TPTP Problem File: SLH0157^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00090_003414__17011306_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1006 ( 309 unt; 182 typ; 0 def)
% Number of atoms : 2268 (1158 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 9230 ( 287 ~; 32 |; 103 &;7523 @)
% ( 0 <=>;1285 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 531 ( 531 >; 0 *; 0 +; 0 <<)
% Number of symbols : 161 ( 158 usr; 16 con; 0-4 aty)
% Number of variables : 2281 ( 44 ^;2172 !; 65 ?;2281 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:35:46.680
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2956882679547061052t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2670972154091845814t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_l2471972001652375325_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
list_list_nat_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
list_P3592885314253461005_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $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__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__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_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__Multiset__Omultiset_Itf__a_J,type,
multiset_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 (158)
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_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia2464479390973590831t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia5361259788508890537t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Coset_Oset__mult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
set_mu8047982887099575916xt_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Divisibility_Oessentially__equal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
essent8953798148185448568xt_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
thf(sy_c_Divisibility_Ofactors_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
factor5638265376665762323xt_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Divisibility_Oirreducible_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
irredu6211895646901577903xt_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid5798828371819920185xt_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Embedded__Algebras_Oring_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_001tf__a_001tf__b,type,
embedded_combine_a_b: partia2175431115845679010xt_a_b > list_a > list_a > a ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001tf__a_001tf__b,type,
embedd2795209813406577254on_a_b: partia2175431115845679010xt_a_b > nat > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oindependent_001tf__a_001tf__b,type,
embedd5208550302661555450nt_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Group_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_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_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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_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_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_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
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_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
append7679239579558125090_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odrop_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
drop_nat_int: nat > list_nat_int > list_nat_int ).
thf(sy_c_List_Odrop_001_062_It__Nat__Onat_Mtf__a_J,type,
drop_nat_a: nat > list_nat_a > list_nat_a ).
thf(sy_c_List_Odrop_001t__Int__Oint,type,
drop_int: nat > list_int > list_int ).
thf(sy_c_List_Odrop_001t__List__Olist_Itf__a_J,type,
drop_list_a: nat > list_list_a > list_list_a ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
drop_P2883665741211355575_a_nat: nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).
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_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
cons_list_nat_int: list_nat_int > list_list_nat_int > list_list_nat_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
cons_l2046435710214046167_a_nat: list_P3592885314253461005_a_nat > list_l2471972001652375325_a_nat > list_l2471972001652375325_a_nat ).
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_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
cons_P5205166803686508359_a_nat: product_prod_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).
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_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
nil_list_nat_int: list_list_nat_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
nil_li191968740515856775_a_nat: list_l2471972001652375325_a_nat ).
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_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
nil_Pr7402525243500994295_a_nat: list_P3592885314253461005_a_nat ).
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_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
hd_Pro8935205257713178578_a_nat: list_P3592885314253461005_a_nat > product_prod_a_nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
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_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
set_Pr924983374503034536_a_nat: list_P3592885314253461005_a_nat > set_Pr4934435412358123699_a_nat ).
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_Onth_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
nth_nat_int: list_nat_int > nat > nat > int ).
thf(sy_c_List_Onth_001_062_It__Nat__Onat_Mtf__a_J,type,
nth_nat_a: list_nat_a > nat > nat > a ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
nth_Pr8461465654520414006_a_nat: list_P3592885314253461005_a_nat > nat > product_prod_a_nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oreplicate_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
replicate_nat_int: nat > ( nat > int ) > list_nat_int ).
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__Int__Oint,type,
replicate_int: nat > int > list_int ).
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_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
replic5595554873386817213_a_nat: nat > product_prod_a_nat > list_P3592885314253461005_a_nat ).
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_Nat_OSuc,type,
suc: nat > nat ).
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_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
size_s984997627204368545_a_nat: list_P3592885314253461005_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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_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_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le8666007276011122963_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > $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_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
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_001tf__a_001tf__b,type,
polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Oroots__on_001tf__a_001tf__b,type,
polyno5714441830345289050on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > multiset_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted__on_001tf__a_001tf__b,type,
polyno2453258491555121552on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno1315193887021588240t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001tf__a_001tf__b,type,
polynomial_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomials_Oring_Ocoeff_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
coeff_6360649920519955023t_unit: partia2670972154091845814t_unit > list_list_a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Ocoeff_001tf__a_001tf__b,type,
coeff_a_b: partia2175431115845679010xt_a_b > list_a > nat > a ).
thf(sy_c_Polynomials_Oring_Ocoeff__rel_001tf__a,type,
coeff_rel_a: list_a > list_a > $o ).
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_Odense__repr_001tf__a_001tf__b,type,
dense_repr_a_b: partia2175431115845679010xt_a_b > list_a > list_P3592885314253461005_a_nat ).
thf(sy_c_Polynomials_Oring_Odense__repr__rel_001tf__a_001tf__b,type,
dense_repr_rel_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
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_Oeval__rel_001tf__a,type,
eval_rel_a: list_a > list_a > $o ).
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_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_Onormalize__rel_001tf__a_001tf__b,type,
normalize_rel_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
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_Oring_Opoly__of__dense_001tf__a_001tf__b,type,
poly_of_dense_a_b: partia2175431115845679010xt_a_b > list_P3592885314253461005_a_nat > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
product_Pair_a_nat: a > nat > product_prod_a_nat ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $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_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_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_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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_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_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_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_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_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_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_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x,type,
x: list_a ).
% Relevant facts (818)
thf(fact_0_b,axiom,
( ( hd_a @ x )
!= ( zero_a_b @ r ) ) ).
% b
thf(fact_1_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_2_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_3_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I: nat] : ( if_a @ ( I = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_4_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_5_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_6_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_7_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_8_assms_I2_J,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ n @ K )
=> ( ( coeff_a_b @ r @ x @ K )
= ( zero_a_b @ r ) ) ) ).
% assms(2)
thf(fact_9_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_10_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_11_coeff__iff__polynomial__cond,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P2 ) ) ) ) ) ).
% coeff_iff_polynomial_cond
thf(fact_12_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_13_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_14_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_15_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_16_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_17_ring_Olead__coeff__simp,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( P != nil_a )
=> ( ( coeff_a_b @ R @ P @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
= ( hd_a @ P ) ) ) ) ).
% ring.lead_coeff_simp
thf(fact_18_ring_Ocoeff__degree,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I2: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ I2 )
=> ( ( coeff_6360649920519955023t_unit @ R @ P @ I2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_degree
thf(fact_19_ring_Ocoeff__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I2: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.coeff_degree
thf(fact_20_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_21_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_22_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_23_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_24_coeff__nth,axiom,
! [I2: nat,P: list_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ P ) )
=> ( ( coeff_a_b @ r @ P @ I2 )
= ( nth_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I2 ) ) ) ) ).
% coeff_nth
thf(fact_25_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 ) )
= ( ^ [I: nat] : ( if_list_a @ ( I = N ) @ A @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_26_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 ) )
= ( ^ [I: nat] : ( if_a @ ( I = N ) @ A @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_27_append__coeff,axiom,
! [P: list_a,Q: list_a] :
( ( coeff_a_b @ r @ ( append_a @ P @ Q ) )
= ( ^ [I: nat] : ( if_a @ ( ord_less_nat @ I @ ( size_size_list_a @ Q ) ) @ ( coeff_a_b @ r @ Q @ I ) @ ( coeff_a_b @ r @ P @ ( minus_minus_nat @ I @ ( size_size_list_a @ Q ) ) ) ) ) ) ).
% append_coeff
thf(fact_28_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_29_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_30_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_31_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P1 )
=> ( ( polynomial_a_b @ R @ K2 @ P2 )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P2 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_32_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_33_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_34_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_35_normalize__coeff,axiom,
! [P: list_a] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_coeff
thf(fact_36_mem__Collect__eq,axiom,
! [A: a,P3: a > $o] :
( ( member_a @ A @ ( collect_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_37_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_38_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_39_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_40_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_41_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_42_normalize__polynomial,axiom,
! [K2: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( normalize_a_b @ r @ P )
= P ) ) ).
% normalize_polynomial
thf(fact_43_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_44_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_45_ring_Onormalize__idem,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( normalize_a_b @ R @ P ) @ Q ) )
= ( normalize_a_b @ R @ ( append_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_46_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_47_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_48_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_49_ring_Onormalize__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ( normalize_a_b @ R @ P )
= P ) ) ) ).
% ring.normalize_polynomial
thf(fact_50_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_51_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_52_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_53_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_54_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_55_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_56_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_57_size__neq__size__imp__neq,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X )
!= ( size_size_multiset_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_58_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_59_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P3 @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P3 @ M ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_60_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P3 @ M ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_61_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_62_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_63_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_64_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_65_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_66_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B: nat] :
( ( P3 @ K )
=> ( ! [Y2: nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P3 @ X3 )
& ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_67_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_68_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_69_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_70_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_71_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_72_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_73_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_74_ring_Oappend__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( append_a @ P @ Q ) )
= ( ^ [I: nat] : ( if_a @ ( ord_less_nat @ I @ ( size_size_list_a @ Q ) ) @ ( coeff_a_b @ R @ Q @ I ) @ ( coeff_a_b @ R @ P @ ( minus_minus_nat @ I @ ( size_size_list_a @ Q ) ) ) ) ) ) ) ).
% ring.append_coeff
thf(fact_75_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_76_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_77_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_78_ring_Ocoeff__nth,axiom,
! [R: partia2175431115845679010xt_a_b,I2: nat,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ P ) )
=> ( ( coeff_a_b @ R @ P @ I2 )
= ( nth_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I2 ) ) ) ) ) ).
% ring.coeff_nth
thf(fact_79_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_80_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_81_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_82_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_83_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_84_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_85_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_86_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_87_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_88_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_89_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_90_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_91_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_92_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_93_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_94_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_95_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_96_ring_Ozero__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( polynomial_a_b @ R @ K2 @ nil_a ) ) ).
% ring.zero_is_polynomial
thf(fact_97_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_98_hd__append2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != nil_int )
=> ( ( hd_int @ ( append_int @ Xs @ Ys ) )
= ( hd_int @ Xs ) ) ) ).
% hd_append2
thf(fact_99_hd__append2,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( Xs != nil_nat_int )
=> ( ( hd_nat_int @ ( append_nat_int @ Xs @ Ys ) )
= ( hd_nat_int @ Xs ) ) ) ).
% hd_append2
thf(fact_100_hd__append2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ( ( hd_Pro8935205257713178578_a_nat @ ( append7679239579558125090_a_nat @ Xs @ Ys ) )
= ( hd_Pro8935205257713178578_a_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_101_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_102_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_103_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_104_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_105_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_106_append__is__Nil__conv,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( ( append_nat_int @ Xs @ Ys )
= nil_nat_int )
= ( ( Xs = nil_nat_int )
& ( Ys = nil_nat_int ) ) ) ).
% append_is_Nil_conv
thf(fact_107_append__is__Nil__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
= nil_Pr7402525243500994295_a_nat )
= ( ( Xs = nil_Pr7402525243500994295_a_nat )
& ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_108_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_109_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_110_Nil__is__append__conv,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( nil_nat_int
= ( append_nat_int @ Xs @ Ys ) )
= ( ( Xs = nil_nat_int )
& ( Ys = nil_nat_int ) ) ) ).
% Nil_is_append_conv
thf(fact_111_Nil__is__append__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( append7679239579558125090_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr7402525243500994295_a_nat )
& ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_112_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_113_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_114_self__append__conv2,axiom,
! [Y: list_nat_int,Xs: list_nat_int] :
( ( Y
= ( append_nat_int @ Xs @ Y ) )
= ( Xs = nil_nat_int ) ) ).
% self_append_conv2
thf(fact_115_self__append__conv2,axiom,
! [Y: list_P3592885314253461005_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( Y
= ( append7679239579558125090_a_nat @ Xs @ Y ) )
= ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv2
thf(fact_116_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_117_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_118_append__self__conv2,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( ( append_nat_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat_int ) ) ).
% append_self_conv2
thf(fact_119_append__self__conv2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv2
thf(fact_120_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_121_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_122_self__append__conv,axiom,
! [Y: list_nat_int,Ys: list_nat_int] :
( ( Y
= ( append_nat_int @ Y @ Ys ) )
= ( Ys = nil_nat_int ) ) ).
% self_append_conv
thf(fact_123_self__append__conv,axiom,
! [Y: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Y
= ( append7679239579558125090_a_nat @ Y @ Ys ) )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv
thf(fact_124_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_125_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_126_append__self__conv,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( ( append_nat_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat_int ) ) ).
% append_self_conv
thf(fact_127_append__self__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv
thf(fact_128_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_129_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_130_append__Nil2,axiom,
! [Xs: list_nat_int] :
( ( append_nat_int @ Xs @ nil_nat_int )
= Xs ) ).
% append_Nil2
thf(fact_131_append__Nil2,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ Xs @ nil_Pr7402525243500994295_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_132_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_133_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_134_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_135_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_136_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_137_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_138_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_139_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_140_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_141_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_142_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_143_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_144_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_145_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_146_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_147_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_148_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_149_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_150_append_Oright__neutral,axiom,
! [A: list_nat_int] :
( ( append_nat_int @ A @ nil_nat_int )
= A ) ).
% append.right_neutral
thf(fact_151_append_Oright__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ A @ nil_Pr7402525243500994295_a_nat )
= A ) ).
% append.right_neutral
thf(fact_152_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_153_length__0__conv,axiom,
! [Xs: list_nat_int] :
( ( ( size_s5718426915756887103at_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat_int ) ) ).
% length_0_conv
thf(fact_154_length__0__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).
% length_0_conv
thf(fact_155_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_156_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_157_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_158_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_159_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_160_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_161_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_162_length__greater__0__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s984997627204368545_a_nat @ Xs ) )
= ( Xs != nil_Pr7402525243500994295_a_nat ) ) ).
% length_greater_0_conv
thf(fact_163_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_164_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_165_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_166_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_167_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_168_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_169_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_170_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_171_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_172_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_173_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_174_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_175_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_176_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P3 @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P3 @ M ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_177_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_178_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_179_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_180_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_181_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_182_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_183_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_184_list_Osize_I3_J,axiom,
( ( size_s5718426915756887103at_int @ nil_nat_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_185_list_Osize_I3_J,axiom,
( ( size_s984997627204368545_a_nat @ nil_Pr7402525243500994295_a_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_186_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_187_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P3 @ I4 ) )
& ( P3 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_188_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_189_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_190_hd__conv__nth,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( hd_int @ Xs )
= ( nth_int @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_191_hd__conv__nth,axiom,
! [Xs: list_nat_int] :
( ( Xs != nil_nat_int )
=> ( ( hd_nat_int @ Xs )
= ( nth_nat_int @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_192_hd__conv__nth,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ( ( hd_Pro8935205257713178578_a_nat @ Xs )
= ( nth_Pr8461465654520414006_a_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_193_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_194_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_195_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_196_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_197_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_198_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_199_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_200_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_201_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_202_append__Nil,axiom,
! [Ys: list_nat_int] :
( ( append_nat_int @ nil_nat_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_203_append__Nil,axiom,
! [Ys: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_204_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_205_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_206_append_Oleft__neutral,axiom,
! [A: list_nat_int] :
( ( append_nat_int @ nil_nat_int @ A )
= A ) ).
% append.left_neutral
thf(fact_207_append_Oleft__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_208_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_209_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_210_eq__Nil__appendI,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat_int @ nil_nat_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_211_eq__Nil__appendI,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_212_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_213_Skolem__list__nth,axiom,
! [K: nat,P3: nat > a > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: a] : ( P3 @ I @ X4 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P3 @ I @ ( nth_a @ Xs3 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_214_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I )
= ( nth_a @ Ys3 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_215_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_216_hd__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( Xs = nil_int )
=> ( ( hd_int @ ( append_int @ Xs @ Ys ) )
= ( hd_int @ Ys ) ) )
& ( ( Xs != nil_int )
=> ( ( hd_int @ ( append_int @ Xs @ Ys ) )
= ( hd_int @ Xs ) ) ) ) ).
% hd_append
thf(fact_217_hd__append,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
( ( ( Xs = nil_nat_int )
=> ( ( hd_nat_int @ ( append_nat_int @ Xs @ Ys ) )
= ( hd_nat_int @ Ys ) ) )
& ( ( Xs != nil_nat_int )
=> ( ( hd_nat_int @ ( append_nat_int @ Xs @ Ys ) )
= ( hd_nat_int @ Xs ) ) ) ) ).
% hd_append
thf(fact_218_hd__append,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( Xs = nil_Pr7402525243500994295_a_nat )
=> ( ( hd_Pro8935205257713178578_a_nat @ ( append7679239579558125090_a_nat @ Xs @ Ys ) )
= ( hd_Pro8935205257713178578_a_nat @ Ys ) ) )
& ( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ( ( hd_Pro8935205257713178578_a_nat @ ( append7679239579558125090_a_nat @ Xs @ Ys ) )
= ( hd_Pro8935205257713178578_a_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_219_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys4 ) )
& ( ( Xs4 = nil_a )
| ( Ys4 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_220_longest__common__prefix,axiom,
! [Xs: list_int,Ys: list_int] :
? [Ps: list_int,Xs4: list_int,Ys4: list_int] :
( ( Xs
= ( append_int @ Ps @ Xs4 ) )
& ( Ys
= ( append_int @ Ps @ Ys4 ) )
& ( ( Xs4 = nil_int )
| ( Ys4 = nil_int )
| ( ( hd_int @ Xs4 )
!= ( hd_int @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_221_longest__common__prefix,axiom,
! [Xs: list_nat_int,Ys: list_nat_int] :
? [Ps: list_nat_int,Xs4: list_nat_int,Ys4: list_nat_int] :
( ( Xs
= ( append_nat_int @ Ps @ Xs4 ) )
& ( Ys
= ( append_nat_int @ Ps @ Ys4 ) )
& ( ( Xs4 = nil_nat_int )
| ( Ys4 = nil_nat_int )
| ( ( hd_nat_int @ Xs4 )
!= ( hd_nat_int @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_222_longest__common__prefix,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
? [Ps: list_P3592885314253461005_a_nat,Xs4: list_P3592885314253461005_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( Xs
= ( append7679239579558125090_a_nat @ Ps @ Xs4 ) )
& ( Ys
= ( append7679239579558125090_a_nat @ Ps @ Ys4 ) )
& ( ( Xs4 = nil_Pr7402525243500994295_a_nat )
| ( Ys4 = nil_Pr7402525243500994295_a_nat )
| ( ( hd_Pro8935205257713178578_a_nat @ Xs4 )
!= ( hd_Pro8935205257713178578_a_nat @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_223_splitted__on__def,axiom,
! [K2: set_a,P: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K2 @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K2 @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_224_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_225_a,axiom,
~ ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ x ) @ one_one_nat ) @ n )
| ( x
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% a
thf(fact_226_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_227_normalize__gives__polynomial,axiom,
! [P: list_a,K2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_gives_polynomial
thf(fact_228_ring_Osubring__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_229_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_230_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_231_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_232_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_233_polynomial__incl,axiom,
! [K2: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 ) ) ).
% polynomial_incl
thf(fact_234_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_235_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_236_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_237_replicate__eq__replicate,axiom,
! [M2: nat,X: a,N: nat,Y: a] :
( ( ( replicate_a @ M2 @ X )
= ( replicate_a @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2 != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_238_length__replicate,axiom,
! [N: nat,X: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_239_drop0,axiom,
( ( drop_a @ zero_zero_nat )
= ( ^ [X2: list_a] : X2 ) ) ).
% drop0
thf(fact_240_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_241_in__set__replicate,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( replicate_list_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_242_in__set__replicate,axiom,
! [X: nat > a,N: nat,Y: nat > a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ ( replicate_nat_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_243_in__set__replicate,axiom,
! [X: a,N: nat,Y: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_244_Bex__set__replicate,axiom,
! [N: nat,A: a,P3: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
& ( P3 @ X2 ) ) )
= ( ( P3 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_245_Ball__set__replicate,axiom,
! [N: nat,A: a,P3: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
=> ( P3 @ X2 ) ) )
= ( ( P3 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_246_replicate__empty,axiom,
! [N: nat,X: int] :
( ( ( replicate_int @ N @ X )
= nil_int )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_247_replicate__empty,axiom,
! [N: nat,X: nat > int] :
( ( ( replicate_nat_int @ N @ X )
= nil_nat_int )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_248_replicate__empty,axiom,
! [N: nat,X: product_prod_a_nat] :
( ( ( replic5595554873386817213_a_nat @ N @ X )
= nil_Pr7402525243500994295_a_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_249_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_250_empty__replicate,axiom,
! [N: nat,X: int] :
( ( nil_int
= ( replicate_int @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_251_empty__replicate,axiom,
! [N: nat,X: nat > int] :
( ( nil_nat_int
= ( replicate_nat_int @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_252_empty__replicate,axiom,
! [N: nat,X: product_prod_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( replic5595554873386817213_a_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_253_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_254_length__drop,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% length_drop
thf(fact_255_hd__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( hd_a @ ( replicate_a @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_256_nth__replicate,axiom,
! [I2: nat,N: nat,X: a] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_a @ ( replicate_a @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_257_drop__replicate,axiom,
! [I2: nat,K: nat,X: a] :
( ( drop_a @ I2 @ ( replicate_a @ K @ X ) )
= ( replicate_a @ ( minus_minus_nat @ K @ I2 ) @ X ) ) ).
% drop_replicate
thf(fact_258_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_259_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_int] :
( ( nil_int
= ( drop_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_260_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat_int] :
( ( nil_nat_int
= ( drop_nat_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s5718426915756887103at_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_261_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( drop_P2883665741211355575_a_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_262_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_263_drop__eq__Nil,axiom,
! [N: nat,Xs: list_int] :
( ( ( drop_int @ N @ Xs )
= nil_int )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_264_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat_int] :
( ( ( drop_nat_int @ N @ Xs )
= nil_nat_int )
= ( ord_less_eq_nat @ ( size_s5718426915756887103at_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_265_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P3592885314253461005_a_nat] :
( ( ( drop_P2883665741211355575_a_nat @ N @ Xs )
= nil_Pr7402525243500994295_a_nat )
= ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_266_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_267_drop__all,axiom,
! [Xs: list_int,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N )
=> ( ( drop_int @ N @ Xs )
= nil_int ) ) ).
% drop_all
thf(fact_268_drop__all,axiom,
! [Xs: list_nat_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s5718426915756887103at_int @ Xs ) @ N )
=> ( ( drop_nat_int @ N @ Xs )
= nil_nat_int ) ) ).
% drop_all
thf(fact_269_drop__all,axiom,
! [Xs: list_P3592885314253461005_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs ) @ N )
=> ( ( drop_P2883665741211355575_a_nat @ N @ Xs )
= nil_Pr7402525243500994295_a_nat ) ) ).
% drop_all
thf(fact_270_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_271_drop__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_272_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_273_assms_I1_J,axiom,
member_list_a @ x @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% assms(1)
thf(fact_274_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_275_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_276_polynomialI,axiom,
! [P: list_a,K2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 )
=> ( ( ( hd_a @ P )
!= ( zero_a_b @ r ) )
=> ( polynomial_a_b @ r @ K2 @ P ) ) ) ).
% polynomialI
thf(fact_277_subset__code_I1_J,axiom,
! [Xs: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B2 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_278_subset__code_I1_J,axiom,
! [Xs: list_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( set_nat_a2 @ Xs ) @ B2 )
= ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( set_nat_a2 @ Xs ) )
=> ( member_nat_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_279_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_280_set__drop__subset,axiom,
! [N: nat,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ N @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_drop_subset
thf(fact_281_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ M2 @ Xs ) ) @ ( set_a2 @ ( drop_a @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_282_replicate__length__same,axiom,
! [Xs: list_a,X: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_283_replicate__eqI,axiom,
! [Xs: list_list_a,N: nat,X: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_list_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_284_replicate__eqI,axiom,
! [Xs: list_nat_a,N: nat,X: nat > a] :
( ( ( size_size_list_nat_a @ Xs )
= N )
=> ( ! [Y2: nat > a] :
( ( member_nat_a @ Y2 @ ( set_nat_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_nat_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_285_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_286_in__set__dropD,axiom,
! [X: list_a,N: nat,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( drop_list_a @ N @ Xs ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_287_in__set__dropD,axiom,
! [X: nat > a,N: nat,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ ( drop_nat_a @ N @ Xs ) ) )
=> ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_288_in__set__dropD,axiom,
! [X: a,N: nat,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( drop_a @ N @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_289_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_290_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_291_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_292_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_293_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_294_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_295_append__replicate__commute,axiom,
! [N: nat,X: a,K: nat] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K @ X ) )
= ( append_a @ ( replicate_a @ K @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_296_drop__0,axiom,
! [Xs: list_a] :
( ( drop_a @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_297_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_298_drop__Nil,axiom,
! [N: nat] :
( ( drop_int @ N @ nil_int )
= nil_int ) ).
% drop_Nil
thf(fact_299_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat_int @ N @ nil_nat_int )
= nil_nat_int ) ).
% drop_Nil
thf(fact_300_drop__Nil,axiom,
! [N: nat] :
( ( drop_P2883665741211355575_a_nat @ N @ nil_Pr7402525243500994295_a_nat )
= nil_Pr7402525243500994295_a_nat ) ).
% drop_Nil
thf(fact_301_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_302_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_303_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_304_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_305_ring_Osubring__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_306_replicate__0,axiom,
! [X: int] :
( ( replicate_int @ zero_zero_nat @ X )
= nil_int ) ).
% replicate_0
thf(fact_307_replicate__0,axiom,
! [X: nat > int] :
( ( replicate_nat_int @ zero_zero_nat @ X )
= nil_nat_int ) ).
% replicate_0
thf(fact_308_replicate__0,axiom,
! [X: product_prod_a_nat] :
( ( replic5595554873386817213_a_nat @ zero_zero_nat @ X )
= nil_Pr7402525243500994295_a_nat ) ).
% replicate_0
thf(fact_309_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_310_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_311_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_312_hd__in__set,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( member_int @ ( hd_int @ Xs ) @ ( set_int2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_313_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_314_hd__in__set,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ( member5724188588386418708_a_nat @ ( hd_Pro8935205257713178578_a_nat @ Xs ) @ ( set_Pr924983374503034536_a_nat @ Xs ) ) ) ).
% hd_in_set
thf(fact_315_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_316_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_317_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_318_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_319_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_320_list_Oset__sel_I1_J,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( A != nil_Pr7402525243500994295_a_nat )
=> ( member5724188588386418708_a_nat @ ( hd_Pro8935205257713178578_a_nat @ A ) @ ( set_Pr924983374503034536_a_nat @ A ) ) ) ).
% list.set_sel(1)
thf(fact_321_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_322_ring_Opolynomial__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 ) ) ) ).
% ring.polynomial_incl
thf(fact_323_ring_Onormalize__def_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ R ) ) @ ( drop_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ P ) ) ) ) ).
% ring.normalize_def'(1)
thf(fact_324_ring_Onormalize__def_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( 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 ) ) ) ) ).
% ring.normalize_def'(1)
thf(fact_325_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_326_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_327_length__pos__if__in__set,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_328_length__pos__if__in__set,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_329_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_330_all__set__conv__all__nth,axiom,
! [Xs: list_a,P3: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P3 @ X2 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( P3 @ ( nth_a @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_331_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P3: list_a > $o,X: list_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P3 @ ( nth_list_a @ Xs @ I3 ) ) )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_332_all__nth__imp__all__set,axiom,
! [Xs: list_nat_a,P3: ( nat > a ) > $o,X: nat > a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat_a @ Xs ) )
=> ( P3 @ ( nth_nat_a @ Xs @ I3 ) ) )
=> ( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_333_all__nth__imp__all__set,axiom,
! [Xs: list_a,P3: a > $o,X: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P3 @ ( nth_a @ Xs @ I3 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_334_in__set__conv__nth,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_335_in__set__conv__nth,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat_a @ Xs ) )
& ( ( nth_nat_a @ Xs @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_336_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_337_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P3: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P3 @ X3 ) )
=> ( P3 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_338_nth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_339_nth__mem,axiom,
! [N: nat,Xs: list_nat_a] :
( ( ord_less_nat @ N @ ( size_size_list_nat_a @ Xs ) )
=> ( member_nat_a @ ( nth_nat_a @ Xs @ N ) @ ( set_nat_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_340_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_341_ring_Onormalize__gives__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 )
=> ( polynomial_a_b @ R @ K2 @ ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.normalize_gives_polynomial
thf(fact_342_ring_Onormalize__replicate__zero,axiom,
! [R: partia2670972154091845814t_unit,N: nat,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P ) )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ).
% ring.normalize_replicate_zero
thf(fact_343_ring_Onormalize__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,P: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P ) )
= ( normalize_a_b @ R @ P ) ) ) ).
% ring.normalize_replicate_zero
thf(fact_344_ring_Oprefix__replicate__zero__coeff,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ P )
= ( coeff_6360649920519955023t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P ) ) ) ) ).
% ring.prefix_replicate_zero_coeff
thf(fact_345_ring_Oprefix__replicate__zero__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ P )
= ( coeff_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P ) ) ) ) ).
% ring.prefix_replicate_zero_coeff
thf(fact_346_ring_Onormalize__def_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ).
% ring.normalize_def'(2)
thf(fact_347_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( hd_a @ ( drop_a @ N @ Xs ) )
= ( nth_a @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_348_polynomial__def,axiom,
( polyno1315193887021588240t_unit
= ( ^ [R2: partia2670972154091845814t_unit,K4: set_list_a,P4: list_list_a] :
( ( P4 = nil_list_a )
| ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ K4 )
& ( ( hd_list_a @ P4 )
!= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ).
% polynomial_def
thf(fact_349_polynomial__def,axiom,
( polynomial_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,K4: set_a,P4: list_a] :
( ( P4 = nil_a )
| ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ K4 )
& ( ( hd_a @ P4 )
!= ( zero_a_b @ R2 ) ) ) ) ) ) ).
% polynomial_def
thf(fact_350_ring_OpolynomialI,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ K2 )
=> ( ( ( hd_list_a @ P )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( polyno1315193887021588240t_unit @ R @ K2 @ P ) ) ) ) ).
% ring.polynomialI
thf(fact_351_ring_OpolynomialI,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 )
=> ( ( ( hd_a @ P )
!= ( zero_a_b @ R ) )
=> ( polynomial_a_b @ R @ K2 @ P ) ) ) ) ).
% ring.polynomialI
thf(fact_352_ring_Onormalize__trick,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ R ) ) @ ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.normalize_trick
thf(fact_353_ring_Onormalize__trick,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( 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 ) ) ) ) ).
% ring.normalize_trick
thf(fact_354_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_355_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_356_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_357_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_358_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_359_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_360_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_361_drop__exp__base,axiom,
! [N: nat,X: a,M2: nat] :
( ( drop_a @ N @ ( polyno2922411391617481336se_a_b @ r @ X @ M2 ) )
= ( polyno2922411391617481336se_a_b @ r @ X @ ( minus_minus_nat @ M2 @ N ) ) ) ).
% drop_exp_base
thf(fact_362_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_363_ring_Osplitted__on__def,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polyno2453258491555121552on_a_b @ R @ K2 @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ R @ K2 @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% ring.splitted_on_def
thf(fact_364_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_365_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 )
=> ~ ! [P5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P5 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_366_poly__add__append__replicate,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_367_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_368_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_369_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_370_list_Oinject,axiom,
! [X21: nat > int,X22: list_nat_int,Y21: nat > int,Y22: list_nat_int] :
( ( ( cons_nat_int @ X21 @ X22 )
= ( cons_nat_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_371_list_Oinject,axiom,
! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat,Y21: product_prod_a_nat,Y22: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X21 @ X22 )
= ( cons_P5205166803686508359_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_degree__oneE,axiom,
! [P: list_a,K2: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ! [B3: a] :
( ( member_a @ B3 @ K2 )
=> ( P
!= ( cons_a @ A3 @ ( cons_a @ B3 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_384_poly__add__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_385_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_386_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_387_append1__eq__conv,axiom,
! [Xs: list_nat_int,X: nat > int,Ys: list_nat_int,Y: nat > int] :
( ( ( append_nat_int @ Xs @ ( cons_nat_int @ X @ nil_nat_int ) )
= ( append_nat_int @ Ys @ ( cons_nat_int @ Y @ nil_nat_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_388_append1__eq__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,X: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ ( cons_P5205166803686508359_a_nat @ X @ nil_Pr7402525243500994295_a_nat ) )
= ( append7679239579558125090_a_nat @ Ys @ ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_389_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_390_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_391_nth__Cons__0,axiom,
! [X: nat > int,Xs: list_nat_int] :
( ( nth_nat_int @ ( cons_nat_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_392_nth__Cons__0,axiom,
! [X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( nth_Pr8461465654520414006_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_393_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_394_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_395_nth__append__length,axiom,
! [Xs: list_nat_int,X: nat > int,Ys: list_nat_int] :
( ( nth_nat_int @ ( append_nat_int @ Xs @ ( cons_nat_int @ X @ Ys ) ) @ ( size_s5718426915756887103at_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_396_nth__append__length,axiom,
! [Xs: list_P3592885314253461005_a_nat,X: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( nth_Pr8461465654520414006_a_nat @ ( append7679239579558125090_a_nat @ Xs @ ( cons_P5205166803686508359_a_nat @ X @ Ys ) ) @ ( size_s984997627204368545_a_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_397_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_398_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_399_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_400_nth__Cons__pos,axiom,
! [N: nat,X: nat > int,Xs: list_nat_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat_int @ ( cons_nat_int @ X @ Xs ) @ N )
= ( nth_nat_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_401_nth__Cons__pos,axiom,
! [N: nat,X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8461465654520414006_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) @ N )
= ( nth_Pr8461465654520414006_a_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_402_ring_Opoly__add_Ocong,axiom,
poly_add_a_b = poly_add_a_b ).
% ring.poly_add.cong
thf(fact_403_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_404_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_405_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_406_not__Cons__self2,axiom,
! [X: nat > int,Xs: list_nat_int] :
( ( cons_nat_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_407_not__Cons__self2,axiom,
! [X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( cons_P5205166803686508359_a_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_408_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_409_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_410_transpose_Ocases,axiom,
! [X: list_list_nat_int] :
( ( X != nil_list_nat_int )
=> ( ! [Xss: list_list_nat_int] :
( X
!= ( cons_list_nat_int @ nil_nat_int @ Xss ) )
=> ~ ! [X3: nat > int,Xs2: list_nat_int,Xss: list_list_nat_int] :
( X
!= ( cons_list_nat_int @ ( cons_nat_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_411_transpose_Ocases,axiom,
! [X: list_l2471972001652375325_a_nat] :
( ( X != nil_li191968740515856775_a_nat )
=> ( ! [Xss: list_l2471972001652375325_a_nat] :
( X
!= ( cons_l2046435710214046167_a_nat @ nil_Pr7402525243500994295_a_nat @ Xss ) )
=> ~ ! [X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Xss: list_l2471972001652375325_a_nat] :
( X
!= ( cons_l2046435710214046167_a_nat @ ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_412_set__ConsD,axiom,
! [Y: list_a,X: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_413_set__ConsD,axiom,
! [Y: nat > a,X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ Y @ ( set_nat_a2 @ ( cons_nat_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat_a @ Y @ ( set_nat_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_414_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_415_set__ConsD,axiom,
! [Y: int,X: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_416_set__ConsD,axiom,
! [Y: nat > int,X: nat > int,Xs: list_nat_int] :
( ( member_nat_int @ Y @ ( set_nat_int2 @ ( cons_nat_int @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat_int @ Y @ ( set_nat_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_417_set__ConsD,axiom,
! [Y: product_prod_a_nat,X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_418_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z2: list_list_a] :
( A
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_419_list_Oset__cases,axiom,
! [E: nat > a,A: list_nat_a] :
( ( member_nat_a @ E @ ( set_nat_a2 @ A ) )
=> ( ! [Z2: list_nat_a] :
( A
!= ( cons_nat_a @ E @ Z2 ) )
=> ~ ! [Z1: nat > a,Z2: list_nat_a] :
( ( A
= ( cons_nat_a @ Z1 @ Z2 ) )
=> ~ ( member_nat_a @ E @ ( set_nat_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_420_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z2: list_a] :
( A
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_421_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int @ E @ ( set_int2 @ A ) )
=> ( ! [Z2: list_int] :
( A
!= ( cons_int @ E @ Z2 ) )
=> ~ ! [Z1: int,Z2: list_int] :
( ( A
= ( cons_int @ Z1 @ Z2 ) )
=> ~ ( member_int @ E @ ( set_int2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_422_list_Oset__cases,axiom,
! [E: nat > int,A: list_nat_int] :
( ( member_nat_int @ E @ ( set_nat_int2 @ A ) )
=> ( ! [Z2: list_nat_int] :
( A
!= ( cons_nat_int @ E @ Z2 ) )
=> ~ ! [Z1: nat > int,Z2: list_nat_int] :
( ( A
= ( cons_nat_int @ Z1 @ Z2 ) )
=> ~ ( member_nat_int @ E @ ( set_nat_int2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_423_list_Oset__cases,axiom,
! [E: product_prod_a_nat,A: list_P3592885314253461005_a_nat] :
( ( member5724188588386418708_a_nat @ E @ ( set_Pr924983374503034536_a_nat @ A ) )
=> ( ! [Z2: list_P3592885314253461005_a_nat] :
( A
!= ( cons_P5205166803686508359_a_nat @ E @ Z2 ) )
=> ~ ! [Z1: product_prod_a_nat,Z2: list_P3592885314253461005_a_nat] :
( ( A
= ( cons_P5205166803686508359_a_nat @ Z1 @ Z2 ) )
=> ~ ( member5724188588386418708_a_nat @ E @ ( set_Pr924983374503034536_a_nat @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_424_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_425_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_426_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_427_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_428_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_429_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] : ( member5724188588386418708_a_nat @ X21 @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_430_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_431_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_432_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_433_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_434_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_435_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_nat,X22: list_P3592885314253461005_a_nat,X21: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ X22 ) )
=> ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_436_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_437_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_438_list_Odistinct_I1_J,axiom,
! [X21: nat > int,X22: list_nat_int] :
( nil_nat_int
!= ( cons_nat_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_439_list_Odistinct_I1_J,axiom,
! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( nil_Pr7402525243500994295_a_nat
!= ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_440_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_441_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_442_list_OdiscI,axiom,
! [List: list_nat_int,X21: nat > int,X22: list_nat_int] :
( ( List
= ( cons_nat_int @ X21 @ X22 ) )
=> ( List != nil_nat_int ) ) ).
% list.discI
thf(fact_443_list_OdiscI,axiom,
! [List: list_P3592885314253461005_a_nat,X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( ( List
= ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) )
=> ( List != nil_Pr7402525243500994295_a_nat ) ) ).
% list.discI
thf(fact_444_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_445_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_446_list_Oexhaust,axiom,
! [Y: list_nat_int] :
( ( Y != nil_nat_int )
=> ~ ! [X212: nat > int,X222: list_nat_int] :
( Y
!= ( cons_nat_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_447_list_Oexhaust,axiom,
! [Y: list_P3592885314253461005_a_nat] :
( ( Y != nil_Pr7402525243500994295_a_nat )
=> ~ ! [X212: product_prod_a_nat,X222: list_P3592885314253461005_a_nat] :
( Y
!= ( cons_P5205166803686508359_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_448_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X3: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_449_min__list_Ocases,axiom,
! [X: list_nat_int] :
( ! [X3: nat > int,Xs2: list_nat_int] :
( X
!= ( cons_nat_int @ X3 @ Xs2 ) )
=> ( X = nil_nat_int ) ) ).
% min_list.cases
thf(fact_450_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_451_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X3: int] :
( X
!= ( cons_int @ X3 @ nil_int ) )
=> ~ ! [X3: int,Y2: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_452_remdups__adj_Ocases,axiom,
! [X: list_nat_int] :
( ( X != nil_nat_int )
=> ( ! [X3: nat > int] :
( X
!= ( cons_nat_int @ X3 @ nil_nat_int ) )
=> ~ ! [X3: nat > int,Y2: nat > int,Xs2: list_nat_int] :
( X
!= ( cons_nat_int @ X3 @ ( cons_nat_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_453_remdups__adj_Ocases,axiom,
! [X: list_P3592885314253461005_a_nat] :
( ( X != nil_Pr7402525243500994295_a_nat )
=> ( ! [X3: product_prod_a_nat] :
( X
!= ( cons_P5205166803686508359_a_nat @ X3 @ nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X3: product_prod_a_nat,Y2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( X
!= ( cons_P5205166803686508359_a_nat @ X3 @ ( cons_P5205166803686508359_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_454_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y5: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_455_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y5: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_456_neq__Nil__conv,axiom,
! [Xs: list_nat_int] :
( ( Xs != nil_nat_int )
= ( ? [Y5: nat > int,Ys3: list_nat_int] :
( Xs
= ( cons_nat_int @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_457_neq__Nil__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
= ( ? [Y5: product_prod_a_nat,Ys3: list_P3592885314253461005_a_nat] :
( Xs
= ( cons_P5205166803686508359_a_nat @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_458_list__induct2_H,axiom,
! [P3: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P3 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P3 @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys5: list_a] : ( P3 @ nil_a @ ( cons_a @ Y2 @ Ys5 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys5: list_a] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_459_list__induct2_H,axiom,
! [P3: list_a > list_int > $o,Xs: list_a,Ys: list_int] :
( ( P3 @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a] : ( P3 @ ( cons_a @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys5: list_int] : ( P3 @ nil_a @ ( cons_int @ Y2 @ Ys5 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_460_list__induct2_H,axiom,
! [P3: list_int > list_a > $o,Xs: list_int,Ys: list_a] :
( ( P3 @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int] : ( P3 @ ( cons_int @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys5: list_a] : ( P3 @ nil_int @ ( cons_a @ Y2 @ Ys5 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_461_list__induct2_H,axiom,
! [P3: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P3 @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int] : ( P3 @ ( cons_int @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys5: list_int] : ( P3 @ nil_int @ ( cons_int @ Y2 @ Ys5 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_462_list__induct2_H,axiom,
! [P3: list_a > list_nat_int > $o,Xs: list_a,Ys: list_nat_int] :
( ( P3 @ nil_a @ nil_nat_int )
=> ( ! [X3: a,Xs2: list_a] : ( P3 @ ( cons_a @ X3 @ Xs2 ) @ nil_nat_int )
=> ( ! [Y2: nat > int,Ys5: list_nat_int] : ( P3 @ nil_a @ ( cons_nat_int @ Y2 @ Ys5 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: nat > int,Ys5: list_nat_int] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat_int @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_463_list__induct2_H,axiom,
! [P3: list_a > list_P3592885314253461005_a_nat > $o,Xs: list_a,Ys: list_P3592885314253461005_a_nat] :
( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X3: a,Xs2: list_a] : ( P3 @ ( cons_a @ X3 @ Xs2 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y2: product_prod_a_nat,Ys5: list_P3592885314253461005_a_nat] : ( P3 @ nil_a @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys5 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: product_prod_a_nat,Ys5: list_P3592885314253461005_a_nat] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_464_list__induct2_H,axiom,
! [P3: list_int > list_nat_int > $o,Xs: list_int,Ys: list_nat_int] :
( ( P3 @ nil_int @ nil_nat_int )
=> ( ! [X3: int,Xs2: list_int] : ( P3 @ ( cons_int @ X3 @ Xs2 ) @ nil_nat_int )
=> ( ! [Y2: nat > int,Ys5: list_nat_int] : ( P3 @ nil_int @ ( cons_nat_int @ Y2 @ Ys5 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: nat > int,Ys5: list_nat_int] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat_int @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_465_list__induct2_H,axiom,
! [P3: list_int > list_P3592885314253461005_a_nat > $o,Xs: list_int,Ys: list_P3592885314253461005_a_nat] :
( ( P3 @ nil_int @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X3: int,Xs2: list_int] : ( P3 @ ( cons_int @ X3 @ Xs2 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y2: product_prod_a_nat,Ys5: list_P3592885314253461005_a_nat] : ( P3 @ nil_int @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys5 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: product_prod_a_nat,Ys5: list_P3592885314253461005_a_nat] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_466_list__induct2_H,axiom,
! [P3: list_nat_int > list_a > $o,Xs: list_nat_int,Ys: list_a] :
( ( P3 @ nil_nat_int @ nil_a )
=> ( ! [X3: nat > int,Xs2: list_nat_int] : ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys5: list_a] : ( P3 @ nil_nat_int @ ( cons_a @ Y2 @ Ys5 ) )
=> ( ! [X3: nat > int,Xs2: list_nat_int,Y2: a,Ys5: list_a] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_467_list__induct2_H,axiom,
! [P3: list_nat_int > list_int > $o,Xs: list_nat_int,Ys: list_int] :
( ( P3 @ nil_nat_int @ nil_int )
=> ( ! [X3: nat > int,Xs2: list_nat_int] : ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys5: list_int] : ( P3 @ nil_nat_int @ ( cons_int @ Y2 @ Ys5 ) )
=> ( ! [X3: nat > int,Xs2: list_nat_int,Y2: int,Ys5: list_int] :
( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_468_list__nonempty__induct,axiom,
! [Xs: list_a,P3: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P3 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_469_list__nonempty__induct,axiom,
! [Xs: list_int,P3: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X3: int] : ( P3 @ ( cons_int @ X3 @ nil_int ) )
=> ( ! [X3: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_470_list__nonempty__induct,axiom,
! [Xs: list_nat_int,P3: list_nat_int > $o] :
( ( Xs != nil_nat_int )
=> ( ! [X3: nat > int] : ( P3 @ ( cons_nat_int @ X3 @ nil_nat_int ) )
=> ( ! [X3: nat > int,Xs2: list_nat_int] :
( ( Xs2 != nil_nat_int )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_471_list__nonempty__induct,axiom,
! [Xs: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > $o] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ( ! [X3: product_prod_a_nat] : ( P3 @ ( cons_P5205166803686508359_a_nat @ X3 @ nil_Pr7402525243500994295_a_nat ) )
=> ( ! [X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_472_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_473_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_474_append__Cons,axiom,
! [X: nat > int,Xs: list_nat_int,Ys: list_nat_int] :
( ( append_nat_int @ ( cons_nat_int @ X @ Xs ) @ Ys )
= ( cons_nat_int @ X @ ( append_nat_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_475_append__Cons,axiom,
! [X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) @ Ys )
= ( cons_P5205166803686508359_a_nat @ X @ ( append7679239579558125090_a_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_476_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_477_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_478_Cons__eq__appendI,axiom,
! [X: nat > int,Xs1: list_nat_int,Ys: list_nat_int,Xs: list_nat_int,Zs: list_nat_int] :
( ( ( cons_nat_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat_int @ Xs1 @ Zs ) )
=> ( ( cons_nat_int @ X @ Xs )
= ( append_nat_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_479_Cons__eq__appendI,axiom,
! [X: product_prod_a_nat,Xs1: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Xs: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append7679239579558125090_a_nat @ Xs1 @ Zs ) )
=> ( ( cons_P5205166803686508359_a_nat @ X @ Xs )
= ( append7679239579558125090_a_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_480_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_481_list_Osel_I1_J,axiom,
! [X21: int,X22: list_int] :
( ( hd_int @ ( cons_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_482_list_Osel_I1_J,axiom,
! [X21: nat > int,X22: list_nat_int] :
( ( hd_nat_int @ ( cons_nat_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_483_list_Osel_I1_J,axiom,
! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( ( hd_Pro8935205257713178578_a_nat @ ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_484_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_485_set__subset__Cons,axiom,
! [Xs: list_nat_int,X: nat > int] : ( ord_le6569500216720880561at_int @ ( set_nat_int2 @ Xs ) @ ( set_nat_int2 @ ( cons_nat_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_486_set__subset__Cons,axiom,
! [Xs: list_P3592885314253461005_a_nat,X: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs ) @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_487_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_488_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P3: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_489_list__induct2,axiom,
! [Xs: list_int,Ys: list_a,P3: list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P3 @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_490_list__induct2,axiom,
! [Xs: list_a,Ys: list_int,P3: list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_491_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P3: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P3 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys5: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_492_list__induct2,axiom,
! [Xs: list_int,Ys: list_nat_int,P3: list_int > list_nat_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_s5718426915756887103at_int @ Ys ) )
=> ( ( P3 @ nil_int @ nil_nat_int )
=> ( ! [X3: int,Xs2: list_int,Y2: nat > int,Ys5: list_nat_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s5718426915756887103at_int @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat_int @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_493_list__induct2,axiom,
! [Xs: list_int,Ys: list_P3592885314253461005_a_nat,P3: list_int > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( P3 @ nil_int @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X3: int,Xs2: list_int,Y2: product_prod_a_nat,Ys5: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_494_list__induct2,axiom,
! [Xs: list_nat_int,Ys: list_int,P3: list_nat_int > list_int > $o] :
( ( ( size_s5718426915756887103at_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_nat_int @ nil_int )
=> ( ! [X3: nat > int,Xs2: list_nat_int,Y2: int,Ys5: list_int] :
( ( ( size_s5718426915756887103at_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_495_list__induct2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_int,P3: list_P3592885314253461005_a_nat > list_int > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_int )
=> ( ! [X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: int,Ys5: list_int] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_496_list__induct2,axiom,
! [Xs: list_nat_int,Ys: list_a,P3: list_nat_int > list_a > $o] :
( ( ( size_s5718426915756887103at_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P3 @ nil_nat_int @ nil_a )
=> ( ! [X3: nat > int,Xs2: list_nat_int,Y2: a,Ys5: list_a] :
( ( ( size_s5718426915756887103at_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_nat_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_497_list__induct2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,P3: list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys5: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( P3 @ Xs2 @ Ys5 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_498_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,P3: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_int @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_499_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_a,P3: list_int > list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_int @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int,Z3: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_500_list__induct3,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_int,P3: list_int > list_a > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_int @ nil_a @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_501_list__induct3,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_a,P3: list_int > list_a > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_int @ nil_a @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_502_list__induct3,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_int,P3: list_a > list_int > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_a @ nil_int @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_503_list__induct3,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_a,P3: list_a > list_int > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_a @ nil_int @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_504_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_int,P3: list_a > list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys5: list_a,Z3: int,Zs2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_505_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P3: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys5: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_506_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_nat_int,P3: list_int > list_int > list_nat_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_s5718426915756887103at_int @ Zs ) )
=> ( ( P3 @ nil_int @ nil_int @ nil_nat_int )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int,Z3: nat > int,Zs2: list_nat_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_s5718426915756887103at_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_nat_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_507_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_P3592885314253461005_a_nat,P3: list_int > list_int > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P3 @ nil_int @ nil_int @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_508_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_a,Ws: list_int,P3: list_int > list_int > list_a > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_int @ nil_int @ nil_a @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int,Z3: a,Zs2: list_a,W: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_509_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_a,Ws: list_a,P3: list_int > list_int > list_a > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_int @ nil_int @ nil_a @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_510_list__induct4,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_int,Ws: list_int,P3: list_int > list_a > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_int @ nil_a @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a,Z3: int,Zs2: list_int,W: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_511_list__induct4,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_int,Ws: list_a,P3: list_int > list_a > list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_int @ nil_a @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a,Z3: int,Zs2: list_int,W: a,Ws2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_512_list__induct4,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_a,Ws: list_int,P3: list_int > list_a > list_a > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_int @ nil_a @ nil_a @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a,Z3: a,Zs2: list_a,W: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_513_list__induct4,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_a,Ws: list_a,P3: list_int > list_a > list_a > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_int @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys5: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_514_list__induct4,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_int,Ws: list_int,P3: list_a > list_int > list_int > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_a @ nil_int @ nil_int @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int,Z3: int,Zs2: list_int,W: int,Ws2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_515_list__induct4,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_int,Ws: list_a,P3: list_a > list_int > list_int > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_int @ nil_int @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int,Z3: int,Zs2: list_int,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_516_list__induct4,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_a,Ws: list_int,P3: list_a > list_int > list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_a @ nil_int @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int,Z3: a,Zs2: list_a,W: int,Ws2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_517_list__induct4,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_a,Ws: list_a,P3: list_a > list_int > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_int @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys5: list_int,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_518_pirreducible__degree,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_519_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K5: a] : ( normalize_a_b @ r @ ( cons_a @ K5 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_520_coeff_Oelims,axiom,
! [X: list_a,Y: nat > a] :
( ( ( coeff_a_b @ r @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y
!= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( Y
!= ( ^ [I: nat] :
( if_a
@ ( I
= ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) )
@ ( hd_a @ ( cons_a @ V @ Va ) )
@ ( coeff_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) @ I ) ) ) ) ) ) ) ).
% coeff.elims
thf(fact_521_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_522_combine__prepend__replicate,axiom,
! [Ks: list_a,Us3: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ Ks ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ ( drop_a @ N @ Us3 ) ) ) ) ) ).
% combine_prepend_replicate
thf(fact_523_const__term__zero,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P5: list_a] :
( ( polynomial_a_b @ r @ K2 @ P5 )
=> ( ( P5 != nil_a )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_524_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_525_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_526_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_527_combine_Osimps_I2_J,axiom,
! [Us3: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us3 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_528_poly__add__closed,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_add_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_add_closed
thf(fact_529_poly__mult__closed,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_530_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_531_poly__coeff__in__carrier,axiom,
! [K2: set_a,P: list_a,I2: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( member_a @ ( coeff_a_b @ r @ P @ I2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_coeff_in_carrier
thf(fact_532_poly__add__zero_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= P ) ) ) ).
% poly_add_zero(1)
thf(fact_533_poly__add__zero_I2_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= P ) ) ) ).
% poly_add_zero(2)
thf(fact_534_poly__mult__l__distr,axiom,
! [K2: set_a,P1: list_a,P2: list_a,P32: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ( polynomial_a_b @ r @ K2 @ P32 )
=> ( ( 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_535_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_536_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_537_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_538_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_539_poly__add__is__polynomial,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_add_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_540_poly__mult__is__polynomial,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_541_normalize_Osimps_I2_J,axiom,
! [V2: a,Va2: list_a] :
( ( ( ( hd_a @ ( cons_a @ V2 @ Va2 ) )
!= ( zero_a_b @ r ) )
=> ( ( normalize_a_b @ r @ ( cons_a @ V2 @ Va2 ) )
= ( cons_a @ V2 @ Va2 ) ) )
& ( ( ( hd_a @ ( cons_a @ V2 @ Va2 ) )
= ( zero_a_b @ r ) )
=> ( ( normalize_a_b @ r @ ( cons_a @ V2 @ Va2 ) )
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va2 ) ) ) ) ) ) ).
% normalize.simps(2)
thf(fact_542_poly__add__replicate__zero_I2_J,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= P ) ) ) ).
% poly_add_replicate_zero(2)
thf(fact_543_poly__add__replicate__zero_I1_J,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= P ) ) ) ).
% poly_add_replicate_zero(1)
thf(fact_544_normalize_Oelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( normalize_a_b @ r @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ~ ( ( ( ( hd_a @ ( cons_a @ V @ Va ) )
!= ( zero_a_b @ r ) )
=> ( Y
= ( cons_a @ V @ Va ) ) )
& ( ( ( hd_a @ ( cons_a @ V @ Va ) )
= ( zero_a_b @ r ) )
=> ( Y
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% normalize.elims
thf(fact_545_append__is__polynomial,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( polynomial_a_b @ r @ K2 @ ( append_a @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% append_is_polynomial
thf(fact_546_combine__replicate,axiom,
! [Us3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( replicate_a @ ( size_size_list_a @ Us3 ) @ ( zero_a_b @ r ) ) @ Us3 )
= ( zero_a_b @ r ) ) ) ).
% combine_replicate
thf(fact_547_combine__append__replicate,axiom,
! [Us3: list_a,Ks: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ) ).
% combine_append_replicate
thf(fact_548_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_549_coeff_Osimps_I2_J,axiom,
! [V2: a,Va2: list_a] :
( ( coeff_a_b @ r @ ( cons_a @ V2 @ Va2 ) )
= ( ^ [I: nat] :
( if_a
@ ( I
= ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V2 @ Va2 ) ) @ one_one_nat ) )
@ ( hd_a @ ( cons_a @ V2 @ Va2 ) )
@ ( coeff_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va2 ) ) @ I ) ) ) ) ).
% coeff.simps(2)
thf(fact_550_combine__append__zero,axiom,
! [Us3: list_a,Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ) ).
% combine_append_zero
thf(fact_551_carrier__polynomial,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).
% carrier_polynomial
thf(fact_552_combine__in__carrier,axiom,
! [Ks: list_a,Us3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_553_carrier__polynomial__shell,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_554_polynomial__in__carrier,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_555_combine__append,axiom,
! [Ks: list_a,Us3: list_a,Ks2: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( 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 @ Us3 ) @ ( embedded_combine_a_b @ r @ Ks2 @ Vs2 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks2 ) @ ( append_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_556_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 ) )
= ( ^ [I: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I ) @ ( coeff_a_b @ r @ P2 @ I ) ) ) ) ) ) ).
% poly_add_coeff
thf(fact_557_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_558_a__lcomm,axiom,
! [X: a,Y: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z4 ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z4 ) ) ) ) ) ) ).
% a_lcomm
thf(fact_559_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_560_a__assoc,axiom,
! [X: a,Y: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z4 )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z4 ) ) ) ) ) ) ).
% a_assoc
thf(fact_561_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_562_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_563_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H2 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_564_local_Ominus__unique,axiom,
! [Y: a,X: a,Y6: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y6 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( 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_565_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_566_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_567_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_568_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_569_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_570_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_571_eval__in__carrier,axiom,
! [P: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_572_eval__poly__in__carrier,axiom,
! [K2: set_a,P: list_a,X: a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_573_combine__eq__eval,axiom,
! [Ks: list_a,X: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X ) ) ).
% combine_eq_eval
thf(fact_574_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 ) )
= ( ^ [I: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I ) @ ( coeff_a_b @ r @ P2 @ I ) ) ) ) ) ).
% poly_add_coeff_aux
thf(fact_575_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_576_eval__poly__add,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_577_eval__poly__add__aux,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_578_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_579_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z4 @ X ) )
= ( Y = Z4 ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_580_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_581_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_582_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_583_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_584_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_585_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_586_is__root__def,axiom,
! [P: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_587_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_588_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_589_a__lcos__mult__one,axiom,
! [M4: set_a] :
( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M4 )
= M4 ) ) ).
% a_lcos_mult_one
thf(fact_590_m__assoc,axiom,
! [X: a,Y: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z4 )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z4 ) ) ) ) ) ) ).
% m_assoc
thf(fact_591_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H2 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_592_l__distr,axiom,
! [X: a,Y: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z4 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z4 ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z4 ) ) ) ) ) ) ).
% l_distr
thf(fact_593_r__distr,axiom,
! [X: a,Y: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z4 @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z4 @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z4 @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_594_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_595_a__lcos__m__assoc,axiom,
! [M4: set_a,G: a,H3: a] :
( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H3 @ M4 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H3 ) @ M4 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_596_combine_Osimps_I1_J,axiom,
! [K: a,Ks: list_a,U: a,Us3: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K @ Ks ) @ ( cons_a @ U @ Us3 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ) ).
% combine.simps(1)
thf(fact_597_combine_Oelims,axiom,
! [X: list_a,Xa: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X @ Xa )
= Y )
=> ( ! [K3: a,Ks3: list_a] :
( ( X
= ( cons_a @ K3 @ Ks3 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_598_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_599_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_600_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_601_INTEG_OR_Odense__repr_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ~ ! [V: int,Va: list_int] :
( X
!= ( cons_int @ V @ Va ) ) ) ).
% INTEG.R.dense_repr.cases
thf(fact_602_INTEG_OP_Odense__repr_Ocases,axiom,
! [X: list_nat_int] :
( ( X != nil_nat_int )
=> ~ ! [V: nat > int,Va: list_nat_int] :
( X
!= ( cons_nat_int @ V @ Va ) ) ) ).
% INTEG.P.dense_repr.cases
thf(fact_603_poly__mult_Oelims,axiom,
! [X: list_a,Xa: list_a,Y: list_a] :
( ( ( poly_mult_a_b @ r @ X @ Xa )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( Y
!= ( poly_add_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V @ Va ) ) ) @ Xa ) @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) @ ( zero_a_b @ r ) ) ) @ ( poly_mult_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) @ Xa ) ) ) ) ) ) ).
% poly_mult.elims
thf(fact_604_poly__mult_Osimps_I2_J,axiom,
! [V2: a,Va2: list_a,P2: list_a] :
( ( poly_mult_a_b @ r @ ( cons_a @ V2 @ Va2 ) @ P2 )
= ( poly_add_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V2 @ Va2 ) ) ) @ P2 ) @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V2 @ Va2 ) ) @ one_one_nat ) @ ( zero_a_b @ r ) ) ) @ ( poly_mult_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va2 ) ) @ P2 ) ) ) ).
% poly_mult.simps(2)
thf(fact_605_eval_Oelims,axiom,
! [X: list_a,Y: a > a] :
( ( ( eval_a_b @ r @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y
!= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( Y
!= ( ^ [X2: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V @ Va ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) @ X2 ) ) ) ) ) ) ) ).
% eval.elims
thf(fact_606_subfield__long__division__theorem__shell,axiom,
! [K2: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ? [Q2: list_a,R3: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ Q2 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_607_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_608_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_609_nat__pow__comm,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_610_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_611_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_612_combine__r__distr,axiom,
! [Ks: list_a,Us3: list_a,K: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( 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 @ Us3 ) )
= ( embedded_combine_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ K ) @ Ks ) @ Us3 ) ) ) ) ) ).
% combine_r_distr
thf(fact_613_eval__append,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ 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 @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_614_eval_Osimps_I2_J,axiom,
! [V2: a,Va2: list_a] :
( ( eval_a_b @ r @ ( cons_a @ V2 @ Va2 ) )
= ( ^ [X2: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V2 @ Va2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V2 @ Va2 ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va2 ) ) @ X2 ) ) ) ) ).
% eval.simps(2)
thf(fact_615_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_616_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_617_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_618_line__extension__smult__closed,axiom,
! [K2: set_a,E2: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K3: a,V: a] :
( ( member_a @ K3 @ K2 )
=> ( ( member_a @ V @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V ) @ E2 ) ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_619_monoid__cancelI,axiom,
( ! [A3: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B3 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B3 ) ) ) ) )
=> ( ! [A3: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B3 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B3 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_620_poly__add__degree__eq,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat )
!= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P2 ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% poly_add_degree_eq
thf(fact_621_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_622_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E2: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K2 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E2 )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_623_poly__add__length__le,axiom,
! [P1: list_a,P2: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P2 ) ) @ ( ord_max_nat @ ( size_size_list_a @ P1 ) @ ( size_size_list_a @ P2 ) ) ) ).
% poly_add_length_le
thf(fact_624_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_625_poly__add__length__eq,axiom,
! [K2: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ( ( size_size_list_a @ P1 )
!= ( size_size_list_a @ P2 ) )
=> ( ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P2 ) )
= ( ord_max_nat @ ( size_size_list_a @ P1 ) @ ( size_size_list_a @ P2 ) ) ) ) ) ) ) ).
% poly_add_length_eq
thf(fact_626_poly__add__degree,axiom,
! [P1: list_a,P2: list_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P2 ) ) @ one_one_nat ) @ ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% poly_add_degree
thf(fact_627_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_628_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_629_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_630_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_631_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_632_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_633_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_634_Span__mem__iff__length__version,axiom,
! [K2: set_a,Us3: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
= ( ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_635_dense__repr__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( dense_repr_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( dense_repr_a_b @ r @ P ) ) ).
% dense_repr_replicate_zero
thf(fact_636_dense__repr_Osimps_I1_J,axiom,
( ( dense_repr_a_b @ r @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ).
% dense_repr.simps(1)
thf(fact_637_dense__repr__normalize,axiom,
! [P: list_a] :
( ( dense_repr_a_b @ r @ ( normalize_a_b @ r @ P ) )
= ( dense_repr_a_b @ r @ P ) ) ).
% dense_repr_normalize
thf(fact_638_Span__in__carrier,axiom,
! [K2: set_a,Us3: list_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_639_Span__subgroup__props_I1_J,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_640_Span__base__incl,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_base_incl
thf(fact_641_Span__same__set,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs2 ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% Span_same_set
thf(fact_642_mono__Span__sublist,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( 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 @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_643_mono__Span__subset,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% mono_Span_subset
thf(fact_644_Span__strict__incl,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( 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 @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Vs2 ) )
& ~ ( member_a @ X3 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ) ).
% Span_strict_incl
thf(fact_645_Span__subgroup__props_I3_J,axiom,
! [K2: set_a,Us3: list_a,V1: a,V22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( ( member_a @ V22 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_subgroup_props(3)
thf(fact_646_Span__subgroup__props_I2_J,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_subgroup_props(2)
thf(fact_647_mono__Span,axiom,
! [K2: set_a,Us3: list_a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ) ).
% mono_Span
thf(fact_648_Span__smult__closed,axiom,
! [K2: set_a,Us3: list_a,K: a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V2 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_649_mono__Span__append_I2_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( 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 @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Vs2 @ Us3 ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_650_mono__Span__append_I1_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( 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 @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_651_monom__decomp,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( P
= ( poly_of_dense_a_b @ r @ ( dense_repr_a_b @ r @ P ) ) ) ) ) ).
% monom_decomp
thf(fact_652_polynomial__dense__repr,axiom,
! [K2: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( ( dense_repr_a_b @ r @ P )
= ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ ( hd_a @ P ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) @ ( dense_repr_a_b @ r @ ( normalize_a_b @ r @ ( tl_a @ P ) ) ) ) ) ) ) ).
% polynomial_dense_repr
thf(fact_653_dense__repr_Oelims,axiom,
! [X: list_a,Y: list_P3592885314253461005_a_nat] :
( ( ( dense_repr_a_b @ r @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ~ ( ( ( ( hd_a @ ( cons_a @ V @ Va ) )
!= ( zero_a_b @ r ) )
=> ( Y
= ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ ( hd_a @ ( cons_a @ V @ Va ) ) @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) ) @ ( dense_repr_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) )
& ( ( ( hd_a @ ( cons_a @ V @ Va ) )
= ( zero_a_b @ r ) )
=> ( Y
= ( dense_repr_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% dense_repr.elims
thf(fact_654_Span__is__subalgebra,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ r ) ) ) ).
% Span_is_subalgebra
thf(fact_655_carrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_656_subalgebra__in__carrier,axiom,
! [K2: set_a,V3: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V3 @ r )
=> ( ord_less_eq_set_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_657_subalgebra__Span__incl,axiom,
! [K2: set_a,V3: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V3 )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ V3 ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_658_Span__subalgebraI,axiom,
! [K2: set_a,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ! [V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V4 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V4 )
=> ( ord_less_eq_set_a @ E2 @ V4 ) ) )
=> ( E2
= ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_659_dense__repr_Osimps_I2_J,axiom,
! [V2: a,Va2: list_a] :
( ( ( ( hd_a @ ( cons_a @ V2 @ Va2 ) )
!= ( zero_a_b @ r ) )
=> ( ( dense_repr_a_b @ r @ ( cons_a @ V2 @ Va2 ) )
= ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ ( hd_a @ ( cons_a @ V2 @ Va2 ) ) @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V2 @ Va2 ) ) @ one_one_nat ) ) @ ( dense_repr_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va2 ) ) ) ) ) )
& ( ( ( hd_a @ ( cons_a @ V2 @ Va2 ) )
= ( zero_a_b @ r ) )
=> ( ( dense_repr_a_b @ r @ ( cons_a @ V2 @ Va2 ) )
= ( dense_repr_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va2 ) ) ) ) ) ) ).
% dense_repr.simps(2)
thf(fact_660_Span__finite__dimension,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_finite_dimension
thf(fact_661_Span__append__eq__set__add,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_662_dense__repr_Opelims,axiom,
! [X: list_a,Y: list_P3592885314253461005_a_nat] :
( ( ( dense_repr_a_b @ r @ X )
= Y )
=> ( ( accp_list_a @ ( dense_repr_rel_a_b @ r ) @ X )
=> ( ( ( X = nil_a )
=> ( ( Y = nil_Pr7402525243500994295_a_nat )
=> ~ ( accp_list_a @ ( dense_repr_rel_a_b @ r ) @ nil_a ) ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( ( ( ( ( hd_a @ ( cons_a @ V @ Va ) )
!= ( zero_a_b @ r ) )
=> ( Y
= ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ ( hd_a @ ( cons_a @ V @ Va ) ) @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) ) @ ( dense_repr_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) )
& ( ( ( hd_a @ ( cons_a @ V @ Va ) )
= ( zero_a_b @ r ) )
=> ( Y
= ( dense_repr_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) )
=> ~ ( accp_list_a @ ( dense_repr_rel_a_b @ r ) @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% dense_repr.pelims
thf(fact_663_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_664_setadd__subset__G,axiom,
! [H: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_665_set__add__comm,axiom,
! [I5: set_a,J3: set_a] :
( ( ord_less_eq_set_a @ I5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I5 @ J3 )
= ( set_add_a_b @ r @ J3 @ I5 ) ) ) ) ).
% set_add_comm
thf(fact_666_set__add__closed,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_667_sum__space__dim_I1_J,axiom,
! [K2: set_a,E2: set_a,F2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( set_add_a_b @ r @ E2 @ F2 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_668_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_669_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E2: set_a,V3: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V3 @ r )
=> ( ( ord_less_eq_set_a @ V3 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V3 ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_670_normalize_Opelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( normalize_a_b @ r @ X )
= Y )
=> ( ( accp_list_a @ ( normalize_rel_a_b @ r ) @ X )
=> ( ( ( X = nil_a )
=> ( ( Y = nil_a )
=> ~ ( accp_list_a @ ( normalize_rel_a_b @ r ) @ nil_a ) ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( ( ( ( ( hd_a @ ( cons_a @ V @ Va ) )
!= ( zero_a_b @ r ) )
=> ( Y
= ( cons_a @ V @ Va ) ) )
& ( ( ( hd_a @ ( cons_a @ V @ Va ) )
= ( zero_a_b @ r ) )
=> ( Y
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) )
=> ~ ( accp_list_a @ ( normalize_rel_a_b @ r ) @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% normalize.pelims
thf(fact_671_coeff_Opelims,axiom,
! [X: list_a,Y: nat > a] :
( ( ( coeff_a_b @ r @ X )
= Y )
=> ( ( accp_list_a @ coeff_rel_a @ X )
=> ( ( ( X = nil_a )
=> ( ( Y
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) )
=> ~ ( accp_list_a @ coeff_rel_a @ nil_a ) ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( ( Y
= ( ^ [I: nat] :
( if_a
@ ( I
= ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) )
@ ( hd_a @ ( cons_a @ V @ Va ) )
@ ( coeff_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) @ I ) ) ) )
=> ~ ( accp_list_a @ coeff_rel_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% coeff.pelims
thf(fact_672_eval_Opelims,axiom,
! [X: list_a,Y: a > a] :
( ( ( eval_a_b @ r @ X )
= Y )
=> ( ( accp_list_a @ eval_rel_a @ X )
=> ( ( ( X = nil_a )
=> ( ( Y
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) )
=> ~ ( accp_list_a @ eval_rel_a @ nil_a ) ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( ( Y
= ( ^ [X2: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V @ Va ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) @ X2 ) ) ) )
=> ~ ( accp_list_a @ eval_rel_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% eval.pelims
thf(fact_673_Span__incl,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ r @ K2 @ ( set_a2 @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_incl
thf(fact_674_set__mult__closed,axiom,
! [H: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ r @ H @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_mult_closed
thf(fact_675_unique__decomposition,axiom,
! [K2: set_a,Us3: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ? [X3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X3 ) @ K2 )
& ( ( size_size_list_a @ X3 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ r @ X3 @ Us3 ) )
& ! [Y3: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ Y3 ) @ K2 )
& ( ( size_size_list_a @ Y3 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ r @ Y3 @ Us3 ) ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% unique_decomposition
thf(fact_676_replacement__theorem,axiom,
! [K2: set_a,Us5: list_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us5 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us5 @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ? [Vs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( set_a2 @ Vs2 ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Us5 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Vs3 @ Us3 ) ) ) ) ) ) ) ).
% replacement_theorem
thf(fact_677_independent__backwards_I2_J,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ).
% independent_backwards(2)
thf(fact_678_li__Nil,axiom,
! [K2: set_a] : ( embedd5208550302661555450nt_a_b @ r @ K2 @ nil_a ) ).
% li_Nil
thf(fact_679_independent__backwards_I3_J,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_backwards(3)
thf(fact_680_independent__backwards_I1_J,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ).
% independent_backwards(1)
thf(fact_681_independent__split_I2_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ) ).
% independent_split(2)
thf(fact_682_independent__split_I1_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 ) ) ) ).
% independent_split(1)
thf(fact_683_independent__in__carrier,axiom,
! [K2: set_a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_in_carrier
thf(fact_684_li__Cons,axiom,
! [U: a,K2: set_a,Us3: list_a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ).
% li_Cons
thf(fact_685_independent__same__set,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs2 ) )
=> ( ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Vs2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% independent_same_set
thf(fact_686_independent_Osimps,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
= ( ? [K4: set_a] :
( ( A1 = K4 )
& ( A22 = nil_a ) )
| ? [U3: a,K4: set_a,Us6: list_a] :
( ( A1 = K4 )
& ( A22
= ( cons_a @ U3 @ Us6 ) )
& ( member_a @ U3 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ U3 @ ( embedded_Span_a_b @ r @ K4 @ Us6 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K4 @ Us6 ) ) ) ) ).
% independent.simps
thf(fact_687_independent_Ocases,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
=> ( ( A22 != nil_a )
=> ~ ! [U2: a,Us4: list_a] :
( ( A22
= ( cons_a @ U2 @ Us4 ) )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U2 @ ( embedded_Span_a_b @ r @ A1 @ Us4 ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ A1 @ Us4 ) ) ) ) ) ) ).
% independent.cases
thf(fact_688_independent__rotate1__aux,axiom,
! [K2: set_a,U: a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ ( append_a @ Us3 @ Vs2 ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ ( append_a @ Us3 @ ( cons_a @ U @ nil_a ) ) @ Vs2 ) ) ) ) ).
% independent_rotate1_aux
thf(fact_689_independent__strict__incl,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ).
% independent_strict_incl
thf(fact_690_filter__base,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ~ ! [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs4 )
=> ( ( embedded_Span_a_b @ r @ K2 @ Vs4 )
!= ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% filter_base
thf(fact_691_independent__replacement,axiom,
! [K2: set_a,U: a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Vs2 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ X3 @ Us3 ) ) ) ) ) ) ) ).
% independent_replacement
thf(fact_692_independent__length__le,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ ( size_size_list_a @ Vs2 ) ) ) ) ) ) ).
% independent_length_le
thf(fact_693_independent__rotate1,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ ( rotate1_a @ Us3 ) @ Vs2 ) ) ) ) ).
% independent_rotate1
thf(fact_694_complete__base,axiom,
! [K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ? [Vs4: list_a] :
( ( ( size_size_list_a @ ( append_a @ Vs4 @ Us3 ) )
= N )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Vs4 @ Us3 ) )
& ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Vs4 @ Us3 ) )
= E2 ) ) ) ) ) ) ).
% complete_base
thf(fact_695_dimension__is__inj,axiom,
! [K2: set_a,N: nat,E2: set_a,M2: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K2 @ E2 )
=> ( N = M2 ) ) ) ) ).
% dimension_is_inj
thf(fact_696_finite__dimensionE_H,axiom,
! [K2: set_a,E2: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ~ ! [N2: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N2 @ K2 @ E2 ) ) ).
% finite_dimensionE'
thf(fact_697_finite__dimensionI,axiom,
! [N: nat,K2: set_a,E2: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 ) ) ).
% finite_dimensionI
thf(fact_698_finite__dimension__def,axiom,
! [K2: set_a,E2: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
= ( ? [N3: nat] : ( embedd2795209813406577254on_a_b @ r @ N3 @ K2 @ E2 ) ) ) ).
% finite_dimension_def
thf(fact_699_space__subgroup__props_I3_J,axiom,
! [K2: set_a,N: nat,E2: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ V1 @ E2 )
=> ( ( member_a @ V22 @ E2 )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ E2 ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_700_space__subgroup__props_I2_J,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( member_a @ ( zero_a_b @ r ) @ E2 ) ) ) ).
% space_subgroup_props(2)
thf(fact_701_telescopic__base__aux,axiom,
! [K2: set_a,F2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F2 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_702_space__subgroup__props_I5_J,axiom,
! [K2: set_a,N: nat,E2: set_a,K: a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V2 @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V2 ) @ E2 ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_703_unique__dimension,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ? [X3: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X3 @ K2 @ E2 )
& ! [Y3: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y3 @ K2 @ E2 )
=> ( Y3 = X3 ) ) ) ) ) ).
% unique_dimension
thf(fact_704_space__subgroup__props_I1_J,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_705_dimensionI,axiom,
! [K2: set_a,Us3: list_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= E2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( size_size_list_a @ Us3 ) @ K2 @ E2 ) ) ) ) ).
% dimensionI
thf(fact_706_independent__length__le__dimension,axiom,
! [K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ N ) ) ) ) ) ).
% independent_length_le_dimension
thf(fact_707_independent__length__eq__dimension,axiom,
! [K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ( ( size_size_list_a @ Us3 )
= N )
= ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= E2 ) ) ) ) ) ) ).
% independent_length_eq_dimension
thf(fact_708_exists__base,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ? [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs4 )
& ( ( size_size_list_a @ Vs4 )
= N )
& ( ( embedded_Span_a_b @ r @ K2 @ Vs4 )
= E2 ) ) ) ) ).
% exists_base
thf(fact_709_dimension__one,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K2 @ K2 ) ) ).
% dimension_one
thf(fact_710_dimension__independent,axiom,
! [K2: set_a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd2795209813406577254on_a_b @ r @ ( size_size_list_a @ Us3 ) @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ).
% dimension_independent
thf(fact_711_dimension__backwards,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ E2 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E3 )
& ~ ( member_a @ X3 @ E3 )
& ( E2
= ( embedd971793762689825387on_a_b @ r @ K2 @ X3 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_712_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_713_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_714_nat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_Suc2
thf(fact_715_Suc__dim,axiom,
! [V2: a,E2: set_a,N: nat,K2: set_a] :
( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V2 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ r @ K2 @ V2 @ E2 ) ) ) ) ) ).
% Suc_dim
thf(fact_716_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_717_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_718_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_719_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_720_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_721_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_722_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_723_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_724_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_725_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_726_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_727_local_Onat__pow__Suc,axiom,
! [X: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ X ) ) ).
% local.nat_pow_Suc
thf(fact_728_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_729_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_730_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_731_dec__induct,axiom,
! [I2: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P3 @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) ) )
=> ( P3 @ J ) ) ) ) ).
% dec_induct
thf(fact_732_inc__induct,axiom,
! [I2: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P3 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_733_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_734_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_735_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_736_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_737_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_738_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_739_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_740_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_741_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_742_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_743_zero__induct,axiom,
! [P3: nat > $o,K: nat] :
( ( P3 @ K )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_744_diff__induct,axiom,
! [P3: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P3 @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P3 @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P3 @ X3 @ Y2 )
=> ( P3 @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P3 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_745_nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) )
=> ( P3 @ N ) ) ) ).
% nat_induct
thf(fact_746_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_747_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_748_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_749_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_750_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_751_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_752_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_753_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_754_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_755_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_756_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_757_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_758_full__nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P3 @ M ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% full_nat_induct
thf(fact_759_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P3 @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_760_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_761_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_762_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_763_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J4: nat] :
( ( M2
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_764_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_765_All__less__Suc2,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P3 @ I ) ) )
= ( ( P3 @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P3 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_766_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_767_Ex__less__Suc2,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P3 @ I ) ) )
= ( ( P3 @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P3 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_768_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_769_strict__inc__induct,axiom,
! [I2: nat,J: nat,P3: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P3 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P3 @ ( suc @ I3 ) )
=> ( P3 @ I3 ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_770_less__Suc__induct,axiom,
! [I2: nat,J: nat,P3: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P3 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P3 @ I3 @ J2 )
=> ( ( P3 @ J2 @ K3 )
=> ( P3 @ I3 @ K3 ) ) ) ) )
=> ( P3 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_771_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_772_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_773_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_774_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_775_All__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P3 @ I ) ) )
= ( ( P3 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P3 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_776_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_777_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_778_Ex__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P3 @ I ) ) )
= ( ( P3 @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P3 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_779_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_780_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_781_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_782_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_783_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_784_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_785_zero__induct__lemma,axiom,
! [P3: nat > $o,K: nat,I2: nat] :
( ( P3 @ K )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_786_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_787_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_788_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_789_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_790_ex__least__nat__less,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P3 @ I4 ) )
& ( P3 @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_791_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_792_nat__induct__non__zero,axiom,
! [N: nat,P3: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P3 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_793_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_794_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_795_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_b @ r ) @ N2 @ F ) ) ).
% bound_upD
thf(fact_796_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_797_Span__subgroup__props_I4_J,axiom,
! [K2: set_a,Us3: list_a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( a_inv_a_b @ r @ V2 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ).
% Span_subgroup_props(4)
thf(fact_798_subring__props_I5_J,axiom,
! [K2: set_a,H3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H3 @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_799_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_800_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_801_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_802_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_803_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_804_a__transpose__inv,axiom,
! [X: a,Y: a,Z4: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z4 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z4 )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_805_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_806_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_807_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_808_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_809_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_810_space__subgroup__props_I4_J,axiom,
! [K2: set_a,N: nat,E2: set_a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ V2 @ E2 )
=> ( member_a @ ( a_inv_a_b @ r @ V2 ) @ E2 ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_811_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_812_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_813_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_814_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_815_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_816_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_817_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_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,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( coeff_a_b @ r @ x @ ( minus_minus_nat @ ( size_size_list_a @ x ) @ one_one_nat ) )
!= ( zero_a_b @ r ) ) ).
%------------------------------------------------------------------------------