TPTP Problem File: SLH0349^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00037_001755__16964098_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1472 ( 368 unt; 194 typ; 0 def)
% Number of atoms : 4273 (1447 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 16450 ( 325 ~; 35 |; 264 &;13263 @)
% ( 0 <=>;2563 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 866 ( 866 >; 0 *; 0 +; 0 <<)
% Number of symbols : 173 ( 170 usr; 20 con; 0-4 aty)
% Number of variables : 3918 ( 206 ^;3537 !; 175 ?;3918 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:30:21.780
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $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_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
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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,
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
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% Explicit typings (170)
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
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thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001tf__a_001tf__b,type,
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thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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thf(sy_c_Embedded__Algebras_Oring_Ocombine_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Oring_Oindependent_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
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thf(sy_c_Group_Omonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_List_Oappend_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
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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_Mtf__a_J,type,
drop_nat_a: nat > list_nat_a > list_nat_a ).
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__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
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_Ofoldr_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_Itf__a_J,type,
foldr_4031981466149041118list_a: ( product_prod_a_nat > list_a > list_a ) > list_P3592885314253461005_a_nat > list_a > list_a ).
thf(sy_c_List_Ofoldr_001tf__a_001t__Set__Oset_Itf__a_J,type,
foldr_a_set_a: ( a > set_a > set_a ) > list_a > set_a > set_a ).
thf(sy_c_List_Ofoldr_001tf__a_001tf__a,type,
foldr_a_a: ( a > a > a ) > list_a > a > a ).
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__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__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
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thf(sy_c_List_Olist_OCons_001tf__a,type,
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nil_Pr7402525243500994295_a_nat: list_P3592885314253461005_a_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
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thf(sy_c_List_Olist_Oset_001tf__a,type,
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thf(sy_c_List_Oreplicate_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > 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_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
bot_bot_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
bot_bo9049108969261143879_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
bot_bo3186585308812441520list_a: set_set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
ord_le5542992221119063950st_a_o: ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_eq_list_a_o: ( list_a > $o ) > ( list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_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_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_Polynomials_Opolynomial_001tf__a_001tf__b,type,
polynomial_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
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_Oeval_001tf__a_001tf__b,type,
eval_a_b: partia2175431115845679010xt_a_b > list_a > a > 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_001tf__a_001tf__b,type,
normalize_a_b: partia2175431115845679010xt_a_b > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001tf__a_001tf__b,type,
poly_add_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__mult_001tf__a_001tf__b,type,
poly_mult_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_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_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001t__Nat__Onat_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc7724251129057698313list_a: ( a > nat > list_a > list_a ) > product_prod_a_nat > list_a > list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
abelian_group_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia2175431115845679010xt_a_b > int > a > a ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
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_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring_Ozero__update_001tf__a_001tf__b,type,
zero_update_a_b: ( a > a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
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_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1274)
thf(fact_0_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_1_finite__subset,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_2_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_3_assms,axiom,
finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ).
% assms
thf(fact_4_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_5_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_6_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_7_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_8_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_9__092_060open_062bounded__degree__polynomials_AR_An_A_092_060subseteq_062_A_123p_O_Aset_Ap_A_092_060subseteq_062_Acarrier_AR_A_092_060and_062_Alength_Ap_A_092_060le_062_An_125_092_060close_062,axiom,
( ord_le8861187494160871172list_a @ ( bounde2262800523058855161ls_a_b @ r @ n )
@ ( collect_list_a
@ ^ [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ n ) ) ) ) ).
% \<open>bounded_degree_polynomials R n \<subseteq> {p. set p \<subseteq> carrier R \<and> length p \<le> n}\<close>
thf(fact_10_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_11_associated__sym,axiom,
! [A2: a,B2: a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 )
=> ( associ5860276527279195403xt_a_b @ r @ B2 @ A2 ) ) ).
% associated_sym
thf(fact_12_monoid__axioms,axiom,
monoid8385113658579753027xt_a_b @ r ).
% monoid_axioms
thf(fact_13_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_14_finite__lists__length__le,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_15_finite__lists__length__le,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_16_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_17_associated__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 )
=> ( ( associ5860276527279195403xt_a_b @ r @ B2 @ C )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ C ) ) ) ) ) ).
% associated_trans
thf(fact_18_assoc__subst,axiom,
! [A2: a,B2: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 )
=> ( ! [A3: a,B3: a] :
( ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A3 @ B3 ) )
=> ( ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B3 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A3 ) @ ( F @ B3 ) ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ) ) ).
% assoc_subst
thf(fact_19_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_20_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_21_finite__Collect__conjI,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_22_finite__Collect__conjI,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_23_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_24_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_25_finite__Collect__disjI,axiom,
! [P2: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_26_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_27_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_28_finite__Collect__subsets,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B4: set_list_a] : ( ord_le8861187494160871172list_a @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_29_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B4: set_a] : ( ord_less_eq_set_a @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_30_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_31_associated__refl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ A2 ) ) ).
% associated_refl
thf(fact_32_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_33_not__finite__existsD,axiom,
! [P2: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
=> ? [X_1: list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_34_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_35_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_36_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_37_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_38_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_39_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_40_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B: set_a,R: list_a > a > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A4: list_a] :
( ( member_list_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_41_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B: set_nat,R: list_a > nat > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A4: list_a] :
( ( member_list_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_42_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_list_a,R: a > list_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_43_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_list_a,R: nat > list_a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_list_a @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_44_pigeonhole__infinite__rel,axiom,
! [A: set_nat_a,B: set_a,R: ( nat > a ) > a > $o] :
( ~ ( finite_finite_nat_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A4: nat > a] :
( ( member_nat_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_45_pigeonhole__infinite__rel,axiom,
! [A: set_nat_a,B: set_nat,R: ( nat > a ) > nat > $o] :
( ~ ( finite_finite_nat_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A4: nat > a] :
( ( member_nat_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_46_finite__has__minimal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ X2 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_47_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_48_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_49_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_50_finite__has__maximal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ A2 @ X2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_51_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_52_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_53_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A2 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_54_rev__finite__subset,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_55_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_56_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_57_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_58_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_59_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_60_exp__base__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X3 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_61_finite__lists__length__eq,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
& ( ( size_s349497388124573686list_a @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_62_finite__lists__length__eq,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
& ( ( size_size_list_a @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_63_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ( size_size_list_nat @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_64_monoid_Oee__sym,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( essent8953798148185448568xt_a_b @ G @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ Bs @ As ) ) ) ) ) ).
% monoid.ee_sym
thf(fact_65_monoid_Oee__refl,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ As @ As ) ) ) ).
% monoid.ee_refl
thf(fact_66_monoid_Oee__trans,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a,Cs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( essent8953798148185448568xt_a_b @ G @ As @ Bs )
=> ( ( essent8953798148185448568xt_a_b @ G @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ As @ Cs ) ) ) ) ) ) ) ).
% monoid.ee_trans
thf(fact_67_List_Ofinite__set,axiom,
! [Xs2: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_68_List_Ofinite__set,axiom,
! [Xs2: list_a] : ( finite_finite_a @ ( set_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_69_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_70_factors__closed,axiom,
! [Fs: list_a,A2: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_71_subalgebra__in__carrier,axiom,
! [K2: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_72_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_73_a__l__coset__subset__G,axiom,
! [H: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X3 @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_74_wfactors__cong__r,axiom,
! [Fs: list_a,A2: a,A5: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A2 )
=> ( ( associ5860276527279195403xt_a_b @ r @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A5 ) ) ) ) ) ) ).
% wfactors_cong_r
thf(fact_75_mem__Collect__eq,axiom,
! [A2: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A2 @ ( collect_nat_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A2: list_a,P2: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X: nat > a] : ( member_nat_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_83_Collect__cong,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_84_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_85_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_86_subset__Idl__subset,axiom,
! [I2: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_87_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_88_factors__wfactors,axiom,
! [As: list_a,A2: a] :
( ( factor5638265376665762323xt_a_b @ r @ As @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ As @ A2 ) ) ) ).
% factors_wfactors
thf(fact_89_wfactors__factors,axiom,
! [As: list_a,A2: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ As @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A6: a] :
( ( factor5638265376665762323xt_a_b @ r @ As @ A6 )
& ( associ5860276527279195403xt_a_b @ r @ A6 @ A2 ) ) ) ) ).
% wfactors_factors
thf(fact_90_monoid_Ofactors__wfactors,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ As @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( wfacto3557276942076956612xt_a_b @ G @ As @ A2 ) ) ) ) ).
% monoid.factors_wfactors
thf(fact_91_monoid_Owfactors__factors,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [A6: a] :
( ( factor5638265376665762323xt_a_b @ G @ As @ A6 )
& ( associ5860276527279195403xt_a_b @ G @ A6 @ A2 ) ) ) ) ) ).
% monoid.wfactors_factors
thf(fact_92_monoid_Ofactors__closed,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fs @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.factors_closed
thf(fact_93_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_94_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_95_monoid_Owfactors__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A2: a,A5: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A2 )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A5 ) ) ) ) ) ) ) ).
% monoid.wfactors_cong_r
thf(fact_96_subset__code_I1_J,axiom,
! [Xs2: list_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( set_nat_a2 @ Xs2 ) @ B )
= ( ! [X: nat > a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs2 ) )
=> ( member_nat_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_97_subset__code_I1_J,axiom,
! [Xs2: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ B )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ( member_list_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_98_subset__code_I1_J,axiom,
! [Xs2: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_99_subset__code_I1_J,axiom,
! [Xs2: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_100_finite__list,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ? [Xs3: list_list_a] :
( ( set_list_a2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_101_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs3: list_a] :
( ( set_a2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_102_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_103_monoid_Oassociated__sym,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B2 )
=> ( associ5860276527279195403xt_a_b @ G @ B2 @ A2 ) ) ) ).
% monoid.associated_sym
thf(fact_104_monoid_Oassociated__trans,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B2 )
=> ( ( associ5860276527279195403xt_a_b @ G @ B2 @ C )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A2 @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_105_monoid_Oassociated__refl,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A2 @ A2 ) ) ) ).
% monoid.associated_refl
thf(fact_106_monoid_Oassoc__subst,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a,F: a > a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B2 )
=> ( ! [A3: a,B3: a] :
( ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ A3 @ B3 ) )
=> ( ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ ( F @ B3 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ ( F @ A3 ) @ ( F @ B3 ) ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_107_monoid_Oee__length,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( essent8953798148185448568xt_a_b @ G @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ) ).
% monoid.ee_length
thf(fact_108_ring_Oexp__base__closed,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ R @ X3 @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_109_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H: set_a,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ G @ X3 @ H ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_110_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_111_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_112_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H ) @ ( genideal_a_b @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_113_ring_Ogenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_114_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_115_a__lcos__m__assoc,axiom,
! [M: set_a,G2: a,H2: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G2 @ ( a_l_coset_a_b @ r @ H2 @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G2 @ H2 ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_116_line__extension__in__carrier,axiom,
! [K2: set_a,A2: a,E: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A2 @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_117_listassoc__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ As ) ) ) ) ).
% listassoc_sym
thf(fact_118_listassoc__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% listassoc_trans
thf(fact_119_a__lcomm,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_120_a__comm,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ Y )
= ( add_a_b @ r @ Y @ X3 ) ) ) ) ).
% a_comm
thf(fact_121_a__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) @ Z )
= ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_122_add_Or__cancel,axiom,
! [A2: a,C: a,B2: a] :
( ( ( add_a_b @ r @ A2 @ C )
= ( add_a_b @ r @ B2 @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) ) ).
% add.r_cancel
thf(fact_123_add_Ol__cancel,axiom,
! [C: a,A2: a,B2: a] :
( ( ( add_a_b @ r @ C @ A2 )
= ( add_a_b @ r @ C @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) ) ).
% add.l_cancel
thf(fact_124_local_Ominus__unique,axiom,
! [Y: a,X3: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_125_add_Or__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_126_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_127_add_Ol__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_128_add_Oinv__comm,axiom,
! [X3: a,Y: a] :
( ( ( add_a_b @ r @ X3 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_129_a__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_130_local_Oadd_Oright__cancel,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X3 )
= ( add_a_b @ r @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_131_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_132_r__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= X3 ) ) ).
% r_zero
thf(fact_133_l__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_zero
thf(fact_134_add_Or__cancel__one_H,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ A2 @ X3 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_135_add_Or__cancel__one,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A2 @ X3 )
= X3 )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_136_add_Ol__cancel__one_H,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ X3 @ A2 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_137_add_Ol__cancel__one,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ A2 )
= X3 )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_138_listassoc__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ As ) ) ).
% listassoc_refl
thf(fact_139_list_Orel__mono,axiom,
! [R: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R @ Ra )
=> ( ord_le5542992221119063950st_a_o @ ( list_all2_a_a @ R ) @ ( list_all2_a_a @ Ra ) ) ) ).
% list.rel_mono
thf(fact_140_list__all2__antisym,axiom,
! [P2: a > a > $o,Q: a > a > $o,Xs2: list_a,Ys: list_a] :
( ! [X2: a,Y3: a] :
( ( P2 @ X2 @ Y3 )
=> ( ( Q @ Y3 @ X2 )
=> ( X2 = Y3 ) ) )
=> ( ( list_all2_a_a @ P2 @ Xs2 @ Ys )
=> ( ( list_all2_a_a @ Q @ Ys @ Xs2 )
=> ( Xs2 = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_141_list__all2__trans,axiom,
! [P1: a > a > $o,P22: a > a > $o,P3: a > a > $o,As: list_a,Bs: list_a,Cs: list_a] :
( ! [A3: a,B3: a,C2: a] :
( ( P1 @ A3 @ B3 )
=> ( ( P22 @ B3 @ C2 )
=> ( P3 @ A3 @ C2 ) ) )
=> ( ( list_all2_a_a @ P1 @ As @ Bs )
=> ( ( list_all2_a_a @ P22 @ Bs @ Cs )
=> ( list_all2_a_a @ P3 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_142_list__all2__refl,axiom,
! [P2: a > a > $o,Xs2: list_a] :
( ! [X2: a] : ( P2 @ X2 @ X2 )
=> ( list_all2_a_a @ P2 @ Xs2 @ Xs2 ) ) ).
% list_all2_refl
thf(fact_143_list__all2__mono,axiom,
! [P2: a > a > $o,Xs2: list_a,Ys: list_a,Q: a > a > $o] :
( ( list_all2_a_a @ P2 @ Xs2 @ Ys )
=> ( ! [Xs3: a,Ys2: a] :
( ( P2 @ Xs3 @ Ys2 )
=> ( Q @ Xs3 @ Ys2 ) )
=> ( list_all2_a_a @ Q @ Xs2 @ Ys ) ) ) ).
% list_all2_mono
thf(fact_144_list__all2__eq,axiom,
( ( ^ [Y4: list_a,Z2: list_a] : ( Y4 = Z2 ) )
= ( list_all2_a_a
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) ) ) ).
% list_all2_eq
thf(fact_145_list_Orel__refl,axiom,
! [Ra: a > a > $o,X3: list_a] :
( ! [X2: a] : ( Ra @ X2 @ X2 )
=> ( list_all2_a_a @ Ra @ X3 @ X3 ) ) ).
% list.rel_refl
thf(fact_146_list_Orel__eq,axiom,
( ( list_all2_a_a
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [Y4: list_a,Z2: list_a] : ( Y4 = Z2 ) ) ) ).
% list.rel_eq
thf(fact_147_ring_Oline__extension_Ocong,axiom,
embedd971793762689825387on_a_b = embedd971793762689825387on_a_b ).
% ring.line_extension.cong
thf(fact_148_list_Orel__cong,axiom,
! [X3: list_nat_a,Ya: list_nat_a,Y: list_nat_a,Xa2: list_nat_a,R: ( nat > a ) > ( nat > a ) > $o,Ra: ( nat > a ) > ( nat > a ) > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: nat > a,Yb: nat > a] :
( ( member_nat_a @ Z3 @ ( set_nat_a2 @ Ya ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Xa2 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_a9087535860334789575_nat_a @ R @ X3 @ Y )
= ( list_a9087535860334789575_nat_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_149_list_Orel__cong,axiom,
! [X3: list_nat_a,Ya: list_nat_a,Y: list_a,Xa2: list_a,R: ( nat > a ) > a > $o,Ra: ( nat > a ) > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: nat > a,Yb: a] :
( ( member_nat_a @ Z3 @ ( set_nat_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_nat_a_a @ R @ X3 @ Y )
= ( list_all2_nat_a_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_150_list_Orel__cong,axiom,
! [X3: list_a,Ya: list_a,Y: list_nat_a,Xa2: list_nat_a,R: a > ( nat > a ) > $o,Ra: a > ( nat > a ) > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: a,Yb: nat > a] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Xa2 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_nat_a @ R @ X3 @ Y )
= ( list_all2_a_nat_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_151_list_Orel__cong,axiom,
! [X3: list_a,Ya: list_a,Y: list_a,Xa2: list_a,R: a > a > $o,Ra: a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_a @ R @ X3 @ Y )
= ( list_all2_a_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_152_list_Orel__mono__strong,axiom,
! [R: ( nat > a ) > ( nat > a ) > $o,X3: list_nat_a,Y: list_nat_a,Ra: ( nat > a ) > ( nat > a ) > $o] :
( ( list_a9087535860334789575_nat_a @ R @ X3 @ Y )
=> ( ! [Z3: nat > a,Yb: nat > a] :
( ( member_nat_a @ Z3 @ ( set_nat_a2 @ X3 ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_a9087535860334789575_nat_a @ Ra @ X3 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_153_list_Orel__mono__strong,axiom,
! [R: ( nat > a ) > a > $o,X3: list_nat_a,Y: list_a,Ra: ( nat > a ) > a > $o] :
( ( list_all2_nat_a_a @ R @ X3 @ Y )
=> ( ! [Z3: nat > a,Yb: a] :
( ( member_nat_a @ Z3 @ ( set_nat_a2 @ X3 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_nat_a_a @ Ra @ X3 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_154_list_Orel__mono__strong,axiom,
! [R: a > ( nat > a ) > $o,X3: list_a,Y: list_nat_a,Ra: a > ( nat > a ) > $o] :
( ( list_all2_a_nat_a @ R @ X3 @ Y )
=> ( ! [Z3: a,Yb: nat > a] :
( ( member_a @ Z3 @ ( set_a2 @ X3 ) )
=> ( ( member_nat_a @ Yb @ ( set_nat_a2 @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_nat_a @ Ra @ X3 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_155_list_Orel__mono__strong,axiom,
! [R: a > a > $o,X3: list_a,Y: list_a,Ra: a > a > $o] :
( ( list_all2_a_a @ R @ X3 @ Y )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( set_a2 @ X3 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_a @ Ra @ X3 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_156_list_Orel__refl__strong,axiom,
! [X3: list_nat_a,Ra: ( nat > a ) > ( nat > a ) > $o] :
( ! [Z3: nat > a] :
( ( member_nat_a @ Z3 @ ( set_nat_a2 @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_a9087535860334789575_nat_a @ Ra @ X3 @ X3 ) ) ).
% list.rel_refl_strong
thf(fact_157_list_Orel__refl__strong,axiom,
! [X3: list_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_a2 @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_all2_a_a @ Ra @ X3 @ X3 ) ) ).
% list.rel_refl_strong
thf(fact_158_list__all2__same,axiom,
! [P2: a > a > $o,Xs2: list_a] :
( ( list_all2_a_a @ P2 @ Xs2 @ Xs2 )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( P2 @ X @ X ) ) ) ) ).
% list_all2_same
thf(fact_159_list__all2__lengthD,axiom,
! [P2: a > a > $o,Xs2: list_a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ Xs2 @ Ys )
=> ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_160_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A2: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K2 @ A2 @ E ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_161_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,M: set_a,G2: a,H2: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ G2 @ ( a_l_coset_a_b @ G @ H2 @ M ) )
= ( a_l_coset_a_b @ G @ ( add_a_b @ G @ G2 @ H2 ) @ M ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_162_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2175431115845679010xt_a_b,M: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ ( zero_a_b @ G ) @ M )
= M ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_163_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_164_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_165_monoid_Olistassoc__sym,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ Bs @ As ) ) ) ) ) ).
% monoid.listassoc_sym
thf(fact_166_monoid_Olistassoc__refl,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ As ) ) ) ).
% monoid.listassoc_refl
thf(fact_167_monoid_Olistassoc__trans,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a,Cs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Cs ) ) ) ) ) ) ) ).
% monoid.listassoc_trans
thf(fact_168_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_169_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_170_ring__iso__imp__img__ring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_ring
thf(fact_171_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_172_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_173_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_174_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_175_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( zero_a_b @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_176_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X3: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X3 )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X3 @ Y2 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_177_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z3 )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y3 )
= ( add_a_b @ R @ Y3 @ X2 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_178_abelian__groupE_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ X2 @ X3 )
= ( zero_a_b @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_179_abelian__groupE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_180_ring_Ounfold__congs_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: partia2175431115845679010xt_a_b,V2: a,F: a > a,F2: a > a] :
( ( R2 = R3 )
=> ( ( ( zero_a_b @ R3 )
= V2 )
=> ( ! [V3: a] :
( ( V3 = V2 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( zero_update_a_b @ F @ R2 )
= ( zero_update_a_b @ F2 @ R3 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_181_ring_Ofold__congs_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: partia2175431115845679010xt_a_b,V2: a,F: a > a,F2: a > a] :
( ( R2 = R3 )
=> ( ( ( zero_a_b @ R3 )
= V2 )
=> ( ! [V3: a] :
( ( V2 = V3 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( zero_update_a_b @ F @ R2 )
= ( zero_update_a_b @ F2 @ R3 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_182_ring_Ois__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( abelian_group_a_b @ R ) ) ).
% ring.is_abelian_group
thf(fact_183_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ G )
=> ( abelian_monoid_a_b @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_184_ring_Ois__monoid,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( monoid8385113658579753027xt_a_b @ R ) ) ).
% ring.is_monoid
thf(fact_185_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_186_semiring_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( monoid8385113658579753027xt_a_b @ R ) ) ).
% semiring.axioms(2)
thf(fact_187_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_188_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_189_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_190_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ Y @ X3 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_191_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_192_abelian__groupE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_193_abelian__groupE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_194_abelian__groupE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_195_abelian__groupE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ Y @ X3 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_196_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_197_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_198_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_199_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y @ Z ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X3 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_200_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X3 @ Y ) @ Z )
= ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_201_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ Y )
= ( add_a_b @ G @ Y @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_202_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_203_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_204_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ Y @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_205_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_206_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_207_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_208_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ Y @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_209_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_210_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_211_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_212_abelian__groupI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Z3: a] :
( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z3 )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y3 )
= ( add_a_b @ R @ Y3 @ X2 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ Xa @ X2 )
= ( zero_a_b @ R ) ) ) )
=> ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_213_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_214_essentially__equalI,axiom,
! [Fs1: list_a,Fs12: list_a,Fs2: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Fs12 @ Fs2 )
=> ( essent8953798148185448568xt_a_b @ r @ Fs1 @ Fs2 ) ) ) ).
% essentially_equalI
thf(fact_215_essentially__equalE,axiom,
! [Fs1: list_a,Fs2: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ Fs1 @ Fs2 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Fs13 @ Fs2 ) ) ) ).
% essentially_equalE
thf(fact_216_add_Oint__pow__distrib,axiom,
! [X3: a,Y: a,I: int] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X3 ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_217_add_Oint__pow__mult__distrib,axiom,
! [X3: a,Y: a,I: int] :
( ( ( add_a_b @ r @ X3 @ Y )
= ( add_a_b @ r @ Y @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X3 ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_218_ring__iso__memE_I3_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X3 @ Y ) )
= ( add_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_219_up__add__closed,axiom,
! [P4: nat > a,Q2: nat > a] :
( ( member_nat_a @ P4 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( add_a_b @ r @ ( P4 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_220_properfactor__cong__l,axiom,
! [X4: a,X3: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X4 @ X3 )
=> ( ( proper19828929941537682xt_a_b @ r @ X3 @ Y )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X4 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_221_properfactor__cong__r,axiom,
! [X3: a,Y: a,Y2: a] :
( ( proper19828929941537682xt_a_b @ r @ X3 @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X3 @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_222_perm__assoc__switch__r,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs ) ) ) ) ) ).
% perm_assoc_switch_r
thf(fact_223_perm__assoc__switch,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs2 @ Cs ) ) ) ) ).
% perm_assoc_switch
thf(fact_224_perm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ 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 ) ) ) ) ).
% perm_closed
thf(fact_225_add_Oint__pow__closed,axiom,
! [X3: a,I: int] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_226_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.int_pow_one
thf(fact_227_perm__setP,axiom,
! [As: list_a,Bs: list_a,P2: set_a > $o] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( P2 @ ( set_a2 @ As ) )
=> ( P2 @ ( set_a2 @ Bs ) ) ) ) ).
% perm_setP
thf(fact_228_monoid_Operm__assoc__switch,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a,Cs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ Bs2 @ Cs ) ) ) ) ) ).
% monoid.perm_assoc_switch
thf(fact_229_monoid_Operm__assoc__switch__r,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a,Cs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ Bs @ Cs )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs ) ) ) ) ) ) ).
% monoid.perm_assoc_switch_r
thf(fact_230_essentially__equal__def,axiom,
( essent8953798148185448568xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,Fs14: list_a,Fs22: list_a] :
? [Fs15: list_a] :
( ( ( mset_a @ Fs14 )
= ( mset_a @ Fs15 ) )
& ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G3 ) @ Fs15 @ Fs22 ) ) ) ) ).
% essentially_equal_def
thf(fact_231_monoid_Oproperfactor__cong__l,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a,X3: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ X4 @ X3 )
=> ( ( proper19828929941537682xt_a_b @ G @ X3 @ Y )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ X4 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_232_monoid_Oproperfactor__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Y2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ X3 @ Y )
=> ( ( associ5860276527279195403xt_a_b @ G @ Y @ Y2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ X3 @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_233_monoid_Operm__closed,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.perm_closed
thf(fact_234_monoid_Oessentially__equalE,axiom,
! [G: partia2175431115845679010xt_a_b,Fs1: list_a,Fs2: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( essent8953798148185448568xt_a_b @ G @ Fs1 @ Fs2 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ Fs13 @ Fs2 ) ) ) ) ).
% monoid.essentially_equalE
thf(fact_235_monoid_Oessentially__equalI,axiom,
! [G: partia2175431115845679010xt_a_b,Fs1: list_a,Fs12: list_a,Fs2: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ Fs12 @ Fs2 )
=> ( essent8953798148185448568xt_a_b @ G @ Fs1 @ Fs2 ) ) ) ) ).
% monoid.essentially_equalI
thf(fact_236_ring__iso__memE_I1_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X3: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_237_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_238_up__minus__closed,axiom,
! [P4: nat > a,Q2: nat > a] :
( ( member_nat_a @ P4 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_minus_a_b @ r @ ( P4 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_239_add_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( add_a_b @ r ) @ As @ ( zero_a_b @ r ) )
= ( foldr_a_a @ ( add_a_b @ r ) @ Bs @ ( zero_a_b @ r ) ) ) ) ) ).
% add.multlist_perm_cong
thf(fact_240_add_Oint__pow__mult,axiom,
! [X3: a,I: int,J: int] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X3 )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X3 ) @ ( add_pow_a_b_int @ r @ J @ X3 ) ) ) ) ).
% add.int_pow_mult
thf(fact_241_irrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X5 ) ) ) ) ).
% irrlist_perm_cong
thf(fact_242_ring_Oup__add__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P4: nat > a,Q2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P4 @ ( up_a_b @ R ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( add_a_b @ R @ ( P4 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_add_closed
thf(fact_243_up__smult__closed,axiom,
! [A2: a,P4: nat > a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P4 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( mult_a_ring_ext_a_b @ r @ A2 @ ( P4 @ I3 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_244_eval__in__carrier,axiom,
! [P4: list_a,X3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P4 @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_245_m__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_246_l__distr,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_247_r__distr,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_248_mult__cong__r,axiom,
! [B2: a,B5: a,A2: a] :
( ( associ5860276527279195403xt_a_b @ r @ B2 @ B5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A2 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B5 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_249_add__pow__ldistr__int,axiom,
! [A2: a,B2: a,K: int] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A2 ) @ B2 )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_250_add__pow__rdistr__int,axiom,
! [A2: a,B2: a,K: int] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A2 @ ( add_pow_a_b_int @ r @ K @ B2 ) )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_251_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A2: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A2 @ E ) )
= ( ? [X: a] :
( ( member_a @ X @ K2 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ A2 ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_252_properfactor__prod__r,axiom,
! [A2: a,B2: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A2 @ B2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A2 @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_253_m__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_254_minus__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_255_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N3: nat] : ( member_a @ ( F @ N3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N4: nat] : ( bound_a @ ( zero_a_b @ R ) @ N4 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_256_l__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_257_r__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_258_r__right__minus__eq,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A2 @ B2 )
= ( zero_a_b @ r ) )
= ( A2 = B2 ) ) ) ) ).
% r_right_minus_eq
thf(fact_259_wfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( wfacto3557276942076956612xt_a_b @ r @ As @ A3 ) ) ) ) ).
% wfactors_prod_exists
thf(fact_260_add_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( add_a_b @ r ) @ Fs @ ( zero_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.multlist_closed
thf(fact_261_bound__below,axiom,
! [Z: a,M2: nat,F: nat > a,N: nat] :
( ( bound_a @ Z @ M2 @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M2 ) ) ) ).
% bound_below
thf(fact_262_foldr__cong,axiom,
! [A2: list_a,B2: list_a,L: list_P3592885314253461005_a_nat,K: list_P3592885314253461005_a_nat,F: product_prod_a_nat > list_a > list_a,G2: product_prod_a_nat > list_a > list_a] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A3: list_a,X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ ( set_Pr924983374503034536_a_nat @ L ) )
=> ( ( F @ X2 @ A3 )
= ( G2 @ X2 @ A3 ) ) )
=> ( ( foldr_4031981466149041118list_a @ F @ L @ A2 )
= ( foldr_4031981466149041118list_a @ G2 @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_263_foldr__cong,axiom,
! [A2: a,B2: a,L: list_a,K: list_a,F: a > a > a,G2: a > a > a] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A3: a,X2: a] :
( ( member_a @ X2 @ ( set_a2 @ L ) )
=> ( ( F @ X2 @ A3 )
= ( G2 @ X2 @ A3 ) ) )
=> ( ( foldr_a_a @ F @ L @ A2 )
= ( foldr_a_a @ G2 @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_264_foldr__cong,axiom,
! [A2: set_a,B2: set_a,L: list_a,K: list_a,F: a > set_a > set_a,G2: a > set_a > set_a] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A3: set_a,X2: a] :
( ( member_a @ X2 @ ( set_a2 @ L ) )
=> ( ( F @ X2 @ A3 )
= ( G2 @ X2 @ A3 ) ) )
=> ( ( foldr_a_set_a @ F @ L @ A2 )
= ( foldr_a_set_a @ G2 @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_265_subalgebra_Osmult__closed,axiom,
! [K2: set_a,V: set_a,R: partia2175431115845679010xt_a_b,K: a,V4: a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V4 @ V )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V4 ) @ V ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_266_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_267_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_268_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_269_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_270_ring__iso__memE_I2_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_271_ring_Oup__minus__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P4: nat > a,Q2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P4 @ ( up_a_b @ R ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_minus_a_b @ R @ ( P4 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_minus_closed
thf(fact_272_ring_Oup__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,P4: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_nat_a @ P4 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( mult_a_ring_ext_a_b @ R @ A2 @ ( P4 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_smult_closed
thf(fact_273_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_274_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_275_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_276_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X3 @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_277_monoid_Omult__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,B2: a,B5: a,A2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ B2 @ B5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A2 @ B2 ) @ ( mult_a_ring_ext_a_b @ G @ A2 @ B5 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_278_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B2: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A2 @ ( add_pow_a_b_int @ R @ K @ B2 ) )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A2 @ B2 ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_279_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B2: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_int @ R @ K @ A2 ) @ B2 )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A2 @ B2 ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_280_monoid_Oproperfactor__prod__r,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A2 @ B2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ A2 @ ( mult_a_ring_ext_a_b @ G @ B2 @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_281_ring_Obound__upD,axiom,
! [R: partia2175431115845679010xt_a_b,F: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ R ) @ N3 @ F ) ) ) ).
% ring.bound_upD
thf(fact_282_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,A2: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A2 @ E ) )
= ( ? [X: a] :
( ( member_a @ X @ K2 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ A2 ) @ Y5 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_283_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_284_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_285_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X3 @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_286_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_287_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_288_abelian__group_Ominus__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_minus_a_b @ G @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_289_monoid_Oirrlist__perm__cong,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X5 ) ) ) ) ) ).
% monoid.irrlist_perm_cong
thf(fact_290_ringI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( ( monoid8385113658579753027xt_a_b @ R )
=> ( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z3 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X2 @ Z3 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z3 @ ( add_a_b @ R @ X2 @ Y3 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z3 @ X2 ) @ ( mult_a_ring_ext_a_b @ R @ Z3 @ Y3 ) ) ) ) ) )
=> ( ring_a_b @ R ) ) ) ) ) ).
% ringI
thf(fact_291_monoid_Owfactors__prod__exists,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
& ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 ) ) ) ) ) ).
% monoid.wfactors_prod_exists
thf(fact_292_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_293_eval__var,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X3 )
= X3 ) ) ).
% eval_var
thf(fact_294_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_295_factorsI,axiom,
! [Fs: list_a,A2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X2 ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A2 )
=> ( factor5638265376665762323xt_a_b @ r @ Fs @ A2 ) ) ) ).
% factorsI
thf(fact_296_ring_Oeval__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,X3: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P4 @ X3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_297_factors__mult,axiom,
! [Fa: list_a,A2: a,Fb: list_a,B2: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A2 )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B2 )
=> ( ( 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 @ A2 @ B2 ) ) ) ) ) ) ).
% factors_mult
thf(fact_298_factors__mult__single,axiom,
! [A2: a,Fb: list_a,B2: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A2 )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A2 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% factors_mult_single
thf(fact_299_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_300_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_301_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_302_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_303_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_304_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_305_inv__unique,axiom,
! [Y: a,X3: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_306_same__append__eq,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_307_append__same__eq,axiom,
! [Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs2 )
= ( append_a @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_308_append__assoc,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs2 @ Ys ) @ Zs )
= ( append_a @ Xs2 @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_309_append_Oassoc,axiom,
! [A2: list_a,B2: list_a,C: list_a] :
( ( append_a @ ( append_a @ A2 @ B2 ) @ C )
= ( append_a @ A2 @ ( append_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_310_length__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_311_append__eq__append__conv,axiom,
! [Xs2: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs2 @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_312_foldr__append,axiom,
! [F: product_prod_a_nat > list_a > list_a,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,A2: list_a] :
( ( foldr_4031981466149041118list_a @ F @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) @ A2 )
= ( foldr_4031981466149041118list_a @ F @ Xs2 @ ( foldr_4031981466149041118list_a @ F @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_313_foldr__append,axiom,
! [F: a > a > a,Xs2: list_a,Ys: list_a,A2: a] :
( ( foldr_a_a @ F @ ( append_a @ Xs2 @ Ys ) @ A2 )
= ( foldr_a_a @ F @ Xs2 @ ( foldr_a_a @ F @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_314_foldr__append,axiom,
! [F: a > set_a > set_a,Xs2: list_a,Ys: list_a,A2: set_a] :
( ( foldr_a_set_a @ F @ ( append_a @ Xs2 @ Ys ) @ A2 )
= ( foldr_a_set_a @ F @ Xs2 @ ( foldr_a_set_a @ F @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_315_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_316_r__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% r_one
thf(fact_317_l__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_one
thf(fact_318_multlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% multlist_closed
thf(fact_319_append__eq__append__conv2,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs2
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs2 @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_320_append__eq__appendI,axiom,
! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_321_Cons__eq__appendI,axiom,
! [X3: a,Xs1: list_a,Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X3 @ Xs2 )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_322_not__Cons__self2,axiom,
! [X3: a,Xs2: list_a] :
( ( cons_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_323_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_324_append__Cons,axiom,
! [X3: a,Xs2: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X3 @ Xs2 ) @ Ys )
= ( cons_a @ X3 @ ( append_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_325_split__list,axiom,
! [X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
=> ? [Ys2: list_nat_a,Zs2: list_nat_a] :
( Xs2
= ( append_nat_a @ Ys2 @ ( cons_nat_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_326_split__list,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_327_split__list__last,axiom,
! [X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
=> ? [Ys2: list_nat_a,Zs2: list_nat_a] :
( ( Xs2
= ( append_nat_a @ Ys2 @ ( cons_nat_a @ X3 @ Zs2 ) ) )
& ~ ( member_nat_a @ X3 @ ( set_nat_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_328_split__list__last,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_329_split__list__prop,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 ) ) ) ).
% split_list_prop
thf(fact_330_split__list__first,axiom,
! [X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
=> ? [Ys2: list_nat_a,Zs2: list_nat_a] :
( ( Xs2
= ( append_nat_a @ Ys2 @ ( cons_nat_a @ X3 @ Zs2 ) ) )
& ~ ( member_nat_a @ X3 @ ( set_nat_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_331_split__list__first,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_332_split__list__propE,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ~ ( P2 @ X2 ) ) ) ).
% split_list_propE
thf(fact_333_append__Cons__eq__iff,axiom,
! [X3: nat > a,Xs2: list_nat_a,Ys: list_nat_a,Xs4: list_nat_a,Ys3: list_nat_a] :
( ~ ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
=> ( ~ ( member_nat_a @ X3 @ ( set_nat_a2 @ Ys ) )
=> ( ( ( append_nat_a @ Xs2 @ ( cons_nat_a @ X3 @ Ys ) )
= ( append_nat_a @ Xs4 @ ( cons_nat_a @ X3 @ Ys3 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_334_append__Cons__eq__iff,axiom,
! [X3: a,Xs2: list_a,Ys: list_a,Xs4: list_a,Ys3: list_a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs2 @ ( cons_a @ X3 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X3 @ Ys3 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_335_in__set__conv__decomp,axiom,
! [X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
= ( ? [Ys4: list_nat_a,Zs3: list_nat_a] :
( Xs2
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_336_in__set__conv__decomp,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_337_split__list__last__prop,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_338_split__list__first__prop,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_339_split__list__last__propE,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_340_split__list__first__propE,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_341_in__set__conv__decomp__last,axiom,
! [X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
= ( ? [Ys4: list_nat_a,Zs3: list_nat_a] :
( ( Xs2
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X3 @ Zs3 ) ) )
& ~ ( member_nat_a @ X3 @ ( set_nat_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_342_in__set__conv__decomp__last,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_343_in__set__conv__decomp__first,axiom,
! [X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) )
= ( ? [Ys4: list_nat_a,Zs3: list_nat_a] :
( ( Xs2
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X3 @ Zs3 ) ) )
& ~ ( member_nat_a @ X3 @ ( set_nat_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_344_in__set__conv__decomp__first,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_345_split__list__last__prop__iff,axiom,
! [Xs2: list_a,P2: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
& ( P2 @ X ) ) )
= ( ? [Ys4: list_a,X: a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_346_split__list__first__prop__iff,axiom,
! [Xs2: list_a,P2: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
& ( P2 @ X ) ) )
= ( ? [Ys4: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_347_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_348_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_349_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_350_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_351_list_Oset__cases,axiom,
! [E2: nat > a,A2: list_nat_a] :
( ( member_nat_a @ E2 @ ( set_nat_a2 @ A2 ) )
=> ( ! [Z22: list_nat_a] :
( A2
!= ( cons_nat_a @ E2 @ Z22 ) )
=> ~ ! [Z1: nat > a,Z22: list_nat_a] :
( ( A2
= ( cons_nat_a @ Z1 @ Z22 ) )
=> ~ ( member_nat_a @ E2 @ ( set_nat_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_352_list_Oset__cases,axiom,
! [E2: a,A2: list_a] :
( ( member_a @ E2 @ ( set_a2 @ A2 ) )
=> ( ! [Z22: list_a] :
( A2
!= ( cons_a @ E2 @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_353_set__ConsD,axiom,
! [Y: nat > a,X3: nat > a,Xs2: list_nat_a] :
( ( member_nat_a @ Y @ ( set_nat_a2 @ ( cons_nat_a @ X3 @ Xs2 ) ) )
=> ( ( Y = X3 )
| ( member_nat_a @ Y @ ( set_nat_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_354_set__ConsD,axiom,
! [Y: a,X3: a,Xs2: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs2 ) ) )
=> ( ( Y = X3 )
| ( member_a @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_355_list__all2__appendI,axiom,
! [P2: a > a > $o,A2: list_a,B2: list_a,C: list_a,D: list_a] :
( ( list_all2_a_a @ P2 @ A2 @ B2 )
=> ( ( list_all2_a_a @ P2 @ C @ D )
=> ( list_all2_a_a @ P2 @ ( append_a @ A2 @ C ) @ ( append_a @ B2 @ D ) ) ) ) ).
% list_all2_appendI
thf(fact_356_list__all2__Cons2,axiom,
! [P2: a > a > $o,Xs2: list_a,Y: a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ Xs2 @ ( cons_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Xs2
= ( cons_a @ Z4 @ Zs3 ) )
& ( P2 @ Z4 @ Y )
& ( list_all2_a_a @ P2 @ Zs3 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_357_list__all2__Cons1,axiom,
! [P2: a > a > $o,X3: a,Xs2: list_a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ ( cons_a @ X3 @ Xs2 ) @ Ys )
= ( ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( P2 @ X3 @ Z4 )
& ( list_all2_a_a @ P2 @ Xs2 @ Zs3 ) ) ) ) ).
% list_all2_Cons1
thf(fact_358_list__all2__Cons,axiom,
! [P2: a > a > $o,X3: a,Xs2: list_a,Y: a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y @ Ys ) )
= ( ( P2 @ X3 @ Y )
& ( list_all2_a_a @ P2 @ Xs2 @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_359_list_Orel__intros_I2_J,axiom,
! [R: a > a > $o,X21: a,Y21: a,X22: list_a,Y22: list_a] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2_a_a @ R @ X22 @ Y22 )
=> ( list_all2_a_a @ R @ ( cons_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_360_list_Orel__inject_I2_J,axiom,
! [R: a > a > $o,X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( list_all2_a_a @ R @ ( cons_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2_a_a @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_361_ring__iso__memE_I4_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_362_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_363_ring_Oring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_364_set__subset__Cons,axiom,
! [Xs2: list_list_a,X3: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ ( cons_list_a @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_365_set__subset__Cons,axiom,
! [Xs2: list_a,X3: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ ( cons_a @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_366_set__subset__Cons,axiom,
! [Xs2: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_367_list__all2__append,axiom,
! [Xs2: list_a,Ys: list_a,P2: a > a > $o,Us: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( list_all2_a_a @ P2 @ ( append_a @ Xs2 @ Us ) @ ( append_a @ Ys @ Vs ) )
= ( ( list_all2_a_a @ P2 @ Xs2 @ Ys )
& ( list_all2_a_a @ P2 @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_368_list__all2__append1,axiom,
! [P2: a > a > $o,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( list_all2_a_a @ P2 @ ( append_a @ Xs2 @ Ys ) @ Zs )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Zs
= ( append_a @ Us2 @ Vs2 ) )
& ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Xs2 ) )
& ( ( size_size_list_a @ Vs2 )
= ( size_size_list_a @ Ys ) )
& ( list_all2_a_a @ P2 @ Xs2 @ Us2 )
& ( list_all2_a_a @ P2 @ Ys @ Vs2 ) ) ) ) ).
% list_all2_append1
thf(fact_369_list__all2__append2,axiom,
! [P2: a > a > $o,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( list_all2_a_a @ P2 @ Xs2 @ ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs2
= ( append_a @ Us2 @ Vs2 ) )
& ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Ys ) )
& ( ( size_size_list_a @ Vs2 )
= ( size_size_list_a @ Zs ) )
& ( list_all2_a_a @ P2 @ Us2 @ Ys )
& ( list_all2_a_a @ P2 @ Vs2 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_370_impossible__Cons,axiom,
! [Xs2: list_a,Ys: list_a,X3: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) )
=> ( Xs2
!= ( cons_a @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_371_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_372_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A2: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A2 )
= ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_373_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,C: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A2 @ C )
= ( mult_a_ring_ext_a_b @ G @ B2 @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_374_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% ring.eval_var
thf(fact_375_ring_Oring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_376_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_377_monoid__cancel_Owfactors__mult__single,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,Fb: list_a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A2 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Fb @ B2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( wfacto3557276942076956612xt_a_b @ G @ ( cons_a @ A2 @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.wfactors_mult_single
thf(fact_378_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a,B5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A2 @ B2 ) @ ( mult_a_ring_ext_a_b @ G @ A2 @ B5 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B2 @ B5 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_379_monoid_Omonoid__cancelI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [A3: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ G @ C2 @ B3 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A3 = B3 ) ) ) ) )
=> ( ! [A3: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ G @ B3 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A3 = B3 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_380_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A2 @ B2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ G @ C @ B2 ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_381_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B2: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
= ( proper19828929941537682xt_a_b @ G @ A2 @ B2 ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_382_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,A5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A2 )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_383_factorsE,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A2: a] :
( ( factor5638265376665762323xt_a_b @ G @ Fs @ A2 )
=> ~ ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X5 ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) )
!= A2 ) ) ) ).
% factorsE
thf(fact_384_factors__def,axiom,
( factor5638265376665762323xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,Fs3: list_a,A4: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs3 ) )
=> ( irredu6211895646901577903xt_a_b @ G3 @ X ) )
& ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G3 ) @ Fs3 @ ( one_a_ring_ext_a_b @ G3 ) )
= A4 ) ) ) ) ).
% factors_def
thf(fact_385_monoid_Ofactors__mult__single,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,Fb: list_a,B2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A2 )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fb @ B2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor5638265376665762323xt_a_b @ G @ ( cons_a @ A2 @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A2 @ B2 ) ) ) ) ) ) ).
% monoid.factors_mult_single
thf(fact_386_monoid_Omultlist__closed,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.multlist_closed
thf(fact_387_monoid_Ofactors__mult,axiom,
! [G: partia2175431115845679010xt_a_b,Fa: list_a,A2: a,Fb: list_a,B2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fa @ A2 )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fb @ B2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor5638265376665762323xt_a_b @ G @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A2 @ B2 ) ) ) ) ) ) ) ).
% monoid.factors_mult
thf(fact_388_wfactors__def,axiom,
( wfacto3557276942076956612xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,Fs3: list_a,A4: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs3 ) )
=> ( irredu6211895646901577903xt_a_b @ G3 @ X ) )
& ( associ5860276527279195403xt_a_b @ G3 @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G3 ) @ Fs3 @ ( one_a_ring_ext_a_b @ G3 ) ) @ A4 ) ) ) ) ).
% wfactors_def
thf(fact_389_wfactorsI,axiom,
! [Fs: list_a,G: partia2175431115845679010xt_a_b,A2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) ) @ A2 )
=> ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A2 ) ) ) ).
% wfactorsI
thf(fact_390_wfactorsE,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A2: a] :
( ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A2 )
=> ~ ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X5 ) )
=> ~ ( associ5860276527279195403xt_a_b @ G @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) ) @ A2 ) ) ) ).
% wfactorsE
thf(fact_391_monoid_OfactorsI,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X2 ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) )
= A2 )
=> ( factor5638265376665762323xt_a_b @ G @ Fs @ A2 ) ) ) ) ).
% monoid.factorsI
thf(fact_392_add__le__imp__le__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_393_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_394_add__le__imp__le__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_395_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_396_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
? [C3: nat] :
( B6
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_397_add__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_398_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_399_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_400_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_401_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_402_add__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_403_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_404_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_405_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_406_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_407_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_408_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_409_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_410_monoid__cancel_Oirrlist__listassoc__cong,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X2 ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X5 ) ) ) ) ) ) ) ).
% monoid_cancel.irrlist_listassoc_cong
thf(fact_411_monoid_Ol__one,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X3 )
= X3 ) ) ) ).
% monoid.l_one
thf(fact_412_monoid_Or__one,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X3 @ ( one_a_ring_ext_a_b @ G ) )
= X3 ) ) ) ).
% monoid.r_one
thf(fact_413_eval__append__aux,axiom,
! [P4: list_a,B2: a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P4 @ ( cons_a @ B2 @ nil_a ) ) @ A2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P4 @ A2 ) @ A2 ) @ B2 ) ) ) ) ) ).
% eval_append_aux
thf(fact_414_monoid_Oone__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% monoid.one_closed
thf(fact_415_eval__append,axiom,
! [P4: list_a,Q2: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P4 @ Q2 ) @ A2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P4 @ A2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ A2 @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ r @ Q2 @ A2 ) ) ) ) ) ) ).
% eval_append
thf(fact_416_mset__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( mset_a @ ( append_a @ Xs2 @ Ys ) )
= ( plus_plus_multiset_a @ ( mset_a @ Xs2 ) @ ( mset_a @ Ys ) ) ) ).
% mset_append
thf(fact_417_perm__append1,axiom,
! [Xs2: list_a,Ys: list_a,L: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( mset_a @ ( append_a @ L @ Xs2 ) )
= ( mset_a @ ( append_a @ L @ Ys ) ) ) ) ).
% perm_append1
thf(fact_418_normalize_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ~ ! [V3: a,Va: list_a] :
( X3
!= ( cons_a @ V3 @ Va ) ) ) ).
% normalize.cases
thf(fact_419_group__commutes__pow,axiom,
! [X3: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_420_nat__pow__comm,axiom,
! [X3: a,N: nat,M2: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_421_pow__mult__distrib,axiom,
! [X3: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_422_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_423_nat__pow__mult,axiom,
! [X3: a,N: nat,M2: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M2 ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( plus_plus_nat @ N @ M2 ) ) ) ) ).
% nat_pow_mult
thf(fact_424_append_Oright__neutral,axiom,
! [A2: list_a] :
( ( append_a @ A2 @ nil_a )
= A2 ) ).
% append.right_neutral
thf(fact_425_append_Oright__neutral,axiom,
! [A2: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ A2 @ nil_Pr7402525243500994295_a_nat )
= A2 ) ).
% append.right_neutral
thf(fact_426_append__Nil2,axiom,
! [Xs2: list_a] :
( ( append_a @ Xs2 @ nil_a )
= Xs2 ) ).
% append_Nil2
thf(fact_427_append__Nil2,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ Xs2 @ nil_Pr7402525243500994295_a_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_428_append__self__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_429_append__self__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv
thf(fact_430_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_431_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_432_append__self__conv2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_a ) ) ).
% append_self_conv2
thf(fact_433_append__self__conv2,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv2
thf(fact_434_self__append__conv2,axiom,
! [Y: list_a,Xs2: list_a] :
( ( Y
= ( append_a @ Xs2 @ Y ) )
= ( Xs2 = nil_a ) ) ).
% self_append_conv2
thf(fact_435_self__append__conv2,axiom,
! [Y: list_P3592885314253461005_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( Y
= ( append7679239579558125090_a_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv2
thf(fact_436_Nil__is__append__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_437_Nil__is__append__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( append7679239579558125090_a_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_Pr7402525243500994295_a_nat )
& ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_438_append__is__Nil__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= nil_a )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_439_append__is__Nil__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys )
= nil_Pr7402525243500994295_a_nat )
= ( ( Xs2 = nil_Pr7402525243500994295_a_nat )
& ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_440_cons__perm__eq,axiom,
! [Z: a,Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Z @ Xs2 ) )
= ( mset_a @ ( cons_a @ Z @ Ys ) ) )
= ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) ) ) ).
% cons_perm_eq
thf(fact_441_list__all2__Nil,axiom,
! [P2: a > product_prod_a_nat > $o,Ys: list_P3592885314253461005_a_nat] :
( ( list_a4219772040603143988_a_nat @ P2 @ nil_a @ Ys )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% list_all2_Nil
thf(fact_442_list__all2__Nil,axiom,
! [P2: product_prod_a_nat > a > $o,Ys: list_a] :
( ( list_a5831681871908439928_nat_a @ P2 @ nil_Pr7402525243500994295_a_nat @ Ys )
= ( Ys = nil_a ) ) ).
% list_all2_Nil
thf(fact_443_list__all2__Nil,axiom,
! [P2: product_prod_a_nat > product_prod_a_nat > $o,Ys: list_P3592885314253461005_a_nat] :
( ( list_a4173326319054506371_a_nat @ P2 @ nil_Pr7402525243500994295_a_nat @ Ys )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% list_all2_Nil
thf(fact_444_list__all2__Nil,axiom,
! [P2: a > a > $o,Ys: list_a] :
( ( list_all2_a_a @ P2 @ nil_a @ Ys )
= ( Ys = nil_a ) ) ).
% list_all2_Nil
thf(fact_445_list__all2__Nil2,axiom,
! [P2: product_prod_a_nat > a > $o,Xs2: list_P3592885314253461005_a_nat] :
( ( list_a5831681871908439928_nat_a @ P2 @ Xs2 @ nil_a )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% list_all2_Nil2
thf(fact_446_list__all2__Nil2,axiom,
! [P2: a > product_prod_a_nat > $o,Xs2: list_a] :
( ( list_a4219772040603143988_a_nat @ P2 @ Xs2 @ nil_Pr7402525243500994295_a_nat )
= ( Xs2 = nil_a ) ) ).
% list_all2_Nil2
thf(fact_447_list__all2__Nil2,axiom,
! [P2: product_prod_a_nat > product_prod_a_nat > $o,Xs2: list_P3592885314253461005_a_nat] :
( ( list_a4173326319054506371_a_nat @ P2 @ Xs2 @ nil_Pr7402525243500994295_a_nat )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% list_all2_Nil2
thf(fact_448_list__all2__Nil2,axiom,
! [P2: a > a > $o,Xs2: list_a] :
( ( list_all2_a_a @ P2 @ Xs2 @ nil_a )
= ( Xs2 = nil_a ) ) ).
% list_all2_Nil2
thf(fact_449_perm__empty,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ nil_Pr7402525243500994295_a_nat )
= ( mset_P502332718515628968_a_nat @ Xs2 ) )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% perm_empty
thf(fact_450_perm__empty,axiom,
! [Xs2: list_a] :
( ( ( mset_a @ nil_a )
= ( mset_a @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% perm_empty
thf(fact_451_perm__empty2,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ Xs2 )
= ( mset_P502332718515628968_a_nat @ nil_Pr7402525243500994295_a_nat ) )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% perm_empty2
thf(fact_452_perm__empty2,axiom,
! [Xs2: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ nil_a ) )
= ( Xs2 = nil_a ) ) ).
% perm_empty2
thf(fact_453_perm__append2__eq,axiom,
! [Xs2: list_a,Zs: list_a,Ys: list_a] :
( ( ( mset_a @ ( append_a @ Xs2 @ Zs ) )
= ( mset_a @ ( append_a @ Ys @ Zs ) ) )
= ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) ) ) ).
% perm_append2_eq
thf(fact_454_perm__append1__eq,axiom,
! [Zs: list_a,Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ ( append_a @ Zs @ Xs2 ) )
= ( mset_a @ ( append_a @ Zs @ Ys ) ) )
= ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) ) ) ).
% perm_append1_eq
thf(fact_455_perm__append2,axiom,
! [Xs2: list_a,Ys: list_a,L: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( mset_a @ ( append_a @ Xs2 @ L ) )
= ( mset_a @ ( append_a @ Ys @ L ) ) ) ) ).
% perm_append2
thf(fact_456_append1__eq__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,X3: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ ( cons_P5205166803686508359_a_nat @ X3 @ nil_Pr7402525243500994295_a_nat ) )
= ( append7679239579558125090_a_nat @ Ys @ ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) )
= ( ( Xs2 = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_457_append1__eq__conv,axiom,
! [Xs2: list_a,X3: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs2 = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_458_perm__sing__eq2,axiom,
! [Y: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) )
= ( mset_P502332718515628968_a_nat @ Ys ) )
= ( Ys
= ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) ) ).
% perm_sing_eq2
thf(fact_459_perm__sing__eq2,axiom,
! [Y: a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Y @ nil_a ) )
= ( mset_a @ Ys ) )
= ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ).
% perm_sing_eq2
thf(fact_460_perm__sing__eq,axiom,
! [Ys: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ Ys )
= ( mset_P502332718515628968_a_nat @ ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) )
= ( Ys
= ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) ) ).
% perm_sing_eq
thf(fact_461_perm__sing__eq,axiom,
! [Ys: list_a,Y: a] :
( ( ( mset_a @ Ys )
= ( mset_a @ ( cons_a @ Y @ nil_a ) ) )
= ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ).
% perm_sing_eq
thf(fact_462_nat__pow__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_463_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_464_transpose_Ocases,axiom,
! [X3: list_l2471972001652375325_a_nat] :
( ( X3 != nil_li191968740515856775_a_nat )
=> ( ! [Xss: list_l2471972001652375325_a_nat] :
( X3
!= ( cons_l2046435710214046167_a_nat @ nil_Pr7402525243500994295_a_nat @ Xss ) )
=> ~ ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Xss: list_l2471972001652375325_a_nat] :
( X3
!= ( cons_l2046435710214046167_a_nat @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_465_transpose_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X3
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs3: list_a,Xss: list_list_a] :
( X3
!= ( cons_list_a @ ( cons_a @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_466_perm__empty__imp,axiom,
! [Ys: list_P3592885314253461005_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ nil_Pr7402525243500994295_a_nat )
= ( mset_P502332718515628968_a_nat @ Ys ) )
=> ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% perm_empty_imp
thf(fact_467_perm__empty__imp,axiom,
! [Ys: list_a] :
( ( ( mset_a @ nil_a )
= ( mset_a @ Ys ) )
=> ( Ys = nil_a ) ) ).
% perm_empty_imp
thf(fact_468_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_469_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_470_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_471_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_472_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_473_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_474_remdups__adj_Ocases,axiom,
! [X3: list_P3592885314253461005_a_nat] :
( ( X3 != nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat] :
( X3
!= ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X2: product_prod_a_nat,Y3: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( X3
!= ( cons_P5205166803686508359_a_nat @ X2 @ ( cons_P5205166803686508359_a_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_475_remdups__adj_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X2: a] :
( X3
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y3: a,Xs3: list_a] :
( X3
!= ( cons_a @ X2 @ ( cons_a @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_476_neq__Nil__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
= ( ? [Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( Xs2
= ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_477_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y5: a,Ys4: list_a] :
( Xs2
= ( cons_a @ Y5 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_478_list__induct2_H,axiom,
! [P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] : ( P2 @ nil_Pr7402525243500994295_a_nat @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_479_list__induct2_H,axiom,
! [P2: list_P3592885314253461005_a_nat > list_a > $o,Xs2: list_P3592885314253461005_a_nat,Ys: list_a] :
( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P2 @ nil_Pr7402525243500994295_a_nat @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_480_list__induct2_H,axiom,
! [P2: list_a > list_P3592885314253461005_a_nat > $o,Xs2: list_a,Ys: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a] : ( P2 @ ( cons_a @ X2 @ Xs3 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] : ( P2 @ nil_a @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_481_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs2: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a] : ( P2 @ ( cons_a @ X2 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P2 @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_482_list__nonempty__induct,axiom,
! [Xs2: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > $o] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( ( Xs3 != nil_Pr7402525243500994295_a_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_483_list__nonempty__induct,axiom,
! [Xs2: list_a,P2: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_484_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_485_append__Nil,axiom,
! [Ys: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_486_append_Oleft__neutral,axiom,
! [A2: list_a] :
( ( append_a @ nil_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_487_append_Oleft__neutral,axiom,
! [A2: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_488_eq__Nil__appendI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_489_eq__Nil__appendI,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_490_list_Octr__transfer_I1_J,axiom,
! [R: a > product_prod_a_nat > $o] : ( list_a4219772040603143988_a_nat @ R @ nil_a @ nil_Pr7402525243500994295_a_nat ) ).
% list.ctr_transfer(1)
thf(fact_491_list_Octr__transfer_I1_J,axiom,
! [R: product_prod_a_nat > a > $o] : ( list_a5831681871908439928_nat_a @ R @ nil_Pr7402525243500994295_a_nat @ nil_a ) ).
% list.ctr_transfer(1)
thf(fact_492_list_Octr__transfer_I1_J,axiom,
! [R: product_prod_a_nat > product_prod_a_nat > $o] : ( list_a4173326319054506371_a_nat @ R @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat ) ).
% list.ctr_transfer(1)
thf(fact_493_list_Octr__transfer_I1_J,axiom,
! [R: a > a > $o] : ( list_all2_a_a @ R @ nil_a @ nil_a ) ).
% list.ctr_transfer(1)
thf(fact_494_monoid_Onat__pow__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.nat_pow_closed
thf(fact_495_monoid_Onat__pow__one,axiom,
! [G: partia2175431115845679010xt_a_b,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( pow_a_1026414303147256608_b_nat @ G @ ( one_a_ring_ext_a_b @ G ) @ N )
= ( one_a_ring_ext_a_b @ G ) ) ) ).
% monoid.nat_pow_one
thf(fact_496_perm__sing__imp,axiom,
! [Ys: list_P3592885314253461005_a_nat,Xs2: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ Ys )
= ( mset_P502332718515628968_a_nat @ Xs2 ) )
=> ( ( Xs2
= ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) )
=> ( Ys
= ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) ) ) ).
% perm_sing_imp
thf(fact_497_perm__sing__imp,axiom,
! [Ys: list_a,Xs2: list_a,Y: a] :
( ( ( mset_a @ Ys )
= ( mset_a @ Xs2 ) )
=> ( ( Xs2
= ( cons_a @ Y @ nil_a ) )
=> ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ) ).
% perm_sing_imp
thf(fact_498_monoid_Onat__pow__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,N: nat,M2: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ M2 ) )
= ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) ) ) ) ) ).
% monoid.nat_pow_comm
thf(fact_499_monoid_Opow__mult__distrib,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ G @ Y @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( pow_a_1026414303147256608_b_nat @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ Y @ N ) ) ) ) ) ) ) ).
% monoid.pow_mult_distrib
thf(fact_500_monoid_Ogroup__commutes__pow,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ G @ Y @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ G @ Y @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) ) ) ) ) ) ) ).
% monoid.group_commutes_pow
thf(fact_501_monoid_Onat__pow__mult,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,N: nat,M2: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ M2 ) )
= ( pow_a_1026414303147256608_b_nat @ G @ X3 @ ( plus_plus_nat @ N @ M2 ) ) ) ) ) ).
% monoid.nat_pow_mult
thf(fact_502_list__induct4,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_503_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_P3592885314253461005_a_nat > list_a > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( 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 ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_504_list__induct4,axiom,
! [Xs2: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_a,P2: list_a > list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_505_list__induct4,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P2: list_a > list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_506_list__induct4,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P2: list_a > list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_507_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_a,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_508_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P2: list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_509_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_510_list__induct4,axiom,
! [Xs2: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P2: list_a > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_511_list__induct4,axiom,
! [Xs2: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P2: list_a > list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_512_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_513_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_a,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_514_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_515_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_a,P2: list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_516_list__induct3,axiom,
! [Xs2: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,P2: list_a > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_517_list__induct3,axiom,
! [Xs2: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_a,P2: list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_518_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_P3592885314253461005_a_nat,P2: list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_519_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_520_list__induct2,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_521_list__induct2,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_a,P2: list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_522_list__induct2,axiom,
! [Xs2: list_a,Ys: list_P3592885314253461005_a_nat,P2: list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_523_list__induct2,axiom,
! [Xs2: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_524_perm__append__single,axiom,
! [A2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( mset_P502332718515628968_a_nat @ ( cons_P5205166803686508359_a_nat @ A2 @ Xs2 ) )
= ( mset_P502332718515628968_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ ( cons_P5205166803686508359_a_nat @ A2 @ nil_Pr7402525243500994295_a_nat ) ) ) ) ).
% perm_append_single
thf(fact_525_perm__append__single,axiom,
! [A2: a,Xs2: list_a] :
( ( mset_a @ ( cons_a @ A2 @ Xs2 ) )
= ( mset_a @ ( append_a @ Xs2 @ ( cons_a @ A2 @ nil_a ) ) ) ) ).
% perm_append_single
thf(fact_526_rev__induct,axiom,
! [P2: list_P3592885314253461005_a_nat > $o,Xs2: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append7679239579558125090_a_nat @ Xs3 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_527_rev__induct,axiom,
! [P2: list_a > $o,Xs2: list_a] :
( ( P2 @ nil_a )
=> ( ! [X2: a,Xs3: list_a] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_528_rev__exhaust,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ~ ! [Ys2: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat] :
( Xs2
!= ( append7679239579558125090_a_nat @ Ys2 @ ( cons_P5205166803686508359_a_nat @ Y3 @ nil_Pr7402525243500994295_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_529_rev__exhaust,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs2
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_530_Cons__eq__append__conv,axiom,
! [X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 )
= ( append7679239579558125090_a_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr7402525243500994295_a_nat )
& ( ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 )
= Zs ) )
| ? [Ys5: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X3 @ Ys5 )
= Ys )
& ( Xs2
= ( append7679239579558125090_a_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_531_Cons__eq__append__conv,axiom,
! [X3: a,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X3 @ Xs2 )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X3 @ Ys5 )
= Ys )
& ( Xs2
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_532_append__eq__Cons__conv,axiom,
! [Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Ys @ Zs )
= ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) )
= ( ( ( Ys = nil_Pr7402525243500994295_a_nat )
& ( Zs
= ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) ) )
| ? [Ys5: list_P3592885314253461005_a_nat] :
( ( Ys
= ( cons_P5205166803686508359_a_nat @ X3 @ Ys5 ) )
& ( ( append7679239579558125090_a_nat @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_533_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X3: a,Xs2: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X3 @ Xs2 ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X3 @ Xs2 ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X3 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_534_rev__nonempty__induct,axiom,
! [Xs2: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > $o] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( ( Xs3 != nil_Pr7402525243500994295_a_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append7679239579558125090_a_nat @ Xs3 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_535_rev__nonempty__induct,axiom,
! [Xs2: list_a,P2: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_536_ring_Odense__repr_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a] :
( ( ring_a_b @ R )
=> ( ( X3 != nil_a )
=> ~ ! [V3: a,Va: list_a] :
( X3
!= ( cons_a @ V3 @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_537_list__all2__induct,axiom,
! [P2: product_prod_a_nat > product_prod_a_nat > $o,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,R: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( list_a4173326319054506371_a_nat @ P2 @ Xs2 @ Ys )
=> ( ( R @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( P2 @ X2 @ Y3 )
=> ( ( list_a4173326319054506371_a_nat @ P2 @ Xs3 @ Ys2 )
=> ( ( R @ Xs3 @ Ys2 )
=> ( R @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( R @ Xs2 @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_538_list__all2__induct,axiom,
! [P2: product_prod_a_nat > a > $o,Xs2: list_P3592885314253461005_a_nat,Ys: list_a,R: list_P3592885314253461005_a_nat > list_a > $o] :
( ( list_a5831681871908439928_nat_a @ P2 @ Xs2 @ Ys )
=> ( ( R @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y3: a,Ys2: list_a] :
( ( P2 @ X2 @ Y3 )
=> ( ( list_a5831681871908439928_nat_a @ P2 @ Xs3 @ Ys2 )
=> ( ( R @ Xs3 @ Ys2 )
=> ( R @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) )
=> ( R @ Xs2 @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_539_list__all2__induct,axiom,
! [P2: a > product_prod_a_nat > $o,Xs2: list_a,Ys: list_P3592885314253461005_a_nat,R: list_a > list_P3592885314253461005_a_nat > $o] :
( ( list_a4219772040603143988_a_nat @ P2 @ Xs2 @ Ys )
=> ( ( R @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs3: list_a,Y3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( P2 @ X2 @ Y3 )
=> ( ( list_a4219772040603143988_a_nat @ P2 @ Xs3 @ Ys2 )
=> ( ( R @ Xs3 @ Ys2 )
=> ( R @ ( cons_a @ X2 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( R @ Xs2 @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_540_list__all2__induct,axiom,
! [P2: a > a > $o,Xs2: list_a,Ys: list_a,R: list_a > list_a > $o] :
( ( list_all2_a_a @ P2 @ Xs2 @ Ys )
=> ( ( R @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a,Y3: a,Ys2: list_a] :
( ( P2 @ X2 @ Y3 )
=> ( ( list_all2_a_a @ P2 @ Xs3 @ Ys2 )
=> ( ( R @ Xs3 @ Ys2 )
=> ( R @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) )
=> ( R @ Xs2 @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_541_list_Orel__induct,axiom,
! [R: product_prod_a_nat > product_prod_a_nat > $o,X3: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat,Q: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( list_a4173326319054506371_a_nat @ R @ X3 @ Y )
=> ( ( Q @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [A21: product_prod_a_nat,A22: list_P3592885314253461005_a_nat,B21: product_prod_a_nat,B22: list_P3592885314253461005_a_nat] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_P5205166803686508359_a_nat @ A21 @ A22 ) @ ( cons_P5205166803686508359_a_nat @ B21 @ B22 ) ) ) )
=> ( Q @ X3 @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_542_list_Orel__induct,axiom,
! [R: product_prod_a_nat > a > $o,X3: list_P3592885314253461005_a_nat,Y: list_a,Q: list_P3592885314253461005_a_nat > list_a > $o] :
( ( list_a5831681871908439928_nat_a @ R @ X3 @ Y )
=> ( ( Q @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [A21: product_prod_a_nat,A22: list_P3592885314253461005_a_nat,B21: a,B22: list_a] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_P5205166803686508359_a_nat @ A21 @ A22 ) @ ( cons_a @ B21 @ B22 ) ) ) )
=> ( Q @ X3 @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_543_list_Orel__induct,axiom,
! [R: a > product_prod_a_nat > $o,X3: list_a,Y: list_P3592885314253461005_a_nat,Q: list_a > list_P3592885314253461005_a_nat > $o] :
( ( list_a4219772040603143988_a_nat @ R @ X3 @ Y )
=> ( ( Q @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [A21: a,A22: list_a,B21: product_prod_a_nat,B22: list_P3592885314253461005_a_nat] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_a @ A21 @ A22 ) @ ( cons_P5205166803686508359_a_nat @ B21 @ B22 ) ) ) )
=> ( Q @ X3 @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_544_list_Orel__induct,axiom,
! [R: a > a > $o,X3: list_a,Y: list_a,Q: list_a > list_a > $o] :
( ( list_all2_a_a @ R @ X3 @ Y )
=> ( ( Q @ nil_a @ nil_a )
=> ( ! [A21: a,A22: list_a,B21: a,B22: list_a] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_a @ A21 @ A22 ) @ ( cons_a @ B21 @ B22 ) ) ) )
=> ( Q @ X3 @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_545_list_Orel__cases,axiom,
! [R: product_prod_a_nat > product_prod_a_nat > $o,A2: list_P3592885314253461005_a_nat,B2: list_P3592885314253461005_a_nat] :
( ( list_a4173326319054506371_a_nat @ R @ A2 @ B2 )
=> ( ( ( A2 = nil_Pr7402525243500994295_a_nat )
=> ( B2 != nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X1: product_prod_a_nat,X23: list_P3592885314253461005_a_nat] :
( ( A2
= ( cons_P5205166803686508359_a_nat @ X1 @ X23 ) )
=> ! [Y1: product_prod_a_nat,Y23: list_P3592885314253461005_a_nat] :
( ( B2
= ( cons_P5205166803686508359_a_nat @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_a4173326319054506371_a_nat @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_546_list_Orel__cases,axiom,
! [R: product_prod_a_nat > a > $o,A2: list_P3592885314253461005_a_nat,B2: list_a] :
( ( list_a5831681871908439928_nat_a @ R @ A2 @ B2 )
=> ( ( ( A2 = nil_Pr7402525243500994295_a_nat )
=> ( B2 != nil_a ) )
=> ~ ! [X1: product_prod_a_nat,X23: list_P3592885314253461005_a_nat] :
( ( A2
= ( cons_P5205166803686508359_a_nat @ X1 @ X23 ) )
=> ! [Y1: a,Y23: list_a] :
( ( B2
= ( cons_a @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_a5831681871908439928_nat_a @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_547_list_Orel__cases,axiom,
! [R: a > product_prod_a_nat > $o,A2: list_a,B2: list_P3592885314253461005_a_nat] :
( ( list_a4219772040603143988_a_nat @ R @ A2 @ B2 )
=> ( ( ( A2 = nil_a )
=> ( B2 != nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X1: a,X23: list_a] :
( ( A2
= ( cons_a @ X1 @ X23 ) )
=> ! [Y1: product_prod_a_nat,Y23: list_P3592885314253461005_a_nat] :
( ( B2
= ( cons_P5205166803686508359_a_nat @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_a4219772040603143988_a_nat @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_548_list_Orel__cases,axiom,
! [R: a > a > $o,A2: list_a,B2: list_a] :
( ( list_all2_a_a @ R @ A2 @ B2 )
=> ( ( ( A2 = nil_a )
=> ( B2 != nil_a ) )
=> ~ ! [X1: a,X23: list_a] :
( ( A2
= ( cons_a @ X1 @ X23 ) )
=> ! [Y1: a,Y23: list_a] :
( ( B2
= ( cons_a @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_all2_a_a @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_549_list_Orel__distinct_I1_J,axiom,
! [R: product_prod_a_nat > a > $o,Y21: a,Y22: list_a] :
~ ( list_a5831681871908439928_nat_a @ R @ nil_Pr7402525243500994295_a_nat @ ( cons_a @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_550_list_Orel__distinct_I1_J,axiom,
! [R: a > a > $o,Y21: a,Y22: list_a] :
~ ( list_all2_a_a @ R @ nil_a @ ( cons_a @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_551_list_Orel__distinct_I2_J,axiom,
! [R: a > product_prod_a_nat > $o,Y21: a,Y22: list_a] :
~ ( list_a4219772040603143988_a_nat @ R @ ( cons_a @ Y21 @ Y22 ) @ nil_Pr7402525243500994295_a_nat ) ).
% list.rel_distinct(2)
thf(fact_552_list_Orel__distinct_I2_J,axiom,
! [R: a > a > $o,Y21: a,Y22: list_a] :
~ ( list_all2_a_a @ R @ ( cons_a @ Y21 @ Y22 ) @ nil_a ) ).
% list.rel_distinct(2)
thf(fact_553_same__length__different,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Xs2 != Ys )
=> ( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ? [Pre: list_P3592885314253461005_a_nat,X2: product_prod_a_nat,Xs5: list_P3592885314253461005_a_nat,Y3: product_prod_a_nat,Ys6: list_P3592885314253461005_a_nat] :
( ( X2 != Y3 )
& ( Xs2
= ( append7679239579558125090_a_nat @ Pre @ ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) @ Xs5 ) ) )
& ( Ys
= ( append7679239579558125090_a_nat @ Pre @ ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ Y3 @ nil_Pr7402525243500994295_a_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_554_same__length__different,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != Ys )
=> ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X2: a,Xs5: list_a,Y3: a,Ys6: list_a] :
( ( X2 != Y3 )
& ( Xs2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs5 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_555_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( eval_a_b @ R @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_556_perm__sym,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( mset_a @ Ys )
= ( mset_a @ Xs2 ) ) ) ).
% perm_sym
thf(fact_557_ex__mset,axiom,
! [X6: multiset_a] :
? [Xs3: list_a] :
( ( mset_a @ Xs3 )
= X6 ) ).
% ex_mset
thf(fact_558_var__def,axiom,
( var_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R4 ) @ ( cons_a @ ( zero_a_b @ R4 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_559_cons__perm__imp__perm,axiom,
! [Z: a,Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Z @ Xs2 ) )
= ( mset_a @ ( cons_a @ Z @ Ys ) ) )
=> ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) ) ) ).
% cons_perm_imp_perm
thf(fact_560_perm__set__eq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( set_a2 @ Xs2 )
= ( set_a2 @ Ys ) ) ) ).
% perm_set_eq
thf(fact_561_perm__length,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) ) ) ).
% perm_length
thf(fact_562_append__perm__imp__perm,axiom,
! [Zs: list_a,Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ ( append_a @ Zs @ Xs2 ) )
= ( mset_a @ ( append_a @ Zs @ Ys ) ) )
=> ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) ) ) ).
% append_perm_imp_perm
thf(fact_563_perm__append__swap,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( mset_a @ ( append_a @ Xs2 @ Ys ) )
= ( mset_a @ ( append_a @ Ys @ Xs2 ) ) ) ).
% perm_append_swap
thf(fact_564_list__all2__reorder__left__invariance,axiom,
! [R: a > a > $o,Xs2: list_a,Ys: list_a,Xs4: list_a] :
( ( list_all2_a_a @ R @ Xs2 @ Ys )
=> ( ( ( mset_a @ Xs4 )
= ( mset_a @ Xs2 ) )
=> ? [Ys6: list_a] :
( ( list_all2_a_a @ R @ Xs4 @ Ys6 )
& ( ( mset_a @ Ys6 )
= ( mset_a @ Ys ) ) ) ) ) ).
% list_all2_reorder_left_invariance
thf(fact_565_ring_Oeval__append,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P4 @ Q2 ) @ A2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P4 @ A2 ) @ ( pow_a_1026414303147256608_b_nat @ R @ A2 @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ R @ Q2 @ A2 ) ) ) ) ) ) ) ).
% ring.eval_append
thf(fact_566_ring_Oeval__append__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,B2: a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P4 @ ( cons_a @ B2 @ nil_a ) ) @ A2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P4 @ A2 ) @ A2 ) @ B2 ) ) ) ) ) ) ).
% ring.eval_append_aux
thf(fact_567_perm__finite,axiom,
! [A: list_a] :
( finite_finite_list_a
@ ( collect_list_a
@ ^ [B4: list_a] :
( ( mset_a @ B4 )
= ( mset_a @ A ) ) ) ) ).
% perm_finite
thf(fact_568_mset__eq__finite,axiom,
! [Xs2: list_a] :
( finite_finite_list_a
@ ( collect_list_a
@ ^ [Ys4: list_a] :
( ( mset_a @ Ys4 )
= ( mset_a @ Xs2 ) ) ) ) ).
% mset_eq_finite
thf(fact_569_monoid_Om__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_570_monoid_Om__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_571_perm__append__Cons,axiom,
! [A2: a,Xs2: list_a,Ys: list_a] :
( ( mset_a @ ( cons_a @ A2 @ ( append_a @ Xs2 @ Ys ) ) )
= ( mset_a @ ( append_a @ Xs2 @ ( cons_a @ A2 @ Ys ) ) ) ) ).
% perm_append_Cons
thf(fact_572_Group_Omonoid_Ointro,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z3 )
= ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
= X2 ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_573_monoid_Oinv__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X3: a,Y2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X3 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y2 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_574_monoid_Oone__unique,axiom,
! [G: partia2175431115845679010xt_a_b,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_575_monoidI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z3 )
= ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
= X2 ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_576_Group_Omonoid__def,axiom,
( monoid8385113658579753027xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b] :
( ! [X: a,Y5: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G3 @ X @ Y5 ) @ ( partia707051561876973205xt_a_b @ G3 ) ) ) )
& ! [X: a,Y5: a,Z4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( mult_a_ring_ext_a_b @ G3 @ ( mult_a_ring_ext_a_b @ G3 @ X @ Y5 ) @ Z4 )
= ( mult_a_ring_ext_a_b @ G3 @ X @ ( mult_a_ring_ext_a_b @ G3 @ Y5 @ Z4 ) ) ) ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G3 ) @ ( partia707051561876973205xt_a_b @ G3 ) )
& ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( mult_a_ring_ext_a_b @ G3 @ ( one_a_ring_ext_a_b @ G3 ) @ X )
= X ) )
& ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( mult_a_ring_ext_a_b @ G3 @ X @ ( one_a_ring_ext_a_b @ G3 ) )
= X ) ) ) ) ) ).
% Group.monoid_def
thf(fact_577_is__root__def,axiom,
! [P4: list_a,X3: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P4 @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P4 @ X3 )
= ( zero_a_b @ r ) )
& ( P4 != nil_a ) ) ) ).
% is_root_def
thf(fact_578_const__term__explicit,axiom,
! [P4: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P4 != nil_a )
=> ( ( ( const_term_a_b @ r @ P4 )
= A2 )
=> ~ ! [P5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P5 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P4
!= ( append_a @ P5 @ ( cons_a @ A2 @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_579_const__term__eq__last,axiom,
! [P4: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P4 @ ( cons_a @ A2 @ nil_a ) ) )
= A2 ) ) ) ).
% const_term_eq_last
thf(fact_580_eval__monom,axiom,
! [B2: a,A2: a,N: nat] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B2 @ N ) @ A2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ ( pow_a_1026414303147256608_b_nat @ r @ A2 @ N ) ) ) ) ) ).
% eval_monom
thf(fact_581_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_582_up__one__closed,axiom,
( member_nat_a
@ ^ [N2: nat] : ( if_a @ ( N2 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_583_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_584_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_585_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_586_const__term__not__zero,axiom,
! [P4: list_a] :
( ( ( const_term_a_b @ r @ P4 )
!= ( zero_a_b @ r ) )
=> ( P4 != nil_a ) ) ).
% const_term_not_zero
thf(fact_587_const__term__def,axiom,
! [P4: list_a] :
( ( const_term_a_b @ r @ P4 )
= ( eval_a_b @ r @ P4 @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_588_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_589_combine__eq__eval,axiom,
! [Ks: list_a,X3: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X3 @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X3 ) ) ).
% combine_eq_eval
thf(fact_590_combine_Oelims,axiom,
! [X3: list_a,Xa2: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X3 @ Xa2 )
= Y )
=> ( ! [K3: a,Ks2: list_a] :
( ( X3
= ( cons_a @ K3 @ Ks2 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) ) ) ) )
=> ( ( ( X3 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_591_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_592_combine__append,axiom,
! [Ks: list_a,Us3: list_a,Ks3: list_a,Vs3: 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 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Vs3 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks3 ) @ ( append_a @ Us3 @ Vs3 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_593_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_594_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_595_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_596_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_597_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_598_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_599_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_600_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_601_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_602_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_603_length__0__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% length_0_conv
thf(fact_604_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_605_local_Onat__pow__0,axiom,
! [X3: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_606_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_607_monom__in__carrier,axiom,
! [A2: a,N: nat] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A2 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_608_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_609_ring_Ocombine_Ocong,axiom,
embedded_combine_a_b = embedded_combine_a_b ).
% ring.combine.cong
thf(fact_610_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_611_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_612_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_613_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_614_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_615_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_616_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_617_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_618_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_619_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_620_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_621_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_622_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_623_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_624_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_625_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_626_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_627_list_Osize_I3_J,axiom,
( ( size_s984997627204368545_a_nat @ nil_Pr7402525243500994295_a_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_628_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_629_ring_Ocombine_Osimps_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ nil_a @ Us3 )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(2)
thf(fact_630_ring_Ocombine_Osimps_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks @ nil_a )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(3)
thf(fact_631_monoid_Onat__pow__0,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( pow_a_1026414303147256608_b_nat @ G @ X3 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G ) ) ) ).
% monoid.nat_pow_0
thf(fact_632_semiring_Onat__pow__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( semiring_a_b @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ ( zero_a_b @ R ) @ N )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_633_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P4 )
!= ( zero_a_b @ R ) )
=> ( P4 != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_634_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P4 )
= ( eval_a_b @ R @ P4 @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_635_ring_Ocombine__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.combine_in_carrier
thf(fact_636_ring_Ocombine_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: a,Ks: list_a,U: a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.combine.simps(1)
thf(fact_637_ring_Ocombine__eq__eval,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,X3: a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks @ ( polyno2922411391617481336se_a_b @ R @ X3 @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ R @ Ks @ X3 ) ) ) ).
% ring.combine_eq_eval
thf(fact_638_ring_Oup__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_nat_a
@ ^ [N2: nat] : ( if_a @ ( N2 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ R ) @ ( zero_a_b @ R ) )
@ ( up_a_b @ R ) ) ) ).
% ring.up_one_closed
thf(fact_639_ring_Omonom__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ R @ A2 @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_640_ring_Ocombine_Oelims,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a,Xa2: list_a,Y: a] :
( ( ring_a_b @ R )
=> ( ( ( embedded_combine_a_b @ R @ X3 @ Xa2 )
= Y )
=> ( ! [K3: a,Ks2: list_a] :
( ( X3
= ( cons_a @ K3 @ Ks2 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K3 @ U2 ) @ ( embedded_combine_a_b @ R @ Ks2 @ Us4 ) ) ) ) )
=> ( ( ( X3 = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% ring.combine.elims
thf(fact_641_ring_Oeval__monom,axiom,
! [R: partia2175431115845679010xt_a_b,B2: a,A2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( monom_a_b @ R @ B2 @ N ) @ A2 )
= ( mult_a_ring_ext_a_b @ R @ B2 @ ( pow_a_1026414303147256608_b_nat @ R @ A2 @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_642_ring_Ocombine__append,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us3: list_a,Ks3: list_a,Vs3: list_a] :
( ( ring_a_b @ R )
=> ( ( ( 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 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( embedded_combine_a_b @ R @ Ks @ Us3 ) @ ( embedded_combine_a_b @ R @ Ks3 @ Vs3 ) )
= ( embedded_combine_a_b @ R @ ( append_a @ Ks @ Ks3 ) @ ( append_a @ Us3 @ Vs3 ) ) ) ) ) ) ) ) ) ).
% ring.combine_append
thf(fact_643_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,X3: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P4 @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P4 @ X3 )
= ( zero_a_b @ R ) )
& ( P4 != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_644_ring_Ocombine__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.combine_append_zero
thf(fact_645_ring_Oconst__term__eq__last,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( const_term_a_b @ R @ ( append_a @ P4 @ ( cons_a @ A2 @ nil_a ) ) )
= A2 ) ) ) ) ).
% ring.const_term_eq_last
thf(fact_646_ring_Oconst__term__explicit,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( P4 != nil_a )
=> ( ( ( const_term_a_b @ R @ P4 )
= A2 )
=> ~ ! [P5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P5 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( P4
!= ( append_a @ P5 @ ( cons_a @ A2 @ nil_a ) ) ) ) ) ) ) ) ).
% ring.const_term_explicit
thf(fact_647_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_648_monic__degree__one__root__condition,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ B2 )
= ( A2 = B2 ) ) ) ).
% monic_degree_one_root_condition
thf(fact_649_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_650_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_651_eval__replicate,axiom,
! [P4: list_a,A2: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P4 ) @ A2 )
= ( eval_a_b @ r @ P4 @ A2 ) ) ) ) ).
% eval_replicate
thf(fact_652_up__a__inv__closed,axiom,
! [P4: nat > a] :
( ( member_nat_a @ P4 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_inv_a_b @ r @ ( P4 @ I3 ) )
@ ( up_a_b @ r ) ) ) ).
% up_a_inv_closed
thf(fact_653_add_Oinv__mult__group,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X3 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_654_add_Oinv__solve__left,axiom,
! [A2: a,B2: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B2 ) @ C ) )
= ( C
= ( add_a_b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_655_add_Oinv__solve__left_H,axiom,
! [A2: a,B2: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B2 ) @ C )
= A2 )
= ( C
= ( add_a_b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_656_add_Oinv__solve__right,axiom,
! [A2: a,B2: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( add_a_b @ r @ B2 @ ( a_inv_a_b @ r @ C ) ) )
= ( B2
= ( add_a_b @ r @ A2 @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_657_add_Oinv__solve__right_H,axiom,
! [A2: a,B2: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B2 @ ( a_inv_a_b @ r @ C ) )
= A2 )
= ( B2
= ( add_a_b @ r @ A2 @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_658_a__transpose__inv,axiom,
! [X3: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X3 @ Y )
= Z )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_659_local_Ominus__add,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_660_r__neg1,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( add_a_b @ r @ X3 @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_661_r__neg2,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_662_l__minus,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) ) ) ) ) ).
% l_minus
thf(fact_663_r__minus,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) ) ) ) ) ).
% r_minus
thf(fact_664_add_Oint__pow__inv,axiom,
! [X3: a,I: int] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X3 ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X3 ) ) ) ) ).
% add.int_pow_inv
thf(fact_665_minus__eq,axiom,
! [X3: a,Y: a] :
( ( a_minus_a_b @ r @ X3 @ Y )
= ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_666_l__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_667_minus__equality,axiom,
! [Y: a,X3: a] :
( ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X3 )
= Y ) ) ) ) ).
% minus_equality
thf(fact_668_r__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ X3 ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_669_local_Omonom__def,axiom,
! [A2: a,N: nat] :
( ( monom_a_b @ r @ A2 @ N )
= ( cons_a @ A2 @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_670_mset__zero__iff__right,axiom,
! [X3: list_P3592885314253461005_a_nat] :
( ( zero_z8410795768267065558_a_nat
= ( mset_P502332718515628968_a_nat @ X3 ) )
= ( X3 = nil_Pr7402525243500994295_a_nat ) ) ).
% mset_zero_iff_right
thf(fact_671_mset__zero__iff__right,axiom,
! [X3: list_a] :
( ( zero_zero_multiset_a
= ( mset_a @ X3 ) )
= ( X3 = nil_a ) ) ).
% mset_zero_iff_right
thf(fact_672_mset__zero__iff,axiom,
! [X3: list_P3592885314253461005_a_nat] :
( ( ( mset_P502332718515628968_a_nat @ X3 )
= zero_z8410795768267065558_a_nat )
= ( X3 = nil_Pr7402525243500994295_a_nat ) ) ).
% mset_zero_iff
thf(fact_673_mset__zero__iff,axiom,
! [X3: list_a] :
( ( ( mset_a @ X3 )
= zero_zero_multiset_a )
= ( X3 = nil_a ) ) ).
% mset_zero_iff
thf(fact_674_replicate__eq__replicate,axiom,
! [M2: nat,X3: a,N: nat,Y: a] :
( ( ( replicate_a @ M2 @ X3 )
= ( replicate_a @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2 != zero_zero_nat )
=> ( X3 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_675_length__replicate,axiom,
! [N: nat,X3: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X3 ) )
= N ) ).
% length_replicate
thf(fact_676_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_677_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_678_Ball__set__replicate,axiom,
! [N: nat,A2: a,P2: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ( P2 @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_679_Bex__set__replicate,axiom,
! [N: nat,A2: a,P2: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ( P2 @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_680_in__set__replicate,axiom,
! [X3: nat > a,N: nat,Y: nat > a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ ( replicate_nat_a @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_681_in__set__replicate,axiom,
! [X3: a,N: nat,Y: a] :
( ( member_a @ X3 @ ( set_a2 @ ( replicate_a @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_682_replicate__empty,axiom,
! [N: nat,X3: product_prod_a_nat] :
( ( ( replic5595554873386817213_a_nat @ N @ X3 )
= nil_Pr7402525243500994295_a_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_683_replicate__empty,axiom,
! [N: nat,X3: a] :
( ( ( replicate_a @ N @ X3 )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_684_empty__replicate,axiom,
! [N: nat,X3: product_prod_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( replic5595554873386817213_a_nat @ N @ X3 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_685_empty__replicate,axiom,
! [N: nat,X3: a] :
( ( nil_a
= ( replicate_a @ N @ X3 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_686_a__inv__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_687_local_Ominus__minus,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X3 ) )
= X3 ) ) ).
% local.minus_minus
thf(fact_688_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_689_add_Oinv__eq__1__iff,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X3 )
= ( zero_a_b @ r ) )
= ( X3
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_690_append__replicate__commute,axiom,
! [N: nat,X3: a,K: nat] :
( ( append_a @ ( replicate_a @ N @ X3 ) @ ( replicate_a @ K @ X3 ) )
= ( append_a @ ( replicate_a @ K @ X3 ) @ ( replicate_a @ N @ X3 ) ) ) ).
% append_replicate_commute
thf(fact_691_replicate__0,axiom,
! [X3: product_prod_a_nat] :
( ( replic5595554873386817213_a_nat @ zero_zero_nat @ X3 )
= nil_Pr7402525243500994295_a_nat ) ).
% replicate_0
thf(fact_692_replicate__0,axiom,
! [X3: a] :
( ( replicate_a @ zero_zero_nat @ X3 )
= nil_a ) ).
% replicate_0
thf(fact_693_replicate__length__same,axiom,
! [Xs2: list_a,X3: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( X2 = X3 ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs2 ) @ X3 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_694_replicate__eqI,axiom,
! [Xs2: list_nat_a,N: nat,X3: nat > a] :
( ( ( size_size_list_nat_a @ Xs2 )
= N )
=> ( ! [Y3: nat > a] :
( ( member_nat_a @ Y3 @ ( set_nat_a2 @ Xs2 ) )
=> ( Y3 = X3 ) )
=> ( Xs2
= ( replicate_nat_a @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_695_replicate__eqI,axiom,
! [Xs2: list_a,N: nat,X3: a] :
( ( ( size_size_list_a @ Xs2 )
= N )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs2 ) )
=> ( Y3 = X3 ) )
=> ( Xs2
= ( replicate_a @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_696_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_697_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X3 ) )
= X3 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_698_ring_Ominus__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% ring.minus_zero
thf(fact_699_replicate__app__Cons__same,axiom,
! [N: nat,X3: a,Xs2: list_a] :
( ( append_a @ ( replicate_a @ N @ X3 ) @ ( cons_a @ X3 @ Xs2 ) )
= ( cons_a @ X3 @ ( append_a @ ( replicate_a @ N @ X3 ) @ Xs2 ) ) ) ).
% replicate_app_Cons_same
thf(fact_700_replicate__add,axiom,
! [N: nat,M2: nat,X3: a] :
( ( replicate_a @ ( plus_plus_nat @ N @ M2 ) @ X3 )
= ( append_a @ ( replicate_a @ N @ X3 ) @ ( replicate_a @ M2 @ X3 ) ) ) ).
% replicate_add
thf(fact_701_abelian__group_Oa__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_inv_a_b @ G @ X3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_702_abelian__group_Ominus__minus,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X3 ) )
= X3 ) ) ) ).
% abelian_group.minus_minus
thf(fact_703_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b,X: a,Y5: a] : ( add_a_b @ R4 @ X @ ( a_inv_a_b @ R4 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_704_mset_Osimps_I1_J,axiom,
( ( mset_P502332718515628968_a_nat @ nil_Pr7402525243500994295_a_nat )
= zero_z8410795768267065558_a_nat ) ).
% mset.simps(1)
thf(fact_705_mset_Osimps_I1_J,axiom,
( ( mset_a @ nil_a )
= zero_zero_multiset_a ) ).
% mset.simps(1)
thf(fact_706_ring_Oup__a__inv__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P4: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P4 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_inv_a_b @ R @ ( P4 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ).
% ring.up_a_inv_closed
thf(fact_707_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_708_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ ( add_a_b @ R @ X3 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_709_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_710_replicate__append__same,axiom,
! [I: nat,X3: product_prod_a_nat] :
( ( append7679239579558125090_a_nat @ ( replic5595554873386817213_a_nat @ I @ X3 ) @ ( cons_P5205166803686508359_a_nat @ X3 @ nil_Pr7402525243500994295_a_nat ) )
= ( cons_P5205166803686508359_a_nat @ X3 @ ( replic5595554873386817213_a_nat @ I @ X3 ) ) ) ).
% replicate_append_same
thf(fact_711_replicate__append__same,axiom,
! [I: nat,X3: a] :
( ( append_a @ ( replicate_a @ I @ X3 ) @ ( cons_a @ X3 @ nil_a ) )
= ( cons_a @ X3 @ ( replicate_a @ I @ X3 ) ) ) ).
% replicate_append_same
thf(fact_712_ring_Or__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( a_inv_a_b @ R @ Y ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_713_ring_Ol__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ Y )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_714_abelian__group_Ominus__add,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X3 @ Y ) )
= ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_715_abelian__group_Or__neg2,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_716_abelian__group_Or__neg1,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ ( add_a_b @ G @ X3 @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_717_abelian__group_Oa__transpose__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ X3 @ Y )
= Z )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_718_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ X3 @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_719_abelian__group_Ominus__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( a_minus_a_b @ G @ X3 @ Y )
= ( add_a_b @ G @ X3 @ ( a_inv_a_b @ G @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_720_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_721_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_722_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_723_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_724_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_725_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_726_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_727_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_728_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( a_inv_a_b @ R @ X3 ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_729_abelian__group_Ol__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ X3 )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_730_abelian__group_Or__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( a_inv_a_b @ G @ X3 ) )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_731_abelian__group_Ominus__equality,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X3 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ X3 )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_732_ring_Omonom__def,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( monom_a_b @ R @ A2 @ N )
= ( cons_a @ A2 @ ( replicate_a @ N @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_def
thf(fact_733_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ A2 ) @ nil_a ) ) @ B2 )
= ( A2 = B2 ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_734_ring_Ocombine__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.combine_replicate
thf(fact_735_ring_Ocombine__append__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a,Ks: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.combine_append_replicate
thf(fact_736_ring_Oeval__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,A2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P4 ) @ A2 )
= ( eval_a_b @ R @ P4 @ A2 ) ) ) ) ) ).
% ring.eval_replicate
thf(fact_737_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_738_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_739_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_740_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_741_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M3 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_742_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_743_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_744_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_745_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_746_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_747_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_748_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_749_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_750_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_751_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_752_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ! [Xa3: a] :
( ( member_a @ Xa3 @ H )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa3 ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_753_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_754_subringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ H ) ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_755_dense__repr__replicate__zero,axiom,
! [N: nat,P4: list_a] :
( ( dense_repr_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P4 ) )
= ( dense_repr_a_b @ r @ P4 ) ) ).
% dense_repr_replicate_zero
thf(fact_756_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_757_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_758_dense__repr_Osimps_I1_J,axiom,
( ( dense_repr_a_b @ r @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ).
% dense_repr.simps(1)
thf(fact_759_size__mset,axiom,
! [Xs2: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% size_mset
thf(fact_760_drop0,axiom,
( ( drop_a @ zero_zero_nat )
= ( ^ [X: list_a] : X ) ) ).
% drop0
thf(fact_761_drop__drop,axiom,
! [N: nat,M2: nat,Xs2: list_a] :
( ( drop_a @ N @ ( drop_a @ M2 @ Xs2 ) )
= ( drop_a @ ( plus_plus_nat @ N @ M2 ) @ Xs2 ) ) ).
% drop_drop
thf(fact_762_set__empty,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( ( set_Pr924983374503034536_a_nat @ Xs2 )
= bot_bo9049108969261143879_a_nat )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% set_empty
thf(fact_763_set__empty,axiom,
! [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= bot_bot_set_a )
= ( Xs2 = nil_a ) ) ).
% set_empty
thf(fact_764_set__empty2,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( bot_bo9049108969261143879_a_nat
= ( set_Pr924983374503034536_a_nat @ Xs2 ) )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% set_empty2
thf(fact_765_set__empty2,axiom,
! [Xs2: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% set_empty2
thf(fact_766_drop__all,axiom,
! [Xs2: list_P3592885314253461005_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) @ N )
=> ( ( drop_P2883665741211355575_a_nat @ N @ Xs2 )
= nil_Pr7402525243500994295_a_nat ) ) ).
% drop_all
thf(fact_767_drop__all,axiom,
! [Xs2: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N )
=> ( ( drop_a @ N @ Xs2 )
= nil_a ) ) ).
% drop_all
thf(fact_768_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_P3592885314253461005_a_nat] :
( ( ( drop_P2883665741211355575_a_nat @ N @ Xs2 )
= nil_Pr7402525243500994295_a_nat )
= ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_769_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_a] :
( ( ( drop_a @ N @ Xs2 )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_770_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( drop_P2883665741211355575_a_nat @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_771_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_772_drop__0,axiom,
! [Xs2: list_a] :
( ( drop_a @ zero_zero_nat @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_773_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_774_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_775_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_776_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_777_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_778_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_779_list__all2__dropI,axiom,
! [P2: a > a > $o,Xs2: list_a,Ys: list_a,N: nat] :
( ( list_all2_a_a @ P2 @ Xs2 @ Ys )
=> ( list_all2_a_a @ P2 @ ( drop_a @ N @ Xs2 ) @ ( drop_a @ N @ Ys ) ) ) ).
% list_all2_dropI
thf(fact_780_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_781_drop__Nil,axiom,
! [N: nat] :
( ( drop_P2883665741211355575_a_nat @ N @ nil_Pr7402525243500994295_a_nat )
= nil_Pr7402525243500994295_a_nat ) ).
% drop_Nil
thf(fact_782_in__set__dropD,axiom,
! [X3: nat > a,N: nat,Xs2: list_nat_a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ ( drop_nat_a @ N @ Xs2 ) ) )
=> ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_783_in__set__dropD,axiom,
! [X3: a,N: nat,Xs2: list_a] :
( ( member_a @ X3 @ ( set_a2 @ ( drop_a @ N @ Xs2 ) ) )
=> ( member_a @ X3 @ ( set_a2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_784_set__drop__subset,axiom,
! [N: nat,Xs2: list_list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( drop_list_a @ N @ Xs2 ) ) @ ( set_list_a2 @ Xs2 ) ) ).
% set_drop_subset
thf(fact_785_set__drop__subset,axiom,
! [N: nat,Xs2: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ N @ Xs2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% set_drop_subset
thf(fact_786_set__drop__subset,axiom,
! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_drop_subset
thf(fact_787_finite__has__minimal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_788_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_789_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_790_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_791_finite__has__maximal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_792_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_793_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_794_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_795_empty__set,axiom,
( bot_bo9049108969261143879_a_nat
= ( set_Pr924983374503034536_a_nat @ nil_Pr7402525243500994295_a_nat ) ) ).
% empty_set
thf(fact_796_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_797_monoid_Ocarrier__not__empty,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( partia707051561876973205xt_a_b @ G )
!= bot_bot_set_a ) ) ).
% monoid.carrier_not_empty
thf(fact_798_ring_Odense__repr_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ) ).
% ring.dense_repr.simps(1)
thf(fact_799_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs2: list_list_a] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( drop_list_a @ M2 @ Xs2 ) ) @ ( set_list_a2 @ ( drop_list_a @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_800_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ M2 @ Xs2 ) ) @ ( set_a2 @ ( drop_a @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_801_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M2 @ Xs2 ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_802_ring_Odense__repr__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P4 ) )
= ( dense_repr_a_b @ R @ P4 ) ) ) ).
% ring.dense_repr_replicate_zero
thf(fact_803_ring_Ocombine__prepend__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us3: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.combine_prepend_replicate
thf(fact_804_ring_OsubringI,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) )
=> ( subring_a_b @ H @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_805_const__term__zero,axiom,
! [K2: set_a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( P4 != nil_a )
=> ( ( ( const_term_a_b @ r @ P4 )
= ( zero_a_b @ r ) )
=> ~ ! [P5: list_a] :
( ( polynomial_a_b @ r @ K2 @ P5 )
=> ( ( P5 != nil_a )
=> ( P4
!= ( append_a @ P5 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_806_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_807_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_808_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_809_subset__antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_810_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_811_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_812_subsetI,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_nat_a @ X2 @ B ) )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_813_subsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% subsetI
thf(fact_814_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_815_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_816_polynomial__incl,axiom,
! [K2: set_a,P4: list_a] :
( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ K2 ) ) ).
% polynomial_incl
thf(fact_817_eval__poly__in__carrier,axiom,
! [K2: set_a,P4: list_a,X3: a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P4 @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_818_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_819_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_820_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_821_append__is__polynomial,axiom,
! [K2: set_a,P4: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( P4 != nil_a )
=> ( polynomial_a_b @ r @ K2 @ ( append_a @ P4 @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% append_is_polynomial
thf(fact_822_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_823_carrier__polynomial,axiom,
! [K2: set_a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P4 ) ) ) ).
% carrier_polynomial
thf(fact_824_polynomial__in__carrier,axiom,
! [K2: set_a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_825_monom__decomp,axiom,
! [K2: set_a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( P4
= ( poly_of_dense_a_b @ r @ ( dense_repr_a_b @ r @ P4 ) ) ) ) ) ).
% monom_decomp
thf(fact_826_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_827_ring_Opolynomial__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ K2 ) ) ) ).
% ring.polynomial_incl
thf(fact_828_ring_Ocarrier__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( polynomial_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) @ P4 ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_829_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q ) )
= ( ! [X: list_a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_830_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_831_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_832_set__eq__subset,axiom,
( ( ^ [Y4: set_list_a,Z2: set_list_a] : ( Y4 = Z2 ) )
= ( ^ [A7: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A7 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_833_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A7: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A7 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_834_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A7: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_835_subset__trans,axiom,
! [A: set_list_a,B: set_list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C4 )
=> ( ord_le8861187494160871172list_a @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_836_subset__trans,axiom,
! [A: set_a,B: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C4 )
=> ( ord_less_eq_set_a @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_837_subset__trans,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_eq_set_nat @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_838_Collect__mono,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_839_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_840_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_841_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_842_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_843_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_844_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A7: set_nat_a,B4: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A7 )
=> ( member_nat_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_845_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B4: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A7 )
=> ( member_list_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_846_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A7 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_847_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A7 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_848_equalityD2,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% equalityD2
thf(fact_849_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_850_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_851_equalityD1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% equalityD1
thf(fact_852_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_853_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_854_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A7: set_nat_a,B4: set_nat_a] :
! [X: nat > a] :
( ( member_nat_a @ X @ A7 )
=> ( member_nat_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_855_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B4: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A7 )
=> ( member_list_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_856_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A7 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_857_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A7 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_858_equalityE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).
% equalityE
thf(fact_859_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_860_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_861_subsetD,axiom,
! [A: set_nat_a,B: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B ) ) ) ).
% subsetD
thf(fact_862_subsetD,axiom,
! [A: set_list_a,B: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B ) ) ) ).
% subsetD
thf(fact_863_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_864_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_865_in__mono,axiom,
! [A: set_nat_a,B: set_nat_a,X3: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ X3 @ A )
=> ( member_nat_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_866_in__mono,axiom,
! [A: set_list_a,B: set_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ X3 @ A )
=> ( member_list_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_867_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_868_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_869_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a,X3: a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P4 @ X3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_870_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A7: set_nat_a,B4: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A7 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_871_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B4: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A7 )
@ ^ [X: list_a] : ( member_list_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_872_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A7 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_873_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A7 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_874_Collect__subset,axiom,
! [A: set_nat_a,P2: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_875_Collect__subset,axiom,
! [A: set_list_a,P2: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_876_Collect__subset,axiom,
! [A: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_877_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_878_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $false ) ) ).
% empty_def
thf(fact_879_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_880_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_881_ring_Opolynomial__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.polynomial_in_carrier
thf(fact_882_ring_Oappend__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( P4 != nil_a )
=> ( polynomial_a_b @ R @ K2 @ ( append_a @ P4 @ ( replicate_a @ N @ ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% ring.append_is_polynomial
thf(fact_883_ring_Oconst__term__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( P4 != nil_a )
=> ( ( ( const_term_a_b @ R @ P4 )
= ( zero_a_b @ R ) )
=> ~ ! [P5: list_a] :
( ( polynomial_a_b @ R @ K2 @ P5 )
=> ( ( P5 != nil_a )
=> ( P4
!= ( append_a @ P5 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_884_subringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subringE(2)
thf(fact_885_subringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_886_subringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_887_ring_Ocarrier__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_888_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_889_poly__mult__prepend__replicate__zero,axiom,
! [P12: list_a,P23: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P12 ) @ P23 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_890_eval__poly__add__aux,axiom,
! [P4: list_a,Q2: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P4 )
= ( size_size_list_a @ Q2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P4 @ Q2 ) @ A2 )
= ( add_a_b @ r @ ( eval_a_b @ r @ P4 @ A2 ) @ ( eval_a_b @ r @ Q2 @ A2 ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_891_poly__mult_Osimps_I1_J,axiom,
! [P23: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P23 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_892_poly__add__closed,axiom,
! [K2: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P12 )
=> ( ( polynomial_a_b @ r @ K2 @ P23 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_closed
thf(fact_893_poly__mult__closed,axiom,
! [K2: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P12 )
=> ( ( polynomial_a_b @ r @ K2 @ P23 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_894_poly__add__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_895_poly__add__comm,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ P23 @ P12 ) ) ) ) ).
% poly_add_comm
thf(fact_896_poly__mult__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_897_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_898_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_899_poly__add__zero_I2_J,axiom,
! [K2: set_a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( poly_add_a_b @ r @ nil_a @ P4 )
= P4 ) ) ) ).
% poly_add_zero(2)
thf(fact_900_poly__add__zero_I1_J,axiom,
! [K2: set_a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( poly_add_a_b @ r @ P4 @ nil_a )
= P4 ) ) ) ).
% poly_add_zero(1)
thf(fact_901_poly__mult__l__distr,axiom,
! [K2: set_a,P12: list_a,P23: list_a,P32: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P12 )
=> ( ( polynomial_a_b @ r @ K2 @ P23 )
=> ( ( polynomial_a_b @ r @ K2 @ P32 )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P12 @ P23 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P32 ) @ ( poly_mult_a_b @ r @ P23 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_l_distr
thf(fact_902_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_903_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_904_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_905_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_906_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_907_poly__mult__zero_I1_J,axiom,
! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P4 )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_908_poly__mult__zero_I2_J,axiom,
! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P4 @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_909_poly__mult__l__distr_H,axiom,
! [P12: list_a,P23: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( 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 @ P12 @ P23 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P32 ) @ ( poly_mult_a_b @ r @ P23 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_910_Idl__subset__ideal_H,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( member_a @ A2 @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_911_poly__add__is__polynomial,axiom,
! [K2: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_912_poly__mult__is__polynomial,axiom,
! [K2: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_913_poly__add__replicate__zero_I2_J,axiom,
! [K2: set_a,P4: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P4 )
= P4 ) ) ) ).
% poly_add_replicate_zero(2)
thf(fact_914_poly__add__replicate__zero_I1_J,axiom,
! [K2: set_a,P4: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( poly_add_a_b @ r @ P4 @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= P4 ) ) ) ).
% poly_add_replicate_zero(1)
thf(fact_915_eval__poly__add,axiom,
! [P4: list_a,Q2: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P4 @ Q2 ) @ A2 )
= ( add_a_b @ r @ ( eval_a_b @ r @ P4 @ A2 ) @ ( eval_a_b @ r @ Q2 @ A2 ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_916_insert__subset,axiom,
! [X3: nat > a,A: set_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X3 @ A ) @ B )
= ( ( member_nat_a @ X3 @ B )
& ( ord_le871467723717165285_nat_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_917_insert__subset,axiom,
! [X3: list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X3 @ A ) @ B )
= ( ( member_list_a @ X3 @ B )
& ( ord_le8861187494160871172list_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_918_insert__subset,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( ( member_a @ X3 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_919_insert__subset,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_920_finite__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A2 @ A ) )
= ( finite_finite_list_a @ A ) ) ).
% finite_insert
thf(fact_921_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_922_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_923_singleton__conv,axiom,
! [A2: list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( X = A2 ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_924_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_925_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X: a] : ( X = A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_926_singleton__conv2,axiom,
! [A2: list_a] :
( ( collect_list_a
@ ( ^ [Y4: list_a,Z2: list_a] : ( Y4 = Z2 )
@ A2 ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_927_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_928_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_929_singleton__insert__inj__eq,axiom,
! [B2: list_a,A2: list_a,A: set_list_a] :
( ( ( insert_list_a @ B2 @ bot_bot_set_list_a )
= ( insert_list_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_930_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_931_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_932_singleton__insert__inj__eq_H,axiom,
! [A2: list_a,A: set_list_a,B2: list_a] :
( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
= ( ( A2 = B2 )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_933_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_934_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_935_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_936_set__replicate,axiom,
! [N: nat,X3: a] :
( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X3 ) )
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% set_replicate
thf(fact_937_Collect__conv__if2,axiom,
! [P2: list_a > $o,A2: list_a] :
( ( ( P2 @ A2 )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A2 = X )
& ( P2 @ X ) ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A2 = X )
& ( P2 @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_938_Collect__conv__if2,axiom,
! [P2: nat > $o,A2: nat] :
( ( ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P2 @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P2 @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_939_Collect__conv__if2,axiom,
! [P2: a > $o,A2: a] :
( ( ( P2 @ A2 )
=> ( ( collect_a
@ ^ [X: a] :
( ( A2 = X )
& ( P2 @ X ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_a
@ ^ [X: a] :
( ( A2 = X )
& ( P2 @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_940_Collect__conv__if,axiom,
! [P2: list_a > $o,A2: list_a] :
( ( ( P2 @ A2 )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A2 )
& ( P2 @ X ) ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A2 )
& ( P2 @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_941_Collect__conv__if,axiom,
! [P2: nat > $o,A2: nat] :
( ( ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P2 @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P2 @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_942_Collect__conv__if,axiom,
! [P2: a > $o,A2: a] :
( ( ( P2 @ A2 )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A2 )
& ( P2 @ X ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A2 )
& ( P2 @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_943_ring_Opoly__mult__l__distr,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P12: list_a,P23: list_a,P32: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P12 )
=> ( ( polynomial_a_b @ R @ K2 @ P23 )
=> ( ( polynomial_a_b @ R @ K2 @ P32 )
=> ( ( poly_mult_a_b @ R @ ( poly_add_a_b @ R @ P12 @ P23 ) @ P32 )
= ( poly_add_a_b @ R @ ( poly_mult_a_b @ R @ P12 @ P32 ) @ ( poly_mult_a_b @ R @ P23 @ P32 ) ) ) ) ) ) ) ) ).
% ring.poly_mult_l_distr
thf(fact_944_insert__compr,axiom,
( insert_nat_a
= ( ^ [A4: nat > a,B4: set_nat_a] :
( collect_nat_a
@ ^ [X: nat > a] :
( ( X = A4 )
| ( member_nat_a @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_945_insert__compr,axiom,
( insert_list_a
= ( ^ [A4: list_a,B4: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( X = A4 )
| ( member_list_a @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_946_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A4 )
| ( member_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_947_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B4: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A4 )
| ( member_a @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_948_insert__Collect,axiom,
! [A2: list_a,P2: list_a > $o] :
( ( insert_list_a @ A2 @ ( collect_list_a @ P2 ) )
= ( collect_list_a
@ ^ [U3: list_a] :
( ( U3 != A2 )
=> ( P2 @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_949_insert__Collect,axiom,
! [A2: nat,P2: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P2 ) )
= ( collect_nat
@ ^ [U3: nat] :
( ( U3 != A2 )
=> ( P2 @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_950_insert__Collect,axiom,
! [A2: a,P2: a > $o] :
( ( insert_a @ A2 @ ( collect_a @ P2 ) )
= ( collect_a
@ ^ [U3: a] :
( ( U3 != A2 )
=> ( P2 @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_951_insert__mono,axiom,
! [C4: set_list_a,D2: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ C4 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A2 @ C4 ) @ ( insert_list_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_952_insert__mono,axiom,
! [C4: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C4 ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_953_insert__mono,axiom,
! [C4: set_nat,D2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C4 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C4 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_954_subset__insert,axiom,
! [X3: nat > a,A: set_nat_a,B: set_nat_a] :
( ~ ( member_nat_a @ X3 @ A )
=> ( ( ord_le871467723717165285_nat_a @ A @ ( insert_nat_a @ X3 @ B ) )
= ( ord_le871467723717165285_nat_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_955_subset__insert,axiom,
! [X3: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a @ X3 @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X3 @ B ) )
= ( ord_le8861187494160871172list_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_956_subset__insert,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_957_subset__insert,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_958_subset__insertI,axiom,
! [B: set_list_a,A2: list_a] : ( ord_le8861187494160871172list_a @ B @ ( insert_list_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_959_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_960_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_961_subset__insertI2,axiom,
! [A: set_list_a,B: set_list_a,B2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_962_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_963_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_964_finite_OinsertI,axiom,
! [A: set_list_a,A2: list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( insert_list_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_965_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_966_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_967_ring_Opoly__mult__l__distr_H,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a,P32: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( 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 @ P12 @ P23 ) @ P32 )
= ( poly_add_a_b @ R @ ( poly_mult_a_b @ R @ P12 @ P32 ) @ ( poly_mult_a_b @ R @ P23 @ P32 ) ) ) ) ) ) ) ).
% ring.poly_mult_l_distr'
thf(fact_968_ring_Opoly__mult_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( poly_mult_a_b @ R @ nil_a @ P23 )
= nil_a ) ) ).
% ring.poly_mult.simps(1)
thf(fact_969_subset__singletonD,axiom,
! [A: set_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) )
=> ( ( A = bot_bot_set_list_a )
| ( A
= ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_970_subset__singletonD,axiom,
! [A: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_971_subset__singletonD,axiom,
! [A: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_972_subset__singleton__iff,axiom,
! [X6: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ X6 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( ( X6 = bot_bot_set_list_a )
| ( X6
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_973_subset__singleton__iff,axiom,
! [X6: set_a,A2: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_974_subset__singleton__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_975_infinite__finite__induct,axiom,
! [P2: set_nat_a > $o,A: set_nat_a] :
( ! [A8: set_nat_a] :
( ~ ( finite_finite_nat_a @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_976_infinite__finite__induct,axiom,
! [P2: set_list_a > $o,A: set_list_a] :
( ! [A8: set_list_a] :
( ~ ( finite_finite_list_a @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_977_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_978_infinite__finite__induct,axiom,
! [P2: set_a > $o,A: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_979_finite__ne__induct,axiom,
! [F4: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F4 )
=> ( ( F4 != bot_bot_set_nat_a )
=> ( ! [X2: nat > a] : ( P2 @ ( insert_nat_a @ X2 @ bot_bot_set_nat_a ) )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_980_finite__ne__induct,axiom,
! [F4: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F4 )
=> ( ( F4 != bot_bot_set_list_a )
=> ( ! [X2: list_a] : ( P2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_981_finite__ne__induct,axiom,
! [F4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_982_finite__ne__induct,axiom,
! [F4: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ! [X2: a] : ( P2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_983_finite__induct,axiom,
! [F4: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F4 )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_984_finite__induct,axiom,
! [F4: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F4 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_985_finite__induct,axiom,
! [F4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_986_finite__induct,axiom,
! [F4: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_987_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A4: set_list_a] :
( ( A4 = bot_bot_set_list_a )
| ? [A7: set_list_a,B6: list_a] :
( ( A4
= ( insert_list_a @ B6 @ A7 ) )
& ( finite_finite_list_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_988_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A7: set_nat,B6: nat] :
( ( A4
= ( insert_nat @ B6 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_989_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A7: set_a,B6: a] :
( ( A4
= ( insert_a @ B6 @ A7 ) )
& ( finite_finite_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_990_finite_Ocases,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( A2 != bot_bot_set_list_a )
=> ~ ! [A8: set_list_a] :
( ? [A3: list_a] :
( A2
= ( insert_list_a @ A3 @ A8 ) )
=> ~ ( finite_finite_list_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_991_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A3: nat] :
( A2
= ( insert_nat @ A3 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_992_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A3: a] :
( A2
= ( insert_a @ A3 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_993_finite__subset__induct_H,axiom,
! [F4: set_nat_a,A: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F4 )
=> ( ( ord_le871467723717165285_nat_a @ F4 @ A )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [A3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A3 @ A )
=> ( ( ord_le871467723717165285_nat_a @ F3 @ A )
=> ( ~ ( member_nat_a @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_994_finite__subset__induct_H,axiom,
! [F4: set_list_a,A: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F4 )
=> ( ( ord_le8861187494160871172list_a @ F4 @ A )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A3 @ A )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A )
=> ( ~ ( member_list_a @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_995_finite__subset__induct_H,axiom,
! [F4: set_a,A: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_996_finite__subset__induct_H,axiom,
! [F4: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_997_finite__subset__induct,axiom,
! [F4: set_nat_a,A: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F4 )
=> ( ( ord_le871467723717165285_nat_a @ F4 @ A )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [A3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A3 @ A )
=> ( ~ ( member_nat_a @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ A3 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_998_finite__subset__induct,axiom,
! [F4: set_list_a,A: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F4 )
=> ( ( ord_le8861187494160871172list_a @ F4 @ A )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A3 @ A )
=> ( ~ ( member_list_a @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ A3 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_999_finite__subset__induct,axiom,
! [F4: set_a,A: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1000_finite__subset__induct,axiom,
! [F4: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1001_ring_Opoly__mult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P12 )
=> ( ( polynomial_a_b @ R @ K2 @ P23 )
=> ( polynomial_a_b @ R @ K2 @ ( poly_mult_a_b @ R @ P12 @ P23 ) ) ) ) ) ) ).
% ring.poly_mult_closed
thf(fact_1002_ring_Opoly__add__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P12 )
=> ( ( polynomial_a_b @ R @ K2 @ P23 )
=> ( polynomial_a_b @ R @ K2 @ ( poly_add_a_b @ R @ P12 @ P23 ) ) ) ) ) ) ).
% ring.poly_add_closed
thf(fact_1003_ring_Opoly__mult__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ R @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.poly_mult_in_carrier
thf(fact_1004_ring_Opoly__add__comm,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P12 @ P23 )
= ( poly_add_a_b @ R @ P23 @ P12 ) ) ) ) ) ).
% ring.poly_add_comm
thf(fact_1005_ring_Opoly__add__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ R @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.poly_add_in_carrier
thf(fact_1006_set__replicate__conv__if,axiom,
! [N: nat,X3: a] :
( ( ( N = zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X3 ) )
= bot_bot_set_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X3 ) )
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1007_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_1008_ring_Ogenideal__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.genideal_zero
thf(fact_1009_ring_Opoly__add__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( poly_add_a_b @ R @ nil_a @ P4 )
= P4 ) ) ) ) ).
% ring.poly_add_zero(2)
thf(fact_1010_ring_Opoly__add__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( poly_add_a_b @ R @ P4 @ nil_a )
= P4 ) ) ) ) ).
% ring.poly_add_zero(1)
thf(fact_1011_ring_Ozeropideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_1012_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_1013_ring_Opoly__mult__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P4 @ nil_a )
= nil_a ) ) ) ).
% ring.poly_mult_zero(2)
thf(fact_1014_ring_Opoly__mult__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ nil_a @ P4 )
= nil_a ) ) ) ).
% ring.poly_mult_zero(1)
thf(fact_1015_ring_Opoly__mult__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K2 )
=> ( polynomial_a_b @ R @ K2 @ ( poly_mult_a_b @ R @ P12 @ P23 ) ) ) ) ) ) ).
% ring.poly_mult_is_polynomial
thf(fact_1016_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( genideal_a_b @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( member_a @ A2 @ ( genideal_a_b @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_1017_ring_Opoly__add__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K2 )
=> ( polynomial_a_b @ R @ K2 @ ( poly_add_a_b @ R @ P12 @ P23 ) ) ) ) ) ) ).
% ring.poly_add_is_polynomial
thf(fact_1018_ring_Ogenideal__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( one_a_ring_ext_a_b @ R ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.genideal_one
thf(fact_1019_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_1020_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_1021_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_1022_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_1023_ring_Opoly__add__replicate__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( poly_add_a_b @ R @ P4 @ ( replicate_a @ N @ ( zero_a_b @ R ) ) )
= P4 ) ) ) ) ).
% ring.poly_add_replicate_zero(1)
thf(fact_1024_ring_Opoly__add__replicate__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( poly_add_a_b @ R @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P4 )
= P4 ) ) ) ) ).
% ring.poly_add_replicate_zero(2)
thf(fact_1025_ring_Oeval__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_add_a_b @ R @ P4 @ Q2 ) @ A2 )
= ( add_a_b @ R @ ( eval_a_b @ R @ P4 @ A2 ) @ ( eval_a_b @ R @ Q2 @ A2 ) ) ) ) ) ) ) ).
% ring.eval_poly_add
thf(fact_1026_ring_Opoly__mult__prepend__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P12 @ P23 )
= ( poly_mult_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P12 ) @ P23 ) ) ) ) ) ).
% ring.poly_mult_prepend_replicate_zero
thf(fact_1027_ring_Oeval__poly__add__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( size_size_list_a @ P4 )
= ( size_size_list_a @ Q2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_add_a_b @ R @ P4 @ Q2 ) @ A2 )
= ( add_a_b @ R @ ( eval_a_b @ R @ P4 @ A2 ) @ ( eval_a_b @ R @ Q2 @ A2 ) ) ) ) ) ) ) ) ).
% ring.eval_poly_add_aux
thf(fact_1028_ring_Omonom__decomp,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( P4
= ( poly_of_dense_a_b @ R @ ( dense_repr_a_b @ R @ P4 ) ) ) ) ) ) ).
% ring.monom_decomp
thf(fact_1029_poly__add__monom,axiom,
! [P4: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A2 @ ( size_size_list_a @ P4 ) ) @ P4 )
= ( cons_a @ A2 @ P4 ) ) ) ) ).
% poly_add_monom
thf(fact_1030_lead__coeff__in__carrier,axiom,
! [K2: set_a,A2: a,P4: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A2 @ P4 ) )
=> ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% lead_coeff_in_carrier
thf(fact_1031_poly__add__append__replicate,axiom,
! [P4: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P4 @ ( replicate_a @ ( size_size_list_a @ Q2 ) @ ( zero_a_b @ r ) ) ) @ Q2 )
= ( normalize_a_b @ r @ ( append_a @ P4 @ Q2 ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_1032_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_1033_local_Onormalize__idem,axiom,
! [P4: list_a,Q2: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P4 ) @ Q2 ) )
= ( normalize_a_b @ r @ ( append_a @ P4 @ Q2 ) ) ) ).
% local.normalize_idem
thf(fact_1034_normalize__polynomial,axiom,
! [K2: set_a,P4: list_a] :
( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( normalize_a_b @ r @ P4 )
= P4 ) ) ).
% normalize_polynomial
thf(fact_1035_dense__repr__normalize,axiom,
! [P4: list_a] :
( ( dense_repr_a_b @ r @ ( normalize_a_b @ r @ P4 ) )
= ( dense_repr_a_b @ r @ P4 ) ) ).
% dense_repr_normalize
thf(fact_1036_normalize__length__le,axiom,
! [P4: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P4 ) ) @ ( size_size_list_a @ P4 ) ) ).
% normalize_length_le
thf(fact_1037_normalize__in__carrier,axiom,
! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P4 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_1038_normalize__gives__polynomial,axiom,
! [P4: list_a,K2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( normalize_a_b @ r @ P4 ) ) ) ).
% normalize_gives_polynomial
thf(fact_1039_normalize__replicate__zero,axiom,
! [N: nat,P4: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P4 ) )
= ( normalize_a_b @ r @ P4 ) ) ).
% normalize_replicate_zero
thf(fact_1040_poly__add__normalize_I3_J,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ ( normalize_a_b @ r @ P23 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_1041_poly__add__normalize_I2_J,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ P12 @ ( normalize_a_b @ r @ P23 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_1042_poly__add__normalize__aux,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_1043_eval__normalize,axiom,
! [P4: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P4 ) @ A2 )
= ( eval_a_b @ r @ P4 @ A2 ) ) ) ) ).
% eval_normalize
thf(fact_1044_poly__mult__normalize,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).
% poly_mult_normalize
thf(fact_1045_poly__add__zero_H_I1_J,axiom,
! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P4 @ nil_a )
= ( normalize_a_b @ r @ P4 ) ) ) ).
% poly_add_zero'(1)
thf(fact_1046_poly__add__zero_H_I2_J,axiom,
! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P4 )
= ( normalize_a_b @ r @ P4 ) ) ) ).
% poly_add_zero'(2)
thf(fact_1047_lead__coeff__not__zero,axiom,
! [K2: set_a,A2: a,P4: list_a] :
( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A2 @ P4 ) )
=> ( member_a @ A2 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_1048_finite__Diff2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_1049_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1050_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_1051_finite__Diff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1052_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1053_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1054_poly__add__replicate__zero_H_I1_J,axiom,
! [P4: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P4 @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P4 ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_1055_poly__add__replicate__zero_H_I2_J,axiom,
! [P4: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P4 )
= ( normalize_a_b @ r @ P4 ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_1056_poly__add__append__zero,axiom,
! [P4: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P4 @ Q2 ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_1057_poly__mult__append__zero,axiom,
! [P4: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q2 )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P4 @ Q2 ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_1058_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1059_Diff__eq__empty__iff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( minus_646659088055828811list_a @ A @ B )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1060_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1061_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1062_finite__Diff__insert,axiom,
! [A: set_list_a,A2: list_a,B: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ B ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1063_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1064_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1065_const__is__polynomial,axiom,
! [A2: a,K2: set_a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K2 @ ( cons_a @ A2 @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_1066_monom__is__polynomial,axiom,
! [K2: set_a,A2: a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K2 @ ( monom_a_b @ r @ A2 @ N ) ) ) ) ).
% monom_is_polynomial
thf(fact_1067_Diff__mono,axiom,
! [A: set_list_a,C4: set_list_a,D2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C4 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B ) @ ( minus_646659088055828811list_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1068_Diff__mono,axiom,
! [A: set_a,C4: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C4 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1069_Diff__mono,axiom,
! [A: set_nat,C4: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1070_Diff__subset,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1071_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1072_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1073_double__diff,axiom,
! [A: set_list_a,B: set_list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C4 )
=> ( ( minus_646659088055828811list_a @ B @ ( minus_646659088055828811list_a @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1074_double__diff,axiom,
! [A: set_a,B: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C4 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1075_double__diff,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1076_diff__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_1077_diff__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_1078_diff__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_1079_diff__eq__diff__less__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1080_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1081_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1082_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1083_set__diff__eq,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A7: set_nat_a,B4: set_nat_a] :
( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A7 )
& ~ ( member_nat_a @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1084_set__diff__eq,axiom,
( minus_646659088055828811list_a
= ( ^ [A7: set_list_a,B4: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A7 )
& ~ ( member_list_a @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1085_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A7 )
& ~ ( member_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1086_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A7: set_a,B4: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A7 )
& ~ ( member_a @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1087_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B6 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1088_diff__le__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_1089_le__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_1090_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1091_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1092_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1093_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1094_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1095_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1096_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1097_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1098_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1099_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1100_subset__Diff__insert,axiom,
! [A: set_nat_a,B: set_nat_a,X3: nat > a,C4: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ ( minus_490503922182417452_nat_a @ B @ ( insert_nat_a @ X3 @ C4 ) ) )
= ( ( ord_le871467723717165285_nat_a @ A @ ( minus_490503922182417452_nat_a @ B @ C4 ) )
& ~ ( member_nat_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1101_subset__Diff__insert,axiom,
! [A: set_list_a,B: set_list_a,X3: list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( minus_646659088055828811list_a @ B @ ( insert_list_a @ X3 @ C4 ) ) )
= ( ( ord_le8861187494160871172list_a @ A @ ( minus_646659088055828811list_a @ B @ C4 ) )
& ~ ( member_list_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1102_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X3: a,C4: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X3 @ C4 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C4 ) )
& ~ ( member_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1103_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X3: nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X3 @ C4 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C4 ) )
& ~ ( member_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1104_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_1105_ring_Onormalize__idem,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( normalize_a_b @ R @ P4 ) @ Q2 ) )
= ( normalize_a_b @ R @ ( append_a @ P4 @ Q2 ) ) ) ) ).
% ring.normalize_idem
thf(fact_1106_ring_Onormalize__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( normalize_a_b @ R @ P4 )
= P4 ) ) ) ).
% ring.normalize_polynomial
thf(fact_1107_ring_Odense__repr__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ ( normalize_a_b @ R @ P4 ) )
= ( dense_repr_a_b @ R @ P4 ) ) ) ).
% ring.dense_repr_normalize
thf(fact_1108_subset__insert__iff,axiom,
! [A: set_nat_a,X3: nat > a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ ( insert_nat_a @ X3 @ B ) )
= ( ( ( member_nat_a @ X3 @ A )
=> ( ord_le871467723717165285_nat_a @ ( minus_490503922182417452_nat_a @ A @ ( insert_nat_a @ X3 @ bot_bot_set_nat_a ) ) @ B ) )
& ( ~ ( member_nat_a @ X3 @ A )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1109_subset__insert__iff,axiom,
! [A: set_list_a,X3: list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X3 @ B ) )
= ( ( ( member_list_a @ X3 @ A )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) @ B ) )
& ( ~ ( member_list_a @ X3 @ A )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1110_subset__insert__iff,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1111_subset__insert__iff,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1112_Diff__single__insert,axiom,
! [A: set_list_a,X3: list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) @ B )
=> ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1113_Diff__single__insert,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1114_Diff__single__insert,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1115_finite__empty__induct,axiom,
! [A: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ A )
=> ( ( P2 @ A )
=> ( ! [A3: nat > a,A8: set_nat_a] :
( ( finite_finite_nat_a @ A8 )
=> ( ( member_nat_a @ A3 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_490503922182417452_nat_a @ A8 @ ( insert_nat_a @ A3 @ bot_bot_set_nat_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1116_finite__empty__induct,axiom,
! [A: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ( P2 @ A )
=> ( ! [A3: list_a,A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ( member_list_a @ A3 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1117_finite__empty__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P2 @ A )
=> ( ! [A3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A3 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1118_finite__empty__induct,axiom,
! [A: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P2 @ A )
=> ( ! [A3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A3 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_minus_set_a @ A8 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1119_infinite__coinduct,axiom,
! [X6: set_list_a > $o,A: set_list_a] :
( ( X6 @ A )
=> ( ! [A8: set_list_a] :
( ( X6 @ A8 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ A8 )
& ( ( X6 @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_1120_infinite__coinduct,axiom,
! [X6: set_nat > $o,A: set_nat] :
( ( X6 @ A )
=> ( ! [A8: set_nat] :
( ( X6 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X6 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1121_infinite__coinduct,axiom,
! [X6: set_a > $o,A: set_a] :
( ( X6 @ A )
=> ( ! [A8: set_a] :
( ( X6 @ A8 )
=> ? [X5: a] :
( ( member_a @ X5 @ A8 )
& ( ( X6 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_1122_infinite__remove,axiom,
! [S: set_list_a,A2: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_1123_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1124_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1125_ring_Onormalize__length__le,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ R @ P4 ) ) @ ( size_size_list_a @ P4 ) ) ) ).
% ring.normalize_length_le
thf(fact_1126_remove__induct,axiom,
! [P2: set_nat_a > $o,B: set_nat_a] :
( ( P2 @ bot_bot_set_nat_a )
=> ( ( ~ ( finite_finite_nat_a @ B )
=> ( P2 @ B ) )
=> ( ! [A8: set_nat_a] :
( ( finite_finite_nat_a @ A8 )
=> ( ( A8 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A8 @ B )
=> ( ! [X5: nat > a] :
( ( member_nat_a @ X5 @ A8 )
=> ( P2 @ ( minus_490503922182417452_nat_a @ A8 @ ( insert_nat_a @ X5 @ bot_bot_set_nat_a ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1127_remove__induct,axiom,
! [P2: set_list_a > $o,B: set_list_a] :
( ( P2 @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B )
=> ( P2 @ B ) )
=> ( ! [A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ( A8 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A8 @ B )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A8 )
=> ( P2 @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1128_remove__induct,axiom,
! [P2: set_a > $o,B: set_a] :
( ( P2 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P2 @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P2 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1129_remove__induct,axiom,
! [P2: set_nat > $o,B: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P2 @ B ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1130_finite__remove__induct,axiom,
! [B: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ B )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [A8: set_nat_a] :
( ( finite_finite_nat_a @ A8 )
=> ( ( A8 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A8 @ B )
=> ( ! [X5: nat > a] :
( ( member_nat_a @ X5 @ A8 )
=> ( P2 @ ( minus_490503922182417452_nat_a @ A8 @ ( insert_nat_a @ X5 @ bot_bot_set_nat_a ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1131_finite__remove__induct,axiom,
! [B: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ B )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ( A8 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A8 @ B )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A8 )
=> ( P2 @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1132_finite__remove__induct,axiom,
! [B: set_a,P2: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P2 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1133_finite__remove__induct,axiom,
! [B: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1134_ring_Onormalize__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ R @ P4 ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_1135_ring_Onormalize__gives__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ K2 )
=> ( polynomial_a_b @ R @ K2 @ ( normalize_a_b @ R @ P4 ) ) ) ) ).
% ring.normalize_gives_polynomial
thf(fact_1136_ring_Onormalize__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P4 ) )
= ( normalize_a_b @ R @ P4 ) ) ) ).
% ring.normalize_replicate_zero
thf(fact_1137_ring_Opoly__add__normalize__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P12 @ P23 )
= ( poly_add_a_b @ R @ ( normalize_a_b @ R @ P12 ) @ P23 ) ) ) ) ) ).
% ring.poly_add_normalize_aux
thf(fact_1138_ring_Opoly__add__normalize_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P12 @ P23 )
= ( poly_add_a_b @ R @ P12 @ ( normalize_a_b @ R @ P23 ) ) ) ) ) ) ).
% ring.poly_add_normalize(2)
thf(fact_1139_ring_Opoly__add__normalize_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P12 @ P23 )
= ( poly_add_a_b @ R @ ( normalize_a_b @ R @ P12 ) @ ( normalize_a_b @ R @ P23 ) ) ) ) ) ) ).
% ring.poly_add_normalize(3)
thf(fact_1140_ring_Opoly__mult__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P12 @ P23 )
= ( poly_mult_a_b @ R @ ( normalize_a_b @ R @ P12 ) @ P23 ) ) ) ) ) ).
% ring.poly_mult_normalize
thf(fact_1141_ring_Oeval__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( normalize_a_b @ R @ P4 ) @ A2 )
= ( eval_a_b @ R @ P4 @ A2 ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_1142_ring_Olead__coeff__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A2: a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ ( cons_a @ A2 @ P4 ) )
=> ( member_a @ A2 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% ring.lead_coeff_not_zero
thf(fact_1143_ring_Opoly__add__zero_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P4 @ nil_a )
= ( normalize_a_b @ R @ P4 ) ) ) ) ).
% ring.poly_add_zero'(1)
thf(fact_1144_ring_Opoly__add__zero_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ nil_a @ P4 )
= ( normalize_a_b @ R @ P4 ) ) ) ) ).
% ring.poly_add_zero'(2)
thf(fact_1145_ring_Oconst__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ R @ K2 @ ( cons_a @ A2 @ nil_a ) ) ) ) ).
% ring.const_is_polynomial
thf(fact_1146_ring_Opoly__add__replicate__zero_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ P4 @ ( replicate_a @ N @ ( zero_a_b @ R ) ) )
= ( normalize_a_b @ R @ P4 ) ) ) ) ).
% ring.poly_add_replicate_zero'(1)
thf(fact_1147_ring_Opoly__add__replicate__zero_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P4 )
= ( normalize_a_b @ R @ P4 ) ) ) ) ).
% ring.poly_add_replicate_zero'(2)
thf(fact_1148_ring_Omonom__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ R @ K2 @ ( monom_a_b @ R @ A2 @ N ) ) ) ) ) ).
% ring.monom_is_polynomial
thf(fact_1149_ring_Olead__coeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A2: a,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ ( cons_a @ A2 @ P4 ) )
=> ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ) ).
% ring.lead_coeff_in_carrier
thf(fact_1150_ring_Opoly__add__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ ( append_a @ Q2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) )
= ( normalize_a_b @ R @ ( append_a @ ( poly_add_a_b @ R @ P4 @ Q2 ) @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ).
% ring.poly_add_append_zero
thf(fact_1151_ring_Opoly__mult__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ Q2 )
= ( normalize_a_b @ R @ ( append_a @ ( poly_mult_a_b @ R @ P4 @ Q2 ) @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ).
% ring.poly_mult_append_zero
thf(fact_1152_ring_Opoly__add__append__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,Q2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_add_a_b @ R @ ( append_a @ P4 @ ( replicate_a @ ( size_size_list_a @ Q2 ) @ ( zero_a_b @ R ) ) ) @ Q2 )
= ( normalize_a_b @ R @ ( append_a @ P4 @ Q2 ) ) ) ) ) ) ).
% ring.poly_add_append_replicate
thf(fact_1153_ring_Opoly__add__monom,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a,A2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ R @ ( monom_a_b @ R @ A2 @ ( size_size_list_a @ P4 ) ) @ P4 )
= ( cons_a @ A2 @ P4 ) ) ) ) ) ).
% ring.poly_add_monom
thf(fact_1154_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K4: a] : ( normalize_a_b @ r @ ( cons_a @ K4 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_1155_le__add__diff__inverse,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1156_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1157_drop__exp__base,axiom,
! [N: nat,X3: a,M2: nat] :
( ( drop_a @ N @ ( polyno2922411391617481336se_a_b @ r @ X3 @ M2 ) )
= ( polyno2922411391617481336se_a_b @ r @ X3 @ ( minus_minus_nat @ M2 @ N ) ) ) ).
% drop_exp_base
thf(fact_1158_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1159_normalize__def_H_I2_J,axiom,
! [P4: list_a] :
( ( normalize_a_b @ r @ P4 )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P4 ) ) ) @ P4 ) ) ).
% normalize_def'(2)
thf(fact_1160_add_Oint__pow__diff,axiom,
! [X3: a,N: int,M2: int] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M2 ) @ X3 )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X3 ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M2 @ X3 ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_1161_normalize__trick,axiom,
! [P4: list_a] :
( P4
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P4 ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P4 ) ) ) ).
% normalize_trick
thf(fact_1162_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_1163_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_1164_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1165_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1166_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1167_length__drop,axiom,
! [N: nat,Xs2: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_1168_normalize__def_H_I1_J,axiom,
! [P4: list_a] :
( P4
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P4 ) ) ) @ ( zero_a_b @ r ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P4 ) ) ) @ P4 ) ) ) ).
% normalize_def'(1)
thf(fact_1169_drop__replicate,axiom,
! [I: nat,K: nat,X3: a] :
( ( drop_a @ I @ ( replicate_a @ K @ X3 ) )
= ( replicate_a @ ( minus_minus_nat @ K @ I ) @ X3 ) ) ).
% drop_replicate
thf(fact_1170_le__add__diff__inverse2,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1171_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1172_drop__append,axiom,
! [N: nat,Xs2: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs2 ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1173_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_1174_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_1175_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_1176_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_1177_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1178_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1179_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_1180_minus__set__def,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A7: set_nat_a,B4: set_nat_a] :
( collect_nat_a
@ ( minus_minus_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A7 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1181_minus__set__def,axiom,
( minus_646659088055828811list_a
= ( ^ [A7: set_list_a,B4: set_list_a] :
( collect_list_a
@ ( minus_minus_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A7 )
@ ^ [X: list_a] : ( member_list_a @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1182_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A7 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1183_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A7: set_a,B4: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X: a] : ( member_a @ X @ A7 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1184_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1185_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1186_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1187_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1188_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1189_ring_Odrop__exp__base,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,X3: a,M2: nat] :
( ( ring_a_b @ R )
=> ( ( drop_a @ N @ ( polyno2922411391617481336se_a_b @ R @ X3 @ M2 ) )
= ( polyno2922411391617481336se_a_b @ R @ X3 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% ring.drop_exp_base
thf(fact_1190_ring_Onormalize__def_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ P4 )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P4 ) ) ) @ P4 ) ) ) ).
% ring.normalize_def'(2)
thf(fact_1191_ring_Opoly__of__const__def,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( poly_of_const_a_b @ R )
= ( ^ [K4: a] : ( normalize_a_b @ R @ ( cons_a @ K4 @ nil_a ) ) ) ) ) ).
% ring.poly_of_const_def
thf(fact_1192_ring_Onormalize__trick,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( P4
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P4 ) ) ) @ ( zero_a_b @ R ) ) @ ( normalize_a_b @ R @ P4 ) ) ) ) ).
% ring.normalize_trick
thf(fact_1193_ring_Onormalize__def_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P4: list_a] :
( ( ring_a_b @ R )
=> ( P4
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P4 ) ) ) @ ( zero_a_b @ R ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P4 ) ) ) @ P4 ) ) ) ) ).
% ring.normalize_def'(1)
thf(fact_1194_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1195_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1196_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1197_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1198_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A2 ) @ K2 )
=> ( member_a @ A2 @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1199_poly__of__dense_Oelims,axiom,
! [X3: list_P3592885314253461005_a_nat,Y: list_a] :
( ( ( poly_of_dense_a_b @ r @ X3 )
= Y )
=> ( Y
= ( foldr_4031981466149041118list_a
@ ( produc7724251129057698313list_a
@ ^ [A4: a,N2: nat] : ( poly_add_a_b @ r @ ( monom_a_b @ r @ A4 @ N2 ) ) )
@ X3
@ nil_a ) ) ) ).
% poly_of_dense.elims
thf(fact_1200_subring__props_I7_J,axiom,
! [K2: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_1201_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_1202_subring__props_I6_J,axiom,
! [K2: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_1203_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1204_subring__props_I5_J,axiom,
! [K2: set_a,H2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H2 @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H2 ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_1205_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_1206_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_1207_line__extension__smult__closed,axiom,
! [K2: set_a,E: set_a,A2: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K3: a,V3: a] :
( ( member_a @ K3 @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A2 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A2 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_1208_poly__of__dense_Osimps,axiom,
! [Dl: list_P3592885314253461005_a_nat] :
( ( poly_of_dense_a_b @ r @ Dl )
= ( foldr_4031981466149041118list_a
@ ( produc7724251129057698313list_a
@ ^ [A4: a,N2: nat] : ( poly_add_a_b @ r @ ( monom_a_b @ r @ A4 @ N2 ) ) )
@ Dl
@ nil_a ) ) ).
% poly_of_dense.simps
thf(fact_1209_subfieldE_I3_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_1210_subfieldE_I5_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_1211_ring_Osubring__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H2 @ K2 )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_1212_ring_Osubring__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_1213_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H12: a,H23: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_1214_subfieldE_I6_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_1215_ring_Osubring__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( K2 != bot_bot_set_a ) ) ) ).
% ring.subring_props(4)
thf(fact_1216_Span__mem__imp__non__trivial__combine,axiom,
! [K2: set_a,Us3: list_a,A2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ~ ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ! [Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ K2 )
=> ( ( ( size_size_list_a @ Ks2 )
= ( size_size_list_a @ Us3 ) )
=> ( ( embedded_combine_a_b @ r @ ( cons_a @ K3 @ Ks2 ) @ ( cons_a @ A2 @ Us3 ) )
!= ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% Span_mem_imp_non_trivial_combine
thf(fact_1217_Span__mem__iff,axiom,
! [K2: set_a,Us3: list_a,A2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
= ( ? [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 )
& ( ( embedded_combine_a_b @ r @ ( cons_a @ X @ Ks4 ) @ ( cons_a @ A2 @ Us3 ) )
= ( zero_a_b @ r ) ) ) ) ) ) ) ) ) ).
% Span_mem_iff
thf(fact_1218_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_1219_mono__Span__subset,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) ) ) ) ) ).
% mono_Span_subset
thf(fact_1220_mono__Span__sublist,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( set_a2 @ Vs3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_1221_Span__same__set,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs3 ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= ( embedded_Span_a_b @ r @ K2 @ Vs3 ) ) ) ) ) ).
% Span_same_set
thf(fact_1222_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_1223_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_1224_Span_Oelims,axiom,
! [X3: set_a,Xa2: list_a,Y: set_a] :
( ( ( embedded_Span_a_b @ r @ X3 @ Xa2 )
= Y )
=> ( Y
= ( foldr_a_set_a @ ( embedd971793762689825387on_a_b @ r @ X3 ) @ Xa2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% Span.elims
thf(fact_1225_Span_Osimps,axiom,
! [K2: set_a,Us3: list_a] :
( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= ( foldr_a_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 ) @ Us3 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% Span.simps
thf(fact_1226_subalgebra__Span__incl,axiom,
! [K2: set_a,V: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ V ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_1227_Span__subalgebraI,axiom,
! [K2: set_a,E: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ E @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E )
=> ( ! [V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V5 )
=> ( ord_less_eq_set_a @ E @ V5 ) ) )
=> ( E
= ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_1228_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_1229_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_1230_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_1231_Span__smult__closed,axiom,
! [K2: set_a,Us3: list_a,K: a,V4: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V4 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V4 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_1232_Span__subgroup__props_I4_J,axiom,
! [K2: set_a,Us3: list_a,V4: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V4 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( a_inv_a_b @ r @ V4 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ).
% Span_subgroup_props(4)
thf(fact_1233_mono__Span__append_I2_J,axiom,
! [K2: set_a,Us3: list_a,Vs3: 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 @ Vs3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Vs3 @ Us3 ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_1234_mono__Span__append_I1_J,axiom,
! [K2: set_a,Us3: list_a,Vs3: 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 @ Vs3 ) @ ( 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 @ Vs3 ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_1235_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_1236_Span__mem__iff__length__version,axiom,
! [K2: set_a,Us3: list_a,A2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( 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 ) )
& ( A2
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_1237_Span__m__inv__simprule,axiom,
! [K2: set_a,Us3: list_a,K: a,A2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A2 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ A2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ) ).
% Span_m_inv_simprule
thf(fact_1238_line__extension__of__combine__set__length__version,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd971793762689825387on_a_b @ r @ K2 @ U
@ ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us3 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 ) ) ) )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ ( cons_a @ U @ Us3 ) ) )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ ( cons_a @ U @ Us3 ) ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 ) ) ) ) ) ) ).
% line_extension_of_combine_set_length_version
thf(fact_1239_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_1240_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F4: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F4 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F4 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F4 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_1241_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_1242_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_1243_Span__eq__combine__set,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 ) ) ) ) ) ) ).
% Span_eq_combine_set
thf(fact_1244_Span__eq__combine__set__length__version,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us3 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 ) ) ) ) ) ) ).
% Span_eq_combine_set_length_version
thf(fact_1245_line__extension__of__combine__set,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd971793762689825387on_a_b @ r @ K2 @ U
@ ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 ) ) ) )
= ( collect_a
@ ^ [Uu: a] :
? [Ks4: list_a] :
( ( Uu
= ( embedded_combine_a_b @ r @ Ks4 @ ( cons_a @ U @ Us3 ) ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 ) ) ) ) ) ) ).
% line_extension_of_combine_set
thf(fact_1246_dependent__imp__non__trivial__combine,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ~ ! [Ks2: list_a] :
( ( ( size_size_list_a @ Ks2 )
= ( size_size_list_a @ Us3 ) )
=> ( ( ( embedded_combine_a_b @ r @ Ks2 @ Us3 )
= ( zero_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ K2 )
=> ( ( set_a2 @ Ks2 )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).
% dependent_imp_non_trivial_combine
thf(fact_1247_Span__append__eq__set__add,axiom,
! [K2: set_a,Us3: list_a,Vs3: 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 @ Vs3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs3 ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_1248_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_1249_li__Nil,axiom,
! [K2: set_a] : ( embedd5208550302661555450nt_a_b @ r @ K2 @ nil_a ) ).
% li_Nil
thf(fact_1250_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_1251_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_1252_set__add__comm,axiom,
! [I2: set_a,J2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J2 )
= ( set_add_a_b @ r @ J2 @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_1253_set__add__closed,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1254_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_1255_independent__split_I1_J,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs3 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs3 ) ) ) ).
% independent_split(1)
thf(fact_1256_independent__split_I2_J,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs3 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ) ).
% independent_split(2)
thf(fact_1257_sum__space__dim_I1_J,axiom,
! [K2: set_a,E: set_a,F4: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F4 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( set_add_a_b @ r @ E @ F4 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1258_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_1259_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_1260_independent__same__set,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs3 ) )
=> ( ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Vs3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs3 ) ) ) ) ) ).
% independent_same_set
thf(fact_1261_independent_Osimps,axiom,
! [A1: set_a,A23: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A23 )
= ( ? [K5: set_a] :
( ( A1 = K5 )
& ( A23 = nil_a ) )
| ? [U3: a,K5: set_a,Us5: list_a] :
( ( A1 = K5 )
& ( A23
= ( cons_a @ U3 @ Us5 ) )
& ( member_a @ U3 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ U3 @ ( embedded_Span_a_b @ r @ K5 @ Us5 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K5 @ Us5 ) ) ) ) ).
% independent.simps
thf(fact_1262_independent_Ocases,axiom,
! [A1: set_a,A23: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A23 )
=> ( ( A23 != nil_a )
=> ~ ! [U2: a,Us4: list_a] :
( ( A23
= ( 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_1263_independent__rotate1__aux,axiom,
! [K2: set_a,U: a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ ( append_a @ Us3 @ Vs3 ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ ( append_a @ Us3 @ ( cons_a @ U @ nil_a ) ) @ Vs3 ) ) ) ) ).
% independent_rotate1_aux
thf(fact_1264_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_1265_independent__replacement,axiom,
! [K2: set_a,U: a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs3 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs3 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ X2 @ Us3 ) ) ) ) ) ) ) ).
% independent_replacement
thf(fact_1266_independent__length__le,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ ( size_size_list_a @ Vs3 ) ) ) ) ) ) ).
% independent_length_le
thf(fact_1267_replacement__theorem,axiom,
! [K2: set_a,Us6: list_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us6 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs3 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us6 @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Vs3 ) )
=> ? [Vs5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs5 ) @ ( set_a2 @ Vs3 ) )
& ( ( size_size_list_a @ Vs5 )
= ( size_size_list_a @ Us6 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Vs5 @ Us3 ) ) ) ) ) ) ) ).
% replacement_theorem
thf(fact_1268_unique__decomposition,axiom,
! [K2: set_a,Us3: list_a,A2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( member_a @ A2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ? [X2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X2 ) @ K2 )
& ( ( size_size_list_a @ X2 )
= ( size_size_list_a @ Us3 ) )
& ( A2
= ( embedded_combine_a_b @ r @ X2 @ Us3 ) )
& ! [Y6: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ Y6 ) @ K2 )
& ( ( size_size_list_a @ Y6 )
= ( size_size_list_a @ Us3 ) )
& ( A2
= ( embedded_combine_a_b @ r @ Y6 @ Us3 ) ) )
=> ( Y6 = X2 ) ) ) ) ) ) ).
% unique_decomposition
thf(fact_1269_independent__rotate1,axiom,
! [K2: set_a,Us3: list_a,Vs3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs3 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ ( rotate1_a @ Us3 ) @ Vs3 ) ) ) ) ).
% independent_rotate1
thf(fact_1270_trivial__combine__imp__independent,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ K2 )
=> ( ( ( embedded_combine_a_b @ r @ Ks2 @ Us3 )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us3 ) @ Ks2 ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% trivial_combine_imp_independent
thf(fact_1271_combine__take,axiom,
! [Us3: list_a,Ks: list_a] :
( ( embedded_combine_a_b @ r @ ( take_a @ ( size_size_list_a @ Us3 ) @ Ks ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ).
% combine_take
thf(fact_1272_non__trivial__combine__imp__dependent,axiom,
! [K2: set_a,Ks: list_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K2 )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us3 )
= ( zero_a_b @ r ) )
=> ( ~ ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us3 ) @ Ks ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ) ) ) ).
% non_trivial_combine_imp_dependent
thf(fact_1273_independent__imp__trivial__combine,axiom,
! [K2: set_a,Us3: list_a,Ks: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K2 )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us3 )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us3 ) @ Ks ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ).
% independent_imp_trivial_combine
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y: a] :
( ( if_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y: a] :
( ( if_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_list_a @ ( bounde2262800523058855161ls_a_b @ r @ n ) ).
%------------------------------------------------------------------------------