TPTP Problem File: SLH0906^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00047_001603__17124946_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1533 ( 483 unt; 258 typ; 0 def)
% Number of atoms : 3954 (1594 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 16054 ( 353 ~; 69 |; 143 &;13380 @)
% ( 0 <=>;2109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 536 ( 536 >; 0 *; 0 +; 0 <<)
% Number of symbols : 228 ( 225 usr; 16 con; 0-4 aty)
% Number of variables : 3456 ( 60 ^;3302 !; 94 ?;3456 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:36:30.212
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
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% Explicit typings (225)
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thf(sy_c_Polynomials_Oring_Opoly__mult_001tf__a_001tf__b,type,
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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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var_a_b: partia2175431115845679010xt_a_b > 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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thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_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 ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
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zero_l7621212060072393831t_unit: partia4556295656693239580t_unit > list_set_list_a ).
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zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
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zero_s2920163772466840039t_unit: partia4960592913263135132t_unit > set_list_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_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
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ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_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 ).
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 ).
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_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
collec5381118732811369429list_a: ( list_set_list_a > $o ) > set_list_set_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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
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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 ).
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subrin3541368690557094692t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
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subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
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subrin5643252653130547402t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_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_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
member352051402189872281list_a: list_list_set_list_a > set_li664707282716828624list_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_It__List__Olist_Itf__a_J_J_J_J,type,
member6124916891863447321list_a: list_set_list_list_a > set_li7845362039408639568list_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_It__List__Olist_Itf__a_J_J_J,type,
member334759470184282131list_a: set_list_list_a > set_set_list_list_a > $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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_x,type,
x: list_a ).
thf(sy_v_y,type,
y: list_a ).
% Relevant facts (1271)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_assms_I2_J,axiom,
member_list_a @ y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% assms(2)
thf(fact_2_assms_I1_J,axiom,
member_list_a @ x @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% assms(1)
thf(fact_3_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_4_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_5_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_6_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_7_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_8_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_9_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_10_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_11_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_12_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_13_add__diff__cancel__left,axiom,
! [C: multiset_list_a,A: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ C @ A ) @ ( plus_p690419498615200257list_a @ C @ B ) )
= ( minus_7431248565939055793list_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_14_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_15_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_16_add__right__cancel,axiom,
! [B: multiset_list_a,A: multiset_list_a,C: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ B @ A )
= ( plus_p690419498615200257list_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_17_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_18_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_19_add__left__cancel,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ A @ B )
= ( plus_p690419498615200257list_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_20_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_21_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_22_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_23_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_24_add__diff__cancel__right_H,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_25_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_26_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_27_add__diff__cancel__right,axiom,
! [A: multiset_list_a,C: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ A @ C ) @ ( plus_p690419498615200257list_a @ B @ C ) )
= ( minus_7431248565939055793list_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_28_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_29_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_30_add__diff__cancel__left_H,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_31_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_32_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_33_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_34_add__right__imp__eq,axiom,
! [B: multiset_list_a,A: multiset_list_a,C: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ B @ A )
= ( plus_p690419498615200257list_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_35_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_36_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_37_add__left__imp__eq,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ A @ B )
= ( plus_p690419498615200257list_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_38_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_39_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_40_add_Oleft__commute,axiom,
! [B: multiset_list_a,A: multiset_list_a,C: multiset_list_a] :
( ( plus_p690419498615200257list_a @ B @ ( plus_p690419498615200257list_a @ A @ C ) )
= ( plus_p690419498615200257list_a @ A @ ( plus_p690419498615200257list_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_41_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_42_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A2: multiset_a,B2: multiset_a] : ( plus_plus_multiset_a @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_43_add_Ocommute,axiom,
( plus_p690419498615200257list_a
= ( ^ [A2: multiset_list_a,B2: multiset_list_a] : ( plus_p690419498615200257list_a @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_44_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_45_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_46_add_Oassoc,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( plus_p690419498615200257list_a @ ( plus_p690419498615200257list_a @ A @ B ) @ C )
= ( plus_p690419498615200257list_a @ A @ ( plus_p690419498615200257list_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_47_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_48_group__cancel_Oadd2,axiom,
! [B3: multiset_a,K: multiset_a,B: multiset_a,A: multiset_a] :
( ( B3
= ( plus_plus_multiset_a @ K @ B ) )
=> ( ( plus_plus_multiset_a @ A @ B3 )
= ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_49_group__cancel_Oadd2,axiom,
! [B3: multiset_list_a,K: multiset_list_a,B: multiset_list_a,A: multiset_list_a] :
( ( B3
= ( plus_p690419498615200257list_a @ K @ B ) )
=> ( ( plus_p690419498615200257list_a @ A @ B3 )
= ( plus_p690419498615200257list_a @ K @ ( plus_p690419498615200257list_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_50_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_51_group__cancel_Oadd1,axiom,
! [A3: multiset_a,K: multiset_a,A: multiset_a,B: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ K @ A ) )
=> ( ( plus_plus_multiset_a @ A3 @ B )
= ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_52_group__cancel_Oadd1,axiom,
! [A3: multiset_list_a,K: multiset_list_a,A: multiset_list_a,B: multiset_list_a] :
( ( A3
= ( plus_p690419498615200257list_a @ K @ A ) )
=> ( ( plus_p690419498615200257list_a @ A3 @ B )
= ( plus_p690419498615200257list_a @ K @ ( plus_p690419498615200257list_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_53_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_54_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_55_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_56_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( plus_p690419498615200257list_a @ ( plus_p690419498615200257list_a @ A @ B ) @ C )
= ( plus_p690419498615200257list_a @ A @ ( plus_p690419498615200257list_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_57_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_58_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A @ C ) @ B )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_59_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multiset_list_a,C: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ A @ C ) @ B )
= ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_60_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_61_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_62_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_63_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_64_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_65_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_66_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_67_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_68_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_69_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_70_mem__Collect__eq,axiom,
! [A: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: list_set_list_a,P: list_set_list_a > $o] :
( ( member5524387281408368019list_a @ A @ ( collec5381118732811369429list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: set_list_a,P: set_list_a > $o] :
( ( member_set_list_a @ A @ ( collect_set_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A3: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A3: set_list_set_list_a] :
( ( collec5381118732811369429list_a
@ ^ [X3: list_set_list_a] : ( member5524387281408368019list_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A3: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_80_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_81_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_82_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_83_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_84_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_85_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_86_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_87_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_88_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_89_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_90_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_91_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_92_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_93_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_94_diff__diff__eq,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ A @ B ) @ C )
= ( minus_7431248565939055793list_a @ A @ ( plus_p690419498615200257list_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_95_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_96_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_97_add__implies__diff,axiom,
! [C: multiset_list_a,B: multiset_list_a,A: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ C @ B )
= A )
=> ( C
= ( minus_7431248565939055793list_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_98_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_99_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_100_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_101_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_102_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_103_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_104_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_105_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_106_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_107_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_108_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_109_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_110_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_111_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_112_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_113_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_114_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_115_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_116_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_117_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_118_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_119_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_120_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_131_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_132_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_133_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_134_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_135_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_136_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_137_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_138_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( 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 ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_139_poly__of__const__in__carrier,axiom,
! [S: a] :
( ( member_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_140_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_141_subalgebra__in__carrier,axiom,
! [K3: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_142_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_143_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_144_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_145_subset__Idl__subset,axiom,
! [I2: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_146_genideal__self,axiom,
! [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S2 @ ( genideal_a_b @ r @ S2 ) ) ) ).
% genideal_self
thf(fact_147_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_148_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_149_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_150_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_151_degree__zero__imp__not__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_152_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,Q2: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno5142720416380192742t_unit @ R @ P2 @ X )
| ( polyno5142720416380192742t_unit @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_153_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia4556295656693239580t_unit,P2: list_list_set_list_a,Q2: list_list_set_list_a,X: list_set_list_a] :
( ( domain2898972329295444579t_unit @ R )
=> ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( member352051402189872281list_a @ Q2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( polyno5576431768617223174t_unit @ R @ ( mult_l7081732281159775569t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno5576431768617223174t_unit @ R @ P2 @ X )
| ( polyno5576431768617223174t_unit @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_154_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,Q2: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member5524387281408368019list_a @ Q2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno4320237611291262604t_unit @ R @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4320237611291262604t_unit @ R @ P2 @ X )
| ( polyno4320237611291262604t_unit @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_155_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
| ( polyno6951661231331188332t_unit @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_156_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( 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 ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_157_splitted__on__def,axiom,
! [K3: set_a,P2: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K3 @ P2 )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K3 @ P2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_158_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_159_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_160_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_161_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_162_a__lcos__m__assoc,axiom,
! [M3: set_a,G: a,H2: a] :
( ( ord_less_eq_set_a @ M3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M3 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M3 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_163_line__extension__in__carrier,axiom,
! [K3: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_164_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_165_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_166_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_167_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_168_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_169_local_Ominus__unique,axiom,
! [Y: a,X: a,Y4: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_170_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_171_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_172_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_173_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_174_eval__in__carrier,axiom,
! [P2: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_175_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_176_add__0,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ zero_z4454100511807792257list_a @ A )
= A ) ).
% add_0
thf(fact_177_add__0,axiom,
! [A: multiset_set_list_a] :
( ( plus_p3298285420967332321list_a @ zero_z7061913751530476641list_a @ A )
= A ) ).
% add_0
thf(fact_178_add__0,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
= A ) ).
% add_0
thf(fact_179_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_180_add__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% add_0
thf(fact_181_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_182_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_183_add__cancel__right__right,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( A
= ( plus_p690419498615200257list_a @ A @ B ) )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_right_right
thf(fact_184_add__cancel__right__right,axiom,
! [A: multiset_set_list_a,B: multiset_set_list_a] :
( ( A
= ( plus_p3298285420967332321list_a @ A @ B ) )
= ( B = zero_z7061913751530476641list_a ) ) ).
% add_cancel_right_right
thf(fact_185_add__cancel__right__right,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ A @ B ) )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_right
thf(fact_186_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_187_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_188_add__cancel__right__left,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( A
= ( plus_p690419498615200257list_a @ B @ A ) )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_right_left
thf(fact_189_add__cancel__right__left,axiom,
! [A: multiset_set_list_a,B: multiset_set_list_a] :
( ( A
= ( plus_p3298285420967332321list_a @ B @ A ) )
= ( B = zero_z7061913751530476641list_a ) ) ).
% add_cancel_right_left
thf(fact_190_add__cancel__right__left,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ B @ A ) )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_left
thf(fact_191_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_192_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_193_add__cancel__left__right,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ A @ B )
= A )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_left_right
thf(fact_194_add__cancel__left__right,axiom,
! [A: multiset_set_list_a,B: multiset_set_list_a] :
( ( ( plus_p3298285420967332321list_a @ A @ B )
= A )
= ( B = zero_z7061913751530476641list_a ) ) ).
% add_cancel_left_right
thf(fact_195_add__cancel__left__right,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B )
= A )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_right
thf(fact_196_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_197_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_198_add__cancel__left__left,axiom,
! [B: multiset_list_a,A: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ B @ A )
= A )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_left_left
thf(fact_199_add__cancel__left__left,axiom,
! [B: multiset_set_list_a,A: multiset_set_list_a] :
( ( ( plus_p3298285420967332321list_a @ B @ A )
= A )
= ( B = zero_z7061913751530476641list_a ) ) ).
% add_cancel_left_left
thf(fact_200_add__cancel__left__left,axiom,
! [B: multiset_nat,A: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B @ A )
= A )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_left
thf(fact_201_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_202_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_203_add_Oright__neutral,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% add.right_neutral
thf(fact_204_add_Oright__neutral,axiom,
! [A: multiset_set_list_a] :
( ( plus_p3298285420967332321list_a @ A @ zero_z7061913751530476641list_a )
= A ) ).
% add.right_neutral
thf(fact_205_add_Oright__neutral,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% add.right_neutral
thf(fact_206_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_207_add_Oright__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.right_neutral
thf(fact_208_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_set_list_a] :
( ( minus_1362096637801929873list_a @ A @ A )
= zero_z7061913751530476641list_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_209_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ A )
= zero_z7348594199698428585et_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_210_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ A )
= zero_z4454100511807792257list_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_211_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_212_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_213_diff__zero,axiom,
! [A: multiset_set_list_a] :
( ( minus_1362096637801929873list_a @ A @ zero_z7061913751530476641list_a )
= A ) ).
% diff_zero
thf(fact_214_diff__zero,axiom,
! [A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% diff_zero
thf(fact_215_diff__zero,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% diff_zero
thf(fact_216_diff__zero,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
= A ) ).
% diff_zero
thf(fact_217_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_218_zero__diff,axiom,
! [A: multiset_set_list_a] :
( ( minus_1362096637801929873list_a @ zero_z7061913751530476641list_a @ A )
= zero_z7061913751530476641list_a ) ).
% zero_diff
thf(fact_219_zero__diff,axiom,
! [A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ A )
= zero_z7348594199698428585et_nat ) ).
% zero_diff
thf(fact_220_zero__diff,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ A )
= zero_z4454100511807792257list_a ) ).
% zero_diff
thf(fact_221_zero__diff,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_222_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_223_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_224_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_225_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_226_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_227_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_228_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_229_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_230_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_231_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_232_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_233_diff__add__zero,axiom,
! [A: multiset_set_list_a,B: multiset_set_list_a] :
( ( minus_1362096637801929873list_a @ A @ ( plus_p3298285420967332321list_a @ A @ B ) )
= zero_z7061913751530476641list_a ) ).
% diff_add_zero
thf(fact_234_diff__add__zero,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B ) )
= zero_z7348594199698428585et_nat ) ).
% diff_add_zero
thf(fact_235_diff__add__zero,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ ( plus_p690419498615200257list_a @ A @ B ) )
= zero_z4454100511807792257list_a ) ).
% diff_add_zero
thf(fact_236_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_237_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_238_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_239_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_240_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_241_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_242_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_243_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_244_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_245_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_246_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_247_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_248_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_249_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).
% ee_refl
thf(fact_250_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_251_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_252_zero__reorient,axiom,
! [X: multiset_list_a] :
( ( zero_z4454100511807792257list_a = X )
= ( X = zero_z4454100511807792257list_a ) ) ).
% zero_reorient
thf(fact_253_zero__reorient,axiom,
! [X: multiset_set_list_a] :
( ( zero_z7061913751530476641list_a = X )
= ( X = zero_z7061913751530476641list_a ) ) ).
% zero_reorient
thf(fact_254_zero__reorient,axiom,
! [X: multiset_nat] :
( ( zero_z7348594199698428585et_nat = X )
= ( X = zero_z7348594199698428585et_nat ) ) ).
% zero_reorient
thf(fact_255_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_256_zero__reorient,axiom,
! [X: multiset_a] :
( ( zero_zero_multiset_a = X )
= ( X = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_257_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_258_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_259_add_Ocomm__neutral,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% add.comm_neutral
thf(fact_260_add_Ocomm__neutral,axiom,
! [A: multiset_set_list_a] :
( ( plus_p3298285420967332321list_a @ A @ zero_z7061913751530476641list_a )
= A ) ).
% add.comm_neutral
thf(fact_261_add_Ocomm__neutral,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% add.comm_neutral
thf(fact_262_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_263_add_Ocomm__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.comm_neutral
thf(fact_264_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ zero_z4454100511807792257list_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_265_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_set_list_a] :
( ( plus_p3298285420967332321list_a @ zero_z7061913751530476641list_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_266_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_267_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_268_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_269_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_270_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_271_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_272_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_273_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_274_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_275_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_276_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_277_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_290_ring_Ois__root_Ocong,axiom,
polyno4320237611291262604t_unit = polyno4320237611291262604t_unit ).
% ring.is_root.cong
thf(fact_291_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_292_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno5142720416380192742t_unit @ R @ P2 @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_293_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia4556295656693239580t_unit,P2: list_list_set_list_a,X: list_set_list_a] :
( ( domain2898972329295444579t_unit @ R )
=> ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno5576431768617223174t_unit @ R @ P2 @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_294_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4320237611291262604t_unit @ R @ P2 @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_295_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_296_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_297_degree__zero__imp__splitted,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ).
% degree_zero_imp_splitted
thf(fact_298_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_299_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_300_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_301_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_302_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_303_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_304_is__root__def,axiom,
! [P2: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P2 @ X )
= ( zero_a_b @ r ) )
& ( P2 != nil_a ) ) ) ).
% is_root_def
thf(fact_305_pdivides__imp__root__sharing,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P2 @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q2 @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_306_zero__pdivides,axiom,
! [P2: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
= ( P2 = nil_a ) ) ).
% zero_pdivides
thf(fact_307_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_308_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_309_union__eq__empty,axiom,
! [M3: multiset_list_a,N4: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ M3 @ N4 )
= zero_z4454100511807792257list_a )
= ( ( M3 = zero_z4454100511807792257list_a )
& ( N4 = zero_z4454100511807792257list_a ) ) ) ).
% union_eq_empty
thf(fact_310_union__eq__empty,axiom,
! [M3: multiset_set_list_a,N4: multiset_set_list_a] :
( ( ( plus_p3298285420967332321list_a @ M3 @ N4 )
= zero_z7061913751530476641list_a )
= ( ( M3 = zero_z7061913751530476641list_a )
& ( N4 = zero_z7061913751530476641list_a ) ) ) ).
% union_eq_empty
thf(fact_311_union__eq__empty,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ M3 @ N4 )
= zero_z7348594199698428585et_nat )
= ( ( M3 = zero_z7348594199698428585et_nat )
& ( N4 = zero_z7348594199698428585et_nat ) ) ) ).
% union_eq_empty
thf(fact_312_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_313_empty__eq__union,axiom,
! [M3: multiset_list_a,N4: multiset_list_a] :
( ( zero_z4454100511807792257list_a
= ( plus_p690419498615200257list_a @ M3 @ N4 ) )
= ( ( M3 = zero_z4454100511807792257list_a )
& ( N4 = zero_z4454100511807792257list_a ) ) ) ).
% empty_eq_union
thf(fact_314_empty__eq__union,axiom,
! [M3: multiset_set_list_a,N4: multiset_set_list_a] :
( ( zero_z7061913751530476641list_a
= ( plus_p3298285420967332321list_a @ M3 @ N4 ) )
= ( ( M3 = zero_z7061913751530476641list_a )
& ( N4 = zero_z7061913751530476641list_a ) ) ) ).
% empty_eq_union
thf(fact_315_empty__eq__union,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( zero_z7348594199698428585et_nat
= ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( ( M3 = zero_z7348594199698428585et_nat )
& ( N4 = zero_z7348594199698428585et_nat ) ) ) ).
% empty_eq_union
thf(fact_316_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_317_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( zero_z4454100511807792257list_a
= ( plus_p690419498615200257list_a @ X @ Y ) )
= ( ( X = zero_z4454100511807792257list_a )
& ( Y = zero_z4454100511807792257list_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_318_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_set_list_a,Y: multiset_set_list_a] :
( ( zero_z7061913751530476641list_a
= ( plus_p3298285420967332321list_a @ X @ Y ) )
= ( ( X = zero_z7061913751530476641list_a )
& ( Y = zero_z7061913751530476641list_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_319_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_nat,Y: multiset_nat] :
( ( zero_z7348594199698428585et_nat
= ( plus_p6334493942879108393et_nat @ X @ Y ) )
= ( ( X = zero_z7348594199698428585et_nat )
& ( Y = zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_320_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_321_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ X @ Y )
= zero_z4454100511807792257list_a )
= ( ( X = zero_z4454100511807792257list_a )
& ( Y = zero_z4454100511807792257list_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_322_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_set_list_a,Y: multiset_set_list_a] :
( ( ( plus_p3298285420967332321list_a @ X @ Y )
= zero_z7061913751530476641list_a )
= ( ( X = zero_z7061913751530476641list_a )
& ( Y = zero_z7061913751530476641list_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_323_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_nat,Y: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ X @ Y )
= zero_z7348594199698428585et_nat )
= ( ( X = zero_z7348594199698428585et_nat )
& ( Y = zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_324_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_325_diff__diff__add__mset,axiom,
! [M3: multiset_a,N4: multiset_a,P: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ P )
= ( minus_3765977307040488491iset_a @ M3 @ ( plus_plus_multiset_a @ N4 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_326_diff__diff__add__mset,axiom,
! [M3: multiset_list_a,N4: multiset_list_a,P: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ M3 @ N4 ) @ P )
= ( minus_7431248565939055793list_a @ M3 @ ( plus_p690419498615200257list_a @ N4 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_327_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_328_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_329_size__empty,axiom,
( ( size_s2335926164413107382list_a @ zero_z4454100511807792257list_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_330_size__empty,axiom,
( ( size_s1226348209404258454list_a @ zero_z7061913751530476641list_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_331_size__empty,axiom,
( ( size_s5917832649809541300et_nat @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% size_empty
thf(fact_332_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_333_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_334_size__eq__0__iff__empty,axiom,
! [M3: multiset_set_list_a] :
( ( ( size_s1226348209404258454list_a @ M3 )
= zero_zero_nat )
= ( M3 = zero_z7061913751530476641list_a ) ) ).
% size_eq_0_iff_empty
thf(fact_335_size__eq__0__iff__empty,axiom,
! [M3: multiset_nat] :
( ( ( size_s5917832649809541300et_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_z7348594199698428585et_nat ) ) ).
% size_eq_0_iff_empty
thf(fact_336_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_337_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_338_ring_Osplitted_Ocong,axiom,
polyno6259083269128200473t_unit = polyno6259083269128200473t_unit ).
% ring.splitted.cong
thf(fact_339_ring_Osplitted_Ocong,axiom,
polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).
% ring.splitted.cong
thf(fact_340_empty__neutral_I1_J,axiom,
! [X: multiset_list_a] :
( ( plus_p690419498615200257list_a @ zero_z4454100511807792257list_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_341_empty__neutral_I1_J,axiom,
! [X: multiset_set_list_a] :
( ( plus_p3298285420967332321list_a @ zero_z7061913751530476641list_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_342_empty__neutral_I1_J,axiom,
! [X: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ X )
= X ) ).
% empty_neutral(1)
thf(fact_343_empty__neutral_I1_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_344_empty__neutral_I2_J,axiom,
! [X: multiset_list_a] :
( ( plus_p690419498615200257list_a @ X @ zero_z4454100511807792257list_a )
= X ) ).
% empty_neutral(2)
thf(fact_345_empty__neutral_I2_J,axiom,
! [X: multiset_set_list_a] :
( ( plus_p3298285420967332321list_a @ X @ zero_z7061913751530476641list_a )
= X ) ).
% empty_neutral(2)
thf(fact_346_empty__neutral_I2_J,axiom,
! [X: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ X @ zero_z7348594199698428585et_nat )
= X ) ).
% empty_neutral(2)
thf(fact_347_empty__neutral_I2_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
= X ) ).
% empty_neutral(2)
thf(fact_348_Multiset_Odiff__add,axiom,
! [M3: multiset_a,N4: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ M3 @ ( plus_plus_multiset_a @ N4 @ Q ) )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_349_Multiset_Odiff__add,axiom,
! [M3: multiset_list_a,N4: multiset_list_a,Q: multiset_list_a] :
( ( minus_7431248565939055793list_a @ M3 @ ( plus_p690419498615200257list_a @ N4 @ Q ) )
= ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ M3 @ N4 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_350_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_351_union__assoc,axiom,
! [M3: multiset_list_a,N4: multiset_list_a,K3: multiset_list_a] :
( ( plus_p690419498615200257list_a @ ( plus_p690419498615200257list_a @ M3 @ N4 ) @ K3 )
= ( plus_p690419498615200257list_a @ M3 @ ( plus_p690419498615200257list_a @ N4 @ K3 ) ) ) ).
% union_assoc
thf(fact_352_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_353_union__lcomm,axiom,
! [M3: multiset_list_a,N4: multiset_list_a,K3: multiset_list_a] :
( ( plus_p690419498615200257list_a @ M3 @ ( plus_p690419498615200257list_a @ N4 @ K3 ) )
= ( plus_p690419498615200257list_a @ N4 @ ( plus_p690419498615200257list_a @ M3 @ K3 ) ) ) ).
% union_lcomm
thf(fact_354_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M4: multiset_a,N5: multiset_a] : ( plus_plus_multiset_a @ N5 @ M4 ) ) ) ).
% union_commute
thf(fact_355_union__commute,axiom,
( plus_p690419498615200257list_a
= ( ^ [M4: multiset_list_a,N5: multiset_list_a] : ( plus_p690419498615200257list_a @ N5 @ M4 ) ) ) ).
% union_commute
thf(fact_356_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_357_union__left__cancel,axiom,
! [K3: multiset_list_a,M3: multiset_list_a,N4: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ K3 @ M3 )
= ( plus_p690419498615200257list_a @ K3 @ N4 ) )
= ( M3 = N4 ) ) ).
% union_left_cancel
thf(fact_358_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_359_diff__union__cancelL,axiom,
! [N4: multiset_list_a,M3: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ N4 @ M3 ) @ N4 )
= M3 ) ).
% diff_union_cancelL
thf(fact_360_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_361_diff__union__cancelR,axiom,
! [M3: multiset_list_a,N4: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ M3 @ N4 ) @ N4 )
= M3 ) ).
% diff_union_cancelR
thf(fact_362_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_363_union__right__cancel,axiom,
! [M3: multiset_list_a,K3: multiset_list_a,N4: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ M3 @ K3 )
= ( plus_p690419498615200257list_a @ N4 @ K3 ) )
= ( M3 = N4 ) ) ).
% union_right_cancel
thf(fact_364_multi__union__self__other__eq,axiom,
! [A3: multiset_a,X4: multiset_a,Y5: multiset_a] :
( ( ( plus_plus_multiset_a @ A3 @ X4 )
= ( plus_plus_multiset_a @ A3 @ Y5 ) )
=> ( X4 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_365_multi__union__self__other__eq,axiom,
! [A3: multiset_list_a,X4: multiset_list_a,Y5: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ A3 @ X4 )
= ( plus_p690419498615200257list_a @ A3 @ Y5 ) )
=> ( X4 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_366_domain_Ozero__pdivides,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( polyno4453881341673752516t_unit @ R @ nil_list_list_a @ P2 )
= ( P2 = nil_list_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_367_domain_Ozero__pdivides,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( polyno9075941895896075626t_unit @ R @ nil_set_list_a @ P2 )
= ( P2 = nil_set_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_368_domain_Ozero__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( polyno8016796738000020810t_unit @ R @ nil_list_a @ P2 )
= ( P2 = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_369_domain_Ozero__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( polyno5814909790663948098es_a_b @ R @ nil_a @ P2 )
= ( P2 = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_370_domain_Ozero__pdivides__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( polyno4453881341673752516t_unit @ R @ nil_list_list_a @ nil_list_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_371_domain_Ozero__pdivides__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( polyno9075941895896075626t_unit @ R @ nil_set_list_a @ nil_set_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_372_domain_Ozero__pdivides__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( polyno8016796738000020810t_unit @ R @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_373_domain_Ozero__pdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( polyno5814909790663948098es_a_b @ R @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_374_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_375_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_376_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a,Q2: list_set_list_list_a,A: set_list_list_a] :
( ( domain7421296078544666595t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( polyno3637028486239637860t_unit @ R @ P2 @ Q2 )
=> ( ( member334759470184282131list_a @ A @ ( partia3317168157747563407t_unit @ R ) )
=> ( ( ( eval_s2941910765802206162t_unit @ R @ P2 @ A )
= ( zero_s2920163772466840039t_unit @ R ) )
=> ( ( eval_s2941910765802206162t_unit @ R @ Q2 @ A )
= ( zero_s2920163772466840039t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_377_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,Q2: list_list_list_a,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno4453881341673752516t_unit @ R @ P2 @ Q2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( eval_l1088911609197519410t_unit @ R @ P2 @ A )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ Q2 @ A )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_378_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia4556295656693239580t_unit,P2: list_list_set_list_a,Q2: list_list_set_list_a,A: list_set_list_a] :
( ( domain2898972329295444579t_unit @ R )
=> ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( polyno8338076773845191652t_unit @ R @ P2 @ Q2 )
=> ( ( member5524387281408368019list_a @ A @ ( partia7265347635606999311t_unit @ R ) )
=> ( ( ( eval_l7642959053407759954t_unit @ R @ P2 @ A )
= ( zero_l7621212060072393831t_unit @ R ) )
=> ( ( eval_l7642959053407759954t_unit @ R @ Q2 @ A )
= ( zero_l7621212060072393831t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_379_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,Q2: list_set_list_a,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno9075941895896075626t_unit @ R @ P2 @ Q2 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( eval_s5133945360527818456t_unit @ R @ P2 @ A )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( eval_s5133945360527818456t_unit @ R @ Q2 @ A )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_380_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( eval_l34571156754992824t_unit @ R @ P2 @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ Q2 @ A )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_381_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,A: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( eval_a_b @ R @ P2 @ A )
= ( zero_a_b @ R ) )
=> ( ( eval_a_b @ R @ Q2 @ A )
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_382_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_383_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_384_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno5970451904377802771t_unit @ R @ P2 ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_385_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia4556295656693239580t_unit,P2: list_list_set_list_a] :
( ( domain2898972329295444579t_unit @ R )
=> ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8445875935890348083t_unit @ R @ P2 ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_386_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno7858167711734664505t_unit @ R @ P2 ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_387_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno6259083269128200473t_unit @ R @ P2 ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_388_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ R @ P2 ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_389_splitted__def,axiom,
! [P2: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P2 )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_390_degree__zero__imp__empty__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_391_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_392_univ__poly__zero__closed,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a] : ( member5524387281408368019list_a @ nil_set_list_a @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_393_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ).
% univ_poly_zero_closed
thf(fact_394_length__0__conv,axiom,
! [Xs: list_set_list_a] :
( ( ( size_s1991367317912710102list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_list_a ) ) ).
% length_0_conv
thf(fact_395_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_396_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_397_degree__oneE,axiom,
! [P2: list_a,K3: set_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ K3 )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ! [B4: a] :
( ( member_a @ B4 @ K3 )
=> ( P2
!= ( cons_a @ A4 @ ( cons_a @ B4 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_398_pdivides__imp__degree__le,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_399_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_400_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_401_univ__poly__is__domain,axiom,
! [K3: set_a] :
( ( subring_a_b @ K3 @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ).
% univ_poly_is_domain
thf(fact_402_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_403_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_404_list_Oinject,axiom,
! [X21: set_list_a,X22: list_set_list_a,Y21: set_list_a,Y22: list_set_list_a] :
( ( ( cons_set_list_a @ X21 @ X22 )
= ( cons_set_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_405_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_406_var__closed_I1_J,axiom,
! [K3: set_a] :
( ( subring_a_b @ K3 @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ).
% var_closed(1)
thf(fact_407_pdivides__zero,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_408_carrier__polynomial__shell,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_409_Multiset_Odiff__cancel,axiom,
! [A3: multiset_set_list_a] :
( ( minus_1362096637801929873list_a @ A3 @ A3 )
= zero_z7061913751530476641list_a ) ).
% Multiset.diff_cancel
thf(fact_410_Multiset_Odiff__cancel,axiom,
! [A3: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A3 @ A3 )
= zero_z7348594199698428585et_nat ) ).
% Multiset.diff_cancel
thf(fact_411_Multiset_Odiff__cancel,axiom,
! [A3: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A3 @ A3 )
= zero_z4454100511807792257list_a ) ).
% Multiset.diff_cancel
thf(fact_412_Multiset_Odiff__cancel,axiom,
! [A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ A3 @ A3 )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_413_diff__empty,axiom,
! [M3: multiset_set_list_a] :
( ( ( minus_1362096637801929873list_a @ M3 @ zero_z7061913751530476641list_a )
= M3 )
& ( ( minus_1362096637801929873list_a @ zero_z7061913751530476641list_a @ M3 )
= zero_z7061913751530476641list_a ) ) ).
% diff_empty
thf(fact_414_diff__empty,axiom,
! [M3: multiset_nat] :
( ( ( minus_8522176038001411705et_nat @ M3 @ zero_z7348594199698428585et_nat )
= M3 )
& ( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ M3 )
= zero_z7348594199698428585et_nat ) ) ).
% diff_empty
thf(fact_415_diff__empty,axiom,
! [M3: multiset_list_a] :
( ( ( minus_7431248565939055793list_a @ M3 @ zero_z4454100511807792257list_a )
= M3 )
& ( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ M3 )
= zero_z4454100511807792257list_a ) ) ).
% diff_empty
thf(fact_416_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_417_Multiset_Odiff__right__commute,axiom,
! [M3: multiset_a,N4: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ Q )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ Q ) @ N4 ) ) ).
% Multiset.diff_right_commute
thf(fact_418_Multiset_Odiff__right__commute,axiom,
! [M3: multiset_list_a,N4: multiset_list_a,Q: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ M3 @ N4 ) @ Q )
= ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ M3 @ Q ) @ N4 ) ) ).
% Multiset.diff_right_commute
thf(fact_419_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_420_list__nonempty__induct,axiom,
! [Xs: list_set_list_a,P: list_set_list_a > $o] :
( ( Xs != nil_set_list_a )
=> ( ! [X2: set_list_a] : ( P @ ( cons_set_list_a @ X2 @ nil_set_list_a ) )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a] :
( ( Xs2 != nil_set_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_421_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_422_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_423_list__induct2_H,axiom,
! [P: list_a > list_set_list_a > $o,Xs: list_a,Ys: list_set_list_a] :
( ( P @ nil_a @ nil_set_list_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_set_list_a )
=> ( ! [Y2: set_list_a,Ys2: list_set_list_a] : ( P @ nil_a @ ( cons_set_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: set_list_a,Ys2: list_set_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_424_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_list_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_425_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_426_list__induct2_H,axiom,
! [P: list_list_a > list_set_list_a > $o,Xs: list_list_a,Ys: list_set_list_a] :
( ( P @ nil_list_a @ nil_set_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_set_list_a )
=> ( ! [Y2: set_list_a,Ys2: list_set_list_a] : ( P @ nil_list_a @ ( cons_set_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: set_list_a,Ys2: list_set_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_set_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_427_list__induct2_H,axiom,
! [P: list_set_list_a > list_a > $o,Xs: list_set_list_a,Ys: list_a] :
( ( P @ nil_set_list_a @ nil_a )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a] : ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_set_list_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_428_list__induct2_H,axiom,
! [P: list_set_list_a > list_list_a > $o,Xs: list_set_list_a,Ys: list_list_a] :
( ( P @ nil_set_list_a @ nil_list_a )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a] : ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_set_list_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_429_list__induct2_H,axiom,
! [P: list_set_list_a > list_set_list_a > $o,Xs: list_set_list_a,Ys: list_set_list_a] :
( ( P @ nil_set_list_a @ nil_set_list_a )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a] : ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ nil_set_list_a )
=> ( ! [Y2: set_list_a,Ys2: list_set_list_a] : ( P @ nil_set_list_a @ ( cons_set_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a,Y2: set_list_a,Ys2: list_set_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ ( cons_set_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_430_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_431_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y6: list_a,Ys3: list_list_a] :
( Xs
= ( cons_list_a @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_432_neq__Nil__conv,axiom,
! [Xs: list_set_list_a] :
( ( Xs != nil_set_list_a )
= ( ? [Y6: set_list_a,Ys3: list_set_list_a] :
( Xs
= ( cons_set_list_a @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_433_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y6: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_434_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X2: list_a] :
( X
!= ( cons_list_a @ X2 @ nil_list_a ) )
=> ~ ! [X2: list_a,Y2: list_a,Xs2: list_list_a] :
( X
!= ( cons_list_a @ X2 @ ( cons_list_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_435_remdups__adj_Ocases,axiom,
! [X: list_set_list_a] :
( ( X != nil_set_list_a )
=> ( ! [X2: set_list_a] :
( X
!= ( cons_set_list_a @ X2 @ nil_set_list_a ) )
=> ~ ! [X2: set_list_a,Y2: set_list_a,Xs2: list_set_list_a] :
( X
!= ( cons_set_list_a @ X2 @ ( cons_set_list_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_436_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y2: a,Xs2: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_437_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 ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_438_transpose_Ocases,axiom,
! [X: list_list_set_list_a] :
( ( X != nil_list_set_list_a )
=> ( ! [Xss: list_list_set_list_a] :
( X
!= ( cons_list_set_list_a @ nil_set_list_a @ Xss ) )
=> ~ ! [X2: set_list_a,Xs2: list_set_list_a,Xss: list_list_set_list_a] :
( X
!= ( cons_list_set_list_a @ ( cons_set_list_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_439_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_440_min__list_Ocases,axiom,
! [X: list_set_list_a] :
( ! [X2: set_list_a,Xs2: list_set_list_a] :
( X
!= ( cons_set_list_a @ X2 @ Xs2 ) )
=> ( X = nil_set_list_a ) ) ).
% min_list.cases
thf(fact_441_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_442_list_Oexhaust,axiom,
! [Y: list_set_list_a] :
( ( Y != nil_set_list_a )
=> ~ ! [X212: set_list_a,X222: list_set_list_a] :
( Y
!= ( cons_set_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_443_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_444_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_445_list_OdiscI,axiom,
! [List: list_set_list_a,X21: set_list_a,X22: list_set_list_a] :
( ( List
= ( cons_set_list_a @ X21 @ X22 ) )
=> ( List != nil_set_list_a ) ) ).
% list.discI
thf(fact_446_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_447_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_448_list_Odistinct_I1_J,axiom,
! [X21: set_list_a,X22: list_set_list_a] :
( nil_set_list_a
!= ( cons_set_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_449_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_450_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_451_not__Cons__self2,axiom,
! [X: set_list_a,Xs: list_set_list_a] :
( ( cons_set_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_452_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_453_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_454_set__ConsD,axiom,
! [Y: list_set_list_a,X: list_set_list_a,Xs: list_list_set_list_a] :
( ( member5524387281408368019list_a @ Y @ ( set_list_set_list_a2 @ ( cons_list_set_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member5524387281408368019list_a @ Y @ ( set_list_set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_455_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_456_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_457_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_458_ring_Oroots_Ocong,axiom,
polyno7858422826990252003t_unit = polyno7858422826990252003t_unit ).
% ring.roots.cong
thf(fact_459_ring_Oroots_Ocong,axiom,
polyno4169377219242390531t_unit = polyno4169377219242390531t_unit ).
% ring.roots.cong
thf(fact_460_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_461_list_Oset__cases,axiom,
! [E2: list_list_a,A: list_list_list_a] :
( ( member_list_list_a @ E2 @ ( set_list_list_a2 @ A ) )
=> ( ! [Z2: list_list_list_a] :
( A
!= ( cons_list_list_a @ E2 @ Z2 ) )
=> ~ ! [Z1: list_list_a,Z2: list_list_list_a] :
( ( A
= ( cons_list_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_list_a @ E2 @ ( set_list_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_462_list_Oset__cases,axiom,
! [E2: list_set_list_a,A: list_list_set_list_a] :
( ( member5524387281408368019list_a @ E2 @ ( set_list_set_list_a2 @ A ) )
=> ( ! [Z2: list_list_set_list_a] :
( A
!= ( cons_list_set_list_a @ E2 @ Z2 ) )
=> ~ ! [Z1: list_set_list_a,Z2: list_list_set_list_a] :
( ( A
= ( cons_list_set_list_a @ Z1 @ Z2 ) )
=> ~ ( member5524387281408368019list_a @ E2 @ ( set_list_set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_463_list_Oset__cases,axiom,
! [E2: set_list_a,A: list_set_list_a] :
( ( member_set_list_a @ E2 @ ( set_set_list_a2 @ A ) )
=> ( ! [Z2: list_set_list_a] :
( A
!= ( cons_set_list_a @ E2 @ Z2 ) )
=> ~ ! [Z1: set_list_a,Z2: list_set_list_a] :
( ( A
= ( cons_set_list_a @ Z1 @ Z2 ) )
=> ~ ( member_set_list_a @ E2 @ ( set_set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_464_list_Oset__cases,axiom,
! [E2: list_a,A: list_list_a] :
( ( member_list_a @ E2 @ ( set_list_a2 @ A ) )
=> ( ! [Z2: list_list_a] :
( A
!= ( cons_list_a @ E2 @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E2 @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_465_list_Oset__cases,axiom,
! [E2: a,A: list_a] :
( ( member_a @ E2 @ ( set_a2 @ A ) )
=> ( ! [Z2: list_a] :
( A
!= ( cons_a @ E2 @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E2 @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_466_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_467_list_Oset__intros_I1_J,axiom,
! [X21: list_set_list_a,X22: list_list_set_list_a] : ( member5524387281408368019list_a @ X21 @ ( set_list_set_list_a2 @ ( cons_list_set_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_468_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_469_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_470_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_471_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_472_list_Oset__intros_I2_J,axiom,
! [Y: list_set_list_a,X22: list_list_set_list_a,X21: list_set_list_a] :
( ( member5524387281408368019list_a @ Y @ ( set_list_set_list_a2 @ X22 ) )
=> ( member5524387281408368019list_a @ Y @ ( set_list_set_list_a2 @ ( cons_list_set_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_473_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_474_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_475_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_476_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2956882679547061052t_unit,K3: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K3 @ R )
=> ( domain9211287710782191037t_unit @ ( univ_p2250591967980070728t_unit @ R @ K3 ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_477_domain_Ouniv__poly__is__domain,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( domain2898972329295444579t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_478_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_479_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_480_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_481_set__subset__Cons,axiom,
! [Xs: list_set_list_a,X: set_list_a] : ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ ( set_set_list_a2 @ ( cons_set_list_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_482_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_483_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_484_univ__poly__zero,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a] :
( ( zero_l7621212060072393831t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) )
= nil_set_list_a ) ).
% univ_poly_zero
thf(fact_485_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K3 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_486_list__induct2,axiom,
! [Xs: list_set_list_a,Ys: list_set_list_a,P: list_set_list_a > list_set_list_a > $o] :
( ( ( size_s1991367317912710102list_a @ Xs )
= ( size_s1991367317912710102list_a @ Ys ) )
=> ( ( P @ nil_set_list_a @ nil_set_list_a )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a,Y2: set_list_a,Ys2: list_set_list_a] :
( ( ( size_s1991367317912710102list_a @ Xs2 )
= ( size_s1991367317912710102list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ ( cons_set_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_487_list__induct2,axiom,
! [Xs: list_set_list_a,Ys: list_a,P: list_set_list_a > list_a > $o] :
( ( ( size_s1991367317912710102list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_set_list_a @ nil_a )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a,Y2: a,Ys2: list_a] :
( ( ( size_s1991367317912710102list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_488_list__induct2,axiom,
! [Xs: list_set_list_a,Ys: list_list_a,P: list_set_list_a > list_list_a > $o] :
( ( ( size_s1991367317912710102list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_set_list_a @ nil_list_a )
=> ( ! [X2: set_list_a,Xs2: list_set_list_a,Y2: list_a,Ys2: list_list_a] :
( ( ( size_s1991367317912710102list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_set_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_489_list__induct2,axiom,
! [Xs: list_a,Ys: list_set_list_a,P: list_a > list_set_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s1991367317912710102list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_set_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: set_list_a,Ys2: list_set_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s1991367317912710102list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_490_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_491_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_set_list_a,P: list_list_a > list_set_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s1991367317912710102list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_set_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: set_list_a,Ys2: list_set_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s1991367317912710102list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_set_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_492_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_493_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_494_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_495_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_set_list_a,P: list_list_a > list_list_a > list_set_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s1991367317912710102list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_set_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a,Z3: set_list_a,Zs2: list_set_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s1991367317912710102list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_set_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_496_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a,Z3: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_497_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_498_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_499_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z3: 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 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_500_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_501_domain_Ovar__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_502_domain_Ovar__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( member_list_a @ ( var_a_b @ R ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_503_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_504_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_505_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_506_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_507_domain_Opdivides__zero,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( polyno8016796738000020810t_unit @ R @ P2 @ nil_list_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_508_domain_Opdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( polyno5814909790663948098es_a_b @ R @ P2 @ nil_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_509_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_510_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_511_domain_Opdivides__imp__degree__le,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pdivides_imp_degree_le
thf(fact_512_domain_Opdivides__imp__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pdivides_imp_degree_le
thf(fact_513_domain_Odegree__zero__imp__empty__roots,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polyno7858422826990252003t_unit @ R @ P2 )
= zero_z4454100511807792257list_a ) ) ) ) ).
% domain.degree_zero_imp_empty_roots
thf(fact_514_domain_Odegree__zero__imp__empty__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ R @ P2 )
= zero_zero_multiset_a ) ) ) ) ).
% domain.degree_zero_imp_empty_roots
thf(fact_515_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_516_univ__poly__a__minus__consistent,axiom,
! [K3: set_a,Q2: list_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_517_pirreducible__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_518_poly__mult__var,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( P2 = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P2 != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ ( var_a_b @ r ) )
= ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_519_eval__as__unique__hom,axiom,
! [K3: set_a,X: a,H2: list_a > a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K3 ) @ r @ H2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K3 )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_a_b @ r ) )
= X )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( H2 @ P2 )
= ( eval_a_b @ r @ P2 @ X ) ) ) ) ) ) ) ) ).
% eval_as_unique_hom
thf(fact_520_pdivides__imp__roots__incl,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_521_pirreducibleE_I1_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_522_determination__of__hom,axiom,
! [K3: set_a,A3: partia2175431115845679010xt_a_b,H2: list_a > a,G: list_a > a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K3 ) @ A3 @ H2 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K3 ) @ A3 @ G )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K3 )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= ( G @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H2 @ ( var_a_b @ r ) )
= ( G @ ( var_a_b @ r ) ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( H2 @ P2 )
= ( G @ P2 ) ) ) ) ) ) ) ) ).
% determination_of_hom
thf(fact_523_subset__mset_Oorder__refl,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_524_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_525_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_526_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_527_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_528_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_529_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_530_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_531_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_532_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_533_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_534_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_535_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_536_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_537_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_538_subset__mset_Oextremum__unique,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
= ( A = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_unique
thf(fact_539_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_a] :
( ( subseteq_mset_a @ N @ zero_zero_multiset_a )
= ( N = zero_zero_multiset_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_540_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_541_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_542_mset__subset__eq__mono__add__left__cancel,axiom,
! [C4: multiset_a,A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C4 @ A3 ) @ ( plus_plus_multiset_a @ C4 @ B3 ) )
= ( subseteq_mset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_543_mset__subset__eq__mono__add__right__cancel,axiom,
! [A3: multiset_a,C4: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C4 ) @ ( plus_plus_multiset_a @ B3 @ C4 ) )
= ( subseteq_mset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_544_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_545_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_546_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ B @ A ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_547_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_548_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
= ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_549_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B @ A ) @ B )
= ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_550_subset__mset_Oadd__diff__assoc,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_551_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A ) @ C )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_552_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B3: multiset_a,A3: multiset_a,C4: multiset_a] :
( ( subseteq_mset_a @ B3 @ A3 )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ B3 ) @ C4 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ C4 ) @ B3 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_553_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_554_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_555_subset__mset_Otrans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_556_subset__mset_Oeq__iff,axiom,
( ( ^ [Y7: multiset_a,Z4: multiset_a] : ( Y7 = Z4 ) )
= ( ^ [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
& ( subseteq_mset_a @ B2 @ A2 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_557_subset__mset_Oantisym,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_558_subset__mset_Oeq__refl,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( X = Y )
=> ( subseteq_mset_a @ X @ Y ) ) ).
% subset_mset.eq_refl
thf(fact_559_subset__mset_Oorder__trans,axiom,
! [X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( subseteq_mset_a @ X @ Y )
=> ( ( subseteq_mset_a @ Y @ Z )
=> ( subseteq_mset_a @ X @ Z ) ) ) ).
% subset_mset.order_trans
thf(fact_560_subset__mset_Oantisym__conv,axiom,
! [Y: multiset_a,X: multiset_a] :
( ( subseteq_mset_a @ Y @ X )
=> ( ( subseteq_mset_a @ X @ Y )
= ( X = Y ) ) ) ).
% subset_mset.antisym_conv
thf(fact_561_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y7: multiset_a,Z4: multiset_a] : ( Y7 = Z4 ) )
= ( ^ [X3: multiset_a,Y6: multiset_a] :
( ( subseteq_mset_a @ X3 @ Y6 )
& ( subseteq_mset_a @ Y6 @ X3 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_562_subset__mset_Oorder__antisym,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X @ Y )
=> ( ( subseteq_mset_a @ Y @ X )
=> ( X = Y ) ) ) ).
% subset_mset.order_antisym
thf(fact_563_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( A = B )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_564_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_565_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( subseteq_mset_a @ C @ B )
=> ( subseteq_mset_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_566_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y7: multiset_a,Z4: multiset_a] : ( Y7 = Z4 ) )
= ( ^ [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ B2 @ A2 )
& ( subseteq_mset_a @ A2 @ B2 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_567_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( subseteq_mset_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_568_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_569_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_570_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_571_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_572_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_573_empty__le,axiom,
! [A3: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A3 ) ).
% empty_le
thf(fact_574_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( A = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_575_subset__mset_Obot__least,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% subset_mset.bot_least
thf(fact_576_subset__mset_Ozero__le,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X ) ).
% subset_mset.zero_le
thf(fact_577_subset__mset_Oadd__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_578_subset__mset_Oless__eqE,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ~ ! [C3: multiset_a] :
( B
!= ( plus_plus_multiset_a @ A @ C3 ) ) ) ).
% subset_mset.less_eqE
thf(fact_579_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A2: multiset_a,B2: multiset_a] :
? [C2: multiset_a] :
( B2
= ( plus_plus_multiset_a @ A2 @ C2 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_580_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_581_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_582_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_583_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_584_mset__subset__eq__add__left,axiom,
! [A3: multiset_a,B3: multiset_a] : ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_add_left
thf(fact_585_mset__subset__eq__mono__add,axiom,
! [A3: multiset_a,B3: multiset_a,C4: multiset_a,D2: multiset_a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ( subseteq_mset_a @ C4 @ D2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C4 ) @ ( plus_plus_multiset_a @ B3 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_586_mset__subset__eq__add__right,axiom,
! [B3: multiset_a,A3: multiset_a] : ( subseteq_mset_a @ B3 @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_add_right
thf(fact_587_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
? [C5: multiset_a] :
( B5
= ( plus_plus_multiset_a @ A5 @ C5 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_588_diff__subset__eq__self,axiom,
! [M3: multiset_a,N4: multiset_a] : ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ M3 ) ).
% diff_subset_eq_self
thf(fact_589_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_590_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_591_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_592_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_593_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_594_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_595_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_596_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_597_append__Cons__eq__iff,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys4: 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 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_598_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys4: 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 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_599_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_600_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_601_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_602_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_603_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_604_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_605_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_606_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_607_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_608_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_609_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_610_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_611_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_612_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_613_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 ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_614_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 ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_615_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_616_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ Y @ zero_zero_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_nonpos_eq_0_iff
thf(fact_617_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ X )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
=> ( ( ( plus_plus_multiset_a @ X @ Y )
= zero_zero_multiset_a )
= ( ( X = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_618_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_619_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_620_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_a,B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
=> ( ( subseteq_mset_a @ B @ A )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_621_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_622_subset__mset_Oadd__increasing,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_623_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ C @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_624_Diff__eq__empty__iff__mset,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ A3 @ B3 )
= zero_zero_multiset_a )
= ( subseteq_mset_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_625_size__mset__mono,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ A3 ) @ ( size_size_multiset_a @ B3 ) ) ) ).
% size_mset_mono
thf(fact_626_subset__mset_Odiff__add,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A ) @ A )
= B ) ) ).
% subset_mset.diff_add
thf(fact_627_subset__mset_Ole__add__diff,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_628_subset__mset_Ole__diff__conv2,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A ) )
= ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_629_subset__mset_Odiff__add__assoc,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B ) @ A )
= ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_630_subset__mset_Odiff__add__assoc2,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C ) @ A )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_631_subset__mset_Odiff__diff__right,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( minus_3765977307040488491iset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_632_subset__mset_Oadd__diff__inverse,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ A @ ( minus_3765977307040488491iset_a @ B @ A ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_633_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ A @ B )
=> ( ( ( minus_3765977307040488491iset_a @ B @ A )
= C )
= ( B
= ( plus_plus_multiset_a @ C @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_634_subset__eq__diff__conv,axiom,
! [A3: multiset_a,C4: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ A3 @ C4 ) @ B3 )
= ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ B3 @ C4 ) ) ) ).
% subset_eq_diff_conv
thf(fact_635_multiset__diff__union__assoc,axiom,
! [C4: multiset_a,B3: multiset_a,A3: multiset_a] :
( ( subseteq_mset_a @ C4 @ B3 )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C4 )
= ( plus_plus_multiset_a @ A3 @ ( minus_3765977307040488491iset_a @ B3 @ C4 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_636_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,X2: a,Xs4: list_a,Y2: a,Ys6: list_a] :
( ( X2 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_637_size__Diff__submset,axiom,
! [M3: multiset_a,M5: multiset_a] :
( ( subseteq_mset_a @ M3 @ M5 )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ M3 ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ M3 ) ) ) ) ).
% size_Diff_submset
thf(fact_638_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ( P2 != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_639_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_640_domain_Odetermination__of__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,A3: partia2175431115845679010xt_a_b,H2: list_a > a,G: list_a > a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K3 ) @ A3 @ H2 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K3 ) @ A3 @ G )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K3 )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= ( G @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H2 @ ( var_a_b @ R ) )
= ( G @ ( var_a_b @ R ) ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( H2 @ P2 )
= ( G @ P2 ) ) ) ) ) ) ) ) ) ).
% domain.determination_of_hom
thf(fact_641_domain_Opdivides__imp__roots__incl,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) @ ( polyno7858422826990252003t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_642_domain_Opdivides__imp__roots__incl,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R @ P2 ) @ ( polynomial_roots_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_643_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,Q2: list_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ Q2 )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_644_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,Q2: list_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_645_domain_Oeval__as__unique__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,X: a,H2: list_a > a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K3 ) @ R @ H2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K3 )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_a_b @ R ) )
= X )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( H2 @ P2 )
= ( eval_a_b @ R @ P2 @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_646_domain_Oeval__as__unique__hom,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,X: list_a,H2: list_list_a > list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ R @ H2 )
=> ( ! [K4: list_a] :
( ( member_list_a @ K4 @ K3 )
=> ( ( H2 @ ( cons_list_a @ K4 @ nil_list_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_li8453953174693405341t_unit @ R ) )
= X )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( H2 @ P2 )
= ( eval_l34571156754992824t_unit @ R @ P2 @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_647_domain_Opoly__mult__var,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ( P2 = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ ( var_li8453953174693405341t_unit @ R ) )
= nil_list_a ) )
& ( ( P2 != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ ( var_li8453953174693405341t_unit @ R ) )
= ( append_list_a @ P2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_648_domain_Opoly__mult__var,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) ) )
=> ( ( ( P2 = nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) @ P2 @ ( var_se6008125447796440765t_unit @ R ) )
= nil_set_list_a ) )
& ( ( P2 != nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) @ P2 @ ( var_se6008125447796440765t_unit @ R ) )
= ( append_set_list_a @ P2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_649_domain_Opoly__mult__var,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ( P2 = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ ( var_a_b @ R ) )
= nil_a ) )
& ( ( P2 != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ ( var_a_b @ R ) )
= ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_650_domain_Opirreducible__roots,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polyno7858422826990252003t_unit @ R @ P2 )
= zero_z4454100511807792257list_a ) ) ) ) ) ).
% domain.pirreducible_roots
thf(fact_651_domain_Opirreducible__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ R @ P2 )
= zero_zero_multiset_a ) ) ) ) ) ).
% domain.pirreducible_roots
thf(fact_652_eval__append__aux,axiom,
! [P2: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P2 @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_653_const__term__explicit,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ r @ P2 )
= A )
=> ~ ! [P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P2
!= ( append_a @ P3 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_654_const__term__eq__last,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P2 @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_655_pirreducible__degree,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_656_degree__one__imp__pirreducible,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_657_combine__append__zero,axiom,
! [Us3: list_a,Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ) ).
% combine_append_zero
thf(fact_658_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_659_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_660_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_661_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_662_subring__props_I7_J,axiom,
! [K3: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H22 @ K3 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K3 ) ) ) ) ).
% subring_props(7)
thf(fact_663_subring__props_I6_J,axiom,
! [K3: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H22 @ K3 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K3 ) ) ) ) ).
% subring_props(6)
thf(fact_664_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_665_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_666_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_667_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_668_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_669_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_670_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_671_combine_Osimps_I3_J,axiom,
! [Ks: list_a] :
( ( embedded_combine_a_b @ r @ Ks @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_672_combine_Osimps_I2_J,axiom,
! [Us3: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us3 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_673_const__term__not__zero,axiom,
! [P2: list_a] :
( ( ( const_term_a_b @ r @ P2 )
!= ( zero_a_b @ r ) )
=> ( P2 != nil_a ) ) ).
% const_term_not_zero
thf(fact_674_line__extension__mem__iff,axiom,
! [U: a,K3: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K3 )
& ? [Y6: a] :
( ( member_a @ Y6 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y6 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_675_const__term__def,axiom,
! [P2: list_a] :
( ( const_term_a_b @ r @ P2 )
= ( eval_a_b @ r @ P2 @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_676_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_677_combine_Osimps_I1_J,axiom,
! [K: a,Ks: list_a,U: a,Us3: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K @ Ks ) @ ( cons_a @ U @ Us3 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ) ).
% combine.simps(1)
thf(fact_678_const__term__simprules_I1_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_679_const__term__simprules__shell_I1_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P2 ) @ K3 ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_680_combine__eq__eval,axiom,
! [Ks: list_a,X: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X ) ) ).
% combine_eq_eval
thf(fact_681_line__extension__smult__closed,axiom,
! [K3: set_a,E: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ! [K4: a,V2: a] :
( ( member_a @ K4 @ K3 )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K3 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K3 @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_682_combine_Oelims,axiom,
! [X: list_a,Xa2: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X @ Xa2 )
= Y )
=> ( ! [K4: a,Ks2: list_a] :
( ( X
= ( cons_a @ K4 @ Ks2 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K4 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_683_const__term__simprules__shell_I2_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ Q2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P2 ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_684_combine__append,axiom,
! [Ks: list_a,Us3: list_a,Ks3: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Vs2 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks3 ) @ ( append_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_685_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_686_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_687_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_688_combine__in__carrier,axiom,
! [Ks: list_a,Us3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_689_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_690_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P2 ) @ K3 ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_691_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( member_a @ ( const_term_a_b @ R @ P2 ) @ K3 ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_692_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ Q2 ) )
= ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P2 ) @ ( const_6738166269504826821t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_693_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( const_term_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ Q2 ) )
= ( add_a_b @ R @ ( const_term_a_b @ R @ P2 ) @ ( const_term_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_694_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_695_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_696_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ Q2 ) )
= ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P2 ) @ ( const_6738166269504826821t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_697_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( const_term_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ Q2 ) )
= ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P2 ) @ ( const_term_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_698_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_699_exists__unique__long__division,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ X2 )
& ! [Y3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ Y3 )
=> ( Y3 = X2 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_700_pprimeE_I3_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q2 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R2 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_701_pprime__iff__pirreducible,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_702_const__term__simprules__shell_I3_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ Q2 ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P2 ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_703_pprimeE_I1_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_704_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_705_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_706_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_707_domain_OpprimeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ( P2 != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_708_domain_OpprimeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_709_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_710_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_711_domain_Oexists__unique__long__division,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( Q2 != nil_list_a )
=> ? [X2: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ X2 )
& ! [Y3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ Y3 )
=> ( Y3 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_712_domain_Oexists__unique__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ X2 )
& ! [Y3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ Y3 )
=> ( Y3 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_713_domain_OpprimeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ Q2 @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
| ( polyno8016796738000020810t_unit @ R @ P2 @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_714_domain_OpprimeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ Q2 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
| ( polyno5814909790663948098es_a_b @ R @ P2 @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_715_long__division__zero_I1_J,axiom,
! [K3: set_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_716_long__division__add_I1_J,axiom,
! [K3: set_a,A: list_a,B: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pdiv_a_b @ r @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_717_long__division__closed_I1_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_718_subfield__long__division__theorem__shell,axiom,
! [K3: set_a,P2: list_a,B: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( 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,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_719_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_720_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_721_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_722_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_723_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_724_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_725_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_726_pmod__const_I1_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P2 @ Q2 )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_727_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_728_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_729_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_730_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_731_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_732_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_733_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_734_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_735_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_736_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_737_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_738_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_739_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_740_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_741_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_742_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_743_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_744_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_745_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_746_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_747_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_748_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_749_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_750_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_751_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_752_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_753_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_754_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_755_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_756_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_757_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_758_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_759_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_760_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys7: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys7 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys7 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_761_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_762_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_763_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_764_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_765_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_766_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_767_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_768_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_769_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_770_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_771_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_772_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_773_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_774_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_775_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_776_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_777_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_778_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_779_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_780_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_781_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_782_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_783_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_784_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_785_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_786_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_787_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_788_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_789_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_790_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_791_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_792_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_793_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_794_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_795_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_796_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_797_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_798_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_799_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_800_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_801_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M7: nat,N3: nat] :
( ( ord_less_nat @ M7 @ N3 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_802_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_803_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_804_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_805_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_806_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_807_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_808_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_809_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_810_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_811_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_812_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_813_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_814_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_815_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_816_nonempty__has__size,axiom,
! [S2: multiset_a] :
( ( S2 != zero_zero_multiset_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_817_domain_Opmod__const_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 )
= nil_list_a ) ) ) ) ) ) ).
% domain.pmod_const(1)
thf(fact_818_domain_Opmod__const_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ R @ P2 @ Q2 )
= nil_a ) ) ) ) ) ) ).
% domain.pmod_const(1)
thf(fact_819_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_820_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_821_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_822_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ nil_a @ Q2 )
= nil_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_823_domain_Olong__division__add_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ A @ B ) @ Q2 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( polyno5893782122288709345t_unit @ R @ A @ Q2 ) @ ( polyno5893782122288709345t_unit @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_824_domain_Olong__division__add_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( polynomial_pdiv_a_b @ R @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_825_domain_Ofield__long__division__theorem__shell,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,B: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ? [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
& ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
& ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.field_long_division_theorem_shell
thf(fact_826_domain_Ofield__long__division__theorem__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,B: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( 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,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.field_long_division_theorem_shell
thf(fact_827_alg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_828_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_829_long__dividesI,axiom,
! [B: list_a,R2: list_a,P2: list_a,Q2: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ B ) @ R2 ) )
=> ( ( ( R2 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_830_pmod__degree,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_831_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_832_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_833_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K4: a,Ks2: list_a,U2: a,Us4: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks2 ) @ ( cons_a @ U2 @ Us4 ) ) )
=> ( ! [Us4: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us4 ) )
=> ~ ! [Ks2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks2 @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_834_long__division__closed_I2_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_835_long__division__zero_I2_J,axiom,
! [K3: set_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_836_long__division__add_I2_J,axiom,
! [K3: set_a,A: list_a,B: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pmod_a_b @ r @ A @ Q2 ) @ ( polynomial_pmod_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_837_long__division__add__iff,axiom,
! [K3: set_a,A: list_a,B: list_a,C: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_838_pmod__zero__iff__pdivides,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ P2 ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_839_exists__long__division,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B4 @ R3 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_840_same__pmod__iff__pdivides,axiom,
! [K3: set_a,A: list_a,B: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K3 ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_841_pdiv__pmod,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q2 @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_842_pmod__const_I2_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= P2 ) ) ) ) ) ).
% pmod_const(2)
thf(fact_843_long__divisionI,axiom,
! [K3: set_a,P2: list_a,Q2: list_a,B: list_a,R2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_844_long__divisionE,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_845_ring_Opmod_Ocong,axiom,
polynomial_pmod_a_b = polynomial_pmod_a_b ).
% ring.pmod.cong
thf(fact_846_ring_Oalg__mult_Ocong,axiom,
polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).
% ring.alg_mult.cong
thf(fact_847_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_848_subset__eq__mset__impl_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X2: a,Xs2: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_849_domain_Olong__divisionE,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_850_domain_Olong__divisionE,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_851_domain_Olong__divisionI,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a,B: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_852_domain_Olong__divisionI,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a,B: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_853_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_854_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_855_domain_Oexists__long__division,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( Q2 != nil_list_a )
=> ~ ! [B4: list_list_a] :
( ( member_list_list_a @ B4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ! [R3: list_list_a] :
( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ~ ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ ( produc8696003437204565271list_a @ B4 @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_856_domain_Oexists__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B4 @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_857_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_858_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ R @ nil_a @ Q2 )
= nil_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_859_domain_Olong__division__add_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ A @ B ) @ Q2 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( polyno1727750685288865234t_unit @ R @ A @ Q2 ) @ ( polyno1727750685288865234t_unit @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_860_domain_Olong__division__add_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( polynomial_pmod_a_b @ R @ A @ Q2 ) @ ( polynomial_pmod_a_b @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_861_domain_Olong__division__add__iff,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ R @ B @ Q2 ) )
= ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ A @ C ) @ Q2 )
= ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_862_domain_Olong__division__add__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,A: list_a,B: list_a,C: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ A @ Q2 )
= ( polynomial_pmod_a_b @ R @ B @ Q2 ) )
= ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ A @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_863_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ R @ Q2 @ P2 ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_864_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ P2 @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ R @ Q2 @ P2 ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_865_domain_Osame__pmod__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ R @ B @ Q2 ) )
= ( polyno8016796738000020810t_unit @ R @ Q2 @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_866_domain_Osame__pmod__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ A @ Q2 )
= ( polynomial_pmod_a_b @ R @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ R @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K3 ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_867_domain_Opdiv__pmod,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ Q2 @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_868_domain_Opdiv__pmod,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ Q2 @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_869_domain_Opmod__const_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 )
= P2 ) ) ) ) ) ) ).
% domain.pmod_const(2)
thf(fact_870_domain_Opmod__const_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ R @ P2 @ Q2 )
= P2 ) ) ) ) ) ) ).
% domain.pmod_const(2)
thf(fact_871_domain_Opmod__degree,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 )
= nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pmod_degree
thf(fact_872_domain_Opmod__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( ( polynomial_pmod_a_b @ R @ P2 @ Q2 )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pmod_degree
thf(fact_873_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ R @ P2 @ X ) )
= ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_874_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ R @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_875_field__long__division__theorem,axiom,
! [K3: set_a,P2: list_a,B: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( polynomial_a_b @ r @ K3 @ B )
=> ( ( B != nil_a )
=> ? [Q3: list_a,R3: list_a] :
( ( polynomial_a_b @ r @ K3 @ Q3 )
& ( polynomial_a_b @ r @ K3 @ R3 )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% field_long_division_theorem
thf(fact_876_roots__inclI_H,axiom,
! [P2: list_a,M: multiset_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A4 ) @ ( count_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ M ) ) ) ).
% roots_inclI'
thf(fact_877_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_878_polynomial__incl,axiom,
! [K3: set_a,P2: list_a] :
( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K3 ) ) ).
% polynomial_incl
thf(fact_879_var__closed_I2_J,axiom,
! [K3: set_a] :
( ( subring_a_b @ K3 @ r )
=> ( polynomial_a_b @ r @ K3 @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_880_eval__poly__in__carrier,axiom,
! [K3: set_a,P2: list_a,X: a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_881_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_882_const__term__zero,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ r @ P2 )
= ( zero_a_b @ r ) )
=> ~ ! [P3: list_a] :
( ( polynomial_a_b @ r @ K3 @ P3 )
=> ( ( P3 != nil_a )
=> ( P2
!= ( append_a @ P3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_883_alg__mult__eq__count__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P2 )
= ( count_a @ ( polynomial_roots_a_b @ r @ P2 ) ) ) ) ).
% alg_mult_eq_count_roots
thf(fact_884_count__empty,axiom,
! [A: a] :
( ( count_a @ zero_zero_multiset_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_885_count__union,axiom,
! [M3: multiset_a,N4: multiset_a,A: a] :
( ( count_a @ ( plus_plus_multiset_a @ M3 @ N4 ) @ A )
= ( plus_plus_nat @ ( count_a @ M3 @ A ) @ ( count_a @ N4 @ A ) ) ) ).
% count_union
thf(fact_886_count__diff,axiom,
! [M3: multiset_a,N4: multiset_a,A: a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) @ A )
= ( minus_minus_nat @ ( count_a @ M3 @ A ) @ ( count_a @ N4 @ A ) ) ) ).
% count_diff
thf(fact_887_zero__is__polynomial,axiom,
! [K3: set_a] : ( polynomial_a_b @ r @ K3 @ nil_a ) ).
% zero_is_polynomial
thf(fact_888_carrier__polynomial,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P2 ) ) ) ).
% carrier_polynomial
thf(fact_889_polynomial__in__carrier,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_890_multiset__eq__iff,axiom,
( ( ^ [Y7: multiset_a,Z4: multiset_a] : ( Y7 = Z4 ) )
= ( ^ [M4: multiset_a,N5: multiset_a] :
! [A2: a] :
( ( count_a @ M4 @ A2 )
= ( count_a @ N5 @ A2 ) ) ) ) ).
% multiset_eq_iff
thf(fact_891_multiset__eqI,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ! [X2: a] :
( ( count_a @ A3 @ X2 )
= ( count_a @ B3 @ X2 ) )
=> ( A3 = B3 ) ) ).
% multiset_eqI
thf(fact_892_count__inject,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( count_a @ X )
= ( count_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_893_mset__subset__eq__count,axiom,
! [A3: multiset_a,B3: multiset_a,A: a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( count_a @ A3 @ A ) @ ( count_a @ B3 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_894_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
! [A2: a] : ( ord_less_eq_nat @ ( count_a @ A5 @ A2 ) @ ( count_a @ B5 @ A2 ) ) ) ) ).
% subseteq_mset_def
thf(fact_895_mset__subset__eqI,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ! [A4: a] : ( ord_less_eq_nat @ ( count_a @ A3 @ A4 ) @ ( count_a @ B3 @ A4 ) )
=> ( subseteq_mset_a @ A3 @ B3 ) ) ).
% mset_subset_eqI
thf(fact_896_zero__multiset_Orep__eq,axiom,
( ( count_a @ zero_zero_multiset_a )
= ( ^ [A2: a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_897_plus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa2: multiset_a] :
( ( count_a @ ( plus_plus_multiset_a @ X @ Xa2 ) )
= ( ^ [A2: a] : ( plus_plus_nat @ ( count_a @ X @ A2 ) @ ( count_a @ Xa2 @ A2 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_898_minus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa2: multiset_a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ X @ Xa2 ) )
= ( ^ [A2: a] : ( minus_minus_nat @ ( count_a @ X @ A2 ) @ ( count_a @ Xa2 @ A2 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_899_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b,K5: set_a,P4: list_a] : ( member_list_a @ P4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R4 @ K5 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_900_domain_Ovar__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( polyno1315193887021588240t_unit @ R @ K3 @ ( var_li8453953174693405341t_unit @ R ) ) ) ) ).
% domain.var_closed(2)
thf(fact_901_domain_Ovar__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( polynomial_a_b @ R @ K3 @ ( var_a_b @ R ) ) ) ) ).
% domain.var_closed(2)
thf(fact_902_domain_Oalg__mult__eq__count__roots,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno4259638811958763678t_unit @ R @ P2 )
= ( count_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) ) ) ) ) ).
% domain.alg_mult_eq_count_roots
thf(fact_903_domain_Oalg__mult__eq__count__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ R @ P2 )
= ( count_a @ ( polynomial_roots_a_b @ R @ P2 ) ) ) ) ) ).
% domain.alg_mult_eq_count_roots
thf(fact_904_domain_Oroots__inclI_H,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,M: multiset_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( P2 != nil_list_a )
=> ( ord_less_eq_nat @ ( polyno4259638811958763678t_unit @ R @ P2 @ A4 ) @ ( count_list_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) @ M ) ) ) ) ).
% domain.roots_inclI'
thf(fact_905_domain_Oroots__inclI_H,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,M: multiset_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( P2 != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ R @ P2 @ A4 ) @ ( count_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R @ P2 ) @ M ) ) ) ) ).
% domain.roots_inclI'
thf(fact_906_domain_Ofield__long__division__theorem,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,B: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P2 )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ B )
=> ( ( B != nil_list_a )
=> ? [Q3: list_list_a,R3: list_list_a] :
( ( polyno1315193887021588240t_unit @ R @ K3 @ Q3 )
& ( polyno1315193887021588240t_unit @ R @ K3 @ R3 )
& ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3 = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.field_long_division_theorem
thf(fact_907_domain_Ofield__long__division__theorem,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,B: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P2 )
=> ( ( polynomial_a_b @ R @ K3 @ B )
=> ( ( B != nil_a )
=> ? [Q3: list_a,R3: list_a] :
( ( polynomial_a_b @ R @ K3 @ Q3 )
& ( polynomial_a_b @ R @ K3 @ R3 )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.field_long_division_theorem
thf(fact_908_pprimeI,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ! [Q3: list_a,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q3 @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R3 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) ) ) ) ) ) ).
% pprimeI
thf(fact_909_eval__append,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P2 @ Q2 ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_910_univ__poly__units_H,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
= ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
& ( P2 != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% univ_poly_units'
thf(fact_911_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_912_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_913_nat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_914_nat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_915_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_916_nat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% nat_pow_mult
thf(fact_917_pirreducibleE_I2_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_918_pprimeE_I2_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_919_pirreducibleE_I3_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q2 @ R2 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_920_pirreducibleI,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ! [Q3: list_a,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q3 @ R3 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_921_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_922_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_923_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_924_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_925_domain_OpprimeE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_926_domain_OpprimeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_927_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( P2
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ Q2 @ R2 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_928_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ Q2 @ R2 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_929_domain_OpirreducibleI,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( P2 != nil_list_a )
=> ( ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ! [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( P2
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ Q3 @ R3 ) )
=> ( ( member_list_list_a @ Q3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
| ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_930_domain_OpirreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ! [Q3: list_a,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ Q3 @ R3 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_931_domain_OpprimeI,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( P2 != nil_list_a )
=> ( ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ! [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ Q3 @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q3 )
| ( polyno8016796738000020810t_unit @ R @ P2 @ R3 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_932_domain_OpprimeI,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ! [Q3: list_a,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ Q3 @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ R @ P2 @ R3 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_933_domain_Ouniv__poly__units_H,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
= ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
& ( P2 != nil_list_a )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% domain.univ_poly_units'
thf(fact_934_domain_Ouniv__poly__units_H,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
= ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
& ( P2 != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% domain.univ_poly_units'
thf(fact_935_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_936_poly__mult__degree__eq,axiom,
! [K3: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P12 )
=> ( ( polynomial_a_b @ r @ K3 @ P23 )
=> ( ( ( ( P12 = nil_a )
| ( P23 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ( P12 = nil_a )
| ( P23 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P12 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P23 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% poly_mult_degree_eq
thf(fact_937_monoid__cancelI,axiom,
( ! [A4: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C3 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C3 @ B4 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C3 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C3 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_938_poly__mult_Osimps_I1_J,axiom,
! [P23: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P23 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_939_poly__mult__closed,axiom,
! [K3: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P12 )
=> ( ( polynomial_a_b @ r @ K3 @ P23 )
=> ( polynomial_a_b @ r @ K3 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_940_poly__mult__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_941_poly__mult__comm,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ P23 @ P12 ) ) ) ) ).
% poly_mult_comm
thf(fact_942_poly__mult__integral,axiom,
! [K3: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P12 )
=> ( ( polynomial_a_b @ r @ K3 @ P23 )
=> ( ( ( poly_mult_a_b @ r @ P12 @ P23 )
= nil_a )
=> ( ( P12 = nil_a )
| ( P23 = nil_a ) ) ) ) ) ) ).
% poly_mult_integral
thf(fact_943_poly__mult__zero_I1_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P2 )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_944_poly__mult__zero_I2_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P2 @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_945_poly__mult__is__polynomial,axiom,
! [K3: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K3 )
=> ( polynomial_a_b @ r @ K3 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_946_poly__mult__monom__assoc,axiom,
! [P2: list_a,Q2: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( 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 ) @ P2 ) @ Q2 )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_947_poly__mult__semiassoc,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( 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 ) @ P2 ) @ Q2 )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_948_eval__poly__mult,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_mult_a_b @ r @ P2 @ Q2 ) @ A )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).
% eval_poly_mult
thf(fact_949_const__term__simprules_I2_J,axiom,
! [P2: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P2 @ Q2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P2 ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_950_poly__mult__append__zero__lcancel,axiom,
! [K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( polynomial_a_b @ r @ K3 @ Q2 )
=> ( ( ( poly_mult_a_b @ r @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q2 )
= ( append_a @ R2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P2 @ Q2 )
= R2 ) ) ) ) ) ).
% poly_mult_append_zero_lcancel
thf(fact_951_poly__mult__append__zero__rcancel,axiom,
! [K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( polynomial_a_b @ r @ K3 @ Q2 )
=> ( ( ( poly_mult_a_b @ r @ P2 @ ( append_a @ Q2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( append_a @ R2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P2 @ Q2 )
= R2 ) ) ) ) ) ).
% poly_mult_append_zero_rcancel
thf(fact_952_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_953_ring_Opoly__mult_Ocong,axiom,
poly_mult_a_b = poly_mult_a_b ).
% ring.poly_mult.cong
thf(fact_954_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_955_domain_Opoly__mult__monom__assoc,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,A: a,N: nat] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( 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 ) @ P2 ) @ Q2 )
= ( poly_mult_a_b @ R @ ( monom_a_b @ R @ A @ N ) @ ( poly_mult_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.poly_mult_monom_assoc
thf(fact_956_domain_Opoly__mult__monom__assoc,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,A: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) @ P2 ) @ Q2 )
= ( poly_m7087347720095500472t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) @ ( poly_m7087347720095500472t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.poly_mult_monom_assoc
thf(fact_957_univ__poly__mult,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) )
= ( poly_mult_a_b @ R ) ) ).
% univ_poly_mult
thf(fact_958_domain_Opoly__mult__comm,axiom,
! [R: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P12 @ P23 )
= ( poly_mult_a_b @ R @ P23 @ P12 ) ) ) ) ) ).
% domain.poly_mult_comm
thf(fact_959_domain_Opoly__mult__comm,axiom,
! [R: partia2670972154091845814t_unit,P12: list_list_a,P23: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P12 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P23 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P12 @ P23 )
= ( poly_m7087347720095500472t_unit @ R @ P23 @ P12 ) ) ) ) ) ).
% domain.poly_mult_comm
thf(fact_960_domain_Opoly__mult__integral,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P12: list_list_a,P23: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P12 )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P23 )
=> ( ( ( poly_m7087347720095500472t_unit @ R @ P12 @ P23 )
= nil_list_a )
=> ( ( P12 = nil_list_a )
| ( P23 = nil_list_a ) ) ) ) ) ) ) ).
% domain.poly_mult_integral
thf(fact_961_domain_Opoly__mult__integral,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P12: list_a,P23: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P12 )
=> ( ( polynomial_a_b @ R @ K3 @ P23 )
=> ( ( ( poly_mult_a_b @ R @ P12 @ P23 )
= nil_a )
=> ( ( P12 = nil_a )
| ( P23 = nil_a ) ) ) ) ) ) ) ).
% domain.poly_mult_integral
thf(fact_962_domain_Opoly__mult__semiassoc,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,A: a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( 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 ) @ P2 ) @ Q2 )
= ( poly_mult_a_b @ R @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.poly_mult_semiassoc
thf(fact_963_domain_Opoly__mult__semiassoc,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ A @ nil_list_a ) @ P2 ) @ Q2 )
= ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ A @ nil_list_a ) @ ( poly_m7087347720095500472t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.poly_mult_semiassoc
thf(fact_964_domain_Opoly__mult__append__zero__rcancel,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P2 )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ Q2 )
=> ( ( ( poly_m7087347720095500472t_unit @ R @ P2 @ ( append_list_a @ Q2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) )
= ( append_list_a @ R2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P2 @ Q2 )
= R2 ) ) ) ) ) ) ).
% domain.poly_mult_append_zero_rcancel
thf(fact_965_domain_Opoly__mult__append__zero__rcancel,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a,P2: list_set_list_a,Q2: list_set_list_a,R2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( ( polyno3115169382166032176t_unit @ R @ K3 @ P2 )
=> ( ( polyno3115169382166032176t_unit @ R @ K3 @ Q2 )
=> ( ( ( poly_m1537542421183396056t_unit @ R @ P2 @ ( append_set_list_a @ Q2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) )
= ( append_set_list_a @ R2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P2 @ Q2 )
= R2 ) ) ) ) ) ) ).
% domain.poly_mult_append_zero_rcancel
thf(fact_966_domain_Opoly__mult__append__zero__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P2 )
=> ( ( polynomial_a_b @ R @ K3 @ Q2 )
=> ( ( ( poly_mult_a_b @ R @ P2 @ ( append_a @ Q2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) )
= ( append_a @ R2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ R @ P2 @ Q2 )
= R2 ) ) ) ) ) ) ).
% domain.poly_mult_append_zero_rcancel
thf(fact_967_domain_Opoly__mult__append__zero__lcancel,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P2 )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ Q2 )
=> ( ( ( poly_m7087347720095500472t_unit @ R @ ( append_list_a @ P2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) @ Q2 )
= ( append_list_a @ R2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P2 @ Q2 )
= R2 ) ) ) ) ) ) ).
% domain.poly_mult_append_zero_lcancel
thf(fact_968_domain_Opoly__mult__append__zero__lcancel,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a,P2: list_set_list_a,Q2: list_set_list_a,R2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( ( polyno3115169382166032176t_unit @ R @ K3 @ P2 )
=> ( ( polyno3115169382166032176t_unit @ R @ K3 @ Q2 )
=> ( ( ( poly_m1537542421183396056t_unit @ R @ ( append_set_list_a @ P2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) @ Q2 )
= ( append_set_list_a @ R2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P2 @ Q2 )
= R2 ) ) ) ) ) ) ).
% domain.poly_mult_append_zero_lcancel
thf(fact_969_domain_Opoly__mult__append__zero__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P2 )
=> ( ( polynomial_a_b @ R @ K3 @ Q2 )
=> ( ( ( poly_mult_a_b @ R @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ Q2 )
= ( append_a @ R2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ R @ P2 @ Q2 )
= R2 ) ) ) ) ) ) ).
% domain.poly_mult_append_zero_lcancel
thf(fact_970_domain_Opoly__mult__degree__eq,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P12: list_list_a,P23: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P12 )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P23 )
=> ( ( ( ( P12 = nil_list_a )
| ( P23 = nil_list_a ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( poly_m7087347720095500472t_unit @ R @ P12 @ P23 ) ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ( P12 = nil_list_a )
| ( P23 = nil_list_a ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( poly_m7087347720095500472t_unit @ R @ P12 @ P23 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P12 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P23 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.poly_mult_degree_eq
thf(fact_971_domain_Opoly__mult__degree__eq,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P12: list_a,P23: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P12 )
=> ( ( polynomial_a_b @ R @ K3 @ P23 )
=> ( ( ( ( P12 = nil_a )
| ( P23 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ R @ P12 @ P23 ) ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ( P12 = nil_a )
| ( P23 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ R @ P12 @ P23 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P12 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P23 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.poly_mult_degree_eq
thf(fact_972_poly__mult__var_H_I2_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P2 @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_973_poly__mult__var_H_I1_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P2 )
= ( normalize_a_b @ r @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_974_poly__mult__append__zero,axiom,
! [P2: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q2 )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P2 @ Q2 ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_975_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_976_normalize__polynomial,axiom,
! [K3: set_a,P2: list_a] :
( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( normalize_a_b @ r @ P2 )
= P2 ) ) ).
% normalize_polynomial
thf(fact_977_local_Onormalize__idem,axiom,
! [P2: list_a,Q2: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P2 ) @ Q2 ) )
= ( normalize_a_b @ r @ ( append_a @ P2 @ Q2 ) ) ) ).
% local.normalize_idem
thf(fact_978_normalize__length__le,axiom,
! [P2: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P2 ) ) @ ( size_size_list_a @ P2 ) ) ).
% normalize_length_le
thf(fact_979_normalize__in__carrier,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P2 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_980_normalize__gives__polynomial,axiom,
! [P2: list_a,K3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K3 )
=> ( polynomial_a_b @ r @ K3 @ ( normalize_a_b @ r @ P2 ) ) ) ).
% normalize_gives_polynomial
thf(fact_981_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K2: a] : ( normalize_a_b @ r @ ( cons_a @ K2 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_982_poly__mult__normalize,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).
% poly_mult_normalize
thf(fact_983_eval__normalize,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P2 ) @ A )
= ( eval_a_b @ r @ P2 @ A ) ) ) ) ).
% eval_normalize
thf(fact_984_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_985_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P2 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( var_se6008125447796440765t_unit @ R ) @ P2 )
= ( normal4052021864830707619t_unit @ R @ ( append_set_list_a @ P2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_986_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( var_a_b @ R ) @ P2 )
= ( normalize_a_b @ R @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_987_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ P2 )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_988_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P2 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P2 @ ( var_se6008125447796440765t_unit @ R ) )
= ( normal4052021864830707619t_unit @ R @ ( append_set_list_a @ P2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_989_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P2 @ ( var_a_b @ R ) )
= ( normalize_a_b @ R @ ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_990_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P2 @ ( var_li8453953174693405341t_unit @ R ) )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_991_poly__mult__one_H_I1_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P2 )
= ( normalize_a_b @ r @ P2 ) ) ) ).
% poly_mult_one'(1)
thf(fact_992_poly__mult__one_H_I2_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P2 @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= ( normalize_a_b @ r @ P2 ) ) ) ).
% poly_mult_one'(2)
thf(fact_993_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_994_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_995_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_996_inv__unique,axiom,
! [Y: a,X: a,Y4: 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 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% inv_unique
thf(fact_997_poly__mult__one_I2_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( poly_mult_a_b @ r @ P2 @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= P2 ) ) ) ).
% poly_mult_one(2)
thf(fact_998_poly__mult__one_I1_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( polynomial_a_b @ r @ K3 @ P2 )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P2 )
= P2 ) ) ) ).
% poly_mult_one(1)
thf(fact_999_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_1000_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_1001_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_1002_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_1003_local_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1004_one__is__polynomial,axiom,
! [K3: set_a] :
( ( subring_a_b @ K3 @ r )
=> ( polynomial_a_b @ r @ K3 @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).
% one_is_polynomial
thf(fact_1005_univ__poly__one,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a] :
( ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ K3 ) )
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ).
% univ_poly_one
thf(fact_1006_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K3 ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_1007_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R4: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R4 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R4 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_1008_var__def,axiom,
( var_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R4 ) @ ( cons_a @ ( zero_a_b @ R4 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_1009_var__def,axiom,
( var_se6008125447796440765t_unit
= ( ^ [R4: partia7496981018696276118t_unit] : ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R4 ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R4 ) @ nil_set_list_a ) ) ) ) ).
% var_def
thf(fact_1010_domain_Oone__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( polyno1315193887021588240t_unit @ R @ K3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_1011_domain_Oone__is__polynomial,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( polyno3115169382166032176t_unit @ R @ K3 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_1012_domain_Oone__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( polynomial_a_b @ R @ K3 @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_1013_domain_Opoly__mult__one_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P2 )
=> ( ( poly_m7087347720095500472t_unit @ R @ P2 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) )
= P2 ) ) ) ) ).
% domain.poly_mult_one(2)
thf(fact_1014_domain_Opoly__mult__one_I2_J,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( ( polyno3115169382166032176t_unit @ R @ K3 @ P2 )
=> ( ( poly_m1537542421183396056t_unit @ R @ P2 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) )
= P2 ) ) ) ) ).
% domain.poly_mult_one(2)
thf(fact_1015_domain_Opoly__mult__one_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P2 )
=> ( ( poly_mult_a_b @ R @ P2 @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) )
= P2 ) ) ) ) ).
% domain.poly_mult_one(2)
thf(fact_1016_domain_Opoly__mult__one_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K3 @ P2 )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) @ P2 )
= P2 ) ) ) ) ).
% domain.poly_mult_one(1)
thf(fact_1017_domain_Opoly__mult__one_I1_J,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( ( polyno3115169382166032176t_unit @ R @ K3 @ P2 )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) @ P2 )
= P2 ) ) ) ) ).
% domain.poly_mult_one(1)
thf(fact_1018_domain_Opoly__mult__one_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( polynomial_a_b @ R @ K3 @ P2 )
=> ( ( poly_mult_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) @ P2 )
= P2 ) ) ) ) ).
% domain.poly_mult_one(1)
thf(fact_1019_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P2 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) @ P2 )
= ( normal4052021864830707619t_unit @ R @ P2 ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_1020_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) @ P2 )
= ( normalize_a_b @ R @ P2 ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_1021_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) @ P2 )
= ( normal637505603836502915t_unit @ R @ P2 ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_1022_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P2 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P2 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) )
= ( normal4052021864830707619t_unit @ R @ P2 ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1023_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P2 @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) )
= ( normalize_a_b @ R @ P2 ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1024_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P2 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) )
= ( normal637505603836502915t_unit @ R @ P2 ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1025_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( zero_a_b @ r ) )
=> ( ( H12
= ( zero_a_b @ r ) )
| ( H23
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_1026_Group_Onat__pow__0,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( pow_li1142815632869257134it_nat @ G2 @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_1027_Group_Onat__pow__0,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_1028_Group_Onat__pow__0,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a] :
( ( pow_se8252319793075206062it_nat @ G2 @ X @ zero_zero_nat )
= ( one_se1127990129394575805t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_1029_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_1030_polynomial__pow__not__zero,axiom,
! [P2: list_a,N: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_1031_subring__polynomial__pow__not__zero,axiom,
! [K3: set_a,P2: list_a,N: nat] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_1032_var__pow__closed,axiom,
! [K3: set_a,N: nat] :
( ( subring_a_b @ K3 @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) ) ) ).
% var_pow_closed
thf(fact_1033_polynomial__pow__division,axiom,
! [P2: list_a,N: nat,M: nat] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_1034_unitary__monom__eq__var__pow,axiom,
! [K3: set_a,N: nat] :
( ( subring_a_b @ K3 @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_1035_pirreducible__pow__pdivides__iff,axiom,
! [K3: set_a,P2: list_a,Q2: list_a,R2: list_a,N: nat] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K3 ) @ Q2 @ R2 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_1036_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ N )
!= nil_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_1037_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ N )
!= nil_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_1038_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ N )
!= nil_list_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_1039_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ N )
!= nil_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_1040_domain_Ovar__pow__closed,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_1041_domain_Ovar__pow__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K3 ) @ ( var_a_b @ R ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_1042_domain_Opolynomial__pow__division,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_1043_domain_Opolynomial__pow__division,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,N: nat,M: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_1044_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( monom_7446464087056152608t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_1045_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia7496981018696276118t_unit,K3: set_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K3 @ R )
=> ( ( monom_317758005976320064t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ N )
= ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ K3 ) @ ( var_se6008125447796440765t_unit @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_1046_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( monom_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K3 ) @ ( var_a_b @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_1047_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
=> ( ~ ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
=> ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ Q2 @ R2 ) )
= ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ N ) @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_1048_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a,R2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
=> ( ~ ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K3 ) @ Q2 @ R2 ) )
= ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ N ) @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_1049_rupture__one__not__zero,axiom,
! [K3: set_a,P2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K3 @ P2 ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K3 @ P2 ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_1050_roots__inclI,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != 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 @ A4 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A4 ) ) @ Q2 ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ) ).
% roots_inclI
thf(fact_1051_subring__props_I5_J,axiom,
! [K3: set_a,H2: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H2 @ K3 )
=> ( member_a @ ( a_inv_a_b @ r @ H2 ) @ K3 ) ) ) ).
% subring_props(5)
thf(fact_1052_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_1053_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_1054_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_1055_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_1056_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_1057_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_1058_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_1059_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_1060_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_1061_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_1062_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_1063_sum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( X
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_1064_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_1065_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_1066_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_1067_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_1068_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_1069_subringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_1070_pdivides__imp__is__root,axiom,
! [P2: list_a,X: a] :
( ( P2 != 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 ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_1071_is__root__imp__pdivides,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 ) ) ) ).
% is_root_imp_pdivides
thf(fact_1072_alg__multE_I1_J,axiom,
! [X: a,P2: list_a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != 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 @ P2 @ X ) ) @ P2 ) ) ) ) ).
% alg_multE(1)
thf(fact_1073_le__alg__mult__imp__pdivides,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ 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 ) @ P2 ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1074_alg__multE_I2_J,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != 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 ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1075_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_1076_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_1077_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_1078_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_1079_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P2 ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_1080_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( const_term_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) )
= ( a_inv_a_b @ R @ ( const_term_a_b @ R @ P2 ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_1081_domain_Opdivides__imp__is__root,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( P2 != nil_set_list_a )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ P2 )
=> ( polyno4320237611291262604t_unit @ R @ P2 @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_1082_domain_Opdivides__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( P2 != nil_list_a )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P2 )
=> ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_1083_domain_Opdivides__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( P2 != 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 ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_1084_domain_Ois__root__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno4320237611291262604t_unit @ R @ P2 @ X )
=> ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ P2 ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_1085_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
=> ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P2 ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_1086_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ P2 ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_1087_domain_Oalg__multE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P2 != nil_set_list_a )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ ( polyno1088517687229135038t_unit @ R @ P2 @ X ) ) @ P2 ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1088_domain_Oalg__multE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R @ P2 @ X ) ) @ P2 ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1089_domain_Oalg__multE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P2 != 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 @ P2 @ X ) ) @ P2 ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1090_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,P2: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R @ P2 @ X ) )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ N ) @ P2 ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1091_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,P2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P2 @ X ) )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ N ) @ P2 ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1092_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,P2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P2 @ 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 ) @ P2 ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1093_domain_Oalg__multE_I2_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,P2: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P2 != nil_set_list_a )
=> ( ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ N ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R @ P2 @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1094_domain_Oalg__multE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,P2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ N ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P2 @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1095_domain_Oalg__multE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,P2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P2 != 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 ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P2 @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1096_domain_Oroots__inclI,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,Q2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member5524387281408368019list_a @ Q2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( Q2 != nil_set_list_a )
=> ( ! [A4: set_list_a] :
( ( member_set_list_a @ A4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( P2 != nil_set_list_a )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ A4 ) @ nil_set_list_a ) ) @ ( polyno1088517687229135038t_unit @ R @ P2 @ A4 ) ) @ Q2 ) ) )
=> ( subset4236506274861796683list_a @ ( polyno4169377219242390531t_unit @ R @ P2 ) @ ( polyno4169377219242390531t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_1097_domain_Oroots__inclI,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( P2 != nil_list_a )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A4 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R @ P2 @ A4 ) ) @ Q2 ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) @ ( polyno7858422826990252003t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_1098_domain_Oroots__inclI,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( P2 != 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 @ A4 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ R @ P2 @ A4 ) ) @ Q2 ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R @ P2 ) @ ( polynomial_roots_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_1099_domain_Orupture__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) )
=> ( ( one_se2489417650821308733t_unit @ ( polyno859807163042199155t_unit @ R @ K3 @ P2 ) )
!= ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ R @ K3 @ P2 ) ) ) ) ) ) ) ).
% domain.rupture_one_not_zero
thf(fact_1100_domain_Orupture__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ R @ K3 @ P2 ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ R @ K3 @ P2 ) ) ) ) ) ) ) ).
% domain.rupture_one_not_zero
thf(fact_1101_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_1102_not__empty__rootsE,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P2 )
!= zero_zero_multiset_a )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A4 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A4 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A4 ) @ nil_a ) ) @ P2 ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_1103_univ__poly__a__inv__consistent,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_1104_univ__poly__a__inv__length,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) )
= ( size_size_list_a @ P2 ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_1105_long__division__a__inv_I2_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1106_long__division__a__inv_I1_J,axiom,
! [K3: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1107_const__term__simprules__shell_I4_J,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P2 ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_1108_univ__poly__a__inv__degree,axiom,
! [K3: set_a,P2: list_a] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K3 ) @ P2 ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_1109_roots__mem__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% roots_mem_iff_is_root
thf(fact_1110_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_1111_count__greater__zero__iff,axiom,
! [M3: multiset_list_a,X: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_list_a @ M3 @ X ) )
= ( member_list_a @ X @ ( set_mset_list_a @ M3 ) ) ) ).
% count_greater_zero_iff
thf(fact_1112_count__greater__zero__iff,axiom,
! [M3: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M3 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M3 ) ) ) ).
% count_greater_zero_iff
thf(fact_1113_count__greater__eq__one__iff,axiom,
! [M3: multiset_list_a,X: list_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_list_a @ M3 @ X ) )
= ( member_list_a @ X @ ( set_mset_list_a @ M3 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_1114_count__greater__eq__one__iff,axiom,
! [M3: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M3 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M3 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_1115_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_1116_count__eq__zero__iff,axiom,
! [M3: multiset_list_a,X: list_a] :
( ( ( count_list_a @ M3 @ X )
= zero_zero_nat )
= ( ~ ( member_list_a @ X @ ( set_mset_list_a @ M3 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1117_count__eq__zero__iff,axiom,
! [M3: multiset_a,X: a] :
( ( ( count_a @ M3 @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M3 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1118_count__inI,axiom,
! [M3: multiset_list_a,X: list_a] :
( ( ( count_list_a @ M3 @ X )
!= zero_zero_nat )
=> ( member_list_a @ X @ ( set_mset_list_a @ M3 ) ) ) ).
% count_inI
thf(fact_1119_count__inI,axiom,
! [M3: multiset_a,X: a] :
( ( ( count_a @ M3 @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M3 ) ) ) ).
% count_inI
thf(fact_1120_mset__subset__eqD,axiom,
! [A3: multiset_list_a,B3: multiset_list_a,X: list_a] :
( ( subseteq_mset_list_a @ A3 @ B3 )
=> ( ( member_list_a @ X @ ( set_mset_list_a @ A3 ) )
=> ( member_list_a @ X @ ( set_mset_list_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_1121_mset__subset__eqD,axiom,
! [A3: multiset_a,B3: multiset_a,X: a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ( member_a @ X @ ( set_mset_a @ A3 ) )
=> ( member_a @ X @ ( set_mset_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_1122_multiset__nonemptyE,axiom,
! [A3: multiset_list_a] :
( ( A3 != zero_z4454100511807792257list_a )
=> ~ ! [X2: list_a] :
~ ( member_list_a @ X2 @ ( set_mset_list_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_1123_multiset__nonemptyE,axiom,
! [A3: multiset_a] :
( ( A3 != zero_zero_multiset_a )
=> ~ ! [X2: a] :
~ ( member_a @ X2 @ ( set_mset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_1124_union__iff,axiom,
! [A: list_a,A3: multiset_list_a,B3: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( plus_p690419498615200257list_a @ A3 @ B3 ) ) )
= ( ( member_list_a @ A @ ( set_mset_list_a @ A3 ) )
| ( member_list_a @ A @ ( set_mset_list_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_1125_union__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A3 ) )
| ( member_a @ A @ ( set_mset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_1126_in__diffD,axiom,
! [A: list_a,M3: multiset_list_a,N4: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ N4 ) ) )
=> ( member_list_a @ A @ ( set_mset_list_a @ M3 ) ) ) ).
% in_diffD
thf(fact_1127_in__diffD,axiom,
! [A: a,M3: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M3 ) ) ) ).
% in_diffD
thf(fact_1128_set__mset__mono,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A3 ) @ ( set_mset_a @ B3 ) ) ) ).
% set_mset_mono
thf(fact_1129_in__diff__count,axiom,
! [A: list_a,M3: multiset_list_a,N4: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ N4 ) ) )
= ( ord_less_nat @ ( count_list_a @ N4 @ A ) @ ( count_list_a @ M3 @ A ) ) ) ).
% in_diff_count
thf(fact_1130_in__diff__count,axiom,
! [A: a,M3: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) ) )
= ( ord_less_nat @ ( count_a @ N4 @ A ) @ ( count_a @ M3 @ A ) ) ) ).
% in_diff_count
thf(fact_1131_count__in__diffI,axiom,
! [N4: multiset_list_a,X: list_a,M3: multiset_list_a] :
( ! [N3: nat] :
( ( count_list_a @ N4 @ X )
!= ( plus_plus_nat @ N3 @ ( count_list_a @ M3 @ X ) ) )
=> ( member_list_a @ X @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ N4 ) ) ) ) ).
% count_in_diffI
thf(fact_1132_count__in__diffI,axiom,
! [N4: multiset_a,X: a,M3: multiset_a] :
( ! [N3: nat] :
( ( count_a @ N4 @ X )
!= ( plus_plus_nat @ N3 @ ( count_a @ M3 @ X ) ) )
=> ( member_a @ X @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ N4 ) ) ) ) ).
% count_in_diffI
thf(fact_1133_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_1134_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_1135_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) )
= ( size_s349497388124573686list_a @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_1136_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) )
= ( size_size_list_a @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_1137_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) @ Q2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_1138_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_1139_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) @ Q2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_1140_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_1141_domain_Oroots__mem__iff__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ X @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) ) )
= ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_1142_domain_Oroots__mem__iff__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ R @ P2 ) ) )
= ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_1143_domain_Ouniv__poly__a__inv__degree,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% domain.univ_poly_a_inv_degree
thf(fact_1144_domain_Ouniv__poly__a__inv__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K3 ) @ P2 ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% domain.univ_poly_a_inv_degree
thf(fact_1145_domain_Onot__empty__rootsE,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ( polyno4169377219242390531t_unit @ R @ P2 )
!= zero_z7061913751530476641list_a )
=> ~ ! [A4: set_list_a] :
( ( member_set_list_a @ A4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ A4 @ ( set_mset_set_list_a @ ( polyno4169377219242390531t_unit @ R @ P2 ) ) )
=> ( ( member5524387281408368019list_a @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ A4 ) @ nil_set_list_a ) ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ~ ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ A4 ) @ nil_set_list_a ) ) @ P2 ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_1146_domain_Onot__empty__rootsE,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ R @ P2 )
!= zero_z4454100511807792257list_a )
=> ~ ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A4 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) ) )
=> ( ( member_list_list_a @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A4 ) @ nil_list_a ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ~ ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A4 ) @ nil_list_a ) ) @ P2 ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_1147_domain_Onot__empty__rootsE,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( polynomial_roots_a_b @ R @ P2 )
!= zero_zero_multiset_a )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A4 @ ( set_mset_a @ ( polynomial_roots_a_b @ R @ P2 ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ A4 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ A4 ) @ nil_a ) ) @ P2 ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_1148_subring__polynomial__pow__degree,axiom,
! [K3: set_a,P2: list_a,N: nat] :
( ( subring_a_b @ K3 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K3 ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K3 ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% subring_polynomial_pow_degree
thf(fact_1149_polynomial__pow__degree,axiom,
! [P2: list_a,N: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% polynomial_pow_degree
thf(fact_1150_nat__pow__pow,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ M )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% nat_pow_pow
thf(fact_1151_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1152_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1153_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1154_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1155_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1156_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1157_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1158_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1159_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1160_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1161_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1162_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1163_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1164_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1165_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1166_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1167_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1168_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1169_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1170_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1171_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1172_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1173_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1174_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1175_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1176_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1177_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1178_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1179_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1180_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1181_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1182_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1183_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1184_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1185_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1186_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_1187_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1188_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1189_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1190_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1191_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1192_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1193_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1194_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1195_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1196_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1197_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1198_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1199_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1200_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1201_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1202_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1203_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1204_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1205_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1206_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1207_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1208_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1209_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1210_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1211_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1212_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1213_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1214_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1215_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1216_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1217_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1218_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1219_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1220_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1221_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1222_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1223_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1224_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1225_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1226_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1227_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1228_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1229_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1230_domain_Opolynomial__pow__degree,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% domain.polynomial_pow_degree
thf(fact_1231_domain_Opolynomial__pow__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% domain.polynomial_pow_degree
thf(fact_1232_domain_Osubring__polynomial__pow__degree,axiom,
! [R: partia2670972154091845814t_unit,K3: set_list_a,P2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K3 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K3 ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K3 ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ).
% domain.subring_polynomial_pow_degree
thf(fact_1233_domain_Osubring__polynomial__pow__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K3: set_a,P2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K3 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K3 ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K3 ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ).
% domain.subring_polynomial_pow_degree
thf(fact_1234_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1235_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1236_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1237_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1238_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1239_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1240_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1241_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1242_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1243_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1244_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1245_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1246_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1247_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1248_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1249_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_1250_multi__self__add__other__not__self,axiom,
! [M3: multiset_a,X: a] :
( M3
!= ( add_mset_a @ X @ M3 ) ) ).
% multi_self_add_other_not_self
thf(fact_1251_add__mset__add__mset__same__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( ( add_mset_a @ A @ A3 )
= ( add_mset_a @ A @ B3 ) )
= ( A3 = B3 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_1252_add__mset__eq__singleton__iff,axiom,
! [X: a,M3: multiset_a,Y: a] :
( ( ( add_mset_a @ X @ M3 )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M3 = zero_zero_multiset_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_1253_single__eq__add__mset,axiom,
! [A: a,B: a,M3: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M3 ) )
= ( ( B = A )
& ( M3 = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_1254_add__mset__eq__single,axiom,
! [B: a,M3: multiset_a,A: a] :
( ( ( add_mset_a @ B @ M3 )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( B = A )
& ( M3 = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_1255_single__eq__single,axiom,
! [A: a,B: a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_1256_union__mset__add__mset__left,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A3 ) @ B3 )
= ( add_mset_a @ A @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_left
thf(fact_1257_union__mset__add__mset__right,axiom,
! [A3: multiset_a,A: a,B3: multiset_a] :
( ( plus_plus_multiset_a @ A3 @ ( add_mset_a @ A @ B3 ) )
= ( add_mset_a @ A @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_right
thf(fact_1258_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_1259_add__mset__subseteq__single__iff,axiom,
! [A: a,M3: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ M3 ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M3 = zero_zero_multiset_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_1260_add__mset__remove__trivial,axiom,
! [X: a,M3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M3 ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M3 ) ).
% add_mset_remove_trivial
thf(fact_1261_diff__add__mset__swap,axiom,
! [B: list_a,A3: multiset_list_a,M3: multiset_list_a] :
( ~ ( member_list_a @ B @ ( set_mset_list_a @ A3 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ M3 ) @ A3 )
= ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ M3 @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_1262_diff__add__mset__swap,axiom,
! [B: a,A3: multiset_a,M3: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M3 ) @ A3 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M3 @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_1263_single__subset__iff,axiom,
! [A: list_a,M3: multiset_list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) @ M3 )
= ( member_list_a @ A @ ( set_mset_list_a @ M3 ) ) ) ).
% single_subset_iff
thf(fact_1264_single__subset__iff,axiom,
! [A: a,M3: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M3 )
= ( member_a @ A @ ( set_mset_a @ M3 ) ) ) ).
% single_subset_iff
thf(fact_1265_insert__DiffM,axiom,
! [X: list_a,M3: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ( add_mset_list_a @ X @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= M3 ) ) ).
% insert_DiffM
thf(fact_1266_insert__DiffM,axiom,
! [X: a,M3: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M3 ) ) ).
% insert_DiffM
thf(fact_1267_diff__union__swap2,axiom,
! [Y: list_a,M3: multiset_list_a,X: list_a] :
( ( member_list_a @ Y @ ( set_mset_list_a @ M3 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ X @ M3 ) @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) )
= ( add_mset_list_a @ X @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_1268_diff__union__swap2,axiom,
! [Y: a,M3: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M3 ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_1269_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M3: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X2: nat,M8: multiset_nat] :
( ( P @ M8 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M8 ) )
=> ( ord_less_eq_nat @ X2 @ Xa ) )
=> ( P @ ( add_mset_nat @ X2 @ M8 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_min
thf(fact_1270_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M3: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X2: nat,M8: multiset_nat] :
( ( P @ M8 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M8 ) )
=> ( ord_less_eq_nat @ Xa @ X2 ) )
=> ( P @ ( add_mset_nat @ X2 @ M8 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_max
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
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 ) ) ).
%------------------------------------------------------------------------------