TPTP Problem File: SLH0519^1.p
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%------------------------------------------------------------------------------
% 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/0001_Lagrange_Interpolation/prob_00245_009620__17351316_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1534 ( 474 unt; 273 typ; 0 def)
% Number of atoms : 3715 (1866 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 16847 ( 416 ~; 41 |; 245 &;14281 @)
% ( 0 <=>;1864 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 745 ( 745 >; 0 *; 0 +; 0 <<)
% Number of symbols : 247 ( 246 usr; 15 con; 0-4 aty)
% Number of variables : 3328 ( 46 ^;3043 !; 239 ?;3328 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:18.632
%------------------------------------------------------------------------------
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polyno8329700637149614481ed_a_b: partia2175431115845679010xt_a_b > list_a > $o ).
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polyno2453258491555121552on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Orupture_001tf__a_001tf__b,type,
polyno5459750281392823787re_a_b: partia2175431115845679010xt_a_b > set_a > list_a > partia7496981018696276118t_unit ).
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const_term_a_b: partia2175431115845679010xt_a_b > list_a > a ).
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eval_l34571156754992824t_unit: partia2670972154091845814t_unit > list_list_a > list_a > list_a ).
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eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).
thf(sy_c_Polynomials_Oring_Omonom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
monom_7446464087056152608t_unit: partia2670972154091845814t_unit > list_a > nat > list_list_a ).
thf(sy_c_Polynomials_Oring_Omonom_001tf__a_001tf__b,type,
monom_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Onormalize_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
normal637505603836502915t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Onormalize_001tf__a_001tf__b,type,
normalize_a_b: partia2175431115845679010xt_a_b > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_a7601779127272115787t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_list_a ).
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poly_add_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
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poly_m7087347720095500472t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_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 ).
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poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_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 ).
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univ_p2250591967980070728t_unit: partia2956882679547061052t_unit > set_list_list_a > partia5333488208502193986t_unit ).
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univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
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var_li3532061862469730199t_unit: partia2956882679547061052t_unit > list_list_list_a ).
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var_li8453953174693405341t_unit: partia2670972154091845814t_unit > list_list_a ).
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var_se6008125447796440765t_unit: partia7496981018696276118t_unit > list_set_list_a ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
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produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
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a_inv_7033018035630854991t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a ).
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thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
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thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
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field_1861437471013600865t_unit: partia2956882679547061052t_unit > $o ).
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field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
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add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
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zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
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zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
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zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
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ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
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ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_r5224476855413033410t_unit: partia5333488208502193986t_unit > list_list_list_a > $o ).
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ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
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ring_r7392830359377363176t_unit: partia4556295656693239580t_unit > list_set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
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ring_r5115406448772830318t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
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ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
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ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
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ring_r1091214237498979717t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_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_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_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_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_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_S,type,
s: set_a ).
thf(sy_v_p____,type,
p: list_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1260)
thf(fact_0_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_2_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K: a] : ( normalize_a_b @ r @ ( cons_a @ K @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_3_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_4__092_060open_062degree_A_Ilagrange__basis__polynomial__aux_AS_J_A_092_060le_062_Acard_AS_092_060close_062,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran9092808442999052491ux_a_b @ r @ s ) ) @ one_one_nat ) @ ( finite_card_a @ s ) ).
% \<open>degree (lagrange_basis_polynomial_aux S) \<le> card S\<close>
thf(fact_5_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_6_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_7_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_8_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_9_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_10_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_11_calculation,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ p ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) ) @ one_one_nat ) ) ).
% calculation
thf(fact_12_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_13_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_14_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_15_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_16_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_17_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_18_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_19_diff__add__zero,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= zero_zero_multiset_a ) ).
% diff_add_zero
thf(fact_20_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_21_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_22_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_23_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_24_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_25_add__right__cancel,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= ( plus_plus_multiset_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_26_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_27_add__left__cancel,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= ( plus_plus_multiset_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_28_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_29_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_30_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_31_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_32_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_33_diff__zero,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
= A ) ).
% diff_zero
thf(fact_34_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_35_zero__diff,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_36_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_37_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_38_add__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% add_0
thf(fact_39_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_40_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_41_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_42_add__cancel__right__right,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ A @ B ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_right
thf(fact_43_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_44_add__cancel__right__left,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ B @ A ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_left
thf(fact_45_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_46_add__cancel__left__right,axiom,
! [A: multiset_a,B: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= A )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_right
thf(fact_47_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_48_add__cancel__left__left,axiom,
! [B: multiset_a,A: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= A )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_left
thf(fact_49_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: set_list_a,P2: set_list_a > $o] :
( ( member_set_list_a @ A @ ( collect_set_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X2: set_list_a] : ( member_set_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_60_add_Oright__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.right_neutral
thf(fact_61_add__diff__cancel__right_H,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_62_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_63_add__diff__cancel__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( minus_3765977307040488491iset_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_64_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_65_add__diff__cancel__left_H,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_66_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_67_add__diff__cancel__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( minus_3765977307040488491iset_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_68_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_69_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_70_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_71_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_72_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_73_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_74_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_75_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_76_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_77_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_78_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_79_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_80_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_81_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_82_assms_I3_J,axiom,
member_a @ x @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ s ) ).
% assms(3)
thf(fact_83_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran2649660974587678107al_a_b = lagran2649660974587678107al_a_b ).
% ring.lagrange_basis_polynomial.cong
thf(fact_84_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_85_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_86_zero__reorient,axiom,
! [X: multiset_a] :
( ( zero_zero_multiset_a = X )
= ( X = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_87_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_88_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_89_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_90_add__right__imp__eq,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= ( plus_plus_multiset_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_91_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_92_add__left__imp__eq,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= ( plus_plus_multiset_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_93_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_94_add_Oleft__commute,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_95_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_96_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A3: multiset_a,B2: multiset_a] : ( plus_plus_multiset_a @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_97_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_98_add_Oassoc,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_99_group__cancel_Oadd2,axiom,
! [B3: nat,K2: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_100_group__cancel_Oadd2,axiom,
! [B3: multiset_a,K2: multiset_a,B: multiset_a,A: multiset_a] :
( ( B3
= ( plus_plus_multiset_a @ K2 @ B ) )
=> ( ( plus_plus_multiset_a @ A @ B3 )
= ( plus_plus_multiset_a @ K2 @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_101_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_102_group__cancel_Oadd1,axiom,
! [A2: multiset_a,K2: multiset_a,A: multiset_a,B: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ K2 @ A ) )
=> ( ( plus_plus_multiset_a @ A2 @ B )
= ( plus_plus_multiset_a @ K2 @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_103_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_104_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_105_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_106_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_107_size__neq__size__imp__neq,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X )
!= ( size_size_multiset_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_108_size__neq__size__imp__neq,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ X )
!= ( size_s2335926164413107382list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_109_size__neq__size__imp__neq,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_110_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_111_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_112_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_113_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_114_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_115_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_116_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_117_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_118_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_119_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_120_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_121_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_122_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_123_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_124_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_125_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_126_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_127_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_128_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_129_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_130_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_131_add_Ocomm__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.comm_neutral
thf(fact_132_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_133_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_134_diff__diff__eq,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A @ B ) @ C )
= ( minus_3765977307040488491iset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_135_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_136_add__implies__diff,axiom,
! [C: multiset_a,B: multiset_a,A: multiset_a] :
( ( ( plus_plus_multiset_a @ C @ B )
= A )
=> ( C
= ( minus_3765977307040488491iset_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_137_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_138_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_139_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_140_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_141_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_142_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_143_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_144_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_145_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_146_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_147_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_148_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_149_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_150_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_151_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_152_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_153_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K: nat] :
( N2
= ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_154_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_155_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_156_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_157_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_158_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_159_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_160_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_161_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_162_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_163_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_164_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_165_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_166_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_167_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_168_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_169_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_170_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_171_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_172_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_173_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_174_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_175_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_176_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_177_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_178_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_179_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_180_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_181_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_182_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_183_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_184_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_185_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_186_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_187_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_188_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_189_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_190_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_191_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_192_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_193_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_194_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_195_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_196_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_197_lagrange__aux__degree,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) ) @ one_one_nat ) @ ( finite_card_a @ S ) ) ) ) ).
% lagrange_aux_degree
thf(fact_198_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_199_card_Oinfinite,axiom,
! [A2: set_list_a] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_card_list_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_200_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_201_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_202__092_060open_062_092_060And_062y_O_Ay_A_092_060in_062_AS_A_092_060Longrightarrow_062_Ax_A_092_060ominus_062_Ay_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( member_a @ ( a_minus_a_b @ r @ x @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% \<open>\<And>y. y \<in> S \<Longrightarrow> x \<ominus> y \<in> carrier R\<close>
thf(fact_203_c,axiom,
member_a @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( units_a_ring_ext_a_b @ r ) ).
% c
thf(fact_204_diff__card__le__card__Diff,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_205_diff__card__le__card__Diff,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_206_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_207_card__Diff__subset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_208_card__Diff__subset,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_209_card__le__sym__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_210_card__le__sym__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_211_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_212_inv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y ) )
=> ( X = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_213_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_214_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_215_finite__Diff2,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( finite_finite_list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_216_finite__Diff2,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_217_finite__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_218_finite__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_219_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_220_Units__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_inv_Units
thf(fact_221_Units__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= X ) ) ).
% Units_inv_inv
thf(fact_222_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_223_length__tl,axiom,
! [Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( tl_list_a @ Xs ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_224_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_225_Units__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_inv_closed
thf(fact_226_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_227_list_Osel_I3_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( tl_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_228_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_229_list_Osel_I2_J,axiom,
( ( tl_list_a @ nil_list_a )
= nil_list_a ) ).
% list.sel(2)
thf(fact_230_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_231_tl__Nil,axiom,
! [Xs: list_list_a] :
( ( ( tl_list_a @ Xs )
= nil_list_a )
= ( ( Xs = nil_list_a )
| ? [X2: list_a] :
( Xs
= ( cons_list_a @ X2 @ nil_list_a ) ) ) ) ).
% tl_Nil
thf(fact_232_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_233_Nil__tl,axiom,
! [Xs: list_list_a] :
( ( nil_list_a
= ( tl_list_a @ Xs ) )
= ( ( Xs = nil_list_a )
| ? [X2: list_a] :
( Xs
= ( cons_list_a @ X2 @ nil_list_a ) ) ) ) ).
% Nil_tl
thf(fact_234_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_235_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_236_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_237_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_238_rev__finite__subset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_239_rev__finite__subset,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_240_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_241_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_242_finite__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( finite_finite_a @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_243_finite__subset,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( finite_finite_list_a @ B3 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_244_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_245_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_246_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_247_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_248_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_249_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_250_finite__has__maximal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ A @ X3 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_251_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_252_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_253_finite__has__minimal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ X3 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_254_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_255_list__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X3: list_a] : ( P2 @ ( cons_list_a @ X3 @ nil_list_a ) )
=> ( ! [X3: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_256_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P2 @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P2 @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_257_list__induct2_H,axiom,
! [P2: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a] : ( P2 @ ( cons_a @ X3 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P2 @ nil_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_258_list__induct2_H,axiom,
! [P2: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P2 @ nil_list_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_259_list__induct2_H,axiom,
! [P2: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P2 @ nil_list_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_260_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_261_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y4: list_a,Ys3: list_list_a] :
( Xs
= ( cons_list_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_262_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_263_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X3: list_a] :
( X
!= ( cons_list_a @ X3 @ nil_list_a ) )
=> ~ ! [X3: list_a,Y2: list_a,Xs2: list_list_a] :
( X
!= ( cons_list_a @ X3 @ ( cons_list_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_264_transpose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X3: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_265_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_266_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_267_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_268_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_269_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_270_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_271_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_272_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ( finite_card_a @ B4 )
= N )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_273_infinite__arbitrarily__large,axiom,
! [A2: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A2 )
=> ? [B4: set_list_a] :
( ( finite_finite_list_a @ B4 )
& ( ( finite_card_list_a @ B4 )
= N )
& ( ord_le8861187494160871172list_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_274_card__subset__eq,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_275_card__subset__eq,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ( finite_card_list_a @ A2 )
= ( finite_card_list_a @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_276_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_277_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P2: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_278_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P2: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_279_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P2: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_280_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_281_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P2: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_282_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P2: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_283_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P2: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_284_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P2: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_285_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P2: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_286_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P2: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_287_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P2: list_list_a > list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_288_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( 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 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_289_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_290_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_291_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_292_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( 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 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_293_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_list_a,P2: list_a > list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: list_a,Zs2: list_list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_294_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a,Z: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_295_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a,Z: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_296_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_297_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a,Z: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_298_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_299_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_300_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_301_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_302_card__mono,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) ) ) ).
% card_mono
thf(fact_303_card__mono,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) ) ) ).
% card_mono
thf(fact_304_card__seteq,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B3 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_305_card__seteq,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ B3 ) @ ( finite_card_list_a @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_306_exists__subset__between,axiom,
! [A2: set_a,N: nat,C4: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C4 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( finite_finite_a @ C4 )
=> ? [B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ C4 )
& ( ( finite_card_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_307_exists__subset__between,axiom,
! [A2: set_list_a,N: nat,C4: set_list_a] :
( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C4 ) )
=> ( ( ord_le8861187494160871172list_a @ A2 @ C4 )
=> ( ( finite_finite_list_a @ C4 )
=> ? [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ C4 )
& ( ( finite_card_list_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_308_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T2: set_a] :
( ( ord_less_eq_set_a @ T2 @ S )
=> ( ( ( finite_card_a @ T2 )
= N )
=> ~ ( finite_finite_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_309_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
=> ~ ! [T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ T2 @ S )
=> ( ( ( finite_card_list_a @ T2 )
= N )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_310_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C4: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C4 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_311_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_list_a,C4: nat] :
( ! [G: set_list_a] :
( ( ord_le8861187494160871172list_a @ G @ F )
=> ( ( finite_finite_list_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G ) @ C4 ) ) )
=> ( ( finite_finite_list_a @ F )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_312_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_313_a__card__cosets__equal,axiom,
! [C: set_a,H: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_314_ring__irreducibleE_I4_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_315_splitted__on__def,axiom,
! [K3: set_a,P: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K3 @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K3 @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_316_carrier__is__subalgebra,axiom,
! [K3: set_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_317_subalgebra__in__carrier,axiom,
! [K3: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V2 @ r )
=> ( ord_less_eq_set_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_318_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_319_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_320_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_321_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_322_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_323_List_Ofinite__set,axiom,
! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_324_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_325_subset__code_I1_J,axiom,
! [Xs: list_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B3 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X2 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_326_subset__code_I1_J,axiom,
! [Xs: list_set_list_a,B3: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ B3 )
= ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( set_set_list_a2 @ Xs ) )
=> ( member_set_list_a @ X2 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_327_subset__code_I1_J,axiom,
! [Xs: list_list_list_a,B3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs ) @ B3 )
= ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ( member_list_list_a @ X2 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_328_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_329_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X2 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_330_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_331_list_Oset__intros_I2_J,axiom,
! [Y: set_list_a,X22: list_set_list_a,X21: set_list_a] :
( ( member_set_list_a @ Y @ ( set_set_list_a2 @ X22 ) )
=> ( member_set_list_a @ Y @ ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_332_list_Oset__intros_I2_J,axiom,
! [Y: list_list_a,X22: list_list_list_a,X21: list_list_a] :
( ( member_list_list_a @ Y @ ( set_list_list_a2 @ X22 ) )
=> ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_333_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_334_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_335_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_336_list_Oset__intros_I1_J,axiom,
! [X21: set_list_a,X22: list_set_list_a] : ( member_set_list_a @ X21 @ ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_337_list_Oset__intros_I1_J,axiom,
! [X21: list_list_a,X22: list_list_list_a] : ( member_list_list_a @ X21 @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_338_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_339_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_340_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z2: list_set_a] :
( A
!= ( cons_set_a @ E @ Z2 ) )
=> ~ ! [Z1: set_a,Z2: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z2 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_341_list_Oset__cases,axiom,
! [E: set_list_a,A: list_set_list_a] :
( ( member_set_list_a @ E @ ( set_set_list_a2 @ A ) )
=> ( ! [Z2: list_set_list_a] :
( A
!= ( cons_set_list_a @ E @ Z2 ) )
=> ~ ! [Z1: set_list_a,Z2: list_set_list_a] :
( ( A
= ( cons_set_list_a @ Z1 @ Z2 ) )
=> ~ ( member_set_list_a @ E @ ( set_set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_342_list_Oset__cases,axiom,
! [E: list_list_a,A: list_list_list_a] :
( ( member_list_list_a @ E @ ( set_list_list_a2 @ A ) )
=> ( ! [Z2: list_list_list_a] :
( A
!= ( cons_list_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_list_a,Z2: list_list_list_a] :
( ( A
= ( cons_list_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_list_a @ E @ ( set_list_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_343_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z2: list_a] :
( A
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_344_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z2: list_list_a] :
( A
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_345_set__ConsD,axiom,
! [Y: set_a,X: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_346_set__ConsD,axiom,
! [Y: set_list_a,X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ Y @ ( set_set_list_a2 @ ( cons_set_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_set_list_a @ Y @ ( set_set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_347_set__ConsD,axiom,
! [Y: list_list_a,X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_list_a @ Y @ ( set_list_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_348_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_349_set__ConsD,axiom,
! [Y: list_a,X: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_350_finite__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_351_finite__list,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ? [Xs2: list_list_a] :
( ( set_list_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_352_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_353_set__subset__Cons,axiom,
! [Xs: list_list_a,X: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_354_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_355_list_Oset__sel_I2_J,axiom,
! [A: list_set_list_a,X: set_list_a] :
( ( A != nil_set_list_a )
=> ( ( member_set_list_a @ X @ ( set_set_list_a2 @ ( tl_set_list_a @ A ) ) )
=> ( member_set_list_a @ X @ ( set_set_list_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_356_list_Oset__sel_I2_J,axiom,
! [A: list_list_list_a,X: list_list_a] :
( ( A != nil_list_list_a )
=> ( ( member_list_list_a @ X @ ( set_list_list_a2 @ ( tl_list_list_a @ A ) ) )
=> ( member_list_list_a @ X @ ( set_list_list_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_357_list_Oset__sel_I2_J,axiom,
! [A: list_a,X: a] :
( ( A != nil_a )
=> ( ( member_a @ X @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_358_list_Oset__sel_I2_J,axiom,
! [A: list_list_a,X: list_a] :
( ( A != nil_list_a )
=> ( ( member_list_a @ X @ ( set_list_a2 @ ( tl_list_a @ A ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_359_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_360_card__length,axiom,
! [Xs: list_list_a] : ( ord_less_eq_nat @ ( finite_card_list_a @ ( set_list_a2 @ Xs ) ) @ ( size_s349497388124573686list_a @ Xs ) ) ).
% card_length
thf(fact_361_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_362_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_363_size__union,axiom,
! [M3: multiset_list_a,N4: multiset_list_a] :
( ( size_s2335926164413107382list_a @ ( plus_p690419498615200257list_a @ M3 @ N4 ) )
= ( plus_plus_nat @ ( size_s2335926164413107382list_a @ M3 ) @ ( size_s2335926164413107382list_a @ N4 ) ) ) ).
% size_union
thf(fact_364_size__union,axiom,
! [M3: multiset_a,N4: multiset_a] :
( ( size_size_multiset_a @ ( plus_plus_multiset_a @ M3 @ N4 ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ M3 ) @ ( size_size_multiset_a @ N4 ) ) ) ).
% size_union
thf(fact_365_size__empty,axiom,
( ( size_s2335926164413107382list_a @ zero_z4454100511807792257list_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_366_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_367_size__eq__0__iff__empty,axiom,
! [M3: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ M3 )
= zero_zero_nat )
= ( M3 = zero_z4454100511807792257list_a ) ) ).
% size_eq_0_iff_empty
thf(fact_368_size__eq__0__iff__empty,axiom,
! [M3: multiset_a] :
( ( ( size_size_multiset_a @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_369_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_370_poly__mult__semiassoc,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_371_x_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V @ Va ) ) ) ).
% x.normalize.cases
thf(fact_372_poly__mult_Osimps_I1_J,axiom,
! [P22: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P22 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_373_diff__diff__add__mset,axiom,
! [M3: multiset_a,N4: multiset_a,P2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ P2 )
= ( minus_3765977307040488491iset_a @ M3 @ ( plus_plus_multiset_a @ N4 @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_374_poly__mult__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_375_poly__mult__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_mult_comm
thf(fact_376_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_377_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_378_poly__mult__normalize,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_normalize
thf(fact_379_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( plus_plus_multiset_a @ X @ Y )
= zero_zero_multiset_a )
= ( ( X = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_380_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ X @ Y ) )
= ( ( X = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_381_empty__eq__union,axiom,
! [M3: multiset_a,N4: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ M3 @ N4 ) )
= ( ( M3 = zero_zero_multiset_a )
& ( N4 = zero_zero_multiset_a ) ) ) ).
% empty_eq_union
thf(fact_382_union__eq__empty,axiom,
! [M3: multiset_a,N4: multiset_a] :
( ( ( plus_plus_multiset_a @ M3 @ N4 )
= zero_zero_multiset_a )
= ( ( M3 = zero_zero_multiset_a )
& ( N4 = zero_zero_multiset_a ) ) ) ).
% union_eq_empty
thf(fact_383_empty__neutral_I2_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
= X ) ).
% empty_neutral(2)
thf(fact_384_empty__neutral_I1_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_385_union__assoc,axiom,
! [M3: multiset_a,N4: multiset_a,K3: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M3 @ N4 ) @ K3 )
= ( plus_plus_multiset_a @ M3 @ ( plus_plus_multiset_a @ N4 @ K3 ) ) ) ).
% union_assoc
thf(fact_386_union__lcomm,axiom,
! [M3: multiset_a,N4: multiset_a,K3: multiset_a] :
( ( plus_plus_multiset_a @ M3 @ ( plus_plus_multiset_a @ N4 @ K3 ) )
= ( plus_plus_multiset_a @ N4 @ ( plus_plus_multiset_a @ M3 @ K3 ) ) ) ).
% union_lcomm
thf(fact_387_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M4: multiset_a,N5: multiset_a] : ( plus_plus_multiset_a @ N5 @ M4 ) ) ) ).
% union_commute
thf(fact_388_union__left__cancel,axiom,
! [K3: multiset_a,M3: multiset_a,N4: multiset_a] :
( ( ( plus_plus_multiset_a @ K3 @ M3 )
= ( plus_plus_multiset_a @ K3 @ N4 ) )
= ( M3 = N4 ) ) ).
% union_left_cancel
thf(fact_389_union__right__cancel,axiom,
! [M3: multiset_a,K3: multiset_a,N4: multiset_a] :
( ( ( plus_plus_multiset_a @ M3 @ K3 )
= ( plus_plus_multiset_a @ N4 @ K3 ) )
= ( M3 = N4 ) ) ).
% union_right_cancel
thf(fact_390_multi__union__self__other__eq,axiom,
! [A2: multiset_a,X4: multiset_a,Y5: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ X4 )
= ( plus_plus_multiset_a @ A2 @ Y5 ) )
=> ( X4 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_391_diff__union__cancelR,axiom,
! [M3: multiset_a,N4: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ M3 @ N4 ) @ N4 )
= M3 ) ).
% diff_union_cancelR
thf(fact_392_diff__union__cancelL,axiom,
! [N4: multiset_a,M3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ N4 @ M3 ) @ N4 )
= M3 ) ).
% diff_union_cancelL
thf(fact_393_Multiset_Odiff__cancel,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ A2 )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_394_diff__empty,axiom,
! [M3: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M3 @ zero_zero_multiset_a )
= M3 )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M3 )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_395_Multiset_Odiff__add,axiom,
! [M3: multiset_a,N4: multiset_a,Q2: multiset_a] :
( ( minus_3765977307040488491iset_a @ M3 @ ( plus_plus_multiset_a @ N4 @ Q2 ) )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ Q2 ) ) ).
% Multiset.diff_add
thf(fact_396_diff__size__le__size__Diff,axiom,
! [M3: multiset_a,M5: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M3 ) @ ( size_size_multiset_a @ M5 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M3 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_397_diff__size__le__size__Diff,axiom,
! [M3: multiset_list_a,M5: multiset_list_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M3 ) @ ( size_s2335926164413107382list_a @ M5 ) ) @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M3 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_398_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_399_poly__mult__monom__assoc,axiom,
! [P: list_a,Q: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_400_poly__mult__one_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(1)
thf(fact_401_poly__mult__one_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(2)
thf(fact_402_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_403_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ P )
= ( ring_ring_prime_a_b @ R2 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_404_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ P )
= ( ring_r6430282645014804837t_unit @ R2 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_405_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( ring_r360171070648044744t_unit @ R2 @ P )
= ( ring_r5437400583859147359t_unit @ R2 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_406_finprod__one__eqI,axiom,
! [A2: set_set_a,F2: set_a > a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro934595834566309783_set_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_407_finprod__one__eqI,axiom,
! [A2: set_list_a,F2: list_a > a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_408_finprod__one__eqI,axiom,
! [A2: set_set_list_a,F2: set_list_a > a] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_409_finprod__one__eqI,axiom,
! [A2: set_list_list_a,F2: list_list_a > a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_410_finprod__one__eqI,axiom,
! [A2: set_a,F2: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_411_inv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% inv_eq_one_eq
thf(fact_412_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_413_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_414_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_415_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_416_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_417_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_418_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_419_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_420_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_421_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_422_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_423_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_424_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_425_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_426_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_427_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_428_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_429_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_430_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_431_finprod__infinite,axiom,
! [A2: set_list_a,F2: list_a > a] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finpro6052973074229812797list_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_432_finprod__infinite,axiom,
! [A2: set_a,F2: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finpro205304725090349623_a_b_a @ r @ F2 @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_433_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_434_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_435_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P2 @ M6 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_436_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( P2 @ M6 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_437_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_438_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_439_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_440_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_441_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_442_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_443_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_444_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_445_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_446_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_447_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_448_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_449_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_450_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_451_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_452_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_453_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_454_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_455_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_456_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P2 @ M6 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_457_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_458_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_459_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_460_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_461_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_462_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_463_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys4: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys4 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys4 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_464_length__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys4: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys4 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P2 @ Ys4 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_465_finite__maxlen,axiom,
! [M3: set_list_a] :
( ( finite_finite_list_a @ M3 )
=> ? [N3: nat] :
! [X5: list_a] :
( ( member_list_a @ X5 @ M3 )
=> ( ord_less_nat @ ( size_size_list_a @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_466_finite__maxlen,axiom,
! [M3: set_list_list_a] :
( ( finite1660835950917165235list_a @ M3 )
=> ? [N3: nat] :
! [X5: list_list_a] :
( ( member_list_list_a @ X5 @ M3 )
=> ( ord_less_nat @ ( size_s349497388124573686list_a @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_467_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_468_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_469_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_470_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_471_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_472_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_473_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_474_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_475_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_476_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_477_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_478_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_479_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_480_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_481_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_482_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_483_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_484_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_485_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_486_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_487_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_488_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_489_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_490_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_491_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_492_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_493_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_494_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_495_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_496_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_497_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_498_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_499_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_500_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_501_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_502_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K2: nat] :
( ! [M7: nat,N3: nat] :
( ( ord_less_nat @ M7 @ N3 )
=> ( ord_less_nat @ ( F2 @ M7 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K2 ) @ ( F2 @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_503_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_504_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_505_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_506_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_507_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_508_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_509_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_510_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_511_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_512_length__pos__if__in__set,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_513_length__pos__if__in__set,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1991367317912710102list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_514_length__pos__if__in__set,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s2403821588304063868list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_515_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_516_length__pos__if__in__set,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_517_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_518_card__ge__0__finite,axiom,
! [A2: set_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_a @ A2 ) )
=> ( finite_finite_list_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_519_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P2 @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_520_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P2 @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_521_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_522_card__less__sym__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) )
=> ( ord_less_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_523_card__less__sym__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B3 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_524_nonempty__has__size,axiom,
! [S: multiset_list_a] :
( ( S != zero_z4454100511807792257list_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s2335926164413107382list_a @ S ) ) ) ).
% nonempty_has_size
thf(fact_525_nonempty__has__size,axiom,
! [S: multiset_a] :
( ( S != zero_zero_multiset_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ S ) ) ) ).
% nonempty_has_size
thf(fact_526_noetherian__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R2 )
=> ( ring_n3639167112692572309ng_a_b @ R2 ) ) ).
% noetherian_domain.axioms(1)
thf(fact_527_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_528_lagrange__aux__poly,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_529__092_060open_062local_Oeval_A_Ilagrange__basis__polynomial_AS_Ax_J_Ax_A_061_A_092_060one_062_092_060close_062,axiom,
( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% \<open>local.eval (lagrange_basis_polynomial S x) x = \<one>\<close>
thf(fact_530_b,axiom,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% b
thf(fact_531_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_532_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_533_eval__poly__of__const,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X ) @ Y )
= X ) ) ).
% eval_poly_of_const
thf(fact_534_eval__in__carrier__2,axiom,
! [X: list_a,Y: a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_535_eval__in__carrier,axiom,
! [P: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_536_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_537_poly__add__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_538_poly__add__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_add_comm
thf(fact_539_eval__normalize,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_540_poly__mult__l__distr_H,axiom,
! [P1: list_a,P22: list_a,P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) @ P3 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P3 ) @ ( poly_mult_a_b @ r @ P22 @ P3 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_541_poly__mult__r__distr_H,axiom,
! [P1: list_a,P22: list_a,P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P22 @ P3 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P22 ) @ ( poly_mult_a_b @ r @ P1 @ P3 ) ) ) ) ) ) ).
% poly_mult_r_distr'
thf(fact_542_poly__add__normalize__aux,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_543_poly__add__normalize_I2_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_544_poly__add__normalize_I3_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_545_x_Oring_Ohom__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.hom_closed
thf(fact_546_finite__psubset__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ! [B5: set_a] :
( ( ord_less_set_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_547_finite__psubset__induct,axiom,
! [A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ! [B5: set_list_a] :
( ( ord_less_set_list_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_548_psubset__card__mono,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_549_psubset__card__mono,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_set_list_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_550_card__psubset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_551_card__psubset,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) )
=> ( ord_less_set_list_a @ A2 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_552_x_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.order_gt_0_iff_finite
thf(fact_553_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_554_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_555_degree__one__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_one_imp_splitted
thf(fact_556_pirreducible__imp__not__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ).
% pirreducible_imp_not_splitted
thf(fact_557_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_558_x_Oexp__base__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.exp_base_closed
thf(fact_559_poly__sub__degree__le,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_560_x_Oa__card__cosets__equal,axiom,
! [C: set_list_a,H: set_list_a] :
( ( member_set_list_a @ C @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( finite_card_list_a @ C )
= ( finite_card_list_a @ H ) ) ) ) ) ).
% x.a_card_cosets_equal
thf(fact_561_x_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_562_x_Ohom__sub,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.hom_sub
thf(fact_563_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_564_field_Opirreducible__imp__not__splitted,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno7858167711734664505t_unit @ R2 @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_565_field_Opirreducible__imp__not__splitted,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno5970451904377802771t_unit @ R2 @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_566_field_Opirreducible__imp__not__splitted,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ R2 @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_567_field_Opirreducible__imp__not__splitted,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno6259083269128200473t_unit @ R2 @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_568_field_Odegree__one__imp__splitted,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno7858167711734664505t_unit @ R2 @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_569_field_Odegree__one__imp__splitted,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno5970451904377802771t_unit @ R2 @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_570_field_Odegree__one__imp__splitted,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ R2 @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_571_field_Odegree__one__imp__splitted,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno6259083269128200473t_unit @ R2 @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_572_splitted__imp__trivial__factors,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q @ P )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ).
% splitted_imp_trivial_factors
thf(fact_573_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_574_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_575_pdivides__imp__splitted,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_576_trivial__factors__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [Q3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q3 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% trivial_factors_imp_splitted
thf(fact_577_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_578_ring_Oexp__base_Ocong,axiom,
polyno3522816881121920896t_unit = polyno3522816881121920896t_unit ).
% ring.exp_base.cong
thf(fact_579_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_580_ring_Oroots__on_Ocong,axiom,
polyno5990348478334826338t_unit = polyno5990348478334826338t_unit ).
% ring.roots_on.cong
thf(fact_581_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_582_ring_Osplitted__on_Ocong,axiom,
polyno1986131841096413848t_unit = polyno1986131841096413848t_unit ).
% ring.splitted_on.cong
thf(fact_583_field_Opdivides__imp__splitted,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( Q != nil_set_list_a )
=> ( ( polyno7858167711734664505t_unit @ R2 @ Q )
=> ( ( polyno9075941895896075626t_unit @ R2 @ P @ Q )
=> ( polyno7858167711734664505t_unit @ R2 @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_584_field_Opdivides__imp__splitted,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R2 @ Q )
=> ( ( polyno4453881341673752516t_unit @ R2 @ P @ Q )
=> ( polyno5970451904377802771t_unit @ R2 @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_585_field_Opdivides__imp__splitted,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R2 @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ R2 @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_586_field_Opdivides__imp__splitted,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R2 @ Q )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
=> ( polyno6259083269128200473t_unit @ R2 @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_587_field_Osplitted__imp__trivial__factors,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( polyno7858167711734664505t_unit @ R2 @ P )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ Q )
=> ( ( polyno9075941895896075626t_unit @ R2 @ Q @ P )
=> ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_588_field_Osplitted__imp__trivial__factors,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R2 @ P )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ Q )
=> ( ( polyno4453881341673752516t_unit @ R2 @ Q @ P )
=> ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_589_field_Osplitted__imp__trivial__factors,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R2 @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ Q @ P )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_590_field_Osplitted__imp__trivial__factors,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R2 @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ Q )
=> ( ( polyno8016796738000020810t_unit @ R2 @ Q @ P )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_591_field_Otrivial__factors__imp__splitted,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ! [Q3: list_set_list_a] :
( ( member5524387281408368019list_a @ Q3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ Q3 )
=> ( ( polyno9075941895896075626t_unit @ R2 @ Q3 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno7858167711734664505t_unit @ R2 @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_592_field_Otrivial__factors__imp__splitted,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ! [Q3: list_list_list_a] :
( ( member5342144027231129785list_a @ Q3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ Q3 )
=> ( ( polyno4453881341673752516t_unit @ R2 @ Q3 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno5970451904377802771t_unit @ R2 @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_593_field_Otrivial__factors__imp__splitted,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ! [Q3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ Q3 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ R2 @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_594_field_Otrivial__factors__imp__splitted,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ! [Q3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ Q3 )
=> ( ( polyno8016796738000020810t_unit @ R2 @ Q3 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno6259083269128200473t_unit @ R2 @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_595_size__roots__le__degree,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% size_roots_le_degree
thf(fact_596_x_Osplitted__def,axiom,
! [P: list_list_a] :
( ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% x.splitted_def
thf(fact_597_x_Osplitted__on__def,axiom,
! [K3: set_list_a,P: list_list_a] :
( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% x.splitted_on_def
thf(fact_598_univ__poly__zero__closed,axiom,
! [R2: partia2175431115845679010xt_a_b,K3: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_599_univ__poly__zero__closed,axiom,
! [R2: partia2670972154091845814t_unit,K3: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_600_splitted__def,axiom,
! [P: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_601_degree__zero__imp__empty__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_602_pirreducible__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_603_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_604_ring_Onormalize_Ocong,axiom,
normal637505603836502915t_unit = normal637505603836502915t_unit ).
% ring.normalize.cong
thf(fact_605_ring_Opoly__mult_Ocong,axiom,
poly_mult_a_b = poly_mult_a_b ).
% ring.poly_mult.cong
thf(fact_606_ring_Opoly__mult_Ocong,axiom,
poly_m7087347720095500472t_unit = poly_m7087347720095500472t_unit ).
% ring.poly_mult.cong
thf(fact_607_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_608_ring_Opoly__of__const_Ocong,axiom,
poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).
% ring.poly_of_const.cong
thf(fact_609_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_610_ring_Omonom_Ocong,axiom,
monom_7446464087056152608t_unit = monom_7446464087056152608t_unit ).
% ring.monom.cong
thf(fact_611_field_Osize__roots__le__degree,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ord_less_eq_nat @ ( size_s1226348209404258454list_a @ ( polyno4169377219242390531t_unit @ R2 @ P ) ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_612_field_Osize__roots__le__degree,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ord_less_eq_nat @ ( size_s8523483970790017596list_a @ ( polyno3707469075594375645t_unit @ R2 @ P ) ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_613_field_Osize__roots__le__degree,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ R2 @ P ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_614_field_Osize__roots__le__degree,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ R2 @ P ) ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_615_univ__poly__one,axiom,
! [R2: partia7496981018696276118t_unit,K3: set_set_list_a] :
( ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K3 ) )
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R2 ) @ nil_set_list_a ) ) ).
% univ_poly_one
thf(fact_616_univ__poly__one,axiom,
! [R2: partia2175431115845679010xt_a_b,K3: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R2 @ K3 ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_617_univ__poly__one,axiom,
! [R2: partia2670972154091845814t_unit,K3: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K3 ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_618_x_Oee__sym,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ As ) ) ) ) ).
% x.ee_sym
thf(fact_619_x_Oee__trans,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ Cs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% x.ee_trans
thf(fact_620_poly__mult__degree__le,axiom,
! [X: list_a,Y: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_621_x_Opoly__of__const__in__carrier,axiom,
! [S2: list_a] :
( ( member_list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_622_x_Opoly__mult_Osimps_I1_J,axiom,
! [P22: list_list_a] :
( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P22 )
= nil_list_a ) ).
% x.poly_mult.simps(1)
thf(fact_623_x_Onormalize_Osimps_I1_J,axiom,
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= nil_list_a ) ).
% x.normalize.simps(1)
thf(fact_624_x_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y6: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y6 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% x.inv_unique
thf(fact_625_x_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.one_unique
thf(fact_626_x_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z3 ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_627_x_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.m_comm
thf(fact_628_x_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z3 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z3 ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_629_x_Onormalize__length__le,axiom,
! [P: list_list_a] : ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ).
% x.normalize_length_le
thf(fact_630_x_Opoly__of__const__def,axiom,
( ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( ^ [K: list_a] : ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K @ nil_list_a ) ) ) ) ).
% x.poly_of_const_def
thf(fact_631_x_Oee__length,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) ) ) ).
% x.ee_length
thf(fact_632_x_Opoly__mult__normalize,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_mult_normalize
thf(fact_633_x_Opoly__mult__in__carrier,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_mult_in_carrier
thf(fact_634_x_Onormalize__in__carrier,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.normalize_in_carrier
thf(fact_635_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_636_x_Opoly__mult__zero_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= nil_list_a ) ) ).
% x.poly_mult_zero(1)
thf(fact_637_x_Opoly__mult__zero_I2_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= nil_list_a ) ) ).
% x.poly_mult_zero(2)
thf(fact_638_associated__polynomials__imp__same__roots,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( plus_plus_multiset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ) ).
% associated_polynomials_imp_same_roots
thf(fact_639_no__roots__imp__same__roots,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% no_roots_imp_same_roots
thf(fact_640_poly__mult__degree__le__1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ).
% poly_mult_degree_le_1
thf(fact_641_x_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.one_closed
thf(fact_642_x_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_one
thf(fact_643_x_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_one
thf(fact_644_x_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_645_x_Omonom__in__carrier,axiom,
! [A: list_a,N: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monom_in_carrier
thf(fact_646_x_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% x.ring.hom_one
thf(fact_647_x_Oee__refl,axiom,
! [As: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ As ) ) ).
% x.ee_refl
thf(fact_648_univ__poly__mult,axiom,
! [R2: partia2175431115845679010xt_a_b,K3: set_a] :
( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K3 ) )
= ( poly_mult_a_b @ R2 ) ) ).
% univ_poly_mult
thf(fact_649_univ__poly__mult,axiom,
! [R2: partia2670972154091845814t_unit,K3: set_list_a] :
( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K3 ) )
= ( poly_m7087347720095500472t_unit @ R2 ) ) ).
% univ_poly_mult
thf(fact_650_field_Ono__roots__imp__same__roots,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( ( polyno4169377219242390531t_unit @ R2 @ P )
= zero_z7061913751530476641list_a )
=> ( ( polyno4169377219242390531t_unit @ R2 @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ P @ Q ) )
= ( polyno4169377219242390531t_unit @ R2 @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_651_field_Ono__roots__imp__same__roots,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( ( polyno3707469075594375645t_unit @ R2 @ P )
= zero_z1542645121299710087list_a )
=> ( ( polyno3707469075594375645t_unit @ R2 @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ P @ Q ) )
= ( polyno3707469075594375645t_unit @ R2 @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_652_field_Ono__roots__imp__same__roots,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( P != nil_a )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ( polynomial_roots_a_b @ R2 @ P )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ R2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q ) )
= ( polynomial_roots_a_b @ R2 @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_653_field_Ono__roots__imp__same__roots,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ R2 @ P )
= zero_z4454100511807792257list_a )
=> ( ( polyno7858422826990252003t_unit @ R2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q ) )
= ( polyno7858422826990252003t_unit @ R2 @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_654_field_Oassociated__polynomials__imp__same__roots,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( Q != nil_set_list_a )
=> ( ( polyno4169377219242390531t_unit @ R2 @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ P @ Q ) )
= ( plus_p3298285420967332321list_a @ ( polyno4169377219242390531t_unit @ R2 @ P ) @ ( polyno4169377219242390531t_unit @ R2 @ Q ) ) ) ) ) ) ) ) ).
% field.associated_polynomials_imp_same_roots
thf(fact_655_field_Oassociated__polynomials__imp__same__roots,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ( polyno3707469075594375645t_unit @ R2 @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ P @ Q ) )
= ( plus_p5485240441447921159list_a @ ( polyno3707469075594375645t_unit @ R2 @ P ) @ ( polyno3707469075594375645t_unit @ R2 @ Q ) ) ) ) ) ) ) ) ).
% field.associated_polynomials_imp_same_roots
thf(fact_656_field_Oassociated__polynomials__imp__same__roots,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( P != nil_a )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polynomial_roots_a_b @ R2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q ) )
= ( plus_plus_multiset_a @ ( polynomial_roots_a_b @ R2 @ P ) @ ( polynomial_roots_a_b @ R2 @ Q ) ) ) ) ) ) ) ) ).
% field.associated_polynomials_imp_same_roots
thf(fact_657_field_Oassociated__polynomials__imp__same__roots,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno7858422826990252003t_unit @ R2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q ) )
= ( plus_p690419498615200257list_a @ ( polyno7858422826990252003t_unit @ R2 @ P ) @ ( polyno7858422826990252003t_unit @ R2 @ Q ) ) ) ) ) ) ) ) ).
% field.associated_polynomials_imp_same_roots
thf(fact_658_x_Omonoid__cancelI,axiom,
( ! [A5: list_a,B6: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A5 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B6 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A5 = B6 ) ) ) ) )
=> ( ! [A5: list_a,B6: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B6 @ C3 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A5 = B6 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_659_x_Oconst__term__simprules_I2_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(2)
thf(fact_660_x_Odegree__oneE,axiom,
! [P: list_list_a,K3: set_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A5: list_a] :
( ( member_list_a @ A5 @ K3 )
=> ( ( A5
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B6: list_a] :
( ( member_list_a @ B6 @ K3 )
=> ( P
!= ( cons_list_a @ A5 @ ( cons_list_a @ B6 @ nil_list_a ) ) ) ) ) ) ) ) ).
% x.degree_oneE
thf(fact_661_degree__oneE,axiom,
! [P: list_a,K3: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A5: a] :
( ( member_a @ A5 @ K3 )
=> ( ( A5
!= ( zero_a_b @ r ) )
=> ! [B6: a] :
( ( member_a @ B6 @ K3 )
=> ( P
!= ( cons_a @ A5 @ ( cons_a @ B6 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_662_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_663_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_664_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_665_ring__irreducibleE_I1_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( R
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_666_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_667_finprod__zero__iff,axiom,
! [A2: set_set_a,F2: set_a > a] :
( ( finite_finite_set_a @ A2 )
=> ( ! [A5: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ( member_a @ ( F2 @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro934595834566309783_set_a @ r @ F2 @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_668_finprod__zero__iff,axiom,
! [A2: set_set_list_a,F2: set_list_a > a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ! [A5: set_list_a] :
( ( member_set_list_a @ A5 @ A2 )
=> ( member_a @ ( F2 @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro3826550488720007709list_a @ r @ F2 @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_669_finprod__zero__iff,axiom,
! [A2: set_list_list_a,F2: list_list_a > a] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ! [A5: list_list_a] :
( ( member_list_list_a @ A5 @ A2 )
=> ( member_a @ ( F2 @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5500967685102550467list_a @ r @ F2 @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_670_finprod__zero__iff,axiom,
! [A2: set_list_a,F2: list_a > a] :
( ( finite_finite_list_a @ A2 )
=> ( ! [A5: list_a] :
( ( member_list_a @ A5 @ A2 )
=> ( member_a @ ( F2 @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F2 @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_671_finprod__zero__iff,axiom,
! [A2: set_a,F2: a > a] :
( ( finite_finite_a @ A2 )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A2 )
=> ( member_a @ ( F2 @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F2 @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_672_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_673_a__lcos__mult__one,axiom,
! [M3: set_a] :
( ( ord_less_eq_set_a @ M3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M3 )
= M3 ) ) ).
% a_lcos_mult_one
thf(fact_674_x_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% x.const_term_not_zero
thf(fact_675_is__root__def,axiom,
! [P: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_676_x_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
% x.ring.zero_closed
thf(fact_677_x_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A5
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ X5 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.cring_fieldI2
thf(fact_678_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_679_x_Oconst__term__simprules_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.const_term_simprules(1)
thf(fact_680_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_681_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_682_x_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.zero_closed
thf(fact_683_x_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_684_x_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_685_x_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A = B ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_686_x_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( zero_a_b @ r ) ) ).
% x.ring.hom_zero
thf(fact_687_univ__poly__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,K3: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R2 @ K3 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_688_univ__poly__zero,axiom,
! [R2: partia2670972154091845814t_unit,K3: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K3 ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_689_ring__prime__def,axiom,
( ring_r5437400583859147359t_unit
= ( ^ [R3: partia2956882679547061052t_unit,A3: list_list_a] :
( ( A3
!= ( zero_l347298301471573063t_unit @ R3 ) )
& ( prime_1232919612140715622t_unit @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_690_ring__prime__def,axiom,
( ring_r1091214237498979717t_unit
= ( ^ [R3: partia7496981018696276118t_unit,A3: set_list_a] :
( ( A3
!= ( zero_s2910681146719230829t_unit @ R3 ) )
& ( prime_5738381090551951334t_unit @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_691_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_692_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( prime_2011924034616061926t_unit @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_693_x_OboundD__carrier,axiom,
! [N: nat,F2: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F2 )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F2 @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_694_boundD__carrier,axiom,
! [N: nat,F2: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F2 )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F2 @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_695_x_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.ring_primeI
thf(fact_696_x_Opoly__mult__append__zero,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ Q )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ).
% x.poly_mult_append_zero
thf(fact_697_same__append__eq,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_698_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_699_append__same__eq,axiom,
! [Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Ys @ Xs )
= ( append_list_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_700_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_701_append__assoc,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( append_list_a @ ( append_list_a @ Xs @ Ys ) @ Zs )
= ( append_list_a @ Xs @ ( append_list_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_702_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_703_append_Oassoc,axiom,
! [A: list_list_a,B: list_list_a,C: list_list_a] :
( ( append_list_a @ ( append_list_a @ A @ B ) @ C )
= ( append_list_a @ A @ ( append_list_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_704_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_705_x_Onormalize__idem,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ Q ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) ) ) ).
% x.normalize_idem
thf(fact_706_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_707_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_708_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_709_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_710_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_711_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_712_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_713_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_714_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_715_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_716_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_717_self__append__conv2,axiom,
! [Y: list_list_a,Xs: list_list_a] :
( ( Y
= ( append_list_a @ Xs @ Y ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_718_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_719_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_720_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_721_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_722_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_723_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_724_x_Oconst__term__explicit,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= A )
=> ~ ! [P4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P
!= ( append_list_a @ P4 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% x.const_term_explicit
thf(fact_725_x_Oconst__term__eq__last,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ).
% x.const_term_eq_last
thf(fact_726_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_727_append1__eq__conv,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_728_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_729_length__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( size_s349497388124573686list_a @ ( append_list_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) ) ) ).
% length_append
thf(fact_730_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_731_tl__append2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( tl_list_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_732_append__eq__append__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,Ts: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Ts ) )
= ( ? [Us2: list_list_a] :
( ( ( Xs
= ( append_list_a @ Zs @ Us2 ) )
& ( ( append_list_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_list_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_733_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_734_append__eq__appendI,axiom,
! [Xs: list_list_a,Xs1: list_list_a,Zs: list_list_a,Ys: list_list_a,Us: list_list_a] :
( ( ( append_list_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_a @ Xs1 @ Us ) )
=> ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_735_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_736_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_737_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_738_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_739_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_740_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_741_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_742_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_743_append__Cons,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
= ( cons_list_a @ X @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_744_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_745_Cons__eq__appendI,axiom,
! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_746_split__list,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_747_split__list,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
=> ? [Ys2: list_set_list_a,Zs2: list_set_list_a] :
( Xs
= ( append_set_list_a @ Ys2 @ ( cons_set_list_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_748_split__list,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ? [Ys2: list_list_list_a,Zs2: list_list_list_a] :
( Xs
= ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_749_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_750_split__list,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_751_split__list__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_752_split__list__last,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
=> ? [Ys2: list_set_list_a,Zs2: list_set_list_a] :
( ( Xs
= ( append_set_list_a @ Ys2 @ ( cons_set_list_a @ X @ Zs2 ) ) )
& ~ ( member_set_list_a @ X @ ( set_set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_753_split__list__last,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ? [Ys2: list_list_list_a,Zs2: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_754_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_755_split__list__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_756_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_757_split__list__prop,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_list_a,X3: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs2 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_758_split__list__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_759_split__list__first,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
=> ? [Ys2: list_set_list_a,Zs2: list_set_list_a] :
( ( Xs
= ( append_set_list_a @ Ys2 @ ( cons_set_list_a @ X @ Zs2 ) ) )
& ~ ( member_set_list_a @ X @ ( set_set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_760_split__list__first,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ? [Ys2: list_list_list_a,Zs2: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_761_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_762_split__list__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_763_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_764_split__list__propE,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_list_a,X3: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs2 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_765_append__Cons__eq__iff,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a,Xs3: list_set_a,Ys5: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ Ys ) )
= ( append_set_a @ Xs3 @ ( cons_set_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_766_append__Cons__eq__iff,axiom,
! [X: set_list_a,Xs: list_set_list_a,Ys: list_set_list_a,Xs3: list_set_list_a,Ys5: list_set_list_a] :
( ~ ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
=> ( ~ ( member_set_list_a @ X @ ( set_set_list_a2 @ Ys ) )
=> ( ( ( append_set_list_a @ Xs @ ( cons_set_list_a @ X @ Ys ) )
= ( append_set_list_a @ Xs3 @ ( cons_set_list_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_767_append__Cons__eq__iff,axiom,
! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,Xs3: list_list_list_a,Ys5: list_list_list_a] :
( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys ) )
=> ( ( ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ Ys ) )
= ( append_list_list_a @ Xs3 @ ( cons_list_list_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_768_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs3 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_769_append__Cons__eq__iff,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys5: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) )
= ( append_list_a @ Xs3 @ ( cons_list_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_770_in__set__conv__decomp,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_771_in__set__conv__decomp,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
= ( ? [Ys3: list_set_list_a,Zs3: list_set_list_a] :
( Xs
= ( append_set_list_a @ Ys3 @ ( cons_set_list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_772_in__set__conv__decomp,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_list_a,Zs3: list_list_list_a] :
( Xs
= ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_773_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_774_in__set__conv__decomp,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_775_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_776_split__list__last__prop,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_list_a,X3: list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_777_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_778_split__list__first__prop,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys2: list_list_a,X3: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_779_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_780_split__list__last__propE,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_list_a,X3: list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_781_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_782_split__list__first__propE,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys2: list_list_a,X3: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_783_in__set__conv__decomp__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_784_in__set__conv__decomp__last,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
= ( ? [Ys3: list_set_list_a,Zs3: list_set_list_a] :
( ( Xs
= ( append_set_list_a @ Ys3 @ ( cons_set_list_a @ X @ Zs3 ) ) )
& ~ ( member_set_list_a @ X @ ( set_set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_785_in__set__conv__decomp__last,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_list_a,Zs3: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_786_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_787_in__set__conv__decomp__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_788_in__set__conv__decomp__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_789_in__set__conv__decomp__first,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
= ( ? [Ys3: list_set_list_a,Zs3: list_set_list_a] :
( ( Xs
= ( append_set_list_a @ Ys3 @ ( cons_set_list_a @ X @ Zs3 ) ) )
& ~ ( member_set_list_a @ X @ ( set_set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_790_in__set__conv__decomp__first,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_list_a,Zs3: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_791_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_792_in__set__conv__decomp__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_793_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_a,X2: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_794_split__list__last__prop__iff,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_list_a,X2: list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_795_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_796_split__list__first__prop__iff,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Ys3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_797_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X3: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_798_rev__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ( P2 @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ X3 @ nil_list_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_799_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_800_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys2: list_list_a,Y2: list_a] :
( Xs
!= ( append_list_a @ Ys2 @ ( cons_list_a @ Y2 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_801_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_802_Cons__eq__append__conv,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X @ Xs )
= Zs ) )
| ? [Ys6: list_list_a] :
( ( ( cons_list_a @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_list_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_803_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_804_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X @ Xs ) ) )
| ? [Ys6: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Ys6 ) )
& ( ( append_list_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_805_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_806_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X3: list_a] : ( P2 @ ( cons_list_a @ X3 @ nil_list_a ) )
=> ( ! [X3: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ X3 @ nil_list_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_807_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_808_tl__append__if,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( Xs = nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( tl_list_a @ Ys ) ) )
& ( ( Xs != nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( tl_list_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_809_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs4: list_a,Y2: a,Ys7: list_a] :
( ( X3 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_810_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X3: list_a,Xs4: list_list_a,Y2: list_a,Ys7: list_list_a] :
( ( X3 != Y2 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X3 @ nil_list_a ) @ Xs4 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y2 @ nil_list_a ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_811_x_Ofactors__mult,axiom,
! [Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fa @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% x.factors_mult
thf(fact_812_x_Oa__lcos__mult__one,axiom,
! [M3: set_list_a] :
( ( ord_le8861187494160871172list_a @ M3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M3 )
= M3 ) ) ).
% x.a_lcos_mult_one
thf(fact_813_x_Opoly__add__append__zero,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ ( append_list_a @ Q @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ).
% x.poly_add_append_zero
thf(fact_814_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_815_x_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_816_x_Opoly__add__comm,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P1 ) ) ) ) ).
% x.poly_add_comm
thf(fact_817_x_Opoly__add__in__carrier,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_add_in_carrier
thf(fact_818_x_Ofactors__closed,axiom,
! [Fs: list_list_a,A: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.factors_closed
thf(fact_819_x_Opoly__add__normalize_I3_J,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.poly_add_normalize(3)
thf(fact_820_x_Opoly__add__normalize_I2_J,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.poly_add_normalize(2)
thf(fact_821_x_Opoly__add__normalize__aux,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_add_normalize_aux
thf(fact_822_x_Opoly__mult__l__distr_H,axiom,
! [P1: list_list_a,P22: list_list_a,P3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) @ P3 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P3 ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P3 ) ) ) ) ) ) ).
% x.poly_mult_l_distr'
thf(fact_823_x_Opoly__add__zero_H_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_zero'(1)
thf(fact_824_x_Opoly__add__zero_H_I2_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_zero'(2)
thf(fact_825_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_826_poly__add__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_827_poly__mult__var_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_828_poly__mult__var_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_829_x_Opoly__add__append__replicate,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( replicate_list_a @ ( size_s349497388124573686list_a @ Q ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) @ Q )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) ) ) ) ) ).
% x.poly_add_append_replicate
thf(fact_830_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_831_x_Onormalize__replicate__zero,axiom,
! [N: nat,P: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ).
% x.normalize_replicate_zero
thf(fact_832_x_Omonom__def,axiom,
! [A: list_a,N: nat] :
( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N )
= ( cons_list_a @ A @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.monom_def
thf(fact_833_replicate__eq__replicate,axiom,
! [M: nat,X: list_a,N: nat,Y: list_a] :
( ( ( replicate_list_a @ M @ X )
= ( replicate_list_a @ N @ Y ) )
= ( ( M = N )
& ( ( M != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_834_replicate__eq__replicate,axiom,
! [M: nat,X: a,N: nat,Y: a] :
( ( ( replicate_a @ M @ X )
= ( replicate_a @ N @ Y ) )
= ( ( M = N )
& ( ( M != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_835_length__replicate,axiom,
! [N: nat,X: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_836_length__replicate,axiom,
! [N: nat,X: list_a] :
( ( size_s349497388124573686list_a @ ( replicate_list_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_837_x_Onormalize__trick,axiom,
! [P: list_list_a] :
( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.normalize_trick
thf(fact_838_x_Opoly__add__replicate__zero_H_I2_J,axiom,
! [P: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_replicate_zero'(2)
thf(fact_839_x_Opoly__add__replicate__zero_H_I1_J,axiom,
! [P: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_replicate_zero'(1)
thf(fact_840_x_Opoly__mult__prepend__replicate__zero,axiom,
! [P1: list_list_a,P22: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_mult_prepend_replicate_zero
thf(fact_841_in__set__replicate,axiom,
! [X: set_a,N: nat,Y: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ ( replicate_set_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_842_in__set__replicate,axiom,
! [X: set_list_a,N: nat,Y: set_list_a] :
( ( member_set_list_a @ X @ ( set_set_list_a2 @ ( replicate_set_list_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_843_in__set__replicate,axiom,
! [X: list_list_a,N: nat,Y: list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ ( replic3997036819131463498list_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_844_in__set__replicate,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( replicate_list_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_845_in__set__replicate,axiom,
! [X: a,N: nat,Y: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_846_Bex__set__replicate,axiom,
! [N: nat,A: list_a,P2: list_a > $o] :
( ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( replicate_list_a @ N @ A ) ) )
& ( P2 @ X2 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_847_Bex__set__replicate,axiom,
! [N: nat,A: a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
& ( P2 @ X2 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_848_Ball__set__replicate,axiom,
! [N: nat,A: list_a,P2: list_a > $o] :
( ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( replicate_list_a @ N @ A ) ) )
=> ( P2 @ X2 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_849_Ball__set__replicate,axiom,
! [N: nat,A: a,P2: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
=> ( P2 @ X2 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_850_replicate__empty,axiom,
! [N: nat,X: list_a] :
( ( ( replicate_list_a @ N @ X )
= nil_list_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_851_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_852_empty__replicate,axiom,
! [N: nat,X: list_a] :
( ( nil_list_a
= ( replicate_list_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_853_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_854_tl__replicate,axiom,
! [N: nat,X: list_a] :
( ( tl_list_a @ ( replicate_list_a @ N @ X ) )
= ( replicate_list_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ).
% tl_replicate
thf(fact_855_tl__replicate,axiom,
! [N: nat,X: a] :
( ( tl_a @ ( replicate_a @ N @ X ) )
= ( replicate_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ).
% tl_replicate
thf(fact_856_append__replicate__commute,axiom,
! [N: nat,X: list_a,K2: nat] :
( ( append_list_a @ ( replicate_list_a @ N @ X ) @ ( replicate_list_a @ K2 @ X ) )
= ( append_list_a @ ( replicate_list_a @ K2 @ X ) @ ( replicate_list_a @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_857_append__replicate__commute,axiom,
! [N: nat,X: a,K2: nat] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K2 @ X ) )
= ( append_a @ ( replicate_a @ K2 @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_858_replicate__length__same,axiom,
! [Xs: list_a,X: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_859_replicate__length__same,axiom,
! [Xs: list_list_a,X: list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_list_a @ ( size_s349497388124573686list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_860_replicate__eqI,axiom,
! [Xs: list_set_a,N: nat,X: set_a] :
( ( ( size_size_list_set_a @ Xs )
= N )
=> ( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_set_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_861_replicate__eqI,axiom,
! [Xs: list_set_list_a,N: nat,X: set_list_a] :
( ( ( size_s1991367317912710102list_a @ Xs )
= N )
=> ( ! [Y2: set_list_a] :
( ( member_set_list_a @ Y2 @ ( set_set_list_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_set_list_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_862_replicate__eqI,axiom,
! [Xs: list_list_list_a,N: nat,X: list_list_a] :
( ( ( size_s2403821588304063868list_a @ Xs )
= N )
=> ( ! [Y2: list_list_a] :
( ( member_list_list_a @ Y2 @ ( set_list_list_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replic3997036819131463498list_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_863_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_864_replicate__eqI,axiom,
! [Xs: list_list_a,N: nat,X: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_list_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_865_replicate__0,axiom,
! [X: list_a] :
( ( replicate_list_a @ zero_zero_nat @ X )
= nil_list_a ) ).
% replicate_0
thf(fact_866_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_867_replicate__app__Cons__same,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( append_list_a @ ( replicate_list_a @ N @ X ) @ ( cons_list_a @ X @ Xs ) )
= ( cons_list_a @ X @ ( append_list_a @ ( replicate_list_a @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_868_replicate__app__Cons__same,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( append_a @ ( replicate_a @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_869_replicate__add,axiom,
! [N: nat,M: nat,X: list_a] :
( ( replicate_list_a @ ( plus_plus_nat @ N @ M ) @ X )
= ( append_list_a @ ( replicate_list_a @ N @ X ) @ ( replicate_list_a @ M @ X ) ) ) ).
% replicate_add
thf(fact_870_replicate__add,axiom,
! [N: nat,M: nat,X: a] :
( ( replicate_a @ ( plus_plus_nat @ N @ M ) @ X )
= ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ M @ X ) ) ) ).
% replicate_add
thf(fact_871_replicate__append__same,axiom,
! [I: nat,X: list_a] :
( ( append_list_a @ ( replicate_list_a @ I @ X ) @ ( cons_list_a @ X @ nil_list_a ) )
= ( cons_list_a @ X @ ( replicate_list_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_872_replicate__append__same,axiom,
! [I: nat,X: a] :
( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_873_Cons__replicate__eq,axiom,
! [X: list_a,Xs: list_list_a,N: nat,Y: list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( replicate_list_a @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_list_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_874_Cons__replicate__eq,axiom,
! [X: a,Xs: list_a,N: nat,Y: a] :
( ( ( cons_a @ X @ Xs )
= ( replicate_a @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_875_var__def,axiom,
( var_li3532061862469730199t_unit
= ( ^ [R3: partia2956882679547061052t_unit] : ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ nil_list_list_a ) ) ) ) ).
% var_def
thf(fact_876_var__def,axiom,
( var_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( zero_a_b @ R3 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_877_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R3: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_878_var__def,axiom,
( var_se6008125447796440765t_unit
= ( ^ [R3: partia7496981018696276118t_unit] : ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ nil_set_list_a ) ) ) ) ).
% var_def
thf(fact_879_x_Ofactors__mult__single,axiom,
! [A: list_a,Fb: list_list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ A @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% x.factors_mult_single
thf(fact_880_x_Oconst__term__simprules_I3_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(3)
thf(fact_881_x_Opirreducible__degree,axiom,
! [K3: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% x.pirreducible_degree
thf(fact_882_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_883_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_884_x_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_885_x_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z3 ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_886_x_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.add.m_comm
thf(fact_887_x_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z3 ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_888_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_889_x_Osubring__props_I2_J,axiom,
! [K3: set_list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K3 ) ) ).
% x.subring_props(2)
thf(fact_890_x_Osubring__props_I7_J,axiom,
! [K3: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K3 )
=> ( ( member_list_a @ H2 @ K3 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K3 ) ) ) ) ).
% x.subring_props(7)
thf(fact_891_x_Osubring__props_I6_J,axiom,
! [K3: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K3 )
=> ( ( member_list_a @ H2 @ K3 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K3 ) ) ) ) ).
% x.subring_props(6)
thf(fact_892_x_Osubring__props_I3_J,axiom,
! [K3: set_list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K3 ) ) ).
% x.subring_props(3)
thf(fact_893_x_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y6: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y6 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_894_x_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_895_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_896_x_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_897_x_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_898_x_Or__distr,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z3 @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z3 @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_899_x_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z3 ) ) ) ) ) ) ).
% x.l_distr
thf(fact_900_x_Osubring__props_I1_J,axiom,
! [K3: set_list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_props(1)
thf(fact_901_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_902_poly__mult__replicate__zero_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= nil_a ) ) ).
% poly_mult_replicate_zero(1)
thf(fact_903_poly__mult__replicate__zero_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= nil_a ) ) ).
% poly_mult_replicate_zero(2)
thf(fact_904_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_905_x_Oa__lcos__m__assoc,axiom,
! [M3: set_list_a,G2: list_a,H3: list_a] :
( ( ord_le8861187494160871172list_a @ M3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G2 @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 @ M3 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G2 @ H3 ) @ M3 ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_906_poly__mult__prepend__replicate__zero,axiom,
! [P1: list_a,P22: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_907_poly__add__replicate__zero_H_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_908_poly__add__replicate__zero_H_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_909_poly__add__degree__le,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_add_degree_le
thf(fact_910_poly__add__append__replicate,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_911_x_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_912_x_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z3 @ X ) )
= ( Y = Z3 ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_913_x_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_zero
thf(fact_914_x_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_zero
thf(fact_915_x_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_916_x_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_917_x_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_918_x_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_919_ring__irreducible__def,axiom,
( ring_r5115406448772830318t_unit
= ( ^ [R3: partia7496981018696276118t_unit,A3: set_list_a] :
( ( A3
!= ( zero_s2910681146719230829t_unit @ R3 ) )
& ( irredu943254396193320253t_unit @ R3 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_920_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R3 ) )
& ( irredu6211895646901577903xt_a_b @ R3 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_921_ring__irreducible__def,axiom,
( ring_r932985474545269838t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( irredu4230924414530676029t_unit @ R3 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_922_ring__irreducible__def,axiom,
( ring_r360171070648044744t_unit
= ( ^ [R3: partia2956882679547061052t_unit,A3: list_list_a] :
( ( A3
!= ( zero_l347298301471573063t_unit @ R3 ) )
& ( irredu4439051761327310013t_unit @ R3 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_923_x_Oline__extension__smult__closed,axiom,
! [K3: set_list_a,E2: set_list_a,A: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K4: list_a,V: list_a] :
( ( member_list_a @ K4 @ K3 )
=> ( ( member_list_a @ V @ E2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K3 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A @ E2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A @ E2 ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_924_x_Osubfield__long__division__theorem__shell,axiom,
! [K3: set_list_a,P: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
=> ? [Q3: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
& ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
& ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) @ B @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% x.subfield_long_division_theorem_shell
thf(fact_925_x_Oeval__append__aux,axiom,
! [P: list_list_a,B: list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ B @ nil_list_a ) ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ A ) @ B ) ) ) ) ) ).
% x.eval_append_aux
thf(fact_926_ring__irreducibleE_I2_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( irredu6211895646901577903xt_a_b @ r @ R ) ) ) ).
% ring_irreducibleE(2)
thf(fact_927_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_928_x_Oeval__var,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.eval_var
thf(fact_929_x_Oline__extension__in__carrier,axiom,
! [K3: set_list_a,A: list_a,E2: set_list_a] :
( ( ord_le8861187494160871172list_a @ K3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A @ E2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_930_x_Oeval_Osimps_I1_J,axiom,
( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.eval.simps(1)
thf(fact_931_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K3: set_list_a,A: list_a,E2: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A @ E2 ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K3 )
& ? [Y4: list_a] :
( ( member_list_a @ Y4 @ E2 )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ A ) @ Y4 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_932_x_Oconst__term__def,axiom,
! [P: list_list_a] :
( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.const_term_def
thf(fact_933_x_Oeval__poly__of__const,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= X ) ) ).
% x.eval_poly_of_const
thf(fact_934_x_Oeval__in__carrier,axiom,
! [P: list_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier
thf(fact_935_x_Oeval__in__carrier__2,axiom,
! [X: list_list_a,Y: list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_936_x_Oeval__normalize,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ A )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) ) ) ) ).
% x.eval_normalize
thf(fact_937_x_Oeval__poly__add,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ).
% x.eval_poly_add
thf(fact_938_x_Oeval__poly__mult,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ).
% x.eval_poly_mult
thf(fact_939_x_Oeval__poly__add__aux,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ) ).
% x.eval_poly_add_aux
thf(fact_940_x_Oeval__replicate,axiom,
! [P: list_list_a,A: list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) @ A )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) ) ) ) ).
% x.eval_replicate
thf(fact_941_x_Ois__root__def,axiom,
! [P: list_list_a,X: list_a] :
( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
= ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( P != nil_list_a ) ) ) ).
% x.is_root_def
thf(fact_942_x_Oeval__append,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( size_s349497388124573686list_a @ Q ) ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ).
% x.eval_append
thf(fact_943_subfield__long__division__theorem__shell,axiom,
! [K3: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ? [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ B @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_944_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_945_univ__poly__is__principal,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ).
% univ_poly_is_principal
thf(fact_946_subring__props_I2_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K3 ) ) ).
% subring_props(2)
thf(fact_947_subring__props_I3_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K3 ) ) ).
% subring_props(3)
thf(fact_948_subring__props_I1_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_949_x_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.nat_pow_zero
thf(fact_950_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_951_x_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ) ).
% x.pow_mult_distrib
thf(fact_952_x_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ).
% x.nat_pow_distrib
thf(fact_953_x_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) ) ) ).
% x.nat_pow_comm
thf(fact_954_x_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) ) ) ) ) ).
% x.group_commutes_pow
thf(fact_955_pprimeE_I1_J,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_956_pprime__iff__pirreducible,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_957_x_Onat__pow__mult,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% x.nat_pow_mult
thf(fact_958_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_959_pprimeE_I3_J,axiom,
! [K3: set_a,P: list_a,Q: list_a,R: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_960_degree__one__imp__pirreducible,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_961_x_Oeval__monom,axiom,
! [B: list_a,A: list_a,N: nat] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ N ) @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) ) ) ) ).
% x.eval_monom
thf(fact_962_pirreducible__pow__pdivides__iff,axiom,
! [K3: set_a,P: list_a,Q: list_a,R: list_a,N: nat] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q @ R ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ P @ N ) @ R ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_963_pirreducible__degree,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_964_x_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.nat_pow_closed
thf(fact_965_x_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.nat_pow_one
thf(fact_966_x_Onat__pow__eone,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ one_one_nat )
= X ) ) ).
% x.nat_pow_eone
thf(fact_967_x_Onat__pow__0,axiom,
! [X: list_a] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.nat_pow_0
thf(fact_968_exists__unique__long__division,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X3 )
& ! [Y3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y3 )
=> ( Y3 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_969_rupture__one__not__zero,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K3 @ P ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K3 @ P ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_970_pmod__const_I1_J,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P @ Q )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_971_long__division__closed_I1_J,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_972_long__division__add_I1_J,axiom,
! [K3: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_973_long__division__zero_I1_J,axiom,
! [K3: set_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_974_rupture__is__field__iff__pirreducible,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K3 @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_975_long__dividesI,axiom,
! [B: list_a,R: list_a,P: list_a,Q: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B ) @ R ) )
=> ( ( ( R = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R ) ) ) ) ) ) ).
% long_dividesI
thf(fact_976_x_Oeval_Oelims,axiom,
! [X: list_list_a,Y: list_a > list_a] :
( ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y )
=> ( ( ( X = nil_list_a )
=> ( Y
!= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ~ ! [V: list_a,Va: list_list_a] :
( ( X
= ( cons_list_a @ V @ Va ) )
=> ( Y
!= ( ^ [X2: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( hd_list_a @ ( cons_list_a @ V @ Va ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( cons_list_a @ V @ Va ) ) @ one_one_nat ) ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( tl_list_a @ ( cons_list_a @ V @ Va ) ) @ X2 ) ) ) ) ) ) ) ).
% x.eval.elims
thf(fact_977_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P12: list_a,P23: list_a] :
( X
!= ( produc6837034575241423639list_a @ P12 @ P23 ) ) ).
% poly_add.cases
thf(fact_978_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P23: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P23 ) )
=> ~ ! [V: a,Va: list_a,P23: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P23 ) ) ) ).
% poly_mult.cases
thf(fact_979_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U2: a,Us3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U2 @ Us3 ) ) )
=> ( ! [Us3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us3 ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_980_x_Onormalize__length__eq,axiom,
! [P: list_list_a] :
( ( ( hd_list_a @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ).
% x.normalize_length_eq
thf(fact_981_exists__long__division,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B6: list_a] :
( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B6 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_982_x_Onormalize__lead__coeff,axiom,
! [P: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( size_s349497388124573686list_a @ P ) )
=> ( ( hd_list_a @ P )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.normalize_lead_coeff
thf(fact_983_x_Onormalize_Osimps_I2_J,axiom,
! [V3: list_a,Va2: list_list_a] :
( ( ( ( hd_list_a @ ( cons_list_a @ V3 @ Va2 ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ V3 @ Va2 ) )
= ( cons_list_a @ V3 @ Va2 ) ) )
& ( ( ( hd_list_a @ ( cons_list_a @ V3 @ Va2 ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ V3 @ Va2 ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( tl_list_a @ ( cons_list_a @ V3 @ Va2 ) ) ) ) ) ) ).
% x.normalize.simps(2)
thf(fact_984_x_Onormalize__length__lt,axiom,
! [P: list_list_a] :
( ( ( hd_list_a @ P )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ P ) )
=> ( ord_less_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ) ) ).
% x.normalize_length_lt
thf(fact_985_x_Onormalize_Oelims,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y )
=> ( ( ( X = nil_list_a )
=> ( Y != nil_list_a ) )
=> ~ ! [V: list_a,Va: list_list_a] :
( ( X
= ( cons_list_a @ V @ Va ) )
=> ~ ( ( ( ( hd_list_a @ ( cons_list_a @ V @ Va ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y
= ( cons_list_a @ V @ Va ) ) )
& ( ( ( hd_list_a @ ( cons_list_a @ V @ Va ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( tl_list_a @ ( cons_list_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% x.normalize.elims
thf(fact_986_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_987_hd__append2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Xs ) ) ) ).
% hd_append2
thf(fact_988_hd__replicate,axiom,
! [N: nat,X: list_a] :
( ( N != zero_zero_nat )
=> ( ( hd_list_a @ ( replicate_list_a @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_989_hd__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( hd_a @ ( replicate_a @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_990_x_Oeval_Osimps_I2_J,axiom,
! [V3: list_a,Va2: list_list_a] :
( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ V3 @ Va2 ) )
= ( ^ [X2: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( hd_list_a @ ( cons_list_a @ V3 @ Va2 ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( cons_list_a @ V3 @ Va2 ) ) @ one_one_nat ) ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( tl_list_a @ ( cons_list_a @ V3 @ Va2 ) ) @ X2 ) ) ) ) ).
% x.eval.simps(2)
thf(fact_991_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_992_hd__Cons__tl,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ Xs ) @ ( tl_list_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_993_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_994_list_Ocollapse,axiom,
! [List: list_list_a] :
( ( List != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ List ) @ ( tl_list_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_995_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_996_shuffles_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
=> ( ! [Xs2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) )
=> ~ ! [X3: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_997_subset__eq__mset__impl_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X3: a,Xs2: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_998_subset__eq__mset__impl_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
=> ~ ! [X3: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_999_list_Oset__sel_I1_J,axiom,
! [A: list_set_a] :
( ( A != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ A ) @ ( set_set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1000_list_Oset__sel_I1_J,axiom,
! [A: list_set_list_a] :
( ( A != nil_set_list_a )
=> ( member_set_list_a @ ( hd_set_list_a @ A ) @ ( set_set_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1001_list_Oset__sel_I1_J,axiom,
! [A: list_list_list_a] :
( ( A != nil_list_list_a )
=> ( member_list_list_a @ ( hd_list_list_a @ A ) @ ( set_list_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1002_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1003_list_Oset__sel_I1_J,axiom,
! [A: list_list_a] :
( ( A != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A ) @ ( set_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1004_hd__in__set,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1005_hd__in__set,axiom,
! [Xs: list_set_list_a] :
( ( Xs != nil_set_list_a )
=> ( member_set_list_a @ ( hd_set_list_a @ Xs ) @ ( set_set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1006_hd__in__set,axiom,
! [Xs: list_list_list_a] :
( ( Xs != nil_list_list_a )
=> ( member_list_list_a @ ( hd_list_list_a @ Xs ) @ ( set_list_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1007_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1008_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1009_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys7: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys7 ) )
& ( ( Xs4 = nil_a )
| ( Ys7 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys7 ) ) ) ) ).
% longest_common_prefix
thf(fact_1010_longest__common__prefix,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
? [Ps: list_list_a,Xs4: list_list_a,Ys7: list_list_a] :
( ( Xs
= ( append_list_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_list_a @ Ps @ Ys7 ) )
& ( ( Xs4 = nil_list_a )
| ( Ys7 = nil_list_a )
| ( ( hd_list_a @ Xs4 )
!= ( hd_list_a @ Ys7 ) ) ) ) ).
% longest_common_prefix
thf(fact_1011_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_1012_hd__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( Xs = nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Ys ) ) )
& ( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_1013_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1014_list_Osel_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( hd_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1015_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_1016_list_Oexpand,axiom,
! [List: list_list_a,List2: list_list_a] :
( ( ( List = nil_list_a )
= ( List2 = nil_list_a ) )
=> ( ( ( List != nil_list_a )
=> ( ( List2 != nil_list_a )
=> ( ( ( hd_list_a @ List )
= ( hd_list_a @ List2 ) )
& ( ( tl_list_a @ List )
= ( tl_list_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_1017_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1018_list_Oexhaust__sel,axiom,
! [List: list_list_a] :
( ( List != nil_list_a )
=> ( List
= ( cons_list_a @ ( hd_list_a @ List ) @ ( tl_list_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1019_x_Opoly__mult_Oelims,axiom,
! [X: list_list_a,Xa2: list_list_a,Y: list_list_a] :
( ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Xa2 )
= Y )
=> ( ( ( X = nil_list_a )
=> ( Y != nil_list_a ) )
=> ~ ! [V: list_a,Va: list_list_a] :
( ( X
= ( cons_list_a @ V @ Va ) )
=> ( Y
!= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( map_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( hd_list_a @ ( cons_list_a @ V @ Va ) ) ) @ Xa2 ) @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( cons_list_a @ V @ Va ) ) @ one_one_nat ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( tl_list_a @ ( cons_list_a @ V @ Va ) ) @ Xa2 ) ) ) ) ) ) ).
% x.poly_mult.elims
thf(fact_1020_x_Opoly__mult_Osimps_I2_J,axiom,
! [V3: list_a,Va2: list_list_a,P22: list_list_a] :
( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ V3 @ Va2 ) @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( map_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( hd_list_a @ ( cons_list_a @ V3 @ Va2 ) ) ) @ P22 ) @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( cons_list_a @ V3 @ Va2 ) ) @ one_one_nat ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( tl_list_a @ ( cons_list_a @ V3 @ Va2 ) ) @ P22 ) ) ) ).
% x.poly_mult.simps(2)
thf(fact_1021_x_Opoly__mult_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [P23: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ P23 ) )
=> ~ ! [V: list_a,Va: list_list_a,P23: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ P23 ) ) ) ).
% x.poly_mult.cases
thf(fact_1022_x_Ocombine_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [K4: list_a,Ks: list_list_a,U2: list_a,Us3: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K4 @ Ks ) @ ( cons_list_a @ U2 @ Us3 ) ) )
=> ( ! [Us3: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Us3 ) )
=> ~ ! [Ks: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% x.combine.cases
thf(fact_1023_normalize__length__eq,axiom,
! [P: list_a] :
( ( ( hd_a @ P )
!= ( zero_a_b @ r ) )
=> ( ( size_size_list_a @ ( normalize_a_b @ r @ P ) )
= ( size_size_list_a @ P ) ) ) ).
% normalize_length_eq
thf(fact_1024_normalize__lead__coeff,axiom,
! [P: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) )
=> ( ( hd_a @ P )
= ( zero_a_b @ r ) ) ) ).
% normalize_lead_coeff
thf(fact_1025_normalize_Osimps_I2_J,axiom,
! [V3: a,Va2: list_a] :
( ( ( ( hd_a @ ( cons_a @ V3 @ Va2 ) )
!= ( zero_a_b @ r ) )
=> ( ( normalize_a_b @ r @ ( cons_a @ V3 @ Va2 ) )
= ( cons_a @ V3 @ Va2 ) ) )
& ( ( ( hd_a @ ( cons_a @ V3 @ Va2 ) )
= ( zero_a_b @ r ) )
=> ( ( normalize_a_b @ r @ ( cons_a @ V3 @ Va2 ) )
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V3 @ Va2 ) ) ) ) ) ) ).
% normalize.simps(2)
thf(fact_1026_normalize__length__lt,axiom,
! [P: list_a] :
( ( ( hd_a @ P )
= ( zero_a_b @ r ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ P ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ) ) ).
% normalize_length_lt
thf(fact_1027_normalize_Oelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( normalize_a_b @ r @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ~ ( ( ( ( hd_a @ ( cons_a @ V @ Va ) )
!= ( zero_a_b @ r ) )
=> ( Y
= ( cons_a @ V @ Va ) ) )
& ( ( ( hd_a @ ( cons_a @ V @ Va ) )
= ( zero_a_b @ r ) )
=> ( Y
= ( normalize_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% normalize.elims
thf(fact_1028_map__eq__conv,axiom,
! [F2: a > list_a,Xs: list_a,G2: a > list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= ( map_a_list_a @ G2 @ Xs ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1029_map__eq__conv,axiom,
! [F2: a > a,Xs: list_a,G2: a > a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G2 @ Xs ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1030_map__eq__conv,axiom,
! [F2: list_a > list_a,Xs: list_list_a,G2: list_a > list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= ( map_list_a_list_a @ G2 @ Xs ) )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1031_list_Omap__disc__iff,axiom,
! [F2: list_a > a,A: list_list_a] :
( ( ( map_list_a_a @ F2 @ A )
= nil_a )
= ( A = nil_list_a ) ) ).
% list.map_disc_iff
thf(fact_1032_list_Omap__disc__iff,axiom,
! [F2: list_a > list_a,A: list_list_a] :
( ( ( map_list_a_list_a @ F2 @ A )
= nil_list_a )
= ( A = nil_list_a ) ) ).
% list.map_disc_iff
thf(fact_1033_list_Omap__disc__iff,axiom,
! [F2: a > list_a,A: list_a] :
( ( ( map_a_list_a @ F2 @ A )
= nil_list_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_1034_list_Omap__disc__iff,axiom,
! [F2: a > a,A: list_a] :
( ( ( map_a_a @ F2 @ A )
= nil_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_1035_Nil__is__map__conv,axiom,
! [F2: list_a > a,Xs: list_list_a] :
( ( nil_a
= ( map_list_a_a @ F2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_map_conv
thf(fact_1036_Nil__is__map__conv,axiom,
! [F2: list_a > list_a,Xs: list_list_a] :
( ( nil_list_a
= ( map_list_a_list_a @ F2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_map_conv
thf(fact_1037_Nil__is__map__conv,axiom,
! [F2: a > list_a,Xs: list_a] :
( ( nil_list_a
= ( map_a_list_a @ F2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_1038_Nil__is__map__conv,axiom,
! [F2: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_1039_map__is__Nil__conv,axiom,
! [F2: list_a > a,Xs: list_list_a] :
( ( ( map_list_a_a @ F2 @ Xs )
= nil_a )
= ( Xs = nil_list_a ) ) ).
% map_is_Nil_conv
thf(fact_1040_map__is__Nil__conv,axiom,
! [F2: list_a > list_a,Xs: list_list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% map_is_Nil_conv
thf(fact_1041_map__is__Nil__conv,axiom,
! [F2: a > list_a,Xs: list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= nil_list_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_1042_map__is__Nil__conv,axiom,
! [F2: a > a,Xs: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_1043_length__map,axiom,
! [F2: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_1044_length__map,axiom,
! [F2: list_a > a,Xs: list_list_a] :
( ( size_size_list_a @ ( map_list_a_a @ F2 @ Xs ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_map
thf(fact_1045_length__map,axiom,
! [F2: a > list_a,Xs: list_a] :
( ( size_s349497388124573686list_a @ ( map_a_list_a @ F2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_1046_length__map,axiom,
! [F2: list_a > list_a,Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( map_list_a_list_a @ F2 @ Xs ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_map
thf(fact_1047_map__append,axiom,
! [F2: list_a > a,Xs: list_list_a,Ys: list_list_a] :
( ( map_list_a_a @ F2 @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( map_list_a_a @ F2 @ Xs ) @ ( map_list_a_a @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_1048_map__append,axiom,
! [F2: list_a > list_a,Xs: list_list_a,Ys: list_list_a] :
( ( map_list_a_list_a @ F2 @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( map_list_a_list_a @ F2 @ Xs ) @ ( map_list_a_list_a @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_1049_map__append,axiom,
! [F2: a > list_a,Xs: list_a,Ys: list_a] :
( ( map_a_list_a @ F2 @ ( append_a @ Xs @ Ys ) )
= ( append_list_a @ ( map_a_list_a @ F2 @ Xs ) @ ( map_a_list_a @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_1050_map__append,axiom,
! [F2: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F2 @ Xs ) @ ( map_a_a @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_1051_map__replicate,axiom,
! [F2: list_a > list_a,N: nat,X: list_a] :
( ( map_list_a_list_a @ F2 @ ( replicate_list_a @ N @ X ) )
= ( replicate_list_a @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_1052_map__replicate,axiom,
! [F2: list_a > a,N: nat,X: list_a] :
( ( map_list_a_a @ F2 @ ( replicate_list_a @ N @ X ) )
= ( replicate_a @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_1053_map__replicate,axiom,
! [F2: a > list_a,N: nat,X: a] :
( ( map_a_list_a @ F2 @ ( replicate_a @ N @ X ) )
= ( replicate_list_a @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_1054_map__replicate,axiom,
! [F2: a > a,N: nat,X: a] :
( ( map_a_a @ F2 @ ( replicate_a @ N @ X ) )
= ( replicate_a @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_1055_list_Omap__sel_I1_J,axiom,
! [A: list_list_a,F2: list_a > a] :
( ( A != nil_list_a )
=> ( ( hd_a @ ( map_list_a_a @ F2 @ A ) )
= ( F2 @ ( hd_list_a @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1056_list_Omap__sel_I1_J,axiom,
! [A: list_list_a,F2: list_a > list_a] :
( ( A != nil_list_a )
=> ( ( hd_list_a @ ( map_list_a_list_a @ F2 @ A ) )
= ( F2 @ ( hd_list_a @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1057_list_Omap__sel_I1_J,axiom,
! [A: list_a,F2: a > list_a] :
( ( A != nil_a )
=> ( ( hd_list_a @ ( map_a_list_a @ F2 @ A ) )
= ( F2 @ ( hd_a @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1058_list_Omap__sel_I1_J,axiom,
! [A: list_a,F2: a > a] :
( ( A != nil_a )
=> ( ( hd_a @ ( map_a_a @ F2 @ A ) )
= ( F2 @ ( hd_a @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1059_hd__map,axiom,
! [Xs: list_list_a,F2: list_a > a] :
( ( Xs != nil_list_a )
=> ( ( hd_a @ ( map_list_a_a @ F2 @ Xs ) )
= ( F2 @ ( hd_list_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_1060_hd__map,axiom,
! [Xs: list_list_a,F2: list_a > list_a] :
( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( map_list_a_list_a @ F2 @ Xs ) )
= ( F2 @ ( hd_list_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_1061_hd__map,axiom,
! [Xs: list_a,F2: a > list_a] :
( ( Xs != nil_a )
=> ( ( hd_list_a @ ( map_a_list_a @ F2 @ Xs ) )
= ( F2 @ ( hd_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_1062_hd__map,axiom,
! [Xs: list_a,F2: a > a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( map_a_a @ F2 @ Xs ) )
= ( F2 @ ( hd_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_1063_list_Omap__sel_I2_J,axiom,
! [A: list_list_a,F2: list_a > a] :
( ( A != nil_list_a )
=> ( ( tl_a @ ( map_list_a_a @ F2 @ A ) )
= ( map_list_a_a @ F2 @ ( tl_list_a @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1064_list_Omap__sel_I2_J,axiom,
! [A: list_list_a,F2: list_a > list_a] :
( ( A != nil_list_a )
=> ( ( tl_list_a @ ( map_list_a_list_a @ F2 @ A ) )
= ( map_list_a_list_a @ F2 @ ( tl_list_a @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1065_list_Omap__sel_I2_J,axiom,
! [A: list_a,F2: a > list_a] :
( ( A != nil_a )
=> ( ( tl_list_a @ ( map_a_list_a @ F2 @ A ) )
= ( map_a_list_a @ F2 @ ( tl_a @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1066_list_Omap__sel_I2_J,axiom,
! [A: list_a,F2: a > a] :
( ( A != nil_a )
=> ( ( tl_a @ ( map_a_a @ F2 @ A ) )
= ( map_a_a @ F2 @ ( tl_a @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1067_ex__map__conv,axiom,
! [Ys: list_a,F2: a > a] :
( ( ? [Xs5: list_a] :
( Ys
= ( map_a_a @ F2 @ Xs5 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ? [Y4: a] :
( X2
= ( F2 @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1068_ex__map__conv,axiom,
! [Ys: list_list_a,F2: list_a > list_a] :
( ( ? [Xs5: list_list_a] :
( Ys
= ( map_list_a_list_a @ F2 @ Xs5 ) ) )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ? [Y4: list_a] :
( X2
= ( F2 @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1069_ex__map__conv,axiom,
! [Ys: list_list_a,F2: a > list_a] :
( ( ? [Xs5: list_a] :
( Ys
= ( map_a_list_a @ F2 @ Xs5 ) ) )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ? [Y4: a] :
( X2
= ( F2 @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1070_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F2: a > list_a,G2: a > list_a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_list_a @ F2 @ Xs )
= ( map_a_list_a @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_1071_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F2: a > a,G2: a > a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_1072_map__cong,axiom,
! [Xs: list_list_a,Ys: list_list_a,F2: list_a > list_a,G2: list_a > list_a] :
( ( Xs = Ys )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_list_a_list_a @ F2 @ Xs )
= ( map_list_a_list_a @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_1073_map__idI,axiom,
! [Xs: list_set_a,F2: set_a > set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_set_a_set_a @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1074_map__idI,axiom,
! [Xs: list_set_list_a,F2: set_list_a > set_list_a] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_se2668659675339852484list_a @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1075_map__idI,axiom,
! [Xs: list_list_list_a,F2: list_list_a > list_list_a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( set_list_list_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_li8713736314956022724list_a @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1076_map__idI,axiom,
! [Xs: list_a,F2: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_a_a @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1077_map__idI,axiom,
! [Xs: list_list_a,F2: list_a > list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_list_a_list_a @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1078_map__ext,axiom,
! [Xs: list_a,F2: a > list_a,G2: a > list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_list_a @ F2 @ Xs )
= ( map_a_list_a @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_1079_map__ext,axiom,
! [Xs: list_a,F2: a > a,G2: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_1080_map__ext,axiom,
! [Xs: list_list_a,F2: list_a > list_a,G2: list_a > list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_list_a_list_a @ F2 @ Xs )
= ( map_list_a_list_a @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_1081_list_Omap__ident__strong,axiom,
! [T3: list_set_a,F2: set_a > set_a] :
( ! [Z: set_a] :
( ( member_set_a @ Z @ ( set_set_a2 @ T3 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_set_a_set_a @ F2 @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1082_list_Omap__ident__strong,axiom,
! [T3: list_set_list_a,F2: set_list_a > set_list_a] :
( ! [Z: set_list_a] :
( ( member_set_list_a @ Z @ ( set_set_list_a2 @ T3 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_se2668659675339852484list_a @ F2 @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1083_list_Omap__ident__strong,axiom,
! [T3: list_list_list_a,F2: list_list_a > list_list_a] :
( ! [Z: list_list_a] :
( ( member_list_list_a @ Z @ ( set_list_list_a2 @ T3 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_li8713736314956022724list_a @ F2 @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1084_list_Omap__ident__strong,axiom,
! [T3: list_a,F2: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ T3 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_a_a @ F2 @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1085_list_Omap__ident__strong,axiom,
! [T3: list_list_a,F2: list_a > list_a] :
( ! [Z: list_a] :
( ( member_list_a @ Z @ ( set_list_a2 @ T3 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_list_a_list_a @ F2 @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1086_list_Oinj__map__strong,axiom,
! [X: list_a,Xa2: list_a,F2: a > list_a,Fa: a > list_a] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( set_a2 @ X ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_list_a @ F2 @ X )
= ( map_a_list_a @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1087_list_Oinj__map__strong,axiom,
! [X: list_a,Xa2: list_a,F2: a > a,Fa: a > a] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( set_a2 @ X ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_a @ F2 @ X )
= ( map_a_a @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1088_list_Oinj__map__strong,axiom,
! [X: list_list_a,Xa2: list_list_a,F2: list_a > list_a,Fa: list_a > list_a] :
( ! [Z: list_a,Za: list_a] :
( ( member_list_a @ Z @ ( set_list_a2 @ X ) )
=> ( ( member_list_a @ Za @ ( set_list_a2 @ Xa2 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_list_a_list_a @ F2 @ X )
= ( map_list_a_list_a @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1089_list_Omap__cong0,axiom,
! [X: list_a,F2: a > list_a,G2: a > list_a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ X ) )
=> ( ( F2 @ Z )
= ( G2 @ Z ) ) )
=> ( ( map_a_list_a @ F2 @ X )
= ( map_a_list_a @ G2 @ X ) ) ) ).
% list.map_cong0
thf(fact_1090_list_Omap__cong0,axiom,
! [X: list_a,F2: a > a,G2: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ X ) )
=> ( ( F2 @ Z )
= ( G2 @ Z ) ) )
=> ( ( map_a_a @ F2 @ X )
= ( map_a_a @ G2 @ X ) ) ) ).
% list.map_cong0
thf(fact_1091_list_Omap__cong0,axiom,
! [X: list_list_a,F2: list_a > list_a,G2: list_a > list_a] :
( ! [Z: list_a] :
( ( member_list_a @ Z @ ( set_list_a2 @ X ) )
=> ( ( F2 @ Z )
= ( G2 @ Z ) ) )
=> ( ( map_list_a_list_a @ F2 @ X )
= ( map_list_a_list_a @ G2 @ X ) ) ) ).
% list.map_cong0
thf(fact_1092_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F2: a > list_a,G2: a > list_a] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G2 @ Z ) ) )
=> ( ( map_a_list_a @ F2 @ X )
= ( map_a_list_a @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1093_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F2: a > a,G2: a > a] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G2 @ Z ) ) )
=> ( ( map_a_a @ F2 @ X )
= ( map_a_a @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1094_list_Omap__cong,axiom,
! [X: list_list_a,Ya: list_list_a,F2: list_a > list_a,G2: list_a > list_a] :
( ( X = Ya )
=> ( ! [Z: list_a] :
( ( member_list_a @ Z @ ( set_list_a2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G2 @ Z ) ) )
=> ( ( map_list_a_list_a @ F2 @ X )
= ( map_list_a_list_a @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1095_map__tl,axiom,
! [F2: list_a > a,Xs: list_list_a] :
( ( map_list_a_a @ F2 @ ( tl_list_a @ Xs ) )
= ( tl_a @ ( map_list_a_a @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_1096_map__tl,axiom,
! [F2: list_a > list_a,Xs: list_list_a] :
( ( map_list_a_list_a @ F2 @ ( tl_list_a @ Xs ) )
= ( tl_list_a @ ( map_list_a_list_a @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_1097_map__tl,axiom,
! [F2: a > list_a,Xs: list_a] :
( ( map_a_list_a @ F2 @ ( tl_a @ Xs ) )
= ( tl_list_a @ ( map_a_list_a @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_1098_map__tl,axiom,
! [F2: a > a,Xs: list_a] :
( ( map_a_a @ F2 @ ( tl_a @ Xs ) )
= ( tl_a @ ( map_a_a @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_1099_list_Osimps_I8_J,axiom,
! [F2: list_a > a] :
( ( map_list_a_a @ F2 @ nil_list_a )
= nil_a ) ).
% list.simps(8)
thf(fact_1100_list_Osimps_I8_J,axiom,
! [F2: list_a > list_a] :
( ( map_list_a_list_a @ F2 @ nil_list_a )
= nil_list_a ) ).
% list.simps(8)
thf(fact_1101_list_Osimps_I8_J,axiom,
! [F2: a > list_a] :
( ( map_a_list_a @ F2 @ nil_a )
= nil_list_a ) ).
% list.simps(8)
thf(fact_1102_list_Osimps_I8_J,axiom,
! [F2: a > a] :
( ( map_a_a @ F2 @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_1103_list_Osimps_I9_J,axiom,
! [F2: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F2 @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F2 @ X21 ) @ ( map_a_a @ F2 @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1104_list_Osimps_I9_J,axiom,
! [F2: a > list_a,X21: a,X22: list_a] :
( ( map_a_list_a @ F2 @ ( cons_a @ X21 @ X22 ) )
= ( cons_list_a @ ( F2 @ X21 ) @ ( map_a_list_a @ F2 @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1105_list_Osimps_I9_J,axiom,
! [F2: list_a > a,X21: list_a,X22: list_list_a] :
( ( map_list_a_a @ F2 @ ( cons_list_a @ X21 @ X22 ) )
= ( cons_a @ ( F2 @ X21 ) @ ( map_list_a_a @ F2 @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1106_list_Osimps_I9_J,axiom,
! [F2: list_a > list_a,X21: list_a,X22: list_list_a] :
( ( map_list_a_list_a @ F2 @ ( cons_list_a @ X21 @ X22 ) )
= ( cons_list_a @ ( F2 @ X21 ) @ ( map_list_a_list_a @ F2 @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1107_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F2: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F2 @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_a_a @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1108_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F2: list_a > a,Ys: list_list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_list_a_a @ F2 @ Ys ) )
=> ? [Z: list_a,Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_list_a_a @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1109_Cons__eq__map__D,axiom,
! [X: list_a,Xs: list_list_a,F2: a > list_a,Ys: list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( map_a_list_a @ F2 @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_a_list_a @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1110_Cons__eq__map__D,axiom,
! [X: list_a,Xs: list_list_a,F2: list_a > list_a,Ys: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( map_list_a_list_a @ F2 @ Ys ) )
=> ? [Z: list_a,Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_list_a_list_a @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1111_map__eq__Cons__D,axiom,
! [F2: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map_a_a @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1112_map__eq__Cons__D,axiom,
! [F2: list_a > a,Xs: list_list_a,Y: a,Ys: list_a] :
( ( ( map_list_a_a @ F2 @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z: list_a,Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map_list_a_a @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1113_map__eq__Cons__D,axiom,
! [F2: a > list_a,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= ( cons_list_a @ Y @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map_a_list_a @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1114_map__eq__Cons__D,axiom,
! [F2: list_a > list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= ( cons_list_a @ Y @ Ys ) )
=> ? [Z: list_a,Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map_list_a_list_a @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1115_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F2: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F2 @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( X
= ( F2 @ Z4 ) )
& ( Xs
= ( map_a_a @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1116_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F2: list_a > a,Ys: list_list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_list_a_a @ F2 @ Ys ) )
= ( ? [Z4: list_a,Zs3: list_list_a] :
( ( Ys
= ( cons_list_a @ Z4 @ Zs3 ) )
& ( X
= ( F2 @ Z4 ) )
& ( Xs
= ( map_list_a_a @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1117_Cons__eq__map__conv,axiom,
! [X: list_a,Xs: list_list_a,F2: a > list_a,Ys: list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( map_a_list_a @ F2 @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( X
= ( F2 @ Z4 ) )
& ( Xs
= ( map_a_list_a @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1118_Cons__eq__map__conv,axiom,
! [X: list_a,Xs: list_list_a,F2: list_a > list_a,Ys: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( map_list_a_list_a @ F2 @ Ys ) )
= ( ? [Z4: list_a,Zs3: list_list_a] :
( ( Ys
= ( cons_list_a @ Z4 @ Zs3 ) )
& ( X
= ( F2 @ Z4 ) )
& ( Xs
= ( map_list_a_list_a @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1119_map__eq__Cons__conv,axiom,
! [F2: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs3 ) )
& ( ( F2 @ Z4 )
= Y )
& ( ( map_a_a @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1120_map__eq__Cons__conv,axiom,
! [F2: list_a > a,Xs: list_list_a,Y: a,Ys: list_a] :
( ( ( map_list_a_a @ F2 @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z4: list_a,Zs3: list_list_a] :
( ( Xs
= ( cons_list_a @ Z4 @ Zs3 ) )
& ( ( F2 @ Z4 )
= Y )
& ( ( map_list_a_a @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1121_map__eq__Cons__conv,axiom,
! [F2: a > list_a,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= ( cons_list_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs3 ) )
& ( ( F2 @ Z4 )
= Y )
& ( ( map_a_list_a @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1122_map__eq__Cons__conv,axiom,
! [F2: list_a > list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= ( cons_list_a @ Y @ Ys ) )
= ( ? [Z4: list_a,Zs3: list_list_a] :
( ( Xs
= ( cons_list_a @ Z4 @ Zs3 ) )
& ( ( F2 @ Z4 )
= Y )
& ( ( map_list_a_list_a @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1123_map__eq__imp__length__eq,axiom,
! [F2: a > list_a,Xs: list_a,G2: a > list_a,Ys: list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= ( map_a_list_a @ G2 @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1124_map__eq__imp__length__eq,axiom,
! [F2: a > a,Xs: list_a,G2: a > a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G2 @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1125_map__eq__imp__length__eq,axiom,
! [F2: a > list_a,Xs: list_a,G2: list_a > list_a,Ys: list_list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= ( map_list_a_list_a @ G2 @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1126_map__eq__imp__length__eq,axiom,
! [F2: a > a,Xs: list_a,G2: list_a > a,Ys: list_list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_list_a_a @ G2 @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1127_map__eq__imp__length__eq,axiom,
! [F2: list_a > a,Xs: list_list_a,G2: a > a,Ys: list_a] :
( ( ( map_list_a_a @ F2 @ Xs )
= ( map_a_a @ G2 @ Ys ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1128_map__eq__imp__length__eq,axiom,
! [F2: list_a > list_a,Xs: list_list_a,G2: a > list_a,Ys: list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= ( map_a_list_a @ G2 @ Ys ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1129_map__eq__imp__length__eq,axiom,
! [F2: list_a > list_a,Xs: list_list_a,G2: list_a > list_a,Ys: list_list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= ( map_list_a_list_a @ G2 @ Ys ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1130_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F2: list_a > a,Xs: list_list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_list_a_a @ F2 @ Xs ) )
= ( ? [Us2: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_list_a_a @ F2 @ Us2 ) )
& ( Zs
= ( map_list_a_a @ F2 @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1131_append__eq__map__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,F2: list_a > list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( map_list_a_list_a @ F2 @ Xs ) )
= ( ? [Us2: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_list_a_list_a @ F2 @ Us2 ) )
& ( Zs
= ( map_list_a_list_a @ F2 @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1132_append__eq__map__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,F2: a > list_a,Xs: list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( map_a_list_a @ F2 @ Xs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_list_a @ F2 @ Us2 ) )
& ( Zs
= ( map_a_list_a @ F2 @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1133_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F2: a > a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_a_a @ F2 @ Xs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_a @ F2 @ Us2 ) )
& ( Zs
= ( map_a_a @ F2 @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1134_map__eq__append__conv,axiom,
! [F2: list_a > a,Xs: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( map_list_a_a @ F2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_list_a_a @ F2 @ Us2 ) )
& ( Zs
= ( map_list_a_a @ F2 @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1135_map__eq__append__conv,axiom,
! [F2: list_a > list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( map_list_a_list_a @ F2 @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ? [Us2: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_list_a_list_a @ F2 @ Us2 ) )
& ( Zs
= ( map_list_a_list_a @ F2 @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1136_map__eq__append__conv,axiom,
! [F2: a > list_a,Xs: list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( map_a_list_a @ F2 @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_list_a @ F2 @ Us2 ) )
& ( Zs
= ( map_a_list_a @ F2 @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1137_map__eq__append__conv,axiom,
! [F2: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_a @ F2 @ Us2 ) )
& ( Zs
= ( map_a_a @ F2 @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1138_x_Olong__dividesI,axiom,
! [B: list_list_a,R: list_list_a,P: list_list_a,Q: list_list_a] :
( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Q @ B ) @ R ) )
=> ( ( ( R = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B @ R ) ) ) ) ) ) ).
% x.long_dividesI
thf(fact_1139_long__divisionI,axiom,
! [K3: set_a,P: list_a,Q: list_a,B: list_a,R: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R ) )
=> ( ( produc6837034575241423639list_a @ B @ R )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1140_x_Opoly__add_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
~ ! [P12: list_list_a,P23: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ P12 @ P23 ) ) ).
% x.poly_add.cases
thf(fact_1141_long__division__closed_I2_J,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1142_long__division__add_I2_J,axiom,
! [K3: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1143_long__division__add__iff,axiom,
! [K3: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_1144_long__division__zero_I2_J,axiom,
! [K3: set_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1145_norm__map__in__poly__ring__carrier,axiom,
! [P: list_a,F2: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F2 @ A5 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F2 @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_1146_pmod__zero__iff__pdivides,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1147_same__pmod__iff__pdivides,axiom,
! [K3: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1148_map__in__poly__ring__carrier,axiom,
! [P: list_a,F2: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F2 @ A5 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A5: a] :
( ( A5
!= ( zero_a_b @ r ) )
=> ( ( F2 @ A5 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F2 @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_1149_pdiv__pmod,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1150_pmod__const_I2_J,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P @ Q )
= P ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1151_pmod__degree,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1152_long__divisionE,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1153_poly__mult_Oelims,axiom,
! [X: list_a,Xa2: list_a,Y: list_a] :
( ( ( poly_mult_a_b @ r @ X @ Xa2 )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ~ ! [V: a,Va: list_a] :
( ( X
= ( cons_a @ V @ Va ) )
=> ( Y
!= ( poly_add_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V @ Va ) ) ) @ Xa2 ) @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V @ Va ) ) @ one_one_nat ) @ ( zero_a_b @ r ) ) ) @ ( poly_mult_a_b @ r @ ( tl_a @ ( cons_a @ V @ Va ) ) @ Xa2 ) ) ) ) ) ) ).
% poly_mult.elims
thf(fact_1154_poly__mult_Osimps_I2_J,axiom,
! [V3: a,Va2: list_a,P22: list_a] :
( ( poly_mult_a_b @ r @ ( cons_a @ V3 @ Va2 ) @ P22 )
= ( poly_add_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V3 @ Va2 ) ) ) @ P22 ) @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V3 @ Va2 ) ) @ one_one_nat ) @ ( zero_a_b @ r ) ) ) @ ( poly_mult_a_b @ r @ ( tl_a @ ( cons_a @ V3 @ Va2 ) ) @ P22 ) ) ) ).
% poly_mult.simps(2)
thf(fact_1155_m__lcomm,axiom,
! [X: a,Y: a,Z3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z3 ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z3 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_1156_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_1157_m__assoc,axiom,
! [X: a,Y: a,Z3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z3 )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z3 ) ) ) ) ) ) ).
% m_assoc
thf(fact_1158_subring__props_I6_J,axiom,
! [K3: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H2 @ K3 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K3 ) ) ) ) ).
% subring_props(6)
thf(fact_1159_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_1160_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_1161_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_1162_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_1163_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_1164_inv__unique,axiom,
! [Y: a,X: a,Y6: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y6 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1165_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_1166_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_1167_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_1168_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_1169_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1170_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1171_inv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_1172_inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_1173_comm__inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ).
% comm_inv_char
thf(fact_1174_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_1175_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_1176_ring__irreducibleE_I5_J,axiom,
! [R: a,A: a,B: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1177_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A5
!= ( zero_a_b @ r ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A5 @ X5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_1178_factors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_1179_eval__poly__mult,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) @ A )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_mult
thf(fact_1180_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_1181_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_1182_poly__mult__monom_H,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% poly_mult_monom'
thf(fact_1183_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1184_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_1185_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_1186_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_1187_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_1188_Units__l__cancel,axiom,
! [X: a,Y: a,Z3: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z3 ) )
= ( Y = Z3 ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1189_Units__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_r_inv
thf(fact_1190_Units__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_l_inv
thf(fact_1191_x_Oring_Ohom__mult,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_mult
thf(fact_1192_degree__one__root_I2_J,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P ) ) @ ( const_term_a_b @ r @ P ) ) @ K3 ) ) ) ) ).
% degree_one_root(2)
thf(fact_1193_alg__multE_I2_J,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1194_subring__props_I5_J,axiom,
! [K3: set_a,H3: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H3 @ K3 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K3 ) ) ) ).
% subring_props(5)
thf(fact_1195_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_1196_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_1197_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_1198_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_1199_square__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_1200_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_1201_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_1202_inv__eq__self,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ r @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% inv_eq_self
thf(fact_1203_const__term__simprules_I2_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_1204_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_1205_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P4 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_1206_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_1207_pdivides__imp__is__root,axiom,
! [P: list_a,X: a] :
( ( P != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_1208_is__root__imp__pdivides,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_1209_alg__multE_I1_J,axiom,
! [X: a,P: list_a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_1210_le__alg__mult__imp__pdivides,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1211_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_1212_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_1213_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_1214_degree__one__root__condition,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( X
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P ) ) @ ( const_term_a_b @ r @ P ) ) ) ) ) ) ) ).
% degree_one_root_condition
thf(fact_1215_degree__one__root_I1_J,axiom,
! [K3: set_a,P: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( eval_a_b @ r @ P @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P ) ) @ ( const_term_a_b @ r @ P ) ) ) )
= ( zero_a_b @ r ) ) ) ) ) ).
% degree_one_root(1)
thf(fact_1216_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_1217_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_1218_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_1219_field_Odegree__one__root__condition,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,X: set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( polyno4320237611291262604t_unit @ R2 @ P @ X )
= ( X
= ( a_inv_5715216516650856053t_unit @ R2 @ ( mult_s7802724872828879953t_unit @ R2 @ ( m_inv_7863988576679134539t_unit @ R2 @ ( hd_set_list_a @ P ) ) @ ( const_3308765751713425893t_unit @ R2 @ P ) ) ) ) ) ) ) ) ).
% field.degree_one_root_condition
thf(fact_1220_field_Odegree__one__root__condition,axiom,
! [R2: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( field_1861437471013600865t_unit @ R2 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( polyno5142720416380192742t_unit @ R2 @ P @ X )
= ( X
= ( a_inv_7033018035630854991t_unit @ R2 @ ( mult_l4853965630390486993t_unit @ R2 @ ( m_inv_1633185356445945547t_unit @ R2 @ ( hd_list_list_a @ P ) ) @ ( const_6243872422735025855t_unit @ R2 @ P ) ) ) ) ) ) ) ) ).
% field.degree_one_root_condition
thf(fact_1221_field_Odegree__one__root__condition,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( field_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( polyno4133073214067823460ot_a_b @ R2 @ P @ X )
= ( X
= ( a_inv_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ ( m_inv_a_ring_ext_a_b @ R2 @ ( hd_a @ P ) ) @ ( const_term_a_b @ R2 @ P ) ) ) ) ) ) ) ) ).
% field.degree_one_root_condition
thf(fact_1222_field_Odegree__one__root__condition,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( polyno6951661231331188332t_unit @ R2 @ P @ X )
= ( X
= ( a_inv_8944721093294617173t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ ( m_inv_2802811658206063947t_unit @ R2 @ ( hd_list_a @ P ) ) @ ( const_6738166269504826821t_unit @ R2 @ P ) ) ) ) ) ) ) ) ).
% field.degree_one_root_condition
thf(fact_1223_degree__one__imp__singleton__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= ( add_mset_a @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P ) ) @ ( const_term_a_b @ r @ P ) ) ) @ zero_zero_multiset_a ) ) ) ) ).
% degree_one_imp_singleton_roots
thf(fact_1224_roots__inclI,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A5 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A5 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% roots_inclI
thf(fact_1225_x_Osubring__props_I5_J,axiom,
! [K3: set_list_a,H3: list_a] :
( ( subfie1779122896746047282t_unit @ K3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H3 @ K3 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) @ K3 ) ) ) ).
% x.subring_props(5)
thf(fact_1226_x_Oa__transpose__inv,axiom,
! [X: list_a,Y: list_a,Z3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= Z3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Z3 )
= Y ) ) ) ) ) ).
% x.a_transpose_inv
thf(fact_1227_x_Oadd_Oinv__mult__group,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ).
% x.add.inv_mult_group
thf(fact_1228_x_Oadd_Oinv__solve__left,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ) ) ) ) ).
% x.add.inv_solve_left
thf(fact_1229_x_Oadd_Oinv__solve__left_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C )
= A )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ) ) ) ) ).
% x.add.inv_solve_left'
thf(fact_1230_x_Oadd_Oinv__solve__right,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% x.add.inv_solve_right
thf(fact_1231_x_Oadd_Oinv__solve__right_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) )
= A )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% x.add.inv_solve_right'
thf(fact_1232_x_Ominus__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% x.minus_add
thf(fact_1233_x_Or__neg1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= Y ) ) ) ).
% x.r_neg1
thf(fact_1234_x_Or__neg2,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% x.r_neg2
thf(fact_1235_x_Ol__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).
% x.l_minus
thf(fact_1236_x_Or__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).
% x.r_minus
thf(fact_1237_long__division__a__inv_I2_J,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1238_x_Ominus__eq,axiom,
! [X: list_a,Y: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ).
% x.minus_eq
thf(fact_1239_long__division__a__inv_I1_J,axiom,
! [K3: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1240_x_Ol__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_neg
thf(fact_1241_x_Ominus__equality,axiom,
! [Y: list_a,X: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% x.minus_equality
thf(fact_1242_x_Or__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_neg
thf(fact_1243_x_Osum__zero__eq__neg,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% x.sum_zero_eq_neg
thf(fact_1244_x_Ocomm__inv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% x.comm_inv_char
thf(fact_1245_x_Oinv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ) ).
% x.inv_char
thf(fact_1246_x_Oinv__unique_H,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ) ).
% x.inv_unique'
thf(fact_1247_x_Oring_Oa__inv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.a_inv_closed
thf(fact_1248_pdivides__imp__roots__incl,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_1249_x_Omonic__degree__one__root__condition,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ).
% x.monic_degree_one_root_condition
thf(fact_1250_monic__degree__one__roots,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polynomial_roots_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) )
= ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ).
% monic_degree_one_roots
thf(fact_1251_degree__one__roots,axiom,
! [A: a,A6: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A6 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( polynomial_roots_a_b @ r @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) )
= ( add_mset_a @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A6 @ B ) ) @ zero_zero_multiset_a ) ) ) ) ) ) ).
% degree_one_roots
thf(fact_1252_poly__mult__degree__one__monic__imp__same__roots,axiom,
! [A: a,P: list_a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ P ) )
= ( add_mset_a @ A @ ( polynomial_roots_a_b @ r @ P ) ) ) ) ) ) ).
% poly_mult_degree_one_monic_imp_same_roots
thf(fact_1253_x_Ominus__minus,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= X ) ) ).
% x.minus_minus
thf(fact_1254_x_Oadd_Oinv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inv_closed
thf(fact_1255_x_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.minus_zero
thf(fact_1256_x_Oinv__neg__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.inv_neg_one
thf(fact_1257_x_Oinv__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.inv_one
thf(fact_1258_x_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.inv_eq_1_iff
thf(fact_1259_x_Oring_Ohom__a__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ x )
= ( a_inv_a_b @ r @ ( eval_a_b @ r @ X @ x ) ) ) ) ).
% x.ring.hom_a_inv
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ p ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) ) @ one_one_nat ) ) @ ( plus_plus_nat @ ( finite_card_a @ s ) @ zero_zero_nat ) ).
%------------------------------------------------------------------------------