( collect_list_list_a
@ ^ [X: list_list_a] :
( ( A = X )
& ( P3 @ X ) ) )
= bot_bo1875519244922727510list_a ) ) ) ).
% Collect_conv_if2
thf(fact_399_Collect__conv__if2,axiom,
! [P3: a > $o,A: a] :
( ( ( P3 @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P3 @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P3 @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P3 @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_400_Collect__conv__if2,axiom,
! [P3: list_a > $o,A: list_a] :
( ( ( P3 @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P3 @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P3 @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P3 @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_401_subset__singletonD,axiom,
! [A3: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ( A3 = bot_bot_set_list_a )
| ( A3
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_402_subset__singletonD,axiom,
! [A3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_403_subset__singleton__iff,axiom,
! [X5: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_404_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_405_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_406_subset__psubset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_407_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_408_psubset__subset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_409_psubset__imp__subset,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_410_Collect__mono__iff,axiom,
! [P3: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P3 ) @ ( collect_list_a @ Q ) )
= ( ! [X: list_a] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_411_Collect__mono__iff,axiom,
! [P3: list_list_a > $o,Q: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P3 ) @ ( collect_list_list_a @ Q ) )
= ( ! [X: list_list_a] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_412_Collect__mono__iff,axiom,
! [P3: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P3 ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_413_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_414_subset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_415_Collect__mono,axiom,
! [P3: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P3 ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_416_Collect__mono,axiom,
! [P3: list_list_a > $o,Q: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P3 ) @ ( collect_list_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_417_Collect__mono,axiom,
! [P3: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P3 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_418_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_419_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B6: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A5 )
=> ( member_list_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_420_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B6: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A5 )
=> ( member_nat_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_421_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B6: set_list_list_a] :
! [T: list_list_a] :
( ( member_list_list_a @ T @ A5 )
=> ( member_list_list_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_422_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_423_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_424_equalityD2,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_425_equalityD1,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_426_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B6: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A5 )
=> ( member_list_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_427_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B6: set_nat_a] :
! [X: nat > a] :
( ( member_nat_a @ X @ A5 )
=> ( member_nat_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_428_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B6: set_list_list_a] :
! [X: list_list_a] :
( ( member_list_list_a @ X @ A5 )
=> ( member_list_list_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_429_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_430_equalityE,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_431_psubsetE,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_432_subsetD,axiom,
! [A3: set_list_a,B4: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B4 )
=> ( ( member_list_a @ C @ A3 )
=> ( member_list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_433_subsetD,axiom,
! [A3: set_nat_a,B4: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A3 @ B4 )
=> ( ( member_nat_a @ C @ A3 )
=> ( member_nat_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_434_subsetD,axiom,
! [A3: set_list_list_a,B4: set_list_list_a,C: list_list_a] :
( ( ord_le8488217952732425610list_a @ A3 @ B4 )
=> ( ( member_list_list_a @ C @ A3 )
=> ( member_list_list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_435_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_436_in__mono,axiom,
! [A3: set_list_a,B4: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B4 )
=> ( ( member_list_a @ X2 @ A3 )
=> ( member_list_a @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_437_in__mono,axiom,
! [A3: set_nat_a,B4: set_nat_a,X2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A3 @ B4 )
=> ( ( member_nat_a @ X2 @ A3 )
=> ( member_nat_a @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_438_in__mono,axiom,
! [A3: set_list_list_a,B4: set_list_list_a,X2: list_list_a] :
( ( ord_le8488217952732425610list_a @ A3 @ B4 )
=> ( ( member_list_list_a @ X2 @ A3 )
=> ( member_list_list_a @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_439_in__mono,axiom,
! [A3: set_a,B4: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_440_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_441_emptyE,axiom,
! [A: list_list_a] :
~ ( member_list_list_a @ A @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_442_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_443_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_444_equals0D,axiom,
! [A3: set_nat_a,A: nat > a] :
( ( A3 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A3 ) ) ).
% equals0D
thf(fact_445_equals0D,axiom,
! [A3: set_list_list_a,A: list_list_a] :
( ( A3 = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A @ A3 ) ) ).
% equals0D
thf(fact_446_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_447_equals0D,axiom,
! [A3: set_list_a,A: list_a] :
( ( A3 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A3 ) ) ).
% equals0D
thf(fact_448_equals0I,axiom,
! [A3: set_nat_a] :
( ! [Y3: nat > a] :
~ ( member_nat_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_449_equals0I,axiom,
! [A3: set_list_list_a] :
( ! [Y3: list_list_a] :
~ ( member_list_list_a @ Y3 @ A3 )
=> ( A3 = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_450_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_451_equals0I,axiom,
! [A3: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_452_ex__in__conv,axiom,
! [A3: set_nat_a] :
( ( ? [X: nat > a] : ( member_nat_a @ X @ A3 ) )
= ( A3 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_453_ex__in__conv,axiom,
! [A3: set_list_list_a] :
( ( ? [X: list_list_a] : ( member_list_list_a @ X @ A3 ) )
= ( A3 != bot_bo1875519244922727510list_a ) ) ).
% ex_in_conv
thf(fact_454_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X: a] : ( member_a @ X @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_455_ex__in__conv,axiom,
! [A3: set_list_a] :
( ( ? [X: list_a] : ( member_list_a @ X @ A3 ) )
= ( A3 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_456_not__psubset__empty,axiom,
! [A3: set_list_a] :
~ ( ord_less_set_list_a @ A3 @ bot_bot_set_list_a ) ).
% not_psubset_empty
thf(fact_457_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_458_bot__set__def,axiom,
( bot_bo1875519244922727510list_a
= ( collect_list_list_a @ bot_bo4370453251804172807st_a_o ) ) ).
% bot_set_def
thf(fact_459_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_460_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_461_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_462_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_463_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 ) )
= ( ^ [I4: nat] : ( if_list_a @ ( I4 = N ) @ A @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_464_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 ) )
= ( ^ [I4: nat] : ( if_a @ ( I4 = N ) @ A @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_465_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B6: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A5 )
@ ^ [X: list_a] : ( member_list_a @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_466_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B6: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A5 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_467_less__eq__set__def,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B6: set_list_list_a] :
( ord_le1801313680655002067st_a_o
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ A5 )
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_468_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_469_Collect__subset,axiom,
! [A3: set_nat_a,P3: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A3 )
& ( P3 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_470_Collect__subset,axiom,
! [A3: set_list_a,P3: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A3 )
& ( P3 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_471_Collect__subset,axiom,
! [A3: set_list_list_a,P3: list_list_a > $o] :
( ord_le8488217952732425610list_a
@ ( collect_list_list_a
@ ^ [X: list_list_a] :
( ( member_list_list_a @ X @ A3 )
& ( P3 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_472_Collect__subset,axiom,
! [A3: set_a,P3: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
& ( P3 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_473_empty__def,axiom,
( bot_bo1875519244922727510list_a
= ( collect_list_list_a
@ ^ [X: list_list_a] : $false ) ) ).
% empty_def
thf(fact_474_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_475_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $false ) ) ).
% empty_def
thf(fact_476_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_477_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2670972154091845814t_unit,H: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_478_ring_Omonom__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ R @ A @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_479_ring_Omonom__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ R @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_480_ring_Omonom__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( monom_4043874212805408666t_unit @ R @ A @ N ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_481_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_482_bound_Ointro,axiom,
! [N: nat,F: nat > a,Z: a] :
( ! [M4: nat] :
( ( ord_less_nat @ N @ M4 )
=> ( ( F @ M4 )
= Z ) )
=> ( bound_a @ Z @ N @ F ) ) ).
% bound.intro
thf(fact_483_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_484_genideal__self_H,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( genideal_a_b @ r @ ( insert_a @ I2 @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_485_ring_Oexp__base__closed,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ R @ X2 @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_486_ring_Oexp__base__closed,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( polyno6819740552565085946t_unit @ R @ X2 @ N ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_487_ring_Oexp__base__closed,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ R @ X2 @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_488_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_489_factors__wfactors,axiom,
! [As: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ As @ A ) ) ) ).
% factors_wfactors
thf(fact_490_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( embedd1768981623711841426t_unit @ K3 @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_491_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_492_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2956882679547061052t_unit,K3: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ K3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( embedd1097489486847499020t_unit @ K3 @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_493_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_494_subset__Idl__subset,axiom,
! [I3: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I3 ) ) ) ) ).
% subset_Idl_subset
thf(fact_495_mem__upI,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit] :
( ! [N3: nat] : ( member_list_a @ ( F @ N3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ? [N4: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N4 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_496_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N3: nat] : ( member_a @ ( F @ N3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N4: nat] : ( bound_a @ ( zero_a_b @ R ) @ N4 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_497_mem__upI,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
( ! [N3: nat] : ( member_list_list_a @ ( F @ N3 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ? [N4: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N4 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_498_psubsetD,axiom,
! [A3: set_list_a,B4: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A3 @ B4 )
=> ( ( member_list_a @ C @ A3 )
=> ( member_list_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_499_psubsetD,axiom,
! [A3: set_nat_a,B4: set_nat_a,C: nat > a] :
( ( ord_less_set_nat_a @ A3 @ B4 )
=> ( ( member_nat_a @ C @ A3 )
=> ( member_nat_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_500_psubsetD,axiom,
! [A3: set_list_list_a,B4: set_list_list_a,C: list_list_a] :
( ( ord_le5338140678153942166list_a @ A3 @ B4 )
=> ( ( member_list_list_a @ C @ A3 )
=> ( member_list_list_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_501_psubsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_502_less__set__def,axiom,
( ord_less_set_list_a
= ( ^ [A5: set_list_a,B6: set_list_a] :
( ord_less_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A5 )
@ ^ [X: list_a] : ( member_list_a @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_503_less__set__def,axiom,
( ord_less_set_nat_a
= ( ^ [A5: set_nat_a,B6: set_nat_a] :
( ord_less_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A5 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_504_less__set__def,axiom,
( ord_le5338140678153942166list_a
= ( ^ [A5: set_list_list_a,B6: set_list_list_a] :
( ord_le3894349897559202247st_a_o
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ A5 )
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_505_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ord_less_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_506_psubset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_507_mem__upD,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit,N: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% mem_upD
thf(fact_508_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_509_mem__upD,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit,N: nat] :
( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) )
=> ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% mem_upD
thf(fact_510_ring_Obound__upD,axiom,
! [R: partia2670972154091845814t_unit,F: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N3 @ F ) ) ) ).
% ring.bound_upD
thf(fact_511_ring_Obound__upD,axiom,
! [R: partia2175431115845679010xt_a_b,F: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ R ) @ N3 @ F ) ) ) ).
% ring.bound_upD
thf(fact_512_ring_Osubset__Idl__subset,axiom,
! [R: partia2670972154091845814t_unit,I3: set_list_a,H2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ H2 ) @ ( genide3243992037924705879t_unit @ R @ I3 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_513_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I3: set_a,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H2 @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H2 ) @ ( genideal_a_b @ R @ I3 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_514_ring_Osubset__Idl__subset,axiom,
! [R: partia2956882679547061052t_unit,I3: set_list_list_a,H2: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ I3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ H2 @ I3 )
=> ( ord_le8488217952732425610list_a @ ( genide2671672708880404049t_unit @ R @ H2 ) @ ( genide2671672708880404049t_unit @ R @ I3 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_515_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_516_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_517_ring_Ogenideal__self,axiom,
! [R: partia2956882679547061052t_unit,S: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ S @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ S @ ( genide2671672708880404049t_unit @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_518_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_519_ring_Ogenideal__self_H,axiom,
! [R: partia2670972154091845814t_unit,I2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I2 @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ I2 @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_520_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I2 @ ( genideal_a_b @ R @ ( insert_a @ I2 @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_521_ring_Ogenideal__self_H,axiom,
! [R: partia2956882679547061052t_unit,I2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ I2 @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ I2 @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_522_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_523_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_524_principalideal_Ogenerate,axiom,
! [I3: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I3 @ R )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
& ( I3
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_525_principalideal_Ogenerate,axiom,
! [I3: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I3 @ R )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I3
= ( genideal_a_b @ R @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_526_principalideal_Ogenerate,axiom,
! [I3: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( princi2534607884127416211t_unit @ I3 @ R )
=> ? [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
& ( I3
= ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ X3 @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_527_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_528_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_529_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) )
= ( member_list_list_a @ A @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_530_bound__def,axiom,
( bound_a
= ( ^ [Z3: a,N2: nat,F3: nat > a] :
! [M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( F3 @ M )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_531_bound_Obound,axiom,
! [Z: a,N: nat,F: nat > a,M2: nat] :
( ( bound_a @ Z @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( ( F @ M2 )
= Z ) ) ) ).
% bound.bound
thf(fact_532_bound__below,axiom,
! [Z: a,M2: nat,F: nat > a,N: nat] :
( ( bound_a @ Z @ M2 @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M2 ) ) ) ).
% bound_below
thf(fact_533_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,V: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd1768981623711841426t_unit @ K3 @ V @ R )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_534_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ V @ R )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_535_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,K3: set_list_list_a,V: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( embedd1097489486847499020t_unit @ K3 @ V @ R )
=> ( ord_le8488217952732425610list_a @ V @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_536_up__minus__closed,axiom,
! [P: nat > a,Q2: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( a_minus_a_b @ r @ ( P @ I4 ) @ ( Q2 @ I4 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_537_wfactors__factors,axiom,
! [As: list_a,A: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A6: a] :
( ( factor5638265376665762323xt_a_b @ r @ As @ A6 )
& ( associ5860276527279195403xt_a_b @ r @ A6 @ A ) ) ) ) ).
% wfactors_factors
thf(fact_538_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_539_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_540_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_541_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_542_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_543_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_544_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_545_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_546_wfactors__cong__r,axiom,
! [Fs: list_a,A: a,A7: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ A7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A7 ) ) ) ) ) ) ).
% wfactors_cong_r
thf(fact_547_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_548_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_549_minus__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_550_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_551_ring__iso__memE_I4_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_552_ring_Oup__minus__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P: nat > a,Q2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P @ ( up_a_b @ R ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( a_minus_a_b @ R @ ( P @ I4 ) @ ( Q2 @ I4 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_minus_closed
thf(fact_553_ring_Ogenideal__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ R ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.genideal_one
thf(fact_554_ring_Ogenideal__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( one_a_ring_ext_a_b @ R ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.genideal_one
thf(fact_555_ring_Ogenideal__one,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( partia2464479390973590831t_unit @ R ) ) ) ).
% ring.genideal_one
thf(fact_556_up__one__closed,axiom,
( member_nat_a
@ ^ [N2: nat] : ( if_a @ ( N2 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_557_ring_Or__right__minus__eq,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 ) )
=> ( ( ( a_minu3984020753470702548t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_558_ring_Or__right__minus__eq,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 ) )
=> ( ( ( a_minus_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_559_ring_Or__right__minus__eq,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( a_minu2241224857956505934t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_560_a__lcos__m__assoc,axiom,
! [M3: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M3 @ ( 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 @ M3 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M3 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_561_line__extension__in__carrier,axiom,
! [K3: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_562_listassoc__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% listassoc_trans
thf(fact_563_listassoc__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ As ) ) ) ) ).
% listassoc_sym
thf(fact_564_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_565_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_566_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_567_a__assoc,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( 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 @ X2 @ Y ) @ Z )
= ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_568_a__comm,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ Y )
= ( add_a_b @ r @ Y @ X2 ) ) ) ) ).
% a_comm
thf(fact_569_a__lcomm,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_570_up__add__closed,axiom,
! [P: nat > a,Q2: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( add_a_b @ r @ ( P @ I4 ) @ ( Q2 @ I4 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_571_add_Oinv__comm,axiom,
! [X2: a,Y: a] :
( ( ( add_a_b @ r @ X2 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_572_add_Ol__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_573_add_Oone__unique,axiom,
! [U2: a] :
( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U2 @ X3 )
= X3 ) )
=> ( U2
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_574_add_Or__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_575_local_Ominus__unique,axiom,
! [Y: a,X2: a,Y6: a] :
( ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ Y6 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_576_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_577_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_578_local_Oadd_Oright__cancel,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X2 )
= ( add_a_b @ r @ Z @ X2 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_579_a__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_580_add_Ol__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ A )
= X2 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_581_add_Ol__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
= ( add_a_b @ r @ X2 @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_582_add_Or__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X2 )
= X2 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_583_add_Or__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
= ( add_a_b @ r @ A @ X2 ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_584_l__zero,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X2 )
= X2 ) ) ).
% l_zero
thf(fact_585_r__zero,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( zero_a_b @ r ) )
= X2 ) ) ).
% r_zero
thf(fact_586_listassoc__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ As ) ) ).
% listassoc_refl
thf(fact_587_ring_Oline__extension_Ocong,axiom,
embedd971793762689825387on_a_b = embedd971793762689825387on_a_b ).
% ring.line_extension.cong
thf(fact_588_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K3 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_589_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_590_ring__iso__memE_I3_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_591_ring__iso__memE_I3_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_592_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_593_ring__iso__memE_I3_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_594_ring__iso__memE_I3_J,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_595_ring__iso__memE_I3_J,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_596_ring_Oup__add__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P: nat > a,Q2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P @ ( up_a_b @ R ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( add_a_b @ R @ ( P @ I4 ) @ ( Q2 @ I4 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_add_closed
thf(fact_597_ring_Oup__add__closed,axiom,
! [R: partia2670972154091845814t_unit,P: nat > list_a,Q2: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_nat_list_a
@ ^ [I4: nat] : ( add_li7652885771158616974t_unit @ R @ ( P @ I4 ) @ ( Q2 @ I4 ) )
@ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ) ).
% ring.up_add_closed
thf(fact_598_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ nil_list_a )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_599_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_600_ring_Oline__extension__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,A: list_a,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ R @ K3 @ A @ E ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_601_ring_Oline__extension__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,K3: set_list_list_a,A: list_list_a,E: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ K3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ E @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( embedd3735808041618263277t_unit @ R @ K3 @ A @ E ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_602_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K3 @ A @ E ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_603_ring_Oup__one__closed,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_nat_list_a
@ ^ [N2: nat] : ( if_list_a @ ( N2 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ R ) @ ( zero_l4142658623432671053t_unit @ R ) )
@ ( up_lis8464167429055313730t_unit @ R ) ) ) ).
% ring.up_one_closed
thf(fact_604_ring_Oup__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_nat_a
@ ^ [N2: nat] : ( if_a @ ( N2 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ R ) @ ( zero_a_b @ R ) )
@ ( up_a_b @ R ) ) ) ).
% ring.up_one_closed
thf(fact_605_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_606_length__greater__0__conv,axiom,
! [Xs: list_nat_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5718426915756887103at_int @ Xs ) )
= ( Xs != nil_nat_int ) ) ).
% length_greater_0_conv
thf(fact_607_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_608_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_609_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_610_length__0__conv,axiom,
! [Xs: list_nat_int] :
( ( ( size_s5718426915756887103at_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat_int ) ) ).
% length_0_conv
thf(fact_611_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_612_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_613_set__empty,axiom,
! [Xs: list_int] :
( ( ( set_int2 @ Xs )
= bot_bot_set_int )
= ( Xs = nil_int ) ) ).
% set_empty
thf(fact_614_set__empty,axiom,
! [Xs: list_nat_int] :
( ( ( set_nat_int2 @ Xs )
= bot_bot_set_nat_int )
= ( Xs = nil_nat_int ) ) ).
% set_empty
thf(fact_615_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_616_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_617_set__empty2,axiom,
! [Xs: list_int] :
( ( bot_bot_set_int
= ( set_int2 @ Xs ) )
= ( Xs = nil_int ) ) ).
% set_empty2
thf(fact_618_set__empty2,axiom,
! [Xs: list_nat_int] :
( ( bot_bot_set_nat_int
= ( set_nat_int2 @ Xs ) )
= ( Xs = nil_nat_int ) ) ).
% set_empty2
thf(fact_619_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_620_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_621_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_622_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_623_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_624_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_625_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_626_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_627_size__neq__size__imp__neq,axiom,
! [X2: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X2 )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_628_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_629_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_630_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_631_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_632_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_633_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_634_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_635_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_636_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_637_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P3 @ M5 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_638_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P3 @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P3 @ M5 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_639_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_640_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B: nat] :
( ( P3 @ K )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P3 @ X3 )
& ! [Y2: nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_641_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_642_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_643_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_644_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_645_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_646_subset__code_I1_J,axiom,
! [Xs: list_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B4 )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_647_subset__code_I1_J,axiom,
! [Xs: list_nat_a,B4: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( set_nat_a2 @ Xs ) @ B4 )
= ( ! [X: nat > a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ( member_nat_a @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_648_subset__code_I1_J,axiom,
! [Xs: list_list_list_a,B4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs ) @ B4 )
= ( ! [X: list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ( member_list_list_a @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_649_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_650_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P3 @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P3 @ M5 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_651_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_652_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_653_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_654_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_655_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_656_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_657_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_658_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_659_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_660_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_661_length__induct,axiom,
! [P3: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P3 @ Ys2 ) )
=> ( P3 @ Xs2 ) )
=> ( P3 @ Xs ) ) ).
% length_induct
thf(fact_662_length__induct,axiom,
! [P3: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys2: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys2 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P3 @ Ys2 ) )
=> ( P3 @ Xs2 ) )
=> ( P3 @ Xs ) ) ).
% length_induct
thf(fact_663_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_664_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_665_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_666_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_667_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_668_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_669_list_Orel__cong,axiom,
! [X2: list_a,Ya: list_a,Y: list_a,Xa: list_a,R: a > a > $o,Ra: a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb: a] :
( ( member_a @ Z4 @ ( set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2_a_a @ R @ X2 @ Y )
= ( list_all2_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_670_list_Orel__cong,axiom,
! [X2: list_list_a,Ya: list_list_a,Y: list_a,Xa: list_a,R: list_a > a > $o,Ra: list_a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: list_a,Yb: a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2_list_a_a @ R @ X2 @ Y )
= ( list_all2_list_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_671_list_Orel__cong,axiom,
! [X2: list_a,Ya: list_a,Y: list_list_a,Xa: list_list_a,R: a > list_a > $o,Ra: a > list_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb: list_a] :
( ( member_a @ Z4 @ ( set_a2 @ Ya ) )
=> ( ( member_list_a @ Yb @ ( set_list_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2_a_list_a @ R @ X2 @ Y )
= ( list_all2_a_list_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_672_list_Orel__cong,axiom,
! [X2: list_list_a,Ya: list_list_a,Y: list_list_a,Xa: list_list_a,R: list_a > list_a > $o,Ra: list_a > list_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: list_a,Yb: list_a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ Ya ) )
=> ( ( member_list_a @ Yb @ ( set_list_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a3802133873445908231list_a @ R @ X2 @ Y )
= ( list_a3802133873445908231list_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_673_list_Orel__cong,axiom,
! [X2: list_nat_a,Ya: list_nat_a,Y: list_a,Xa: list_a,R: ( nat > a ) > a > $o,Ra: ( nat > a ) > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: nat > a,Yb: a] :
( ( member_nat_a @ Z4 @ ( set_nat_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2_nat_a_a @ R @ X2 @ Y )
= ( list_all2_nat_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_674_list_Orel__cong,axiom,
! [X2: list_list_list_a,Ya: list_list_list_a,Y: list_a,Xa: list_a,R: list_list_a > a > $o,Ra: list_list_a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: list_list_a,Yb: a] :
( ( member_list_list_a @ Z4 @ ( set_list_list_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a2339687664058784123st_a_a @ R @ X2 @ Y )
= ( list_a2339687664058784123st_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_675_list_Orel__cong,axiom,
! [X2: list_a,Ya: list_a,Y: list_nat_a,Xa: list_nat_a,R: a > ( nat > a ) > $o,Ra: a > ( nat > a ) > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb: nat > a] :
( ( member_a @ Z4 @ ( set_a2 @ Ya ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2_a_nat_a @ R @ X2 @ Y )
= ( list_all2_a_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_676_list_Orel__cong,axiom,
! [X2: list_a,Ya: list_a,Y: list_list_list_a,Xa: list_list_list_a,R: a > list_list_a > $o,Ra: a > list_list_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb: list_list_a] :
( ( member_a @ Z4 @ ( set_a2 @ Ya ) )
=> ( ( member_list_list_a @ Yb @ ( set_list_list_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a1736694032391712403list_a @ R @ X2 @ Y )
= ( list_a1736694032391712403list_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_677_list_Orel__cong,axiom,
! [X2: list_list_a,Ya: list_list_a,Y: list_nat_a,Xa: list_nat_a,R: list_a > ( nat > a ) > $o,Ra: list_a > ( nat > a ) > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: list_a,Yb: nat > a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ Ya ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a1636186360766657384_nat_a @ R @ X2 @ Y )
= ( list_a1636186360766657384_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_678_list_Orel__cong,axiom,
! [X2: list_list_a,Ya: list_list_a,Y: list_list_list_a,Xa: list_list_list_a,R: list_a > list_list_a > $o,Ra: list_a > list_list_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: list_a,Yb: list_list_a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ Ya ) )
=> ( ( member_list_list_a @ Yb @ ( set_list_list_a2 @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a1787013445579514253list_a @ R @ X2 @ Y )
= ( list_a1787013445579514253list_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_679_list_Orel__mono__strong,axiom,
! [R: a > a > $o,X2: list_a,Y: list_a,Ra: a > a > $o] :
( ( list_all2_a_a @ R @ X2 @ Y )
=> ( ! [Z4: a,Yb: a] :
( ( member_a @ Z4 @ ( set_a2 @ X2 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2_a_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_680_list_Orel__mono__strong,axiom,
! [R: list_a > a > $o,X2: list_list_a,Y: list_a,Ra: list_a > a > $o] :
( ( list_all2_list_a_a @ R @ X2 @ Y )
=> ( ! [Z4: list_a,Yb: a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ X2 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2_list_a_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_681_list_Orel__mono__strong,axiom,
! [R: a > list_a > $o,X2: list_a,Y: list_list_a,Ra: a > list_a > $o] :
( ( list_all2_a_list_a @ R @ X2 @ Y )
=> ( ! [Z4: a,Yb: list_a] :
( ( member_a @ Z4 @ ( set_a2 @ X2 ) )
=> ( ( member_list_a @ Yb @ ( set_list_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2_a_list_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_682_list_Orel__mono__strong,axiom,
! [R: list_a > list_a > $o,X2: list_list_a,Y: list_list_a,Ra: list_a > list_a > $o] :
( ( list_a3802133873445908231list_a @ R @ X2 @ Y )
=> ( ! [Z4: list_a,Yb: list_a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ X2 ) )
=> ( ( member_list_a @ Yb @ ( set_list_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a3802133873445908231list_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_683_list_Orel__mono__strong,axiom,
! [R: ( nat > a ) > a > $o,X2: list_nat_a,Y: list_a,Ra: ( nat > a ) > a > $o] :
( ( list_all2_nat_a_a @ R @ X2 @ Y )
=> ( ! [Z4: nat > a,Yb: a] :
( ( member_nat_a @ Z4 @ ( set_nat_a2 @ X2 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2_nat_a_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_684_list_Orel__mono__strong,axiom,
! [R: list_list_a > a > $o,X2: list_list_list_a,Y: list_a,Ra: list_list_a > a > $o] :
( ( list_a2339687664058784123st_a_a @ R @ X2 @ Y )
=> ( ! [Z4: list_list_a,Yb: a] :
( ( member_list_list_a @ Z4 @ ( set_list_list_a2 @ X2 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a2339687664058784123st_a_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_685_list_Orel__mono__strong,axiom,
! [R: a > ( nat > a ) > $o,X2: list_a,Y: list_nat_a,Ra: a > ( nat > a ) > $o] :
( ( list_all2_a_nat_a @ R @ X2 @ Y )
=> ( ! [Z4: a,Yb: nat > a] :
( ( member_a @ Z4 @ ( set_a2 @ X2 ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2_a_nat_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_686_list_Orel__mono__strong,axiom,
! [R: a > list_list_a > $o,X2: list_a,Y: list_list_list_a,Ra: a > list_list_a > $o] :
( ( list_a1736694032391712403list_a @ R @ X2 @ Y )
=> ( ! [Z4: a,Yb: list_list_a] :
( ( member_a @ Z4 @ ( set_a2 @ X2 ) )
=> ( ( member_list_list_a @ Yb @ ( set_list_list_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a1736694032391712403list_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_687_list_Orel__mono__strong,axiom,
! [R: list_a > ( nat > a ) > $o,X2: list_list_a,Y: list_nat_a,Ra: list_a > ( nat > a ) > $o] :
( ( list_a1636186360766657384_nat_a @ R @ X2 @ Y )
=> ( ! [Z4: list_a,Yb: nat > a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ X2 ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a1636186360766657384_nat_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_688_list_Orel__mono__strong,axiom,
! [R: list_a > list_list_a > $o,X2: list_list_a,Y: list_list_list_a,Ra: list_a > list_list_a > $o] :
( ( list_a1787013445579514253list_a @ R @ X2 @ Y )
=> ( ! [Z4: list_a,Yb: list_list_a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ X2 ) )
=> ( ( member_list_list_a @ Yb @ ( set_list_list_a2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a1787013445579514253list_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_689_list_Orel__refl__strong,axiom,
! [X2: list_list_a,Ra: list_a > list_a > $o] :
( ! [Z4: list_a] :
( ( member_list_a @ Z4 @ ( set_list_a2 @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_a3802133873445908231list_a @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_690_list_Orel__refl__strong,axiom,
! [X2: list_nat_a,Ra: ( nat > a ) > ( nat > a ) > $o] :
( ! [Z4: nat > a] :
( ( member_nat_a @ Z4 @ ( set_nat_a2 @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_a9087535860334789575_nat_a @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_691_list_Orel__refl__strong,axiom,
! [X2: list_list_list_a,Ra: list_list_a > list_list_a > $o] :
( ! [Z4: list_list_a] :
( ( member_list_list_a @ Z4 @ ( set_list_list_a2 @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_a6511221006964818183list_a @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_692_list_Orel__refl__strong,axiom,
! [X2: list_a,Ra: a > a > $o] :
( ! [Z4: a] :
( ( member_a @ Z4 @ ( set_a2 @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_all2_a_a @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_693_list__all2__same,axiom,
! [P3: a > a > $o,Xs: list_a] :
( ( list_all2_a_a @ P3 @ Xs @ Xs )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P3 @ X @ X ) ) ) ) ).
% list_all2_same
thf(fact_694_list__all2__lengthD,axiom,
! [P3: a > a > $o,Xs: list_a,Ys: list_a] :
( ( list_all2_a_a @ P3 @ Xs @ Ys )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_695_list__all2__lengthD,axiom,
! [P3: a > list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( list_all2_a_list_a @ P3 @ Xs @ Ys )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_696_list__all2__lengthD,axiom,
! [P3: list_a > a > $o,Xs: list_list_a,Ys: list_a] :
( ( list_all2_list_a_a @ P3 @ Xs @ Ys )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_697_list__all2__lengthD,axiom,
! [P3: list_a > list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( list_a3802133873445908231list_a @ P3 @ Xs @ Ys )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_698_empty__set,axiom,
( bot_bot_set_int
= ( set_int2 @ nil_int ) ) ).
% empty_set
thf(fact_699_empty__set,axiom,
( bot_bot_set_nat_int
= ( set_nat_int2 @ nil_nat_int ) ) ).
% empty_set
thf(fact_700_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_701_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_702_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_703_list_Osize_I3_J,axiom,
( ( size_s5718426915756887103at_int @ nil_nat_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_704_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_705_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_706_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P3 @ I5 ) )
& ( P3 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_707_length__pos__if__in__set,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_708_length__pos__if__in__set,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s2403821588304063868list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_709_length__pos__if__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_710_length__pos__if__in__set,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_711_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_712_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_713_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_714_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( add_a_b @ r @ X3 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_715_poly__add__coeff,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( coeff_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I4 ) @ ( coeff_a_b @ r @ P2 @ I4 ) ) ) ) ) ) ).
% poly_add_coeff
thf(fact_716_ring_Oring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ ( zero_l4142658623432671053t_unit @ R ) )
= X2 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_717_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ ( zero_a_b @ R ) )
= X2 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_718_ring_Oring__simprules_I15_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X2 @ ( zero_l347298301471573063t_unit @ R ) )
= X2 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_719_up__a__inv__closed,axiom,
! [P: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( a_inv_a_b @ r @ ( P @ I4 ) )
@ ( up_a_b @ r ) ) ) ).
% up_a_inv_closed
thf(fact_720_add_Oinv__mult__group,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X2 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_721_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_722_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_723_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_724_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_725_a__transpose__inv,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X2 @ Y )
= Z )
=> ( ( member_a @ X2 @ ( 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 @ X2 ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_726_local_Ominus__add,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_727_r__neg1,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ ( add_a_b @ r @ X2 @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_728_r__neg2,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_729_minus__eq,axiom,
! [X2: a,Y: a] :
( ( a_minus_a_b @ r @ X2 @ Y )
= ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_730_l__neg,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ X2 )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_731_minus__equality,axiom,
! [Y: a,X2: a] :
( ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X2 )
= Y ) ) ) ) ).
% minus_equality
thf(fact_732_r__neg,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ X2 ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_733_poly__add__comm,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ P2 @ P1 ) ) ) ) ).
% poly_add_comm
thf(fact_734_poly__add__in__carrier,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P2 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_735_poly__add__coeff__aux,axiom,
! [P2: list_a,P1: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P2 ) @ ( size_size_list_a @ P1 ) )
=> ( ( coeff_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I4 ) @ ( coeff_a_b @ r @ P2 @ I4 ) ) ) ) ) ).
% poly_add_coeff_aux
thf(fact_736_a__inv__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_737_local_Ominus__minus,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X2 ) )
= X2 ) ) ).
% local.minus_minus
thf(fact_738_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_739_add_Oinv__eq__1__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X2 )
= ( zero_a_b @ r ) )
= ( X2
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_740_ring_Opoly__add_Ocong,axiom,
poly_add_a_b = poly_add_a_b ).
% ring.poly_add.cong
thf(fact_741_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_742_univ__poly__add,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) )
= ( poly_a7601779127272115787t_unit @ R ) ) ).
% univ_poly_add
thf(fact_743_univ__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) )
= ( poly_add_a_b @ R ) ) ).
% univ_poly_add
thf(fact_744_ring_Oring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_745_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_746_ring_Oring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_747_ring_Oring__simprules_I20_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) )
= X2 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_748_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X2 ) )
= X2 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_749_ring_Oring__simprules_I20_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) )
= X2 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_750_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_751_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_752_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X: a,Y4: a] : ( add_a_b @ R2 @ X @ ( a_inv_a_b @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_753_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X: list_a,Y4: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X @ ( a_inv_8944721093294617173t_unit @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_754_ring_Oring__simprules_I19_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_755_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X2 ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_756_ring_Oring__simprules_I19_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) @ ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_757_ring_Oring__simprules_I18_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_758_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X2 ) @ ( add_a_b @ R @ X2 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_759_ring_Oring__simprules_I18_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_760_ring_Oring__simprules_I17_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_761_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_762_ring_Oring__simprules_I17_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_763_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X2 @ Y )
= ( add_a_b @ R @ X2 @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_764_ring_Oring__simprules_I14_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_minu3984020753470702548t_unit @ R @ X2 @ Y )
= ( add_li7652885771158616974t_unit @ R @ X2 @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_765_ring_Oup__a__inv__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( a_inv_a_b @ R @ ( P @ I4 ) )
@ ( up_a_b @ R ) ) ) ) ).
% ring.up_a_inv_closed
thf(fact_766_ring_Oring__simprules_I16_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_767_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ ( a_inv_a_b @ R @ X2 ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_768_ring_Oring__simprules_I16_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X2 @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_769_ring_Oring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_770_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X2 ) @ X2 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_771_ring_Oring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) @ X2 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_772_ring_Oconst__term__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( P != nil_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_773_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P )
!= ( zero_a_b @ R ) )
=> ( P != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_774_ring_Opoly__add__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.poly_add_in_carrier
thf(fact_775_ring_Opoly__add__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ R @ P1 @ P2 ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.poly_add_in_carrier
thf(fact_776_ring_Opoly__add__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P2: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( poly_a7341706734723628101t_unit @ R @ P1 @ P2 ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.poly_add_in_carrier
thf(fact_777_ring_Opoly__add__comm,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 )
= ( poly_a7601779127272115787t_unit @ R @ P2 @ P1 ) ) ) ) ) ).
% ring.poly_add_comm
thf(fact_778_ring_Opoly__add__comm,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P1 @ P2 )
= ( poly_add_a_b @ R @ P2 @ P1 ) ) ) ) ) ).
% ring.poly_add_comm
thf(fact_779_ring_Opoly__add__comm,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P2: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ P1 @ P2 )
= ( poly_a7341706734723628101t_unit @ R @ P2 @ P1 ) ) ) ) ) ).
% ring.poly_add_comm
thf(fact_780_ring_Opoly__add__coeff__aux,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,P1: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P2 ) @ ( size_s349497388124573686list_a @ P1 ) )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_li7652885771158616974t_unit @ R @ ( coeff_6360649920519955023t_unit @ R @ P1 @ I4 ) @ ( coeff_6360649920519955023t_unit @ R @ P2 @ I4 ) ) ) ) ) ) ).
% ring.poly_add_coeff_aux
thf(fact_781_ring_Opoly__add__coeff__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,P1: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ P2 ) @ ( size_size_list_a @ P1 ) )
=> ( ( coeff_a_b @ R @ ( poly_add_a_b @ R @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_a_b @ R @ ( coeff_a_b @ R @ P1 @ I4 ) @ ( coeff_a_b @ R @ P2 @ I4 ) ) ) ) ) ) ).
% ring.poly_add_coeff_aux
thf(fact_782_ring_Opoly__add__coeff,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_li7652885771158616974t_unit @ R @ ( coeff_6360649920519955023t_unit @ R @ P1 @ I4 ) @ ( coeff_6360649920519955023t_unit @ R @ P2 @ I4 ) ) ) ) ) ) ) ).
% ring.poly_add_coeff
thf(fact_783_ring_Opoly__add__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( coeff_a_b @ R @ ( poly_add_a_b @ R @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_a_b @ R @ ( coeff_a_b @ R @ P1 @ I4 ) @ ( coeff_a_b @ R @ P2 @ I4 ) ) ) ) ) ) ) ).
% ring.poly_add_coeff
thf(fact_784_ring_Opoly__add__coeff,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P2: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( coeff_1681977662540381769t_unit @ R @ ( poly_a7341706734723628101t_unit @ R @ P1 @ P2 ) )
= ( ^ [I4: nat] : ( add_li174743652000525320t_unit @ R @ ( coeff_1681977662540381769t_unit @ R @ P1 @ I4 ) @ ( coeff_1681977662540381769t_unit @ R @ P2 @ I4 ) ) ) ) ) ) ) ).
% ring.poly_add_coeff
thf(fact_785_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_786_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_787_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_788_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_789_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_790_ring__hom__closed,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_791_ring__hom__closed,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_792_ring__hom__closed,axiom,
! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X2: list_a] :
( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_793_ring__hom__closed,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_794_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_795_ring__hom__closed,axiom,
! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X2: a] :
( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_796_ring__hom__closed,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_797_ring__hom__closed,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_798_ring__hom__closed,axiom,
! [H: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X2: list_list_a] :
( ( member8231385768148312316list_a @ H @ ( ring_h8129544334414776832t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_799_ring__hom__one,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_800_ring_Ounfold__congs_I4_J,axiom,
! [R3: partia2175431115845679010xt_a_b,R4: partia2175431115845679010xt_a_b,V2: a,F: a > a,F4: a > a] :
( ( R3 = R4 )
=> ( ( ( zero_a_b @ R4 )
= V2 )
=> ( ! [V3: a] :
( ( V3 = V2 )
=> ( ( F @ V3 )
= ( F4 @ V3 ) ) )
=> ( ( zero_update_a_b @ F @ R3 )
= ( zero_update_a_b @ F4 @ R4 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_801_ring_Ounfold__congs_I4_J,axiom,
! [R3: partia2670972154091845814t_unit,R4: partia2670972154091845814t_unit,V2: list_a,F: list_a > list_a,F4: list_a > list_a] :
( ( R3 = R4 )
=> ( ( ( zero_l4142658623432671053t_unit @ R4 )
= V2 )
=> ( ! [V3: list_a] :
( ( V3 = V2 )
=> ( ( F @ V3 )
= ( F4 @ V3 ) ) )
=> ( ( zero_u1196785550890449590t_unit @ F @ R3 )
= ( zero_u1196785550890449590t_unit @ F4 @ R4 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_802_ring_Ofold__congs_I4_J,axiom,
! [R3: partia2175431115845679010xt_a_b,R4: partia2175431115845679010xt_a_b,V2: a,F: a > a,F4: a > a] :
( ( R3 = R4 )
=> ( ( ( zero_a_b @ R4 )
= V2 )
=> ( ! [V3: a] :
( ( V2 = V3 )
=> ( ( F @ V3 )
= ( F4 @ V3 ) ) )
=> ( ( zero_update_a_b @ F @ R3 )
= ( zero_update_a_b @ F4 @ R4 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_803_ring_Ofold__congs_I4_J,axiom,
! [R3: partia2670972154091845814t_unit,R4: partia2670972154091845814t_unit,V2: list_a,F: list_a > list_a,F4: list_a > list_a] :
( ( R3 = R4 )
=> ( ( ( zero_l4142658623432671053t_unit @ R4 )
= V2 )
=> ( ! [V3: list_a] :
( ( V2 = V3 )
=> ( ( F @ V3 )
= ( F4 @ V3 ) ) )
=> ( ( zero_u1196785550890449590t_unit @ F @ R3 )
= ( zero_u1196785550890449590t_unit @ F4 @ R4 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_804_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_805_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_806_ring_Oring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_807_ring_Oring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_808_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_809_ring_Oring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_810_ring_Oring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( 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 @ X2 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_811_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( 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 @ X2 @ Y ) @ Z )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_812_ring_Oring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_813_ring_Oring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X2 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_814_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y )
= ( add_a_b @ R @ Y @ X2 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_815_ring_Oring__simprules_I10_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X2 @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X2 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_816_ring_Oring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_817_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_818_ring_Oring__simprules_I22_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_819_ring_Oring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_820_ring_Oring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_821_ring_Oring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_822_ring__hom__add,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_823_ring__hom__add,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_824_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_825_ring__hom__add,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_826_ring__hom__add,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_827_ring__hom__add,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_828_ring_Oring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_829_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_830_ring_Oring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ R @ X2 @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_831_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_832_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_833_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_834_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_835_ring_Oring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_836_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_837_ring_Oring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_838_eval__poly__add__aux,axiom,
! [P: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q2 ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_839_eval__poly__add,axiom,
! [P: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q2 ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_840_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_841_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_842_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_843_Units__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_844_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_845_normalize__coeff,axiom,
! [P: list_a] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_coeff
thf(fact_846_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_847_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_848_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_849_unit__wfactors,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ nil_a @ A ) ) ).
% unit_wfactors
thf(fact_850_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_851_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_852_eval__in__carrier,axiom,
! [P: list_a,X2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_853_poly__add__normalize__aux,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P2 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_854_poly__add__normalize_I2_J,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P2 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_855_poly__add__normalize_I3_J,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P2 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_856_eval__normalize,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_857_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_858_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_859_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_860_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_861_ring_Oeval__normalize,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) @ A )
= ( eval_l34571156754992824t_unit @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_862_ring_Oeval__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( normalize_a_b @ R @ P ) @ A )
= ( eval_a_b @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_863_ring_Oeval__normalize,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,A: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( normal1297324897130370429t_unit @ R @ P ) @ A )
= ( eval_l1088911609197519410t_unit @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_864_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ nil_a )
= nil_a ) ) ).
% ring.normalize.simps(1)
thf(fact_865_ring_Onormalize__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ P )
= ( coeff_a_b @ R @ ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.normalize_coeff
thf(fact_866_ring_OUnits__minus__one__closed,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) @ ( units_2932844235741507942t_unit @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_867_ring_OUnits__minus__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) @ ( units_a_ring_ext_a_b @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_868_ring_Onormalize__length__le,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) @ ( size_size_list_a @ P ) ) ) ).
% ring.normalize_length_le
thf(fact_869_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( eval_l34571156754992824t_unit @ R @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_870_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( eval_a_b @ R @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_871_ring_Oconst__term__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( const_6738166269504826821t_unit @ R @ P )
= ( eval_l34571156754992824t_unit @ R @ P @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_872_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P )
= ( eval_a_b @ R @ P @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_873_ring_Onormalize__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_874_ring_Onormalize__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ R @ P ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_875_ring_Onormalize__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( normal1297324897130370429t_unit @ R @ P ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_876_ring_Oeval__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P @ X2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_877_ring_Oeval__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_878_ring_Oeval__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( eval_l1088911609197519410t_unit @ R @ P @ X2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_879_ring_Opoly__add__normalize_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 )
= ( poly_a7601779127272115787t_unit @ R @ ( normal637505603836502915t_unit @ R @ P1 ) @ ( normal637505603836502915t_unit @ R @ P2 ) ) ) ) ) ) ).
% ring.poly_add_normalize(3)
thf(fact_880_ring_Opoly__add__normalize_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P1 @ P2 )
= ( poly_add_a_b @ R @ ( normalize_a_b @ R @ P1 ) @ ( normalize_a_b @ R @ P2 ) ) ) ) ) ) ).
% ring.poly_add_normalize(3)
thf(fact_881_ring_Opoly__add__normalize_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P2: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ P1 @ P2 )
= ( poly_a7341706734723628101t_unit @ R @ ( normal1297324897130370429t_unit @ R @ P1 ) @ ( normal1297324897130370429t_unit @ R @ P2 ) ) ) ) ) ) ).
% ring.poly_add_normalize(3)
thf(fact_882_ring_Opoly__add__normalize_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 )
= ( poly_a7601779127272115787t_unit @ R @ P1 @ ( normal637505603836502915t_unit @ R @ P2 ) ) ) ) ) ) ).
% ring.poly_add_normalize(2)
thf(fact_883_ring_Opoly__add__normalize_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P1 @ P2 )
= ( poly_add_a_b @ R @ P1 @ ( normalize_a_b @ R @ P2 ) ) ) ) ) ) ).
% ring.poly_add_normalize(2)
thf(fact_884_ring_Opoly__add__normalize_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P2: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ P1 @ P2 )
= ( poly_a7341706734723628101t_unit @ R @ P1 @ ( normal1297324897130370429t_unit @ R @ P2 ) ) ) ) ) ) ).
% ring.poly_add_normalize(2)
thf(fact_885_ring_Opoly__add__normalize__aux,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P2 )
= ( poly_a7601779127272115787t_unit @ R @ ( normal637505603836502915t_unit @ R @ P1 ) @ P2 ) ) ) ) ) ).
% ring.poly_add_normalize_aux
thf(fact_886_ring_Opoly__add__normalize__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P1 @ P2 )
= ( poly_add_a_b @ R @ ( normalize_a_b @ R @ P1 ) @ P2 ) ) ) ) ) ).
% ring.poly_add_normalize_aux
thf(fact_887_ring_Opoly__add__normalize__aux,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P2: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ P1 @ P2 )
= ( poly_a7341706734723628101t_unit @ R @ ( normal1297324897130370429t_unit @ R @ P1 ) @ P2 ) ) ) ) ) ).
% ring.poly_add_normalize_aux
thf(fact_888_ring_Opoly__add__zero_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P @ nil_list_a )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(1)
thf(fact_889_ring_Opoly__add__zero_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P @ nil_a )
= ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(1)
thf(fact_890_ring_Opoly__add__zero_H_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ P @ nil_list_list_a )
= ( normal1297324897130370429t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(1)
thf(fact_891_ring_Opoly__add__zero_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ nil_list_a @ P )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(2)
thf(fact_892_ring_Opoly__add__zero_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ nil_a @ P )
= ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(2)
thf(fact_893_ring_Opoly__add__zero_H_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ nil_list_list_a @ P )
= ( normal1297324897130370429t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(2)
thf(fact_894_ring_Oeval__poly__add,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q2: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P @ Q2 ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( eval_l34571156754992824t_unit @ R @ Q2 @ A ) ) ) ) ) ) ) ).
% ring.eval_poly_add
thf(fact_895_ring_Oeval__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q2: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_add_a_b @ R @ P @ Q2 ) @ A )
= ( add_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( eval_a_b @ R @ Q2 @ A ) ) ) ) ) ) ) ).
% ring.eval_poly_add
thf(fact_896_ring_Oeval__poly__add,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q2: list_list_list_a,A: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( poly_a7341706734723628101t_unit @ R @ P @ Q2 ) @ A )
= ( add_li174743652000525320t_unit @ R @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) @ ( eval_l1088911609197519410t_unit @ R @ Q2 @ A ) ) ) ) ) ) ) ).
% ring.eval_poly_add
thf(fact_897_ring_Oeval__poly__add__aux,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q2: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P @ Q2 ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( eval_l34571156754992824t_unit @ R @ Q2 @ A ) ) ) ) ) ) ) ) ).
% ring.eval_poly_add_aux
thf(fact_898_ring_Oeval__poly__add__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q2: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_add_a_b @ R @ P @ Q2 ) @ A )
= ( add_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( eval_a_b @ R @ Q2 @ A ) ) ) ) ) ) ) ) ).
% ring.eval_poly_add_aux
thf(fact_899_ring_Oeval__poly__add__aux,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q2: list_list_list_a,A: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( size_s2403821588304063868list_a @ P )
= ( size_s2403821588304063868list_a @ Q2 ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( poly_a7341706734723628101t_unit @ R @ P @ Q2 ) @ A )
= ( add_li174743652000525320t_unit @ R @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) @ ( eval_l1088911609197519410t_unit @ R @ Q2 @ A ) ) ) ) ) ) ) ) ).
% ring.eval_poly_add_aux
thf(fact_900_is__root__def,axiom,
! [P: list_a,X2: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
= ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X2 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_901_eval__var,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X2 )
= X2 ) ) ).
% eval_var
thf(fact_902_poly__add__replicate__zero_H_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_903_poly__add__replicate__zero_H_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_904_normalize__length__lt,axiom,
! [P: list_a] :
( ( ( hd_a @ P )
= ( zero_a_b @ r ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ P ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ) ) ).
% normalize_length_lt
thf(fact_905_normalize__length__eq,axiom,
! [P: list_a] :
( ( ( hd_a @ P )
!= ( zero_a_b @ r ) )
=> ( ( size_size_list_a @ ( normalize_a_b @ r @ P ) )
= ( size_size_list_a @ P ) ) ) ).
% normalize_length_eq
thf(fact_906_normalize__lead__coeff,axiom,
! [P: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) )
=> ( ( hd_a @ P )
= ( zero_a_b @ r ) ) ) ).
% normalize_lead_coeff
thf(fact_907_length__replicate,axiom,
! [N: nat,X2: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_908_length__replicate,axiom,
! [N: nat,X2: list_a] :
( ( size_s349497388124573686list_a @ ( replicate_list_a @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_909_Ball__set__replicate,axiom,
! [N: nat,A: a,P3: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
=> ( P3 @ X ) ) )
= ( ( P3 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_910_Bex__set__replicate,axiom,
! [N: nat,A: a,P3: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
& ( P3 @ X ) ) )
= ( ( P3 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_911_in__set__replicate,axiom,
! [X2: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( replicate_list_a @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_912_in__set__replicate,axiom,
! [X2: nat > a,N: nat,Y: nat > a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ ( replicate_nat_a @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_913_in__set__replicate,axiom,
! [X2: list_list_a,N: nat,Y: list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ ( replic3997036819131463498list_a @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_914_in__set__replicate,axiom,
! [X2: a,N: nat,Y: a] :
( ( member_a @ X2 @ ( set_a2 @ ( replicate_a @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_915_replicate__zero__coeff,axiom,
! [N: nat] :
( ( coeff_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% replicate_zero_coeff
thf(fact_916_set__replicate,axiom,
! [N: nat,X2: a] :
( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X2 ) )
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% set_replicate
thf(fact_917_set__replicate,axiom,
! [N: nat,X2: list_a] :
( ( N != zero_zero_nat )
=> ( ( set_list_a2 @ ( replicate_list_a @ N @ X2 ) )
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ).
% set_replicate
thf(fact_918_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_919_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_920_hd__in__set,axiom,
! [Xs: list_nat_a] :
( ( Xs != nil_nat_a )
=> ( member_nat_a @ ( hd_nat_a @ Xs ) @ ( set_nat_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_921_hd__in__set,axiom,
! [Xs: list_list_list_a] :
( ( Xs != nil_list_list_a )
=> ( member_list_list_a @ ( hd_list_list_a @ Xs ) @ ( set_list_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_922_hd__in__set,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( member_int @ ( hd_int @ Xs ) @ ( set_int2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_923_hd__in__set,axiom,
! [Xs: list_nat_int] :
( ( Xs != nil_nat_int )
=> ( member_nat_int @ ( hd_nat_int @ Xs ) @ ( set_nat_int2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_924_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_925_list_Oset__sel_I1_J,axiom,
! [A: list_list_a] :
( ( A != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A ) @ ( set_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_926_list_Oset__sel_I1_J,axiom,
! [A: list_nat_a] :
( ( A != nil_nat_a )
=> ( member_nat_a @ ( hd_nat_a @ A ) @ ( set_nat_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_927_list_Oset__sel_I1_J,axiom,
! [A: list_list_list_a] :
( ( A != nil_list_list_a )
=> ( member_list_list_a @ ( hd_list_list_a @ A ) @ ( set_list_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_928_list_Oset__sel_I1_J,axiom,
! [A: list_int] :
( ( A != nil_int )
=> ( member_int @ ( hd_int @ A ) @ ( set_int2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_929_list_Oset__sel_I1_J,axiom,
! [A: list_nat_int] :
( ( A != nil_nat_int )
=> ( member_nat_int @ ( hd_nat_int @ A ) @ ( set_nat_int2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_930_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_931_replicate__eqI,axiom,
! [Xs: list_nat_a,N: nat,X2: nat > a] :
( ( ( size_size_list_nat_a @ Xs )
= N )
=> ( ! [Y3: nat > a] :
( ( member_nat_a @ Y3 @ ( set_nat_a2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replicate_nat_a @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_932_replicate__eqI,axiom,
! [Xs: list_list_list_a,N: nat,X2: list_list_a] :
( ( ( size_s2403821588304063868list_a @ Xs )
= N )
=> ( ! [Y3: list_list_a] :
( ( member_list_list_a @ Y3 @ ( set_list_list_a2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replic3997036819131463498list_a @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_933_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X2: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replicate_a @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_934_replicate__eqI,axiom,
! [Xs: list_list_a,N: nat,X2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replicate_list_a @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_935_replicate__length__same,axiom,
! [Xs: list_a,X2: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( X3 = X2 ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_936_replicate__length__same,axiom,
! [Xs: list_list_a,X2: list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( X3 = X2 ) )
=> ( ( replicate_list_a @ ( size_s349497388124573686list_a @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_937_ring_Oreplicate__zero__coeff,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.replicate_zero_coeff
thf(fact_938_ring_Oreplicate__zero__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( replicate_a @ N @ ( zero_a_b @ R ) ) )
= ( ^ [Uu: nat] : ( zero_a_b @ R ) ) ) ) ).
% ring.replicate_zero_coeff
thf(fact_939_set__replicate__conv__if,axiom,
! [N: nat,X2: a] :
( ( ( N = zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X2 ) )
= bot_bot_set_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X2 ) )
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% set_replicate_conv_if
thf(fact_940_set__replicate__conv__if,axiom,
! [N: nat,X2: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( set_list_a2 @ ( replicate_list_a @ N @ X2 ) )
= bot_bot_set_list_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_list_a2 @ ( replicate_list_a @ N @ X2 ) )
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% set_replicate_conv_if
thf(fact_941_ring_Onormalize__length__eq,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( hd_list_a @ P )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ) ).
% ring.normalize_length_eq
thf(fact_942_ring_Onormalize__length__eq,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( hd_a @ P )
!= ( zero_a_b @ R ) )
=> ( ( size_size_list_a @ ( normalize_a_b @ R @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% ring.normalize_length_eq
thf(fact_943_ring_Oeval__var,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.eval_var
thf(fact_944_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X2 )
= X2 ) ) ) ).
% ring.eval_var
thf(fact_945_ring_Oeval__var,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( var_li3532061862469730199t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.eval_var
thf(fact_946_ring_Onormalize__lead__coeff,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( size_s349497388124573686list_a @ P ) )
=> ( ( hd_list_a @ P )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.normalize_lead_coeff
thf(fact_947_ring_Onormalize__lead__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) @ ( size_size_list_a @ P ) )
=> ( ( hd_a @ P )
= ( zero_a_b @ R ) ) ) ) ).
% ring.normalize_lead_coeff
thf(fact_948_ring_Onormalize__length__lt,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( hd_list_a @ P )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ P ) )
=> ( ord_less_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ) ) ) ).
% ring.normalize_length_lt
thf(fact_949_ring_Onormalize__length__lt,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( hd_a @ P )
= ( zero_a_b @ R ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ P ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) @ ( size_size_list_a @ P ) ) ) ) ) ).
% ring.normalize_length_lt
thf(fact_950_ring_Opoly__add__replicate__zero_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(2)
thf(fact_951_ring_Opoly__add__replicate__zero_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P )
= ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(2)
thf(fact_952_ring_Opoly__add__replicate__zero_H_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ ( replic3997036819131463498list_a @ N @ ( zero_l347298301471573063t_unit @ R ) ) @ P )
= ( normal1297324897130370429t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(2)
thf(fact_953_ring_Opoly__add__replicate__zero_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(1)
thf(fact_954_ring_Opoly__add__replicate__zero_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P @ ( replicate_a @ N @ ( zero_a_b @ R ) ) )
= ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(1)
thf(fact_955_ring_Opoly__add__replicate__zero_H_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_a7341706734723628101t_unit @ R @ P @ ( replic3997036819131463498list_a @ N @ ( zero_l347298301471573063t_unit @ R ) ) )
= ( normal1297324897130370429t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(1)
thf(fact_956_ring_Ois__root__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X2 )
= ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( eval_l34571156754992824t_unit @ R @ P @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) )
& ( P != nil_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_957_ring_Ois__root__def,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X2 )
= ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
& ( ( eval_l1088911609197519410t_unit @ R @ P @ X2 )
= ( zero_l347298301471573063t_unit @ R ) )
& ( P != nil_list_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_958_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X2 )
= ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P @ X2 )
= ( zero_a_b @ R ) )
& ( P != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_959_poly__add__append__replicate,axiom,
! [P: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q2 ) @ ( zero_a_b @ r ) ) ) @ Q2 )
= ( normalize_a_b @ r @ ( append_a @ P @ Q2 ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_960_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_961_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_962_combine__replicate,axiom,
! [Us: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( replicate_a @ ( size_size_list_a @ Us ) @ ( zero_a_b @ r ) ) @ Us )
= ( zero_a_b @ r ) ) ) ).
% combine_replicate
thf(fact_963_normalize_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [V3: a,Va: list_a] :
( X2
!= ( cons_a @ V3 @ Va ) ) ) ).
% normalize.cases
thf(fact_964_local_Onormalize__idem,axiom,
! [P: list_a,Q2: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q2 ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q2 ) ) ) ).
% local.normalize_idem
thf(fact_965_combine_Osimps_I2_J,axiom,
! [Us: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_966_combine_Osimps_I3_J,axiom,
! [Ks: list_a] :
( ( embedded_combine_a_b @ r @ Ks @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_967_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_968_prefix__replicate__zero__coeff,axiom,
! [P: list_a,N: nat] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) ) ) ).
% prefix_replicate_zero_coeff
thf(fact_969_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_970_combine__eq__eval,axiom,
! [Ks: list_a,X2: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X2 @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X2 ) ) ).
% combine_eq_eval
thf(fact_971_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us2 )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_972_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us2 )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us2 )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_973_combine__append,axiom,
! [Ks: list_a,Us: list_a,Ks2: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us ) @ ( embedded_combine_a_b @ r @ Ks2 @ Vs2 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks2 ) @ ( append_a @ Us @ Vs2 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_974_combine__append__replicate,axiom,
! [Us: list_a,Ks: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) @ Us )
= ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ).
% combine_append_replicate
thf(fact_975_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_976_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P6: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P6 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P6 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_977_combine__append__zero,axiom,
! [Us: list_a,Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us )
= ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ).
% combine_append_zero
thf(fact_978_poly__add__append__zero,axiom,
! [P: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q2 ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_979_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_980_list_Osimps_I15_J,axiom,
! [X21: int,X22: list_int] :
( ( set_int2 @ ( cons_int @ X21 @ X22 ) )
= ( insert_int @ X21 @ ( set_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_981_list_Osimps_I15_J,axiom,
! [X21: nat > int,X22: list_nat_int] :
( ( set_nat_int2 @ ( cons_nat_int @ X21 @ X22 ) )
= ( insert_nat_int @ X21 @ ( set_nat_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_982_combine__in__carrier,axiom,
! [Ks: list_a,Us: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_983_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X3: int,Xs3: list_int,Y3: int,Ys3: list_int] :
( ( X3 != Y3 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X3 @ nil_int ) @ Xs3 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_984_same__length__different,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( Xs != Ys )
=> ( ( ( size_s5718426915756887103at_int @ Xs )
= ( size_s5718426915756887103at_int @ Ys ) )
=> ? [Pre: list_nat_int,X3: nat > int,Xs3: list_nat_int,Y3: nat > int,Ys3: list_nat_int] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat_int @ Pre @ ( append_nat_int @ ( cons_nat_int @ X3 @ nil_nat_int ) @ Xs3 ) ) )
& ( Ys
= ( append_nat_int @ Pre @ ( append_nat_int @ ( cons_nat_int @ Y3 @ nil_nat_int ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_985_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs3: list_a,Y3: a,Ys3: list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_986_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X3: list_a,Xs3: list_list_a,Y3: list_a,Ys3: list_list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X3 @ nil_list_a ) @ Xs3 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y3 @ nil_list_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_987_ring_Ocombine_Ocong,axiom,
embedded_combine_a_b = embedded_combine_a_b ).
% ring.combine.cong
thf(fact_988_split__list__first__prop__iff,axiom,
! [Xs: list_a,P3: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P3 @ X ) ) )
= ( ? [Ys4: list_a,X: a] :
( ? [Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs ) ) )
& ( P3 @ X )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys4 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_989_split__list__first__prop__iff,axiom,
! [Xs: list_int,P3: int > $o] :
( ( ? [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
& ( P3 @ X ) ) )
= ( ? [Ys4: list_int,X: int] :
( ? [Zs: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs ) ) )
& ( P3 @ X )
& ! [Y4: int] :
( ( member_int @ Y4 @ ( set_int2 @ Ys4 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_990_split__list__first__prop__iff,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ( ? [X: nat > int] :
( ( member_nat_int @ X @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X ) ) )
= ( ? [Ys4: list_nat_int,X: nat > int] :
( ? [Zs: list_nat_int] :
( Xs
= ( append_nat_int @ Ys4 @ ( cons_nat_int @ X @ Zs ) ) )
& ( P3 @ X )
& ! [Y4: nat > int] :
( ( member_nat_int @ Y4 @ ( set_nat_int2 @ Ys4 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_991_split__list__last__prop__iff,axiom,
! [Xs: list_a,P3: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P3 @ X ) ) )
= ( ? [Ys4: list_a,X: a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs ) ) )
& ( P3 @ X )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_992_split__list__last__prop__iff,axiom,
! [Xs: list_int,P3: int > $o] :
( ( ? [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
& ( P3 @ X ) ) )
= ( ? [Ys4: list_int,X: int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs ) ) )
& ( P3 @ X )
& ! [Y4: int] :
( ( member_int @ Y4 @ ( set_int2 @ Zs ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_993_split__list__last__prop__iff,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ( ? [X: nat > int] :
( ( member_nat_int @ X @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X ) ) )
= ( ? [Ys4: list_nat_int,X: nat > int,Zs: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys4 @ ( cons_nat_int @ X @ Zs ) ) )
& ( P3 @ X )
& ! [Y4: nat > int] :
( ( member_nat_int @ Y4 @ ( set_nat_int2 @ Zs ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_994_in__set__conv__decomp__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_995_in__set__conv__decomp__first,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
= ( ? [Ys4: list_nat_a,Zs: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X2 @ Zs ) ) )
& ~ ( member_nat_a @ X2 @ ( set_nat_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_996_in__set__conv__decomp__first,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_list_a,Zs: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys4 @ ( cons_list_list_a @ X2 @ Zs ) ) )
& ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_997_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_998_in__set__conv__decomp__first,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys4: list_int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X2 @ Zs ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_999_in__set__conv__decomp__first,axiom,
! [X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
= ( ? [Ys4: list_nat_int,Zs: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys4 @ ( cons_nat_int @ X2 @ Zs ) ) )
& ~ ( member_nat_int @ X2 @ ( set_nat_int2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1000_in__set__conv__decomp__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1001_in__set__conv__decomp__last,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
= ( ? [Ys4: list_nat_a,Zs: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X2 @ Zs ) ) )
& ~ ( member_nat_a @ X2 @ ( set_nat_a2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1002_in__set__conv__decomp__last,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_list_a,Zs: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys4 @ ( cons_list_list_a @ X2 @ Zs ) ) )
& ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1003_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1004_in__set__conv__decomp__last,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys4: list_int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X2 @ Zs ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1005_in__set__conv__decomp__last,axiom,
! [X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
= ( ? [Ys4: list_nat_int,Zs: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys4 @ ( cons_nat_int @ X2 @ Zs ) ) )
& ~ ( member_nat_int @ X2 @ ( set_nat_int2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1006_split__list__first__propE,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Ys5 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1007_split__list__first__propE,axiom,
! [Xs: list_int,P3: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Ys5 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1008_split__list__first__propE,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ? [X6: nat > int] :
( ( member_nat_int @ X6 @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_nat_int,X3: nat > int] :
( ? [Zs2: list_nat_int] :
( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa3: nat > int] :
( ( member_nat_int @ Xa3 @ ( set_nat_int2 @ Ys5 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1009_split__list__last__propE,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1010_split__list__last__propE,axiom,
! [Xs: list_int,P3: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_int,X3: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Zs2 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1011_split__list__last__propE,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ? [X6: nat > int] :
( ( member_nat_int @ X6 @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_nat_int,X3: nat > int,Zs2: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa3: nat > int] :
( ( member_nat_int @ Xa3 @ ( set_nat_int2 @ Zs2 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1012_split__list__first__prop,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Ys5 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_1013_split__list__first__prop,axiom,
! [Xs: list_int,P3: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Ys5 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_1014_split__list__first__prop,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ? [X6: nat > int] :
( ( member_nat_int @ X6 @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_nat_int,X3: nat > int] :
( ? [Zs2: list_nat_int] :
( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa3: nat > int] :
( ( member_nat_int @ Xa3 @ ( set_nat_int2 @ Ys5 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_1015_split__list__last__prop,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_1016_split__list__last__prop,axiom,
! [Xs: list_int,P3: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_int,X3: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Zs2 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_1017_split__list__last__prop,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ? [X6: nat > int] :
( ( member_nat_int @ X6 @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_nat_int,X3: nat > int,Zs2: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa3: nat > int] :
( ( member_nat_int @ Xa3 @ ( set_nat_int2 @ Zs2 ) )
=> ~ ( P3 @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_1018_in__set__conv__decomp,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs: list_list_a] :
( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1019_in__set__conv__decomp,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
= ( ? [Ys4: list_nat_a,Zs: list_nat_a] :
( Xs
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1020_in__set__conv__decomp,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_list_a,Zs: list_list_list_a] :
( Xs
= ( append_list_list_a @ Ys4 @ ( cons_list_list_a @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1021_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1022_in__set__conv__decomp,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys4: list_int,Zs: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1023_in__set__conv__decomp,axiom,
! [X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
= ( ? [Ys4: list_nat_int,Zs: list_nat_int] :
( Xs
= ( append_nat_int @ Ys4 @ ( cons_nat_int @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1024_append__Cons__eq__iff,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a,Xs4: list_list_a,Ys6: list_list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ Ys ) )
= ( append_list_a @ Xs4 @ ( cons_list_a @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1025_append__Cons__eq__iff,axiom,
! [X2: nat > a,Xs: list_nat_a,Ys: list_nat_a,Xs4: list_nat_a,Ys6: list_nat_a] :
( ~ ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
=> ( ~ ( member_nat_a @ X2 @ ( set_nat_a2 @ Ys ) )
=> ( ( ( append_nat_a @ Xs @ ( cons_nat_a @ X2 @ Ys ) )
= ( append_nat_a @ Xs4 @ ( cons_nat_a @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1026_append__Cons__eq__iff,axiom,
! [X2: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,Xs4: list_list_list_a,Ys6: list_list_list_a] :
( ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ( ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Ys ) )
=> ( ( ( append_list_list_a @ Xs @ ( cons_list_list_a @ X2 @ Ys ) )
= ( append_list_list_a @ Xs4 @ ( cons_list_list_a @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1027_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys6: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1028_append__Cons__eq__iff,axiom,
! [X2: int,Xs: list_int,Ys: list_int,Xs4: list_int,Ys6: list_int] :
( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X2 @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) )
= ( append_int @ Xs4 @ ( cons_int @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1029_append__Cons__eq__iff,axiom,
! [X2: nat > int,Xs: list_nat_int,Ys: list_nat_int,Xs4: list_nat_int,Ys6: list_nat_int] :
( ~ ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
=> ( ~ ( member_nat_int @ X2 @ ( set_nat_int2 @ Ys ) )
=> ( ( ( append_nat_int @ Xs @ ( cons_nat_int @ X2 @ Ys ) )
= ( append_nat_int @ Xs4 @ ( cons_nat_int @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1030_split__list__propE,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1031_split__list__propE,axiom,
! [Xs: list_int,P3: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1032_split__list__propE,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ? [X6: nat > int] :
( ( member_nat_int @ X6 @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys5: list_nat_int,X3: nat > int] :
( ? [Zs2: list_nat_int] :
( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1033_split__list__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys5: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys5 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1034_split__list__first,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
=> ? [Ys5: list_nat_a,Zs2: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys5 @ ( cons_nat_a @ X2 @ Zs2 ) ) )
& ~ ( member_nat_a @ X2 @ ( set_nat_a2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1035_split__list__first,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ? [Ys5: list_list_list_a,Zs2: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys5 @ ( cons_list_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1036_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys5: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1037_split__list__first,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys5: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X2 @ Zs2 ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1038_split__list__first,axiom,
! [X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
=> ? [Ys5: list_nat_int,Zs2: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X2 @ Zs2 ) ) )
& ~ ( member_nat_int @ X2 @ ( set_nat_int2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1039_split__list__prop,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1040_split__list__prop,axiom,
! [Xs: list_int,P3: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_int,X3: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1041_split__list__prop,axiom,
! [Xs: list_nat_int,P3: ( nat > int ) > $o] :
( ? [X6: nat > int] :
( ( member_nat_int @ X6 @ ( set_nat_int2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys5: list_nat_int,X3: nat > int] :
( ? [Zs2: list_nat_int] :
( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1042_split__list__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys5: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys5 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1043_split__list__last,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
=> ? [Ys5: list_nat_a,Zs2: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys5 @ ( cons_nat_a @ X2 @ Zs2 ) ) )
& ~ ( member_nat_a @ X2 @ ( set_nat_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1044_split__list__last,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ? [Ys5: list_list_list_a,Zs2: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys5 @ ( cons_list_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1045_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys5: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1046_split__list__last,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys5: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X2 @ Zs2 ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1047_split__list__last,axiom,
! [X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
=> ? [Ys5: list_nat_int,Zs2: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X2 @ Zs2 ) ) )
& ~ ( member_nat_int @ X2 @ ( set_nat_int2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1048_split__list,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys5: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys5 @ ( cons_list_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1049_split__list,axiom,
! [X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
=> ? [Ys5: list_nat_a,Zs2: list_nat_a] :
( Xs
= ( append_nat_a @ Ys5 @ ( cons_nat_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1050_split__list,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ? [Ys5: list_list_list_a,Zs2: list_list_list_a] :
( Xs
= ( append_list_list_a @ Ys5 @ ( cons_list_list_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1051_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys5: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1052_split__list,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys5: list_int,Zs2: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1053_split__list,axiom,
! [X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ X2 @ ( set_nat_int2 @ Xs ) )
=> ? [Ys5: list_nat_int,Zs2: list_nat_int] :
( Xs
= ( append_nat_int @ Ys5 @ ( cons_nat_int @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1054_set__ConsD,axiom,
! [Y: list_a,X2: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1055_set__ConsD,axiom,
! [Y: nat > a,X2: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ Y @ ( set_nat_a2 @ ( cons_nat_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat_a @ Y @ ( set_nat_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1056_set__ConsD,axiom,
! [Y: list_list_a,X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_list_a @ Y @ ( set_list_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1057_set__ConsD,axiom,
! [Y: a,X2: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1058_set__ConsD,axiom,
! [Y: int,X2: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1059_set__ConsD,axiom,
! [Y: nat > int,X2: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ Y @ ( set_nat_int2 @ ( cons_nat_int @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat_int @ Y @ ( set_nat_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1060_list_Oset__cases,axiom,
! [E2: list_a,A: list_list_a] :
( ( member_list_a @ E2 @ ( set_list_a2 @ A ) )
=> ( ! [Z22: list_list_a] :
( A
!= ( cons_list_a @ E2 @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E2 @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1061_list_Oset__cases,axiom,
! [E2: nat > a,A: list_nat_a] :
( ( member_nat_a @ E2 @ ( set_nat_a2 @ A ) )
=> ( ! [Z22: list_nat_a] :
( A
!= ( cons_nat_a @ E2 @ Z22 ) )
=> ~ ! [Z1: nat > a,Z22: list_nat_a] :
( ( A
= ( cons_nat_a @ Z1 @ Z22 ) )
=> ~ ( member_nat_a @ E2 @ ( set_nat_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1062_list_Oset__cases,axiom,
! [E2: list_list_a,A: list_list_list_a] :
( ( member_list_list_a @ E2 @ ( set_list_list_a2 @ A ) )
=> ( ! [Z22: list_list_list_a] :
( A
!= ( cons_list_list_a @ E2 @ Z22 ) )
=> ~ ! [Z1: list_list_a,Z22: list_list_list_a] :
( ( A
= ( cons_list_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_list_a @ E2 @ ( set_list_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1063_list_Oset__cases,axiom,
! [E2: a,A: list_a] :
( ( member_a @ E2 @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E2 @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1064_list_Oset__cases,axiom,
! [E2: int,A: list_int] :
( ( member_int @ E2 @ ( set_int2 @ A ) )
=> ( ! [Z22: list_int] :
( A
!= ( cons_int @ E2 @ Z22 ) )
=> ~ ! [Z1: int,Z22: list_int] :
( ( A
= ( cons_int @ Z1 @ Z22 ) )
=> ~ ( member_int @ E2 @ ( set_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1065_list_Oset__cases,axiom,
! [E2: nat > int,A: list_nat_int] :
( ( member_nat_int @ E2 @ ( set_nat_int2 @ A ) )
=> ( ! [Z22: list_nat_int] :
( A
!= ( cons_nat_int @ E2 @ Z22 ) )
=> ~ ! [Z1: nat > int,Z22: list_nat_int] :
( ( A
= ( cons_nat_int @ Z1 @ Z22 ) )
=> ~ ( member_nat_int @ E2 @ ( set_nat_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1066_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1067_list_Oset__intros_I1_J,axiom,
! [X21: nat > a,X22: list_nat_a] : ( member_nat_a @ X21 @ ( set_nat_a2 @ ( cons_nat_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1068_list_Oset__intros_I1_J,axiom,
! [X21: list_list_a,X22: list_list_list_a] : ( member_list_list_a @ X21 @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1069_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1070_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1071_list_Oset__intros_I1_J,axiom,
! [X21: nat > int,X22: list_nat_int] : ( member_nat_int @ X21 @ ( set_nat_int2 @ ( cons_nat_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1072_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1073_list_Oset__intros_I2_J,axiom,
! [Y: nat > a,X22: list_nat_a,X21: nat > a] :
( ( member_nat_a @ Y @ ( set_nat_a2 @ X22 ) )
=> ( member_nat_a @ Y @ ( set_nat_a2 @ ( cons_nat_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1074_list_Oset__intros_I2_J,axiom,
! [Y: list_list_a,X22: list_list_list_a,X21: list_list_a] :
( ( member_list_list_a @ Y @ ( set_list_list_a2 @ X22 ) )
=> ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1075_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1076_list_Oset__intros_I2_J,axiom,
! [Y: int,X22: list_int,X21: int] :
( ( member_int @ Y @ ( set_int2 @ X22 ) )
=> ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1077_list_Oset__intros_I2_J,axiom,
! [Y: nat > int,X22: list_nat_int,X21: nat > int] :
( ( member_nat_int @ Y @ ( set_nat_int2 @ X22 ) )
=> ( member_nat_int @ Y @ ( set_nat_int2 @ ( cons_nat_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1078_ring_Ocombine__append__zero,axiom,
! [R: partia2670972154091845814t_unit,Us: list_list_a,Ks: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd2435972518007585703t_unit @ R @ ( append_list_a @ Ks @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) @ Us )
= ( embedd2435972518007585703t_unit @ R @ Ks @ Us ) ) ) ) ).
% ring.combine_append_zero
thf(fact_1079_ring_Ocombine__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,Us: list_a,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ Us )
= ( embedded_combine_a_b @ R @ Ks @ Us ) ) ) ) ).
% ring.combine_append_zero
thf(fact_1080_ring_Ocombine__append__zero,axiom,
! [R: partia2956882679547061052t_unit,Us: list_list_list_a,Ks: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Us ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( embedd7921816065501514657t_unit @ R @ ( append_list_list_a @ Ks @ ( cons_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ nil_list_list_a ) ) @ Us )
= ( embedd7921816065501514657t_unit @ R @ Ks @ Us ) ) ) ) ).
% ring.combine_append_zero
thf(fact_1081_set__subset__Cons,axiom,
! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1082_set__subset__Cons,axiom,
! [Xs: list_nat_int,X2: nat > int] : ( ord_le6569500216720880561at_int @ ( set_nat_int2 @ Xs ) @ ( set_nat_int2 @ ( cons_nat_int @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1083_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1084_list__all2__append,axiom,
! [Xs: list_a,Ys: list_a,P3: a > a > $o,Us2: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( list_all2_a_a @ P3 @ ( append_a @ Xs @ Us2 ) @ ( append_a @ Ys @ Vs ) )
= ( ( list_all2_a_a @ P3 @ Xs @ Ys )
& ( list_all2_a_a @ P3 @ Us2 @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_1085_list__all2__append,axiom,
! [Xs: list_a,Ys: list_list_a,P3: a > list_a > $o,Us2: list_a,Vs: list_list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( list_all2_a_list_a @ P3 @ ( append_a @ Xs @ Us2 ) @ ( append_list_a @ Ys @ Vs ) )
= ( ( list_all2_a_list_a @ P3 @ Xs @ Ys )
& ( list_all2_a_list_a @ P3 @ Us2 @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_1086_list__all2__append,axiom,
! [Xs: list_list_a,Ys: list_a,P3: list_a > a > $o,Us2: list_list_a,Vs: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( list_all2_list_a_a @ P3 @ ( append_list_a @ Xs @ Us2 ) @ ( append_a @ Ys @ Vs ) )
= ( ( list_all2_list_a_a @ P3 @ Xs @ Ys )
& ( list_all2_list_a_a @ P3 @ Us2 @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_1087_list__all2__append,axiom,
! [Xs: list_list_a,Ys: list_list_a,P3: list_a > list_a > $o,Us2: list_list_a,Vs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( list_a3802133873445908231list_a @ P3 @ ( append_list_a @ Xs @ Us2 ) @ ( append_list_a @ Ys @ Vs ) )
= ( ( list_a3802133873445908231list_a @ P3 @ Xs @ Ys )
& ( list_a3802133873445908231list_a @ P3 @ Us2 @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_1088_list__all2__append1,axiom,
! [P3: a > a > $o,Xs: list_a,Ys: list_a,Zs3: list_a] :
( ( list_all2_a_a @ P3 @ ( append_a @ Xs @ Ys ) @ Zs3 )
= ( ? [Us3: list_a,Vs3: list_a] :
( ( Zs3
= ( append_a @ Us3 @ Vs3 ) )
& ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Xs ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Ys ) )
& ( list_all2_a_a @ P3 @ Xs @ Us3 )
& ( list_all2_a_a @ P3 @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_1089_list__all2__append1,axiom,
! [P3: list_a > a > $o,Xs: list_list_a,Ys: list_list_a,Zs3: list_a] :
( ( list_all2_list_a_a @ P3 @ ( append_list_a @ Xs @ Ys ) @ Zs3 )
= ( ? [Us3: list_a,Vs3: list_a] :
( ( Zs3
= ( append_a @ Us3 @ Vs3 ) )
& ( ( size_size_list_a @ Us3 )
= ( size_s349497388124573686list_a @ Xs ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_s349497388124573686list_a @ Ys ) )
& ( list_all2_list_a_a @ P3 @ Xs @ Us3 )
& ( list_all2_list_a_a @ P3 @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_1090_list__all2__append1,axiom,
! [P3: a > list_a > $o,Xs: list_a,Ys: list_a,Zs3: list_list_a] :
( ( list_all2_a_list_a @ P3 @ ( append_a @ Xs @ Ys ) @ Zs3 )
= ( ? [Us3: list_list_a,Vs3: list_list_a] :
( ( Zs3
= ( append_list_a @ Us3 @ Vs3 ) )
& ( ( size_s349497388124573686list_a @ Us3 )
= ( size_size_list_a @ Xs ) )
& ( ( size_s349497388124573686list_a @ Vs3 )
= ( size_size_list_a @ Ys ) )
& ( list_all2_a_list_a @ P3 @ Xs @ Us3 )
& ( list_all2_a_list_a @ P3 @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_1091_list__all2__append1,axiom,
! [P3: list_a > list_a > $o,Xs: list_list_a,Ys: list_list_a,Zs3: list_list_a] :
( ( list_a3802133873445908231list_a @ P3 @ ( append_list_a @ Xs @ Ys ) @ Zs3 )
= ( ? [Us3: list_list_a,Vs3: list_list_a] :
( ( Zs3
= ( append_list_a @ Us3 @ Vs3 ) )
& ( ( size_s349497388124573686list_a @ Us3 )
= ( size_s349497388124573686list_a @ Xs ) )
& ( ( size_s349497388124573686list_a @ Vs3 )
= ( size_s349497388124573686list_a @ Ys ) )
& ( list_a3802133873445908231list_a @ P3 @ Xs @ Us3 )
& ( list_a3802133873445908231list_a @ P3 @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_1092_list__all2__append2,axiom,
! [P3: a > list_a > $o,Xs: list_a,Ys: list_list_a,Zs3: list_list_a] :
( ( list_all2_a_list_a @ P3 @ Xs @ ( append_list_a @ Ys @ Zs3 ) )
= ( ? [Us3: list_a,Vs3: list_a] :
( ( Xs
= ( append_a @ Us3 @ Vs3 ) )
& ( ( size_size_list_a @ Us3 )
= ( size_s349497388124573686list_a @ Ys ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
& ( list_all2_a_list_a @ P3 @ Us3 @ Ys )
& ( list_all2_a_list_a @ P3 @ Vs3 @ Zs3 ) ) ) ) ).
% list_all2_append2
thf(fact_1093_list__all2__append2,axiom,
! [P3: list_a > a > $o,Xs: list_list_a,Ys: list_a,Zs3: list_a] :
( ( list_all2_list_a_a @ P3 @ Xs @ ( append_a @ Ys @ Zs3 ) )
= ( ? [Us3: list_list_a,Vs3: list_list_a] :
( ( Xs
= ( append_list_a @ Us3 @ Vs3 ) )
& ( ( size_s349497388124573686list_a @ Us3 )
= ( size_size_list_a @ Ys ) )
& ( ( size_s349497388124573686list_a @ Vs3 )
= ( size_size_list_a @ Zs3 ) )
& ( list_all2_list_a_a @ P3 @ Us3 @ Ys )
& ( list_all2_list_a_a @ P3 @ Vs3 @ Zs3 ) ) ) ) ).
% list_all2_append2
thf(fact_1094_list__all2__append2,axiom,
! [P3: list_a > list_a > $o,Xs: list_list_a,Ys: list_list_a,Zs3: list_list_a] :
( ( list_a3802133873445908231list_a @ P3 @ Xs @ ( append_list_a @ Ys @ Zs3 ) )
= ( ? [Us3: list_list_a,Vs3: list_list_a] :
( ( Xs
= ( append_list_a @ Us3 @ Vs3 ) )
& ( ( size_s349497388124573686list_a @ Us3 )
= ( size_s349497388124573686list_a @ Ys ) )
& ( ( size_s349497388124573686list_a @ Vs3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
& ( list_a3802133873445908231list_a @ P3 @ Us3 @ Ys )
& ( list_a3802133873445908231list_a @ P3 @ Vs3 @ Zs3 ) ) ) ) ).
% list_all2_append2
thf(fact_1095_poly__add__monom,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P ) ) @ P )
= ( cons_a @ A @ P ) ) ) ) ).
% poly_add_monom
thf(fact_1096_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K4: a] : ( normalize_a_b @ r @ ( cons_a @ K4 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_1097_eval__append__aux,axiom,
! [P: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_1098_poly__mult__append__zero,axiom,
! [P: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q2 )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q2 ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_1099_m__assoc,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( 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 @ X2 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_1100_poly__mult_Osimps_I1_J,axiom,
! [P2: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P2 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_1101_l__distr,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( 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 @ X2 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_1102_r__distr,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( 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 @ X2 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X2 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_1103_l__minus,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) ) ) ) ) ).
% l_minus
thf(fact_1104_r__minus,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) ) ) ) ) ).
% r_minus
thf(fact_1105_inv__unique,axiom,
! [Y: a,X2: a,Y6: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y6 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1106_one__unique,axiom,
! [U2: a] :
( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U2 @ X3 )
= X3 ) )
=> ( U2
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_1107_mult__cong__r,axiom,
! [B: a,B7: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B7 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1108_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_1109_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_1110_line__extension__mem__iff,axiom,
! [U2: a,K3: set_a,A: a,E: set_a] :
( ( member_a @ U2 @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) )
= ( ? [X: a] :
( ( member_a @ X @ K3 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U2
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ A ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_1111_Units__inv__comm,axiom,
! [X2: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_1112_up__smult__closed,axiom,
! [A: a,P: nat > a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I4: nat] : ( mult_a_ring_ext_a_b @ r @ A @ ( P @ I4 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_1113_poly__mult__in__carrier,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_1114_combine_Osimps_I1_J,axiom,
! [K: a,Ks: list_a,U2: a,Us: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K @ Ks ) @ ( cons_a @ U2 @ Us ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K @ U2 ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ).
% combine.simps(1)
thf(fact_1115_Units__l__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1116_Units__r__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1117_associatedI2,axiom,
! [U2: a,A: a,B: a] :
( ( member_a @ U2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_1118_associatedI2_H,axiom,
! [A: a,B: a,U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U2 ) )
=> ( ( member_a @ U2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_1119_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_1120_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_1121_poly__mult__l__distr_H,axiom,
! [P1: list_a,P2: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P2 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_1122_poly__mult__normalize,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P2 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P2 ) ) ) ) ).
% poly_mult_normalize
thf(fact_1123_combine_Oelims,axiom,
! [X2: list_a,Xa: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X2 @ Xa )
= Y )
=> ( ! [K2: a,Ks3: list_a] :
( ( X2
= ( cons_a @ K2 @ Ks3 ) )
=> ! [U3: a,Us4: list_a] :
( ( Xa
= ( cons_a @ U3 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K2 @ U3 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X2 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_1124_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_1125_poly__mult__prepend__replicate__zero,axiom,
! [P1: list_a,P2: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P2 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P1 ) @ P2 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_1126_m__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_1127_Units__m__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1128_l__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X2 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_1129_r__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_1130_l__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X2 )
= X2 ) ) ).
% l_one
thf(fact_1131_r__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( one_a_ring_ext_a_b @ r ) )
= X2 ) ) ).
% r_one
thf(fact_1132_Units__l__cancel,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1133_eval__append,axiom,
! [P: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q2 ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_1134_monoid__cancelI,axiom,
( ! [A2: a,B2: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C4 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C4 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C4 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C4 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_1135_combine__prepend__replicate,axiom,
! [Ks: list_a,Us: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ Ks ) @ Us )
= ( embedded_combine_a_b @ r @ Ks @ ( drop_a @ N @ Us ) ) ) ) ) ).
% combine_prepend_replicate
thf(fact_1136_Units__pow__closed,axiom,
! [X2: a,D2: nat] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ D2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_1137_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_1138_drop__exp__base,axiom,
! [N: nat,X2: a,M2: nat] :
( ( drop_a @ N @ ( polyno2922411391617481336se_a_b @ r @ X2 @ M2 ) )
= ( polyno2922411391617481336se_a_b @ r @ X2 @ ( minus_minus_nat @ M2 @ N ) ) ) ).
% drop_exp_base
thf(fact_1139_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1140_pow__mult__distrib,axiom,
! [X2: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_1141_nat__pow__comm,axiom,
! [X2: a,N: nat,M2: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_1142_group__commutes__pow,axiom,
! [X2: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_1143_normalize__def_H_I2_J,axiom,
! [P: list_a] :
( ( normalize_a_b @ r @ P )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ P ) ) ).
% normalize_def'(2)
thf(fact_1144_append__coeff,axiom,
! [P: list_a,Q2: list_a] :
( ( coeff_a_b @ r @ ( append_a @ P @ Q2 ) )
= ( ^ [I4: nat] : ( if_a @ ( ord_less_nat @ I4 @ ( size_size_list_a @ Q2 ) ) @ ( coeff_a_b @ r @ Q2 @ I4 ) @ ( coeff_a_b @ r @ P @ ( minus_minus_nat @ I4 @ ( size_size_list_a @ Q2 ) ) ) ) ) ) ).
% append_coeff
thf(fact_1145_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_1146_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_1147_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1148_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1149_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1150_normalize__def_H_I1_J,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ P ) ) ) ).
% normalize_def'(1)
thf(fact_1151_nat__pow__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_1152_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_1153_local_Onat__pow__0,axiom,
! [X2: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1154_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1155_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_1156_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1157_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1158_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1159_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1160_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1161_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_1162_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1163_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1164_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1165_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_1166_poly__degree__bound__from__coeff,axiom,
! [X2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ N @ K2 )
=> ( ( coeff_a_b @ r @ X2 @ K2 )
= ( zero_a_b @ r ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ X2 ) @ one_one_nat ) @ N )
| ( X2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% poly_degree_bound_from_coeff
thf(fact_1167_degree__oneE,axiom,
! [P: list_a,K3: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A2: a] :
( ( member_a @ A2 @ K3 )
=> ( ( A2
!= ( zero_a_b @ r ) )
=> ! [B2: a] :
( ( member_a @ B2 @ K3 )
=> ( P
!= ( cons_a @ A2 @ ( cons_a @ B2 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1168_build__poly__degree,axiom,
! [F: nat > a,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( bounde1002222742488328185ly_a_b @ r @ F @ N ) ) @ one_one_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ).
% build_poly_degree
thf(fact_1169_coeff__degree,axiom,
! [P: list_a,I2: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I2 )
=> ( ( coeff_a_b @ r @ P @ I2 )
= ( zero_a_b @ r ) ) ) ).
% coeff_degree
thf(fact_1170_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1171_nat__pow__eone,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ one_one_nat )
= X2 ) ) ).
% nat_pow_eone
thf(fact_1172_lead__coeff__simp,axiom,
! [P: list_a] :
( ( P != nil_a )
=> ( ( coeff_a_b @ r @ P @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
= ( hd_a @ P ) ) ) ).
% lead_coeff_simp
thf(fact_1173_units__of__pow,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X2 @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ).
% units_of_pow
thf(fact_1174_eval_Oelims,axiom,
! [X2: list_a,Y: a > a] :
( ( ( eval_a_b @ r @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y
!= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) )
=> ~ ! [V3: a,Va: list_a] :
( ( X2
= ( cons_a @ V3 @ Va ) )
=> ( Y
!= ( ^ [X: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V3 @ Va ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V3 @ Va ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V3 @ Va ) ) @ X ) ) ) ) ) ) ) ).
% eval.elims
thf(fact_1175_eval_Osimps_I2_J,axiom,
! [V4: a,Va2: list_a] :
( ( eval_a_b @ r @ ( cons_a @ V4 @ Va2 ) )
= ( ^ [X: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V4 @ Va2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V4 @ Va2 ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V4 @ Va2 ) ) @ X ) ) ) ) ).
% eval.simps(2)
thf(fact_1176_normalize_Osimps_I2_J,axiom,
! [V4: a,Va2: list_a] :
( ( ( ( hd_a @ ( cons_a @ V4 @ Va2 ) )
!= ( zero_a_b @ r ) )
=> ( ( normalize_a_b @ r @ ( cons_a @ V4 @ Va2 ) )
= ( cons_a @ V4 @ Va2 ) ) )
& ( ( ( hd_a @ ( cons_a @ V4 @ Va2 ) )
= ( zero_a_b @ r ) )
=> ( ( normalize_a_b @ r @ ( cons_a @ V4 @ Va2 ) )
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V4 @ Va2 ) ) ) ) ) ) ).
% normalize.simps(2)
thf(fact_1177_normalize_Oelims,axiom,
! [X2: list_a,Y: list_a] :
( ( ( normalize_a_b @ r @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y != nil_a ) )
=> ~ ! [V3: a,Va: list_a] :
( ( X2
= ( cons_a @ V3 @ Va ) )
=> ~ ( ( ( ( hd_a @ ( cons_a @ V3 @ Va ) )
!= ( zero_a_b @ r ) )
=> ( Y
= ( cons_a @ V3 @ Va ) ) )
& ( ( ( hd_a @ ( cons_a @ V3 @ Va ) )
= ( zero_a_b @ r ) )
=> ( Y
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ) ).
% normalize.elims
thf(fact_1178_coeff_Osimps_I2_J,axiom,
! [V4: a,Va2: list_a] :
( ( coeff_a_b @ r @ ( cons_a @ V4 @ Va2 ) )
= ( ^ [I4: nat] :
( if_a
@ ( I4
= ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V4 @ Va2 ) ) @ one_one_nat ) )
@ ( hd_a @ ( cons_a @ V4 @ Va2 ) )
@ ( coeff_a_b @ r @ ( tl_a @ ( cons_a @ V4 @ Va2 ) ) @ I4 ) ) ) ) ).
% coeff.simps(2)
thf(fact_1179_coeff_Oelims,axiom,
! [X2: list_a,Y: nat > a] :
( ( ( coeff_a_b @ r @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y
!= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) )
=> ~ ! [V3: a,Va: list_a] :
( ( X2
= ( cons_a @ V3 @ Va ) )
=> ( Y
!= ( ^ [I4: nat] :
( if_a
@ ( I4
= ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V3 @ Va ) ) @ one_one_nat ) )
@ ( hd_a @ ( cons_a @ V3 @ Va ) )
@ ( coeff_a_b @ r @ ( tl_a @ ( cons_a @ V3 @ Va ) ) @ I4 ) ) ) ) ) ) ) ).
% coeff.elims
thf(fact_1180_poly__mult_Oelims,axiom,
! [X2: list_a,Xa: list_a,Y: list_a] :
( ( ( poly_mult_a_b @ r @ X2 @ Xa )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y != nil_a ) )
=> ~ ! [V3: a,Va: list_a] :
( ( X2
= ( cons_a @ V3 @ Va ) )
=> ( Y
!= ( poly_add_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V3 @ Va ) ) ) @ Xa ) @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V3 @ Va ) ) @ one_one_nat ) @ ( zero_a_b @ r ) ) ) @ ( poly_mult_a_b @ r @ ( tl_a @ ( cons_a @ V3 @ Va ) ) @ Xa ) ) ) ) ) ) ).
% poly_mult.elims
thf(fact_1181_poly__mult_Osimps_I2_J,axiom,
! [V4: a,Va2: list_a,P2: list_a] :
( ( poly_mult_a_b @ r @ ( cons_a @ V4 @ Va2 ) @ P2 )
= ( poly_add_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V4 @ Va2 ) ) ) @ P2 ) @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V4 @ Va2 ) ) @ one_one_nat ) @ ( zero_a_b @ r ) ) ) @ ( poly_mult_a_b @ r @ ( tl_a @ ( cons_a @ V4 @ Va2 ) ) @ P2 ) ) ) ).
% poly_mult.simps(2)
thf(fact_1182_scalar__coeff,axiom,
! [A: a,P: list_a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( coeff_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) )
= ( ^ [I4: nat] : ( mult_a_ring_ext_a_b @ r @ A @ ( coeff_a_b @ r @ P @ I4 ) ) ) ) ) ).
% scalar_coeff
thf(fact_1183_combine__r__distr,axiom,
! [Ks: list_a,Us: list_a,K: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K @ ( embedded_combine_a_b @ r @ Ks @ Us ) )
= ( embedded_combine_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ K ) @ Ks ) @ Us ) ) ) ) ) ).
% combine_r_distr
thf(fact_1184_combine__l__distr,axiom,
! [Ks: list_a,Us: list_a,U2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us ) @ U2 )
= ( embedded_combine_a_b @ r @ Ks
@ ( map_a_a
@ ^ [U4: a] : ( mult_a_ring_ext_a_b @ r @ U4 @ U2 )
@ Us ) ) ) ) ) ) ).
% combine_l_distr
thf(fact_1185_subfield__long__division__theorem__shell,axiom,
! [K3: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ? [Q3: list_a,R5: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( member_list_a @ R5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ B @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R5 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1186_factors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_1187_subring__props_I7_J,axiom,
! [K3: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H22 @ K3 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K3 ) ) ) ) ).
% subring_props(7)
thf(fact_1188_subring__props_I2_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K3 ) ) ).
% subring_props(2)
thf(fact_1189_subring__props_I6_J,axiom,
! [K3: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H22 @ K3 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K3 ) ) ) ) ).
% subring_props(6)
thf(fact_1190_subring__props_I4_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( K3 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1191_subring__props_I5_J,axiom,
! [K3: set_a,H: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H @ K3 )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K3 ) ) ) ).
% subring_props(5)
thf(fact_1192_subring__props_I3_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K3 ) ) ).
% subring_props(3)
thf(fact_1193_subring__props_I1_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1194_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_1195_line__extension__smult__closed,axiom,
! [K3: set_a,E: set_a,A: a,K: a,U2: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ! [K2: a,V3: a] :
( ( member_a @ K2 @ K3 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K3 )
=> ( ( member_a @ U2 @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U2 ) @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_1196_subfield__m__inv__simprule,axiom,
! [K3: set_a,K: a,A: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K3 @ ( 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 ) @ K3 )
=> ( member_a @ A @ K3 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1197_wfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( wfacto3557276942076956612xt_a_b @ r @ As @ A2 ) ) ) ) ).
% wfactors_prod_exists
thf(fact_1198_INTEG_OR_Odense__repr_Ocases,axiom,
! [X2: list_int] :
( ( X2 != nil_int )
=> ~ ! [V3: int,Va: list_int] :
( X2
!= ( cons_int @ V3 @ Va ) ) ) ).
% INTEG.R.dense_repr.cases
thf(fact_1199_INTEG_OP_Odense__repr_Ocases,axiom,
! [X2: list_nat_int] :
( ( X2 != nil_nat_int )
=> ~ ! [V3: nat > int,Va: list_nat_int] :
( X2
!= ( cons_nat_int @ V3 @ Va ) ) ) ).
% INTEG.P.dense_repr.cases
thf(fact_1200_pirreducible__degree,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1201_Span__mem__imp__non__trivial__combine,axiom,
! [K3: set_a,Us: list_a,A: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ~ ! [K2: a] :
( ( member_a @ K2 @ ( minus_minus_set_a @ K3 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ! [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K3 )
=> ( ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us ) )
=> ( ( embedded_combine_a_b @ r @ ( cons_a @ K2 @ Ks3 ) @ ( cons_a @ A @ Us ) )
!= ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% Span_mem_imp_non_trivial_combine
thf(fact_1202_Span__in__carrier,axiom,
! [K3: set_a,Us: list_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_1203_mono__Span__subset,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) ) ) ) ) ).
% mono_Span_subset
thf(fact_1204_mono__Span__sublist,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( set_a2 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_1205_Span__same__set,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us )
= ( set_a2 @ Vs2 ) )
=> ( ( embedded_Span_a_b @ r @ K3 @ Us )
= ( embedded_Span_a_b @ r @ K3 @ Vs2 ) ) ) ) ) ).
% Span_same_set
thf(fact_1206_Span__base__incl,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ).
% Span_base_incl
thf(fact_1207_Span__subgroup__props_I1_J,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_1208_Span__strict__incl,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Vs2 ) )
& ~ ( member_a @ X3 @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ) ) ).
% Span_strict_incl
thf(fact_1209_subalgebra__Span__incl,axiom,
! [K3: set_a,V: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ V @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ V )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ V ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_1210_Span__subalgebraI,axiom,
! [K3: set_a,E: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ E @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ E )
=> ( ! [V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V5 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ V5 )
=> ( ord_less_eq_set_a @ E @ V5 ) ) )
=> ( E
= ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_1211_Span__subgroup__props_I3_J,axiom,
! [K3: set_a,Us: list_a,V1: a,V22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ( ( member_a @ V22 @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ) ).
% Span_subgroup_props(3)
thf(fact_1212_Span__subgroup__props_I2_J,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ).
% Span_subgroup_props(2)
thf(fact_1213_mono__Span,axiom,
! [K3: set_a,Us: list_a,U2: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) ) ) ) ) ) ).
% mono_Span
thf(fact_1214_Span__smult__closed,axiom,
! [K3: set_a,Us: list_a,K: a,V4: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K3 )
=> ( ( member_a @ V4 @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V4 ) @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_1215_Span__subgroup__props_I4_J,axiom,
! [K3: set_a,Us: list_a,V4: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V4 @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ( member_a @ ( a_inv_a_b @ r @ V4 ) @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ).
% Span_subgroup_props(4)
thf(fact_1216_mono__Span__append_I1_J,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ ( append_a @ Us @ Vs2 ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_1217_mono__Span__append_I2_J,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ ( append_a @ Vs2 @ Us ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_1218_Span__is__subalgebra,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ r ) ) ) ).
% Span_is_subalgebra
thf(fact_1219_Span__mem__iff__length__version,axiom,
! [K3: set_a,Us: list_a,A: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
= ( ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us ) )
& ( A
= ( embedded_combine_a_b @ r @ Ks4 @ Us ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_1220_Span__m__inv__simprule,axiom,
! [K3: set_a,Us: list_a,K: a,A: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K3 @ ( 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 ) @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ( member_a @ A @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ) ) ).
% Span_m_inv_simprule
thf(fact_1221_Span__mem__iff,axiom,
! [K3: set_a,Us: list_a,A: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
= ( ? [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ K3 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 )
& ( ( embedded_combine_a_b @ r @ ( cons_a @ X @ Ks4 ) @ ( cons_a @ A @ Us ) )
= ( zero_a_b @ r ) ) ) ) ) ) ) ) ) ).
% Span_mem_iff
thf(fact_1222_poly__of__const__over__subfield,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K3 )
= ( collect_list_a
@ ^ [P7: list_a] :
( ( member_list_a @ P7 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ P7 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% poly_of_const_over_subfield
thf(fact_1223_line__extension__of__combine__set__length__version,axiom,
! [K3: set_a,U2: a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd971793762689825387on_a_b @ r @ K3 @ U2
@ ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us ) )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 ) ) ) )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ ( cons_a @ U2 @ Us ) ) )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ ( cons_a @ U2 @ Us ) ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 ) ) ) ) ) ) ).
% line_extension_of_combine_set_length_version
thf(fact_1224_univ__poly__subfield__of__consts,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K3 ) @ ( univ_poly_a_b @ r @ K3 ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_1225_Span__eq__combine__set,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K3 @ Us )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 ) ) ) ) ) ) ).
% Span_eq_combine_set
thf(fact_1226_Span__eq__combine__set__length__version,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K3 @ Us )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us ) )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 ) ) ) ) ) ) ).
% Span_eq_combine_set_length_version
thf(fact_1227_line__extension__of__combine__set,axiom,
! [K3: set_a,U2: a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd971793762689825387on_a_b @ r @ K3 @ U2
@ ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 ) ) ) )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ ( cons_a @ U2 @ Us ) ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K3 ) ) ) ) ) ) ).
% line_extension_of_combine_set
thf(fact_1228_univ__poly__units,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) )
= ( collect_list_a
@ ^ [Uu: list_a] :
? [K4: a] :
( ( Uu
= ( cons_a @ K4 @ nil_a ) )
& ( member_a @ K4 @ ( minus_minus_set_a @ K3 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ) ).
% univ_poly_units
thf(fact_1229_Span__finite__dimension,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd8708762675212832759on_a_b @ r @ K3 @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ).
% Span_finite_dimension
thf(fact_1230_add_Osurj__const__mult,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_1231_telescopic__base__dim_I1_J,axiom,
! [K3: set_a,F5: set_a,E: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( subfield_a_b @ F5 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ F5 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F5 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K3 @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_1232_finite__dimension__imp__subalgebra,axiom,
! [K3: set_a,E: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ E )
=> ( embedd9027525575939734154ra_a_b @ K3 @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_1233_subalbegra__incl__imp__finite__dimension,axiom,
! [K3: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K3 @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_1234_dependent__imp__non__trivial__combine,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ~ ! [Ks3: list_a] :
( ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us ) )
=> ( ( ( embedded_combine_a_b @ r @ Ks3 @ Us )
= ( zero_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K3 )
=> ( ( set_a2 @ Ks3 )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).
% dependent_imp_non_trivial_combine
thf(fact_1235_Span__append__eq__set__add,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K3 @ ( append_a @ Us @ Vs2 ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_1236_independent__backwards_I2_J,axiom,
! [K3: set_a,U2: a,Us: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us ) ) ).
% independent_backwards(2)
thf(fact_1237_li__Nil,axiom,
! [K3: set_a] : ( embedd5208550302661555450nt_a_b @ r @ K3 @ nil_a ) ).
% li_Nil
thf(fact_1238_independent__backwards_I3_J,axiom,
! [K3: set_a,U2: a,Us: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) )
=> ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_backwards(3)
thf(fact_1239_set__add__closed,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1240_set__add__comm,axiom,
! [I3: set_a,J3: set_a] :
( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I3 @ J3 )
= ( set_add_a_b @ r @ J3 @ I3 ) ) ) ) ).
% set_add_comm
thf(fact_1241_setadd__subset__G,axiom,
! [H2: set_a,K3: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1242_independent__backwards_I1_J,axiom,
! [K3: set_a,U2: a,Us: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) )
=> ~ ( member_a @ U2 @ ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ).
% independent_backwards(1)
thf(fact_1243_independent__split_I2_J,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ Us @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us ) ) ) ).
% independent_split(2)
thf(fact_1244_independent__split_I1_J,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ Us @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ Vs2 ) ) ) ).
% independent_split(1)
thf(fact_1245_sum__space__dim_I1_J,axiom,
! [K3: set_a,E: set_a,F5: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ F5 )
=> ( embedd8708762675212832759on_a_b @ r @ K3 @ ( set_add_a_b @ r @ E @ F5 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1246_independent__in__carrier,axiom,
! [K3: set_a,Us: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_in_carrier
thf(fact_1247_li__Cons,axiom,
! [U2: a,K3: set_a,Us: list_a] :
( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U2 @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) ) ) ) ) ).
% li_Cons
thf(fact_1248_independent__same__set,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ( set_a2 @ Us )
= ( set_a2 @ Vs2 ) )
=> ( ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ Vs2 ) ) ) ) ) ).
% independent_same_set
thf(fact_1249_independent_Ocases,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
=> ( ( A22 != nil_a )
=> ~ ! [U3: a,Us4: list_a] :
( ( A22
= ( cons_a @ U3 @ Us4 ) )
=> ( ( member_a @ U3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U3 @ ( embedded_Span_a_b @ r @ A1 @ Us4 ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ A1 @ Us4 ) ) ) ) ) ) ).
% independent.cases
thf(fact_1250_independent_Osimps,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
= ( ? [K5: set_a] :
( ( A1 = K5 )
& ( A22 = nil_a ) )
| ? [U: a,K5: set_a,Us5: list_a] :
( ( A1 = K5 )
& ( A22
= ( cons_a @ U @ Us5 ) )
& ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K5 @ Us5 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K5 @ Us5 ) ) ) ) ).
% independent.simps
thf(fact_1251_independent__rotate1__aux,axiom,
! [K3: set_a,U2: a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ ( append_a @ Us @ Vs2 ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ ( append_a @ Us @ ( cons_a @ U2 @ nil_a ) ) @ Vs2 ) ) ) ) ).
% independent_rotate1_aux
thf(fact_1252_independent__strict__incl,axiom,
! [K3: set_a,U2: a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) )
=> ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) ) ) ) ) ).
% independent_strict_incl
thf(fact_1253_filter__base,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ~ ! [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Vs4 )
=> ( ( embedded_Span_a_b @ r @ K3 @ Vs4 )
!= ( embedded_Span_a_b @ r @ K3 @ Us ) ) ) ) ) ) ).
% filter_base
thf(fact_1254_independent__replacement,axiom,
! [K3: set_a,U2: a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ ( cons_a @ U2 @ Us ) ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Vs2 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( cons_a @ X3 @ Us ) ) ) ) ) ) ) ).
% independent_replacement
thf(fact_1255_independent__length__le,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us ) @ ( size_size_list_a @ Vs2 ) ) ) ) ) ) ).
% independent_length_le
thf(fact_1256_replacement__theorem,axiom,
! [K3: set_a,Us6: list_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ Us6 @ Us ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K3 @ ( append_a @ Us6 @ Us ) ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) )
=> ? [Vs5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs5 ) @ ( set_a2 @ Vs2 ) )
& ( ( size_size_list_a @ Vs5 )
= ( size_size_list_a @ Us6 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ Vs5 @ Us ) ) ) ) ) ) ) ).
% replacement_theorem
thf(fact_1257_unique__decomposition,axiom,
! [K3: set_a,Us: list_a,A: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K3 @ Us ) )
=> ? [X3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X3 ) @ K3 )
& ( ( size_size_list_a @ X3 )
= ( size_size_list_a @ Us ) )
& ( A
= ( embedded_combine_a_b @ r @ X3 @ Us ) )
& ! [Y2: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ Y2 ) @ K3 )
& ( ( size_size_list_a @ Y2 )
= ( size_size_list_a @ Us ) )
& ( A
= ( embedded_combine_a_b @ r @ Y2 @ Us ) ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% unique_decomposition
thf(fact_1258_independent__rotate1,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ Us @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ ( rotate1_a @ Us ) @ Vs2 ) ) ) ) ).
% independent_rotate1
thf(fact_1259_trivial__combine__imp__independent,axiom,
! [K3: set_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K3 )
=> ( ( ( embedded_combine_a_b @ r @ Ks3 @ Us )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us ) @ Ks3 ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us ) ) ) ) ).
% trivial_combine_imp_independent
thf(fact_1260_combine__take,axiom,
! [Us: list_a,Ks: list_a] :
( ( embedded_combine_a_b @ r @ ( take_a @ ( size_size_list_a @ Us ) @ Ks ) @ Us )
= ( embedded_combine_a_b @ r @ Ks @ Us ) ) ).
% combine_take
thf(fact_1261_non__trivial__combine__imp__dependent,axiom,
! [K3: set_a,Ks: list_a,Us: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K3 )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us )
= ( zero_a_b @ r ) )
=> ( ~ ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us ) @ Ks ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us ) ) ) ) ) ).
% non_trivial_combine_imp_dependent
thf(fact_1262_independent__imp__trivial__combine,axiom,
! [K3: set_a,Us: list_a,Ks: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ Us )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K3 )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us ) @ Ks ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ).
% independent_imp_trivial_combine
thf(fact_1263_combine__normalize,axiom,
! [Ks: list_a,Us: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us )
= A )
=> ~ ! [Ks5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us ) @ Ks ) ) @ ( set_a2 @ Ks5 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks5 ) @ ( sup_sup_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us ) @ Ks ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( ( size_size_list_a @ Ks5 )
= ( size_size_list_a @ Us ) )
=> ( ( embedded_combine_a_b @ r @ Ks5 @ Us )
!= A ) ) ) ) ) ) ) ).
% combine_normalize
thf(fact_1264_independent__split_I3_J,axiom,
! [K3: set_a,Us: list_a,Vs2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K3 @ ( append_a @ Us @ Vs2 ) )
=> ( ( inf_inf_set_a @ ( embedded_Span_a_b @ r @ K3 @ Us ) @ ( embedded_Span_a_b @ r @ K3 @ Vs2 ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% independent_split(3)
thf(fact_1265_subalgebra__inter,axiom,
! [K3: set_a,V: set_a,V6: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ V6 @ r )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( inf_inf_set_a @ V @ V6 ) @ r ) ) ) ).
% subalgebra_inter
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_list_a @ ( bounde1002222742488328185ly_a_b @ r @ f @ n )
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ X ) @ n ) ) ) ) ).
%------------------------------------------------------------------------------