TPTP Problem File: SLH0903^1.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_00186_006814__17277800_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1119 ( 313 unt; 181 typ; 0 def)
% Number of atoms : 2728 (1104 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12112 ( 239 ~; 38 |; 76 &;10203 @)
% ( 0 <=>;1556 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 527 ( 527 >; 0 *; 0 +; 0 <<)
% Number of symbols : 161 ( 158 usr; 13 con; 0-4 aty)
% Number of variables : 2691 ( 137 ^;2517 !; 37 ?;2691 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:37:46.186
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001tf__a_001tf__b,type,
polyno5814909790663948098es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno3522816881121920896t_unit: partia2670972154091845814t_unit > list_a > nat > list_list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001tf__a_001tf__b,type,
polyno2922411391617481336se_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001tf__a_001tf__b,type,
polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Oroots__on_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno5990348478334826338t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > multiset_list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Oroots__on_001tf__a_001tf__b,type,
polyno5714441830345289050on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > multiset_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted_001tf__a_001tf__b,type,
polyno8329700637149614481ed_a_b: partia2175431115845679010xt_a_b > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted__on_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno1986131841096413848t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted__on_001tf__a_001tf__b,type,
polyno2453258491555121552on_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno1315193887021588240t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001tf__a_001tf__b,type,
polynomial_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomials_Oring_Ocoeff_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
coeff_6360649920519955023t_unit: partia2670972154091845814t_unit > list_list_a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Ocoeff_001tf__a_001tf__b,type,
coeff_a_b: partia2175431115845679010xt_a_b > list_a > nat > a ).
thf(sy_c_Polynomials_Oring_Oconst__term_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
const_6738166269504826821t_unit: partia2670972154091845814t_unit > list_list_a > list_a ).
thf(sy_c_Polynomials_Oring_Oconst__term_001tf__a_001tf__b,type,
const_term_a_b: partia2175431115845679010xt_a_b > list_a > a ).
thf(sy_c_Polynomials_Oring_Odense__repr_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
dense_5814815041220002634t_unit: partia2670972154091845814t_unit > list_list_a > list_P1129550237270585747_a_nat ).
thf(sy_c_Polynomials_Oring_Odense__repr_001tf__a_001tf__b,type,
dense_repr_a_b: partia2175431115845679010xt_a_b > list_a > list_P3592885314253461005_a_nat ).
thf(sy_c_Polynomials_Oring_Omonom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
monom_7446464087056152608t_unit: partia2670972154091845814t_unit > list_a > nat > list_list_a ).
thf(sy_c_Polynomials_Oring_Omonom_001tf__a_001tf__b,type,
monom_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Onormalize_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
normal637505603836502915t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Onormalize_001tf__a_001tf__b,type,
normalize_a_b: partia2175431115845679010xt_a_b > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_a7601779127272115787t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001tf__a_001tf__b,type,
poly_add_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__mult_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_m7087347720095500472t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__mult_001tf__a_001tf__b,type,
poly_mult_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__dense_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_o2635896782027652242t_unit: partia2670972154091845814t_unit > list_P1129550237270585747_a_nat > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__dense_001tf__a_001tf__b,type,
poly_of_dense_a_b: partia2175431115845679010xt_a_b > list_P3592885314253461005_a_nat > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
univ_p2250591967980070728t_unit: partia2956882679547061052t_unit > set_list_list_a > partia5333488208502193986t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
a_minu4820293213911669576t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_minu2241224857956505934t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oabelian__group_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
abelia2778853791629620336t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Oabelian__group_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia3891852623213500406t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
abelian_group_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia226231641709521465t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
domain7810152921033798211t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_l1939023646219158831t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > 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_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
collect_nat_list_a: ( ( nat > list_a ) > $o ) > set_nat_list_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_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_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
% Relevant facts (932)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_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_2_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_3_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X: set_list_a] : ( ord_le8861187494160871172list_a @ X @ X ) ).
% order_refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% dual_order.refl
thf(fact_9_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_10_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_11_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_12_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_13_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_14_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_15_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_16_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_17_order__antisym__conv,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( ord_le8861187494160871172list_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_18_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_19_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_20_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_21_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_22_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_23_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_24_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_25_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_26_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_27_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_28_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_29_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_30_ord__eq__le__subst,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_31_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_32_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_33_ord__eq__le__subst,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_34_ord__eq__le__subst,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_35_ord__eq__le__subst,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_36_ord__eq__le__subst,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_37_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_38_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_39_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_40_order__eq__refl,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( X = Y )
=> ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_41_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_42_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_43_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_44_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_45_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_46_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_47_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_48_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_49_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_50_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_51_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_52_order__subst1,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_53_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_54_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_55_order__subst1,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_56_order__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_57_order__subst1,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_58_order__subst1,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_59_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_60_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_61_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
& ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_62_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_63_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_64_antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_65_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_66_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_67_dual__order_Otrans,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_68_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_69_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_70_dual__order_Oantisym,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_71_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_72_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_73_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A2 )
& ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_74_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_75_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_76_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_77_order__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z2 )
=> ( ord_le8861187494160871172list_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_78_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_79_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_80_order_Otrans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% order.trans
thf(fact_81_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_82_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_83_order__antisym,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_84_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_85_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_86_ord__le__eq__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( B = C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_87_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_88_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_89_ord__eq__le__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( A = B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_90_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_91_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_92_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_93_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_94_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_95_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_96_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_97_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_98_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_99_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_100_size__neq__size__imp__neq,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ X )
!= ( size_s2335926164413107382list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_101_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A: nat > list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A @ ( collect_nat_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A4: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A4: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A4: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A4: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_111_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_112_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_113_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_114_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_115_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_116_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_117_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_118_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_119_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_120_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_121_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_122_normalize__def_H_I2_J,axiom,
! [P: list_a] :
( ( normalize_a_b @ r @ P )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ P ) ) ).
% normalize_def'(2)
thf(fact_123_dense__repr__normalize,axiom,
! [P: list_a] :
( ( dense_repr_a_b @ r @ ( normalize_a_b @ r @ P ) )
= ( dense_repr_a_b @ r @ P ) ) ).
% dense_repr_normalize
thf(fact_124_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_125_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_126_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_127_normalize__polynomial,axiom,
! [K2: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( normalize_a_b @ r @ P )
= P ) ) ).
% normalize_polynomial
thf(fact_128_normalize__coeff,axiom,
! [P: list_a] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_coeff
thf(fact_129_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_130_coeff__iff__length__cond,axiom,
! [P1: list_a,P22: list_a] :
( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P22 ) )
=> ( ( P1 = P22 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P22 ) ) ) ) ).
% coeff_iff_length_cond
thf(fact_131_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_132_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_133_coeff__iff__polynomial__cond,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( ( P1 = P22 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P22 ) ) ) ) ) ).
% coeff_iff_polynomial_cond
thf(fact_134_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_135_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_136_drop__exp__base,axiom,
! [N: nat,X: a,M: nat] :
( ( drop_a @ N @ ( polyno2922411391617481336se_a_b @ r @ X @ M ) )
= ( polyno2922411391617481336se_a_b @ r @ X @ ( minus_minus_nat @ M @ N ) ) ) ).
% drop_exp_base
thf(fact_137_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_138_drop__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_139_drop__append,axiom,
! [N: nat,Xs: list_list_a,Ys: list_list_a] :
( ( drop_list_a @ N @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( drop_list_a @ N @ Xs ) @ ( drop_list_a @ ( minus_minus_nat @ N @ ( size_s349497388124573686list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_140_drop__all,axiom,
! [Xs: list_P3592885314253461005_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs ) @ N )
=> ( ( drop_P2883665741211355575_a_nat @ N @ Xs )
= nil_Pr7402525243500994295_a_nat ) ) ).
% drop_all
thf(fact_141_drop__all,axiom,
! [Xs: list_P1129550237270585747_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s4007175526842055207_a_nat @ Xs ) @ N )
=> ( ( drop_P8823092557689526717_a_nat @ N @ Xs )
= nil_Pr6246850598307483389_a_nat ) ) ).
% drop_all
thf(fact_142_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_143_drop__all,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( drop_list_a @ N @ Xs )
= nil_list_a ) ) ).
% drop_all
thf(fact_144_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P3592885314253461005_a_nat] :
( ( ( drop_P2883665741211355575_a_nat @ N @ Xs )
= nil_Pr7402525243500994295_a_nat )
= ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_145_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P1129550237270585747_a_nat] :
( ( ( drop_P8823092557689526717_a_nat @ N @ Xs )
= nil_Pr6246850598307483389_a_nat )
= ( ord_less_eq_nat @ ( size_s4007175526842055207_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_146_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_147_drop__eq__Nil,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( drop_list_a @ N @ Xs )
= nil_list_a )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_148_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( drop_P2883665741211355575_a_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_149_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P1129550237270585747_a_nat] :
( ( nil_Pr6246850598307483389_a_nat
= ( drop_P8823092557689526717_a_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s4007175526842055207_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_150_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_151_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_list_a] :
( ( nil_list_a
= ( drop_list_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_152_length__drop,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% length_drop
thf(fact_153_length__drop,axiom,
! [N: nat,Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( drop_list_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% length_drop
thf(fact_154_splitted__on__def,axiom,
! [K2: set_a,P: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K2 @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K2 @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_155_ring_Onormalize__def_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ P )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) ) @ P ) ) ) ).
% ring.normalize_def'(2)
thf(fact_156_ring_Onormalize__def_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ P )
= ( drop_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ P ) ) ) ).
% ring.normalize_def'(2)
thf(fact_157_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_158_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_159_dense__repr_Osimps_I1_J,axiom,
( ( dense_repr_a_b @ r @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ).
% dense_repr.simps(1)
thf(fact_160_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_161_append_Oassoc,axiom,
! [A: list_list_a,B: list_list_a,C: list_list_a] :
( ( append_list_a @ ( append_list_a @ A @ B ) @ C )
= ( append_list_a @ A @ ( append_list_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_162_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_163_append__assoc,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( append_list_a @ ( append_list_a @ Xs @ Ys ) @ Zs )
= ( append_list_a @ Xs @ ( append_list_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_164_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_165_append__same__eq,axiom,
! [Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Ys @ Xs )
= ( append_list_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_166_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_167_same__append__eq,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_168_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_169_append_Oright__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ A @ nil_Pr7402525243500994295_a_nat )
= A ) ).
% append.right_neutral
thf(fact_170_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_171_append_Oright__neutral,axiom,
! [A: list_P1129550237270585747_a_nat] :
( ( append6709258537191095464_a_nat @ A @ nil_Pr6246850598307483389_a_nat )
= A ) ).
% append.right_neutral
thf(fact_172_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_173_append__Nil2,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ Xs @ nil_Pr7402525243500994295_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_174_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_175_append__Nil2,axiom,
! [Xs: list_P1129550237270585747_a_nat] :
( ( append6709258537191095464_a_nat @ Xs @ nil_Pr6246850598307483389_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_176_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_177_append__self__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv
thf(fact_178_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_179_append__self__conv,axiom,
! [Xs: list_P1129550237270585747_a_nat,Ys: list_P1129550237270585747_a_nat] :
( ( ( append6709258537191095464_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr6246850598307483389_a_nat ) ) ).
% append_self_conv
thf(fact_180_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_181_self__append__conv,axiom,
! [Y: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Y
= ( append7679239579558125090_a_nat @ Y @ Ys ) )
= ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv
thf(fact_182_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_183_self__append__conv,axiom,
! [Y: list_P1129550237270585747_a_nat,Ys: list_P1129550237270585747_a_nat] :
( ( Y
= ( append6709258537191095464_a_nat @ Y @ Ys ) )
= ( Ys = nil_Pr6246850598307483389_a_nat ) ) ).
% self_append_conv
thf(fact_184_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_185_append__self__conv2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv2
thf(fact_186_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_187_append__self__conv2,axiom,
! [Xs: list_P1129550237270585747_a_nat,Ys: list_P1129550237270585747_a_nat] :
( ( ( append6709258537191095464_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr6246850598307483389_a_nat ) ) ).
% append_self_conv2
thf(fact_188_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_189_self__append__conv2,axiom,
! [Y: list_P3592885314253461005_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( Y
= ( append7679239579558125090_a_nat @ Xs @ Y ) )
= ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv2
thf(fact_190_self__append__conv2,axiom,
! [Y: list_list_a,Xs: list_list_a] :
( ( Y
= ( append_list_a @ Xs @ Y ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_191_self__append__conv2,axiom,
! [Y: list_P1129550237270585747_a_nat,Xs: list_P1129550237270585747_a_nat] :
( ( Y
= ( append6709258537191095464_a_nat @ Xs @ Y ) )
= ( Xs = nil_Pr6246850598307483389_a_nat ) ) ).
% self_append_conv2
thf(fact_192_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_193_Nil__is__append__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( append7679239579558125090_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr7402525243500994295_a_nat )
& ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_194_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_195_Nil__is__append__conv,axiom,
! [Xs: list_P1129550237270585747_a_nat,Ys: list_P1129550237270585747_a_nat] :
( ( nil_Pr6246850598307483389_a_nat
= ( append6709258537191095464_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr6246850598307483389_a_nat )
& ( Ys = nil_Pr6246850598307483389_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_196_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_197_append__is__Nil__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
= nil_Pr7402525243500994295_a_nat )
= ( ( Xs = nil_Pr7402525243500994295_a_nat )
& ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_198_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_199_append__is__Nil__conv,axiom,
! [Xs: list_P1129550237270585747_a_nat,Ys: list_P1129550237270585747_a_nat] :
( ( ( append6709258537191095464_a_nat @ Xs @ Ys )
= nil_Pr6246850598307483389_a_nat )
= ( ( Xs = nil_Pr6246850598307483389_a_nat )
& ( Ys = nil_Pr6246850598307483389_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_200_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_201_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_202_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_203_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_204_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_205_append__eq__appendI,axiom,
! [Xs: list_list_a,Xs1: list_list_a,Zs: list_list_a,Ys: list_list_a,Us: list_list_a] :
( ( ( append_list_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_a @ Xs1 @ Us ) )
=> ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_206_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_207_append__eq__append__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,Ts: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Ts ) )
= ( ? [Us2: list_list_a] :
( ( ( Xs
= ( append_list_a @ Zs @ Us2 ) )
& ( ( append_list_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_list_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_208_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_209_ring_Onormalize_Ocong,axiom,
normal637505603836502915t_unit = normal637505603836502915t_unit ).
% ring.normalize.cong
thf(fact_210_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_211_ring_Ocoeff_Ocong,axiom,
coeff_6360649920519955023t_unit = coeff_6360649920519955023t_unit ).
% ring.coeff.cong
thf(fact_212_ring_Odense__repr_Ocong,axiom,
dense_repr_a_b = dense_repr_a_b ).
% ring.dense_repr.cong
thf(fact_213_ring_Odense__repr_Ocong,axiom,
dense_5814815041220002634t_unit = dense_5814815041220002634t_unit ).
% ring.dense_repr.cong
thf(fact_214_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_215_append__Nil,axiom,
! [Ys: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_216_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_217_append__Nil,axiom,
! [Ys: list_P1129550237270585747_a_nat] :
( ( append6709258537191095464_a_nat @ nil_Pr6246850598307483389_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_218_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_219_append_Oleft__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_220_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_221_append_Oleft__neutral,axiom,
! [A: list_P1129550237270585747_a_nat] :
( ( append6709258537191095464_a_nat @ nil_Pr6246850598307483389_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_222_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_223_eq__Nil__appendI,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_224_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_225_eq__Nil__appendI,axiom,
! [Xs: list_P1129550237270585747_a_nat,Ys: list_P1129550237270585747_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append6709258537191095464_a_nat @ nil_Pr6246850598307483389_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_226_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_227_drop__Nil,axiom,
! [N: nat] :
( ( drop_P2883665741211355575_a_nat @ N @ nil_Pr7402525243500994295_a_nat )
= nil_Pr7402525243500994295_a_nat ) ).
% drop_Nil
thf(fact_228_drop__Nil,axiom,
! [N: nat] :
( ( drop_list_a @ N @ nil_list_a )
= nil_list_a ) ).
% drop_Nil
thf(fact_229_drop__Nil,axiom,
! [N: nat] :
( ( drop_P8823092557689526717_a_nat @ N @ nil_Pr6246850598307483389_a_nat )
= nil_Pr6246850598307483389_a_nat ) ).
% drop_Nil
thf(fact_230_ring_Ozero__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( polynomial_a_b @ R @ K2 @ nil_a ) ) ).
% ring.zero_is_polynomial
thf(fact_231_ring_Ozero__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( polyno1315193887021588240t_unit @ R @ K2 @ nil_list_a ) ) ).
% ring.zero_is_polynomial
thf(fact_232_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ nil_a )
= nil_a ) ) ).
% ring.normalize.simps(1)
thf(fact_233_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ nil_list_a )
= nil_list_a ) ) ).
% ring.normalize.simps(1)
thf(fact_234_ring_Ocoeff__iff__length__cond,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P22 ) )
=> ( ( P1 = P22 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P22 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_235_ring_Ocoeff__iff__length__cond,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( size_s349497388124573686list_a @ P1 )
= ( size_s349497388124573686list_a @ P22 ) )
=> ( ( P1 = P22 )
= ( ( coeff_6360649920519955023t_unit @ R @ P1 )
= ( coeff_6360649920519955023t_unit @ R @ P22 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_236_ring_Onormalize__idem,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( normalize_a_b @ R @ P ) @ Q ) )
= ( normalize_a_b @ R @ ( append_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_237_ring_Onormalize__idem,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( normal637505603836502915t_unit @ R @ P ) @ Q ) )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_238_ring_Odense__repr_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ) ).
% ring.dense_repr.simps(1)
thf(fact_239_ring_Odense__repr_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( dense_5814815041220002634t_unit @ R @ nil_list_a )
= nil_Pr6246850598307483389_a_nat ) ) ).
% ring.dense_repr.simps(1)
thf(fact_240_ring_Onormalize__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ( normalize_a_b @ R @ P )
= P ) ) ) ).
% ring.normalize_polynomial
thf(fact_241_ring_Onormalize__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ( normal637505603836502915t_unit @ R @ P )
= P ) ) ) ).
% ring.normalize_polynomial
thf(fact_242_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P1 )
=> ( ( polynomial_a_b @ R @ K2 @ P22 )
=> ( ( P1 = P22 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P22 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_243_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P1 )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P22 )
=> ( ( P1 = P22 )
= ( ( coeff_6360649920519955023t_unit @ R @ P1 )
= ( coeff_6360649920519955023t_unit @ R @ P22 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_244_ring_Onormalize__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ P )
= ( coeff_a_b @ R @ ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.normalize_coeff
thf(fact_245_ring_Onormalize__coeff,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ P )
= ( coeff_6360649920519955023t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.normalize_coeff
thf(fact_246_ring_Odense__repr__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ ( normalize_a_b @ R @ P ) )
= ( dense_repr_a_b @ R @ P ) ) ) ).
% ring.dense_repr_normalize
thf(fact_247_ring_Odense__repr__normalize,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( dense_5814815041220002634t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) )
= ( dense_5814815041220002634t_unit @ R @ P ) ) ) ).
% ring.dense_repr_normalize
thf(fact_248_ring_Onormalize__length__le,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) @ ( size_size_list_a @ P ) ) ) ).
% ring.normalize_length_le
thf(fact_249_ring_Onormalize__length__le,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ) ).
% ring.normalize_length_le
thf(fact_250_ring_Osplitted__on__def,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polyno2453258491555121552on_a_b @ R @ K2 @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ R @ K2 @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% ring.splitted_on_def
thf(fact_251_ring_Osplitted__on__def,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno1986131841096413848t_unit @ R @ K2 @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ R @ K2 @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ).
% ring.splitted_on_def
thf(fact_252_append__coeff,axiom,
! [P: list_a,Q: list_a] :
( ( coeff_a_b @ r @ ( append_a @ P @ Q ) )
= ( ^ [I2: nat] : ( if_a @ ( ord_less_nat @ I2 @ ( size_size_list_a @ Q ) ) @ ( coeff_a_b @ r @ Q @ I2 ) @ ( coeff_a_b @ r @ P @ ( minus_minus_nat @ I2 @ ( size_size_list_a @ Q ) ) ) ) ) ) ).
% append_coeff
thf(fact_253_ring_Odrop__exp__base,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,X: a,M: nat] :
( ( ring_a_b @ R )
=> ( ( drop_a @ N @ ( polyno2922411391617481336se_a_b @ R @ X @ M ) )
= ( polyno2922411391617481336se_a_b @ R @ X @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% ring.drop_exp_base
thf(fact_254_ring_Odrop__exp__base,axiom,
! [R: partia2670972154091845814t_unit,N: nat,X: list_a,M: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( drop_list_a @ N @ ( polyno3522816881121920896t_unit @ R @ X @ M ) )
= ( polyno3522816881121920896t_unit @ R @ X @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% ring.drop_exp_base
thf(fact_255_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_256_subalgebra__in__carrier,axiom,
! [K2: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_257_carrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_258_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_259_x_Onormalize__def_H_I2_J,axiom,
! [P: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( drop_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) @ P ) ) ).
% x.normalize_def'(2)
thf(fact_260_x_Onormalize__length__le,axiom,
! [P: list_list_a] : ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ).
% x.normalize_length_le
thf(fact_261_x_Ocoeff__iff__polynomial__cond,axiom,
! [K2: set_list_a,P1: list_list_a,P22: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P1 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P22 )
=> ( ( P1 = P22 )
= ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 )
= ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.coeff_iff_polynomial_cond
thf(fact_262_x_Odense__repr_Osimps_I1_J,axiom,
( ( dense_5814815041220002634t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= nil_Pr6246850598307483389_a_nat ) ).
% x.dense_repr.simps(1)
thf(fact_263_x_Oring__axioms,axiom,
ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.ring_axioms
thf(fact_264_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_265_x_Ocoeff__iff__length__cond,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ( size_s349497388124573686list_a @ P1 )
= ( size_s349497388124573686list_a @ P22 ) )
=> ( ( P1 = P22 )
= ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 )
= ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ).
% x.coeff_iff_length_cond
thf(fact_266_x_Onormalize__polynomial,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= P ) ) ).
% x.normalize_polynomial
thf(fact_267_x_Onormalize__coeff,axiom,
! [P: list_list_a] :
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.normalize_coeff
thf(fact_268_x_Odense__repr__normalize,axiom,
! [P: list_list_a] :
( ( dense_5814815041220002634t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
= ( dense_5814815041220002634t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ).
% x.dense_repr_normalize
thf(fact_269_x_Onormalize_Osimps_I1_J,axiom,
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= nil_list_a ) ).
% x.normalize.simps(1)
thf(fact_270_x_Onormalize__idem,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ Q ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) ) ) ).
% x.normalize_idem
thf(fact_271_x_Odrop__exp__base,axiom,
! [N: nat,X: list_a,M: nat] :
( ( drop_list_a @ N @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) )
= ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( minus_minus_nat @ M @ N ) ) ) ).
% x.drop_exp_base
thf(fact_272_x_Oappend__coeff,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) )
= ( ^ [I2: nat] : ( if_list_a @ ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Q ) ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ I2 ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( minus_minus_nat @ I2 @ ( size_s349497388124573686list_a @ Q ) ) ) ) ) ) ).
% x.append_coeff
thf(fact_273_x_Osplitted__on__def,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% x.splitted_on_def
thf(fact_274_x_Ozero__is__polynomial,axiom,
! [K2: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ nil_list_a ) ).
% x.zero_is_polynomial
thf(fact_275_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_276_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_277_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_278_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_279_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_280_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P2 @ Y5 ) )
=> ( P2 @ X2 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_281_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_282_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_283_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_284_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_285_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_286_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_287_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_288_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_289_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_290_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_291_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_292_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_293_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_294_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_295_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_296_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_297_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_298_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_299_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_300_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_301_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_302_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_303_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_304_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_305_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_306_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_307_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_308_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_309_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_310_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_311_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_312_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_313_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_314_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_315_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_316_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_317_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_318_leD,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ~ ( ord_less_set_list_a @ X @ Y ) ) ).
% leD
thf(fact_319_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_320_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_321_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_322_nless__le,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ~ ( ord_less_set_list_a @ A @ B ) )
= ( ~ ( ord_le8861187494160871172list_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_323_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_324_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_325_antisym__conv1,axiom,
! [X: set_list_a,Y: set_list_a] :
( ~ ( ord_less_set_list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_326_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_327_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_328_antisym__conv2,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ~ ( ord_less_set_list_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_329_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_330_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_331_less__le__not__le,axiom,
( ord_less_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ~ ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_332_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_333_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_334_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_335_order_Oorder__iff__strict,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A2: set_list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_336_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_337_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_338_order_Ostrict__iff__order,axiom,
( ord_less_set_list_a
= ( ^ [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_339_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_340_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_341_order_Ostrict__trans1,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_342_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_343_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_344_order_Ostrict__trans2,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_345_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_346_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_347_order_Ostrict__iff__not,axiom,
( ord_less_set_list_a
= ( ^ [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
& ~ ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_348_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_349_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_350_dual__order_Oorder__iff__strict,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B2: set_list_a,A2: set_list_a] :
( ( ord_less_set_list_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_351_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_352_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_353_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_list_a
= ( ^ [B2: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_354_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_dual__order_Ostrict__trans1,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_less_set_list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_357_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_358_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_359_dual__order_Ostrict__trans2,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_360_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_361_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_362_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_list_a
= ( ^ [B2: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A2 )
& ~ ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_363_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_364_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_365_order_Ostrict__implies__order,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_366_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_367_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_368_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_369_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_370_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_371_order__le__less,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_less_set_list_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_372_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_373_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_374_order__less__le,axiom,
( ord_less_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_375_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_376_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_377_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_378_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_379_order__less__imp__le,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_380_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_381_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_382_order__le__neq__trans,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_383_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_384_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_385_order__neq__le__trans,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A != B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_386_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_387_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_388_order__le__less__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_less_set_list_a @ Y @ Z2 )
=> ( ord_less_set_list_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_389_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_390_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_391_order__less__le__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z2 )
=> ( ord_less_set_list_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_392_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_393_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_394_order__le__less__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_395_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_396_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_397_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_398_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_399_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_400_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_401_order__le__less__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_402_order__le__less__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_403_order__le__less__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_404_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_405_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_406_order__less__le__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_less_set_list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_407_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_408_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_409_order__less__le__subst1,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( ord_less_set_list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_410_order__less__le__subst1,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_411_order__less__le__subst1,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_412_order__less__le__subst1,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_413_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_414_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_415_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_416_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_417_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_418_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_419_order__le__imp__less__or__eq,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_less_set_list_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_420_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_421_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_422_length__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys2: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys2 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_423_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_434_ring_Oexp__base_Ocong,axiom,
polyno3522816881121920896t_unit = polyno3522816881121920896t_unit ).
% ring.exp_base.cong
thf(fact_435_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_436_ring_Oroots__on_Ocong,axiom,
polyno5990348478334826338t_unit = polyno5990348478334826338t_unit ).
% ring.roots_on.cong
thf(fact_437_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_438_ring_Osplitted__on_Ocong,axiom,
polyno1986131841096413848t_unit = polyno1986131841096413848t_unit ).
% ring.splitted_on.cong
thf(fact_439_ring_Oappend__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( append_a @ P @ Q ) )
= ( ^ [I2: nat] : ( if_a @ ( ord_less_nat @ I2 @ ( size_size_list_a @ Q ) ) @ ( coeff_a_b @ R @ Q @ I2 ) @ ( coeff_a_b @ R @ P @ ( minus_minus_nat @ I2 @ ( size_size_list_a @ Q ) ) ) ) ) ) ) ).
% ring.append_coeff
thf(fact_440_ring_Oappend__coeff,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( append_list_a @ P @ Q ) )
= ( ^ [I2: nat] : ( if_list_a @ ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Q ) ) @ ( coeff_6360649920519955023t_unit @ R @ Q @ I2 ) @ ( coeff_6360649920519955023t_unit @ R @ P @ ( minus_minus_nat @ I2 @ ( size_s349497388124573686list_a @ Q ) ) ) ) ) ) ) ).
% ring.append_coeff
thf(fact_441_x_Ocoeff__degree,axiom,
! [P: list_list_a,I: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ I )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.coeff_degree
thf(fact_442_coeff__degree,axiom,
! [P: list_a,I: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I )
=> ( ( coeff_a_b @ r @ P @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_degree
thf(fact_443_x_Oee__length,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) ) ) ).
% x.ee_length
thf(fact_444_x_Ocoeff__nth,axiom,
! [I: nat,P: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ P ) )
=> ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ I )
= ( nth_list_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ I ) ) ) ) ).
% x.coeff_nth
thf(fact_445_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_446_coeff__nth,axiom,
! [I: nat,P: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ P ) )
=> ( ( coeff_a_b @ r @ P @ I )
= ( nth_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I ) ) ) ) ).
% coeff_nth
thf(fact_447_x_Ocoeff__length,axiom,
! [P: list_list_a,I: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ I )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.coeff_length
thf(fact_448_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ G @ X @ H ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_449_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2670972154091845814t_unit,H: set_list_a,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ G @ X @ H ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_450_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2956882679547061052t_unit,H: set_list_list_a,X: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ord_le8488217952732425610list_a @ ( a_l_co3970804650394549132t_unit @ G @ X @ H ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_451_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_452_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_453_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2956882679547061052t_unit,K2: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ K2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( embedd1097489486847499020t_unit @ K2 @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_454_x_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_455_x_Osubalgebra__in__carrier,axiom,
! [K2: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_in_carrier
thf(fact_456_x_Ocarrier__is__subalgebra,axiom,
! [K2: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.carrier_is_subalgebra
thf(fact_457_x_Ois__abelian__group,axiom,
abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.is_abelian_group
thf(fact_458_x_Oa__lcos__mult__one,axiom,
! [M4: set_list_a] :
( ( ord_le8861187494160871172list_a @ M4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M4 )
= M4 ) ) ).
% x.a_lcos_mult_one
thf(fact_459_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_460_coeff__length,axiom,
! [P: list_a,I: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I )
=> ( ( coeff_a_b @ r @ P @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_length
thf(fact_461_a__lcos__mult__one,axiom,
! [M4: set_a] :
( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M4 )
= M4 ) ) ).
% a_lcos_mult_one
thf(fact_462_x_Ocoeff_Osimps_I1_J,axiom,
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.coeff.simps(1)
thf(fact_463_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_464_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_465_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_466_x_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.zero_closed
thf(fact_467_lagrange__aux__poly,axiom,
! [S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_468_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_469_nth__equalityI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I3 )
= ( nth_list_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_470_Skolem__list__nth,axiom,
! [K: nat,P2: nat > a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: a] : ( P2 @ I2 @ X5 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P2 @ I2 @ ( nth_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_471_Skolem__list__nth,axiom,
! [K: nat,P2: nat > list_a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: list_a] : ( P2 @ I2 @ X5 ) ) )
= ( ? [Xs3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P2 @ I2 @ ( nth_list_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_472_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_a,Z: list_a] : ( Y3 = Z ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I2 )
= ( nth_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_473_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_list_a,Z: list_list_a] : ( Y3 = Z ) )
= ( ^ [Xs3: list_list_a,Ys3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs3 )
= ( size_s349497388124573686list_a @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ( ( nth_list_a @ Xs3 @ I2 )
= ( nth_list_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_474_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_475_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_476_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,K3: set_a,P5: list_a] : ( member_list_a @ P5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K3 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_477_univ__poly__carrier,axiom,
( polyno1315193887021588240t_unit
= ( ^ [R2: partia2670972154091845814t_unit,K3: set_list_a,P5: list_list_a] : ( member_list_list_a @ P5 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K3 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_478_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2175431115845679010xt_a_b,M4: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ ( zero_a_b @ G ) @ M4 )
= M4 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_479_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2670972154091845814t_unit,M4: set_list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ M4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_l_co7008843373686234386t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ M4 )
= M4 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_480_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2956882679547061052t_unit,M4: set_list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( ord_le8488217952732425610list_a @ M4 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( a_l_co3970804650394549132t_unit @ G @ ( zero_l347298301471573063t_unit @ G ) @ M4 )
= M4 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_481_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_482_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ nil_list_a )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_483_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_484_nth__append,axiom,
! [N: nat,Xs: list_list_a,Ys: list_list_a] :
( ( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ Xs @ Ys ) @ N )
= ( nth_list_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ Xs @ Ys ) @ N )
= ( nth_list_a @ Ys @ ( minus_minus_nat @ N @ ( size_s349497388124573686list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_485_ring_Ocoeff__length,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I )
=> ( ( coeff_a_b @ R @ P @ I )
= ( zero_a_b @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_486_ring_Ocoeff__length,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ R @ P @ I )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_487_ring_Ocoeff__nth,axiom,
! [R: partia2175431115845679010xt_a_b,I: nat,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ P ) )
=> ( ( coeff_a_b @ R @ P @ I )
= ( nth_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I ) ) ) ) ) ).
% ring.coeff_nth
thf(fact_488_ring_Ocoeff__nth,axiom,
! [R: partia2670972154091845814t_unit,I: nat,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ P ) )
=> ( ( coeff_6360649920519955023t_unit @ R @ P @ I )
= ( nth_list_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ I ) ) ) ) ) ).
% ring.coeff_nth
thf(fact_489_ring_Ocoeff__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I )
=> ( ( coeff_a_b @ R @ P @ I )
= ( zero_a_b @ R ) ) ) ) ).
% ring.coeff_degree
thf(fact_490_ring_Ocoeff__degree,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ R @ P @ I )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_degree
thf(fact_491_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_492_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,V: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd1768981623711841426t_unit @ K2 @ V @ R )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_493_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,K2: set_list_list_a,V: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( embedd1097489486847499020t_unit @ K2 @ V @ R )
=> ( ord_le8488217952732425610list_a @ V @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_494_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_495_x_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_496_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_497_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_498_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_499_poly__sub__degree__le,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_500_x_Omonom__coeff,axiom,
! [A: list_a,N: nat] :
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) )
= ( ^ [I2: nat] : ( if_list_a @ ( I2 = N ) @ A @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.monom_coeff
thf(fact_501_x_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_502_x_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A = B ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_503_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_504_ring_Omonom_Ocong,axiom,
monom_7446464087056152608t_unit = monom_7446464087056152608t_unit ).
% ring.monom.cong
thf(fact_505_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_506_ring_Omonom__coeff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) )
= ( ^ [I2: nat] : ( if_list_a @ ( I2 = N ) @ A @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_507_ring_Omonom__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( monom_a_b @ R @ A @ N ) )
= ( ^ [I2: nat] : ( if_a @ ( I2 = N ) @ A @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_508_domain_Opoly__sub__degree__le,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_list_a,N: nat,Y: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_509_domain_Opoly__sub__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,N: nat,Y: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_510_domain_Opoly__sub__degree__le,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,N: nat,Y: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_511_domain_Olagrange__aux__poly,axiom,
! [R: partia2670972154091845814t_unit,S2: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ S2 )
=> ( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( lagran3534788790333317459t_unit @ R @ S2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_512_domain_Olagrange__aux__poly,axiom,
! [R: partia2956882679547061052t_unit,S2: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite1660835950917165235list_a @ S2 )
=> ( ( ord_le8488217952732425610list_a @ S2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( lagran8640377047181650765t_unit @ R @ S2 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_513_domain_Olagrange__aux__poly,axiom,
! [R: partia2175431115845679010xt_a_b,S2: set_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ R @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_514_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_515_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_516_x_Odegree__oneE,axiom,
! [P: list_list_a,K2: set_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B3: list_a] :
( ( member_list_a @ B3 @ K2 )
=> ( P
!= ( cons_list_a @ A3 @ ( cons_list_a @ B3 @ nil_list_a ) ) ) ) ) ) ) ) ).
% x.degree_oneE
thf(fact_517_degree__oneE,axiom,
! [P: list_a,K2: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ! [B3: a] :
( ( member_a @ B3 @ K2 )
=> ( P
!= ( cons_a @ A3 @ ( cons_a @ B3 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_518_poly__add__degree__le,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_add_degree_le
thf(fact_519_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_520_x_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V2: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V2 @ Va ) ) ) ).
% x.normalize.cases
thf(fact_521_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I2: nat] : ( if_a @ ( I2 = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_522_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_523_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_524_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_525_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_526_x_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z2 ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_527_x_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.add.m_comm
thf(fact_528_x_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z2 ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_529_x_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_530_x_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_531_x_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_532_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_533_x_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_534_x_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y6: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y6 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_535_x_Oa__lcos__m__assoc,axiom,
! [M4: set_list_a,G2: list_a,H2: list_a] :
( ( ord_le8861187494160871172list_a @ M4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G2 @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 @ M4 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G2 @ H2 ) @ M4 ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_536_append1__eq__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,X: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs @ ( cons_P5205166803686508359_a_nat @ X @ nil_Pr7402525243500994295_a_nat ) )
= ( append7679239579558125090_a_nat @ Ys @ ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_537_append1__eq__conv,axiom,
! [Xs: list_P1129550237270585747_a_nat,X: produc424395135190311811_a_nat,Ys: list_P1129550237270585747_a_nat,Y: produc424395135190311811_a_nat] :
( ( ( append6709258537191095464_a_nat @ Xs @ ( cons_P8101317568005672781_a_nat @ X @ nil_Pr6246850598307483389_a_nat ) )
= ( append6709258537191095464_a_nat @ Ys @ ( cons_P8101317568005672781_a_nat @ Y @ nil_Pr6246850598307483389_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_538_append1__eq__conv,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_539_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_540_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_541_nth__append__length,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a] :
( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) ) @ ( size_s349497388124573686list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_542_x_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_543_x_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_544_x_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_545_x_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_546_x_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_547_x_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_548_x_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_zero
thf(fact_549_x_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_zero
thf(fact_550_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_551_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_552_transpose_Ocases,axiom,
! [X: list_l2471972001652375325_a_nat] :
( ( X != nil_li191968740515856775_a_nat )
=> ( ! [Xss: list_l2471972001652375325_a_nat] :
( X
!= ( cons_l2046435710214046167_a_nat @ nil_Pr7402525243500994295_a_nat @ Xss ) )
=> ~ ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Xss: list_l2471972001652375325_a_nat] :
( X
!= ( cons_l2046435710214046167_a_nat @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_553_transpose_Ocases,axiom,
! [X: list_l4436818523590721827_a_nat] :
( ( X != nil_li7055775836061915149_a_nat )
=> ( ! [Xss: list_l4436818523590721827_a_nat] :
( X
!= ( cons_l3733157488909561949_a_nat @ nil_Pr6246850598307483389_a_nat @ Xss ) )
=> ~ ! [X2: produc424395135190311811_a_nat,Xs2: list_P1129550237270585747_a_nat,Xss: list_l4436818523590721827_a_nat] :
( X
!= ( cons_l3733157488909561949_a_nat @ ( cons_P8101317568005672781_a_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_554_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_555_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_556_list_Odistinct_I1_J,axiom,
! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( nil_Pr7402525243500994295_a_nat
!= ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_557_list_Odistinct_I1_J,axiom,
! [X21: produc424395135190311811_a_nat,X22: list_P1129550237270585747_a_nat] :
( nil_Pr6246850598307483389_a_nat
!= ( cons_P8101317568005672781_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_558_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_559_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_560_list_OdiscI,axiom,
! [List: list_P3592885314253461005_a_nat,X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( ( List
= ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) )
=> ( List != nil_Pr7402525243500994295_a_nat ) ) ).
% list.discI
thf(fact_561_list_OdiscI,axiom,
! [List: list_P1129550237270585747_a_nat,X21: produc424395135190311811_a_nat,X22: list_P1129550237270585747_a_nat] :
( ( List
= ( cons_P8101317568005672781_a_nat @ X21 @ X22 ) )
=> ( List != nil_Pr6246850598307483389_a_nat ) ) ).
% list.discI
thf(fact_562_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_563_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_564_list_Oexhaust,axiom,
! [Y: list_P3592885314253461005_a_nat] :
( ( Y != nil_Pr7402525243500994295_a_nat )
=> ~ ! [X212: product_prod_a_nat,X222: list_P3592885314253461005_a_nat] :
( Y
!= ( cons_P5205166803686508359_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_565_list_Oexhaust,axiom,
! [Y: list_P1129550237270585747_a_nat] :
( ( Y != nil_Pr6246850598307483389_a_nat )
=> ~ ! [X212: produc424395135190311811_a_nat,X222: list_P1129550237270585747_a_nat] :
( Y
!= ( cons_P8101317568005672781_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_566_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_567_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_568_remdups__adj_Ocases,axiom,
! [X: list_P3592885314253461005_a_nat] :
( ( X != nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat] :
( X
!= ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X2: product_prod_a_nat,Y2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( X
!= ( cons_P5205166803686508359_a_nat @ X2 @ ( cons_P5205166803686508359_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_569_remdups__adj_Ocases,axiom,
! [X: list_P1129550237270585747_a_nat] :
( ( X != nil_Pr6246850598307483389_a_nat )
=> ( ! [X2: produc424395135190311811_a_nat] :
( X
!= ( cons_P8101317568005672781_a_nat @ X2 @ nil_Pr6246850598307483389_a_nat ) )
=> ~ ! [X2: produc424395135190311811_a_nat,Y2: produc424395135190311811_a_nat,Xs2: list_P1129550237270585747_a_nat] :
( X
!= ( cons_P8101317568005672781_a_nat @ X2 @ ( cons_P8101317568005672781_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_570_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_571_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_572_neq__Nil__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
= ( ? [Y4: product_prod_a_nat,Ys3: list_P3592885314253461005_a_nat] :
( Xs
= ( cons_P5205166803686508359_a_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_573_neq__Nil__conv,axiom,
! [Xs: list_P1129550237270585747_a_nat] :
( ( Xs != nil_Pr6246850598307483389_a_nat )
= ( ? [Y4: produc424395135190311811_a_nat,Ys3: list_P1129550237270585747_a_nat] :
( Xs
= ( cons_P8101317568005672781_a_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_574_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y4: list_a,Ys3: list_list_a] :
( Xs
= ( cons_list_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_575_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_576_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P2 @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys4: list_a] : ( P2 @ nil_a @ ( cons_a @ Y2 @ Ys4 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_577_list__induct2_H,axiom,
! [P2: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys4: list_a] : ( P2 @ nil_list_a @ ( cons_a @ Y2 @ Ys4 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys4: list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_578_list__induct2_H,axiom,
! [P2: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a] : ( P2 @ ( cons_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys4: list_list_a] : ( P2 @ nil_a @ ( cons_list_a @ Y2 @ Ys4 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys4: list_list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_579_list__induct2_H,axiom,
! [P2: list_P3592885314253461005_a_nat > list_a > $o,Xs: list_P3592885314253461005_a_nat,Ys: list_a] :
( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys4: list_a] : ( P2 @ nil_Pr7402525243500994295_a_nat @ ( cons_a @ Y2 @ Ys4 ) )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_580_list__induct2_H,axiom,
! [P2: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys4: list_list_a] : ( P2 @ nil_list_a @ ( cons_list_a @ Y2 @ Ys4 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys4: list_list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_581_list__induct2_H,axiom,
! [P2: list_a > list_P3592885314253461005_a_nat > $o,Xs: list_a,Ys: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs2: list_a] : ( P2 @ ( cons_a @ X2 @ Xs2 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] : ( P2 @ nil_a @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_582_list__induct2_H,axiom,
! [P2: list_P3592885314253461005_a_nat > list_list_a > $o,Xs: list_P3592885314253461005_a_nat,Ys: list_list_a] :
( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_list_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys4: list_list_a] : ( P2 @ nil_Pr7402525243500994295_a_nat @ ( cons_list_a @ Y2 @ Ys4 ) )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: list_a,Ys4: list_list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_583_list__induct2_H,axiom,
! [P2: list_P1129550237270585747_a_nat > list_a > $o,Xs: list_P1129550237270585747_a_nat,Ys: list_a] :
( ( P2 @ nil_Pr6246850598307483389_a_nat @ nil_a )
=> ( ! [X2: produc424395135190311811_a_nat,Xs2: list_P1129550237270585747_a_nat] : ( P2 @ ( cons_P8101317568005672781_a_nat @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys4: list_a] : ( P2 @ nil_Pr6246850598307483389_a_nat @ ( cons_a @ Y2 @ Ys4 ) )
=> ( ! [X2: produc424395135190311811_a_nat,Xs2: list_P1129550237270585747_a_nat,Y2: a,Ys4: list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P8101317568005672781_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_584_list__induct2_H,axiom,
! [P2: list_list_a > list_P3592885314253461005_a_nat > $o,Xs: list_list_a,Ys: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_list_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] : ( P2 @ nil_list_a @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_585_list__induct2_H,axiom,
! [P2: list_a > list_P1129550237270585747_a_nat > $o,Xs: list_a,Ys: list_P1129550237270585747_a_nat] :
( ( P2 @ nil_a @ nil_Pr6246850598307483389_a_nat )
=> ( ! [X2: a,Xs2: list_a] : ( P2 @ ( cons_a @ X2 @ Xs2 ) @ nil_Pr6246850598307483389_a_nat )
=> ( ! [Y2: produc424395135190311811_a_nat,Ys4: list_P1129550237270585747_a_nat] : ( P2 @ nil_a @ ( cons_P8101317568005672781_a_nat @ Y2 @ Ys4 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: produc424395135190311811_a_nat,Ys4: list_P1129550237270585747_a_nat] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P8101317568005672781_a_nat @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_586_list__nonempty__induct,axiom,
! [Xs: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > $o] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_587_list__nonempty__induct,axiom,
! [Xs: list_P1129550237270585747_a_nat,P2: list_P1129550237270585747_a_nat > $o] :
( ( Xs != nil_Pr6246850598307483389_a_nat )
=> ( ! [X2: produc424395135190311811_a_nat] : ( P2 @ ( cons_P8101317568005672781_a_nat @ X2 @ nil_Pr6246850598307483389_a_nat ) )
=> ( ! [X2: produc424395135190311811_a_nat,Xs2: list_P1129550237270585747_a_nat] :
( ( Xs2 != nil_Pr6246850598307483389_a_nat )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_P8101317568005672781_a_nat @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_588_list__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P2 @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_589_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_590_append__Cons,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
= ( cons_list_a @ X @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_591_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_592_Cons__eq__appendI,axiom,
! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_593_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_594_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_595_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P2: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys4: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_596_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P2: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys4: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_597_list__induct2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,P2: list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_598_list__induct2,axiom,
! [Xs: list_a,Ys: list_P3592885314253461005_a_nat,P2: list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_599_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P2: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys4: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_600_list__induct2,axiom,
! [Xs: list_P1129550237270585747_a_nat,Ys: list_a,P2: list_P1129550237270585747_a_nat > list_a > $o] :
( ( ( size_s4007175526842055207_a_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_Pr6246850598307483389_a_nat @ nil_a )
=> ( ! [X2: produc424395135190311811_a_nat,Xs2: list_P1129550237270585747_a_nat,Y2: a,Ys4: list_a] :
( ( ( size_s4007175526842055207_a_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P8101317568005672781_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_601_list__induct2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_list_a,P2: list_P3592885314253461005_a_nat > list_list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_list_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: list_a,Ys4: list_list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_602_list__induct2,axiom,
! [Xs: list_a,Ys: list_P1129550237270585747_a_nat,P2: list_a > list_P1129550237270585747_a_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s4007175526842055207_a_nat @ Ys ) )
=> ( ( P2 @ nil_a @ nil_Pr6246850598307483389_a_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: produc424395135190311811_a_nat,Ys4: list_P1129550237270585747_a_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s4007175526842055207_a_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P8101317568005672781_a_nat @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_603_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_P3592885314253461005_a_nat,P2: list_list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_604_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_605_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P2: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_606_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P2: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys4: list_list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_607_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P2: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_608_list__induct3,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_a,P2: list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_609_list__induct3,axiom,
! [Xs: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_a,P2: list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_610_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_P3592885314253461005_a_nat,P2: list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_611_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P2: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys4: list_list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys4 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_612_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P2: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys4: list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_613_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P2: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys4: list_list_a,Z3: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_614_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_615_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_616_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_list_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_617_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys4: list_list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_618_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_619_list__induct4,axiom,
! [Xs: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_a,P2: list_a > list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_620_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P2: list_a > list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_621_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P2: list_a > list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_622_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_list_a,P2: list_a > list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys4: list_a,Z3: list_a,Zs2: list_list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_list_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_623_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys4: list_list_a,Z3: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_624_x_Opoly__of__const__in__carrier,axiom,
! [S: list_a] :
( ( member_list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_625_x_Opoly__of__const__def,axiom,
( ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( ^ [K4: list_a] : ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K4 @ nil_list_a ) ) ) ) ).
% x.poly_of_const_def
thf(fact_626_x_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N3 @ F ) ) ).
% x.bound_upD
thf(fact_627_x_Onormalize__def_H_I1_J,axiom,
! [P: list_list_a] :
( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( drop_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) @ P ) ) ) ).
% x.normalize_def'(1)
thf(fact_628_x_Osubset__Idl__subset,axiom,
! [I4: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I4 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_629_x_Ogenideal__self,axiom,
! [S2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ).
% x.genideal_self
thf(fact_630_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_631_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_632_a__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z2 )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% a_assoc
thf(fact_633_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_634_a__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z2 ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z2 ) ) ) ) ) ) ).
% a_lcomm
thf(fact_635_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_636_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_637_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_638_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_639_local_Ominus__unique,axiom,
! [Y: a,X: a,Y6: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y6 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_640_a__lcos__m__assoc,axiom,
! [M4: set_a,G2: a,H2: a] :
( ( ord_less_eq_set_a @ M4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G2 @ ( a_l_coset_a_b @ r @ H2 @ M4 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G2 @ H2 ) @ M4 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_641_x_Onormalize__replicate__zero,axiom,
! [N: nat,P: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ).
% x.normalize_replicate_zero
thf(fact_642_x_Oprefix__replicate__zero__coeff,axiom,
! [P: list_list_a,N: nat] :
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) ) ) ).
% x.prefix_replicate_zero_coeff
thf(fact_643_x_Odense__repr__replicate__zero,axiom,
! [N: nat,P: list_list_a] :
( ( dense_5814815041220002634t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) )
= ( dense_5814815041220002634t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ).
% x.dense_repr_replicate_zero
thf(fact_644_x_Omonom__def,axiom,
! [A: list_a,N: nat] :
( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N )
= ( cons_list_a @ A @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.monom_def
thf(fact_645_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_646_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_647_x_Onormalize__trick,axiom,
! [P: list_list_a] :
( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.normalize_trick
thf(fact_648_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_649_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_650_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_651_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_652_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_653_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_654_x_Oreplicate__zero__coeff,axiom,
! [N: nat] :
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.replicate_zero_coeff
thf(fact_655_x_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% x.const_term_not_zero
thf(fact_656_x_Oappend__is__polynomial,axiom,
! [K2: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( P != nil_list_a )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( append_list_a @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% x.append_is_polynomial
thf(fact_657_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_658_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_659_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K4: a] : ( normalize_a_b @ r @ ( cons_a @ K4 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_660_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_661_prefix__replicate__zero__coeff,axiom,
! [P: list_a,N: nat] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) ) ) ).
% prefix_replicate_zero_coeff
thf(fact_662_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_663_dense__repr__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( dense_repr_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( dense_repr_a_b @ r @ P ) ) ).
% dense_repr_replicate_zero
thf(fact_664_x_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.carrier_is_subring
thf(fact_665_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_666_x_Opoly__coeff__in__carrier,axiom,
! [K2: set_list_a,P: list_list_a,I: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ I ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_coeff_in_carrier
thf(fact_667_normalize__def_H_I1_J,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ P ) ) ) ).
% normalize_def'(1)
thf(fact_668_x_Oconst__term__zero,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ! [P6: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P6 )
=> ( ( P6 != nil_list_a )
=> ( P
!= ( append_list_a @ P6 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% x.const_term_zero
thf(fact_669_replicate__zero__coeff,axiom,
! [N: nat] :
( ( coeff_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% replicate_zero_coeff
thf(fact_670_x_Ocarrier__polynomial,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% x.carrier_polynomial
thf(fact_671_x_Ocarrier__polynomial__shell,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.carrier_polynomial_shell
thf(fact_672_x_Omonom__decomp,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( P
= ( poly_o2635896782027652242t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( dense_5814815041220002634t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ) ).
% x.monom_decomp
thf(fact_673_x_Oconst__term__explicit,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= A )
=> ~ ! [P6: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P6 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P
!= ( append_list_a @ P6 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% x.const_term_explicit
thf(fact_674_x_Oconst__term__eq__last,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ).
% x.const_term_eq_last
thf(fact_675_x_Oline__extension__in__carrier,axiom,
! [K2: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_676_map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_677_x_Oee__trans,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ Cs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% x.ee_trans
thf(fact_678_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_679_univ__poly__is__abelian__monoid,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_abelian_monoid
thf(fact_680_univ__poly__is__domain,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_domain
thf(fact_681_x_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.abelian_monoid_axioms
thf(fact_682_const__term__simprules__shell_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K2 ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_683_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_684_var__closed_I2_J,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( polynomial_a_b @ r @ K2 @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_685_univ__poly__is__ring,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_ring
thf(fact_686_univ__poly__is__abelian__group,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_abelian_group
thf(fact_687_var__closed_I1_J,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ).
% var_closed(1)
thf(fact_688_poly__coeff__in__carrier,axiom,
! [K2: set_a,P: list_a,I: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_coeff_in_carrier
thf(fact_689_const__term__simprules__shell_I3_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_690_univ__poly__a__minus__consistent,axiom,
! [K2: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_691_x_Opolynomial__incl,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ K2 ) ) ).
% x.polynomial_incl
thf(fact_692_const__term__zero,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P6: list_a] :
( ( polynomial_a_b @ r @ K2 @ P6 )
=> ( ( P6 != nil_a )
=> ( P
!= ( append_a @ P6 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_693_append__is__polynomial,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( polynomial_a_b @ r @ K2 @ ( append_a @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% append_is_polynomial
thf(fact_694_x_Onormalize__in__carrier,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.normalize_in_carrier
thf(fact_695_x_Ocoeff__in__carrier,axiom,
! [P: list_list_a,I: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ I ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.coeff_in_carrier
thf(fact_696_x_Oconst__term__simprules_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.const_term_simprules(1)
thf(fact_697_x_Onormalize__gives__polynomial,axiom,
! [P: list_list_a,K2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ K2 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.normalize_gives_polynomial
thf(fact_698_x_Oexp__base__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.exp_base_closed
thf(fact_699_x_Oee__sym,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ As ) ) ) ) ).
% x.ee_sym
thf(fact_700_map__norm__in__poly__ring__carrier,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_701_norm__map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_702_carrier__polynomial,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).
% carrier_polynomial
thf(fact_703_carrier__polynomial__shell,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_704_x_Omonom__in__carrier,axiom,
! [A: list_a,N: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monom_in_carrier
thf(fact_705_x_Oee__refl,axiom,
! [As: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ As ) ) ).
% x.ee_refl
thf(fact_706_x_Opolynomial__in__carrier,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.polynomial_in_carrier
thf(fact_707_monom__decomp,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( P
= ( poly_of_dense_a_b @ r @ ( dense_repr_a_b @ r @ P ) ) ) ) ) ).
% monom_decomp
thf(fact_708_x_Opoly__add__append__replicate,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( replicate_list_a @ ( size_s349497388124573686list_a @ Q ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) @ Q )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) ) ) ) ) ).
% x.poly_add_append_replicate
thf(fact_709_x_Opoly__add__append__zero,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ ( append_list_a @ Q @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ).
% x.poly_add_append_zero
thf(fact_710_x_Opoly__mult__append__zero,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ Q )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ).
% x.poly_mult_append_zero
thf(fact_711_x_Ocombine__prepend__replicate,axiom,
! [Ks: list_list_a,Us3: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Ks ) @ Us3 )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ ( drop_list_a @ N @ Us3 ) ) ) ) ) ).
% x.combine_prepend_replicate
thf(fact_712_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( 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 @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_713_polynomial__incl,axiom,
! [K2: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 ) ) ).
% polynomial_incl
thf(fact_714_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_715_coeff__in__carrier,axiom,
! [P: list_a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% coeff_in_carrier
thf(fact_716_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_717_normalize__gives__polynomial,axiom,
! [P: list_a,K2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_gives_polynomial
thf(fact_718_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_719_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_720_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_721_x_Opoly__mult_Osimps_I1_J,axiom,
! [P22: list_list_a] :
( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P22 )
= nil_list_a ) ).
% x.poly_mult.simps(1)
thf(fact_722_x_Ocombine_Osimps_I3_J,axiom,
! [Ks: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ nil_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.combine.simps(3)
thf(fact_723_x_Ocombine_Osimps_I2_J,axiom,
! [Us3: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Us3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.combine.simps(2)
thf(fact_724_x_Opoly__add__closed,axiom,
! [K2: set_list_a,P1: list_list_a,P22: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P1 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P22 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) ) ) ) ).
% x.poly_add_closed
thf(fact_725_x_Opoly__mult__closed,axiom,
! [K2: set_list_a,P1: list_list_a,P22: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P1 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P22 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) ) ) ) ).
% x.poly_mult_closed
thf(fact_726_x_Opoly__add__in__carrier,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_add_in_carrier
thf(fact_727_x_Opoly__add__comm,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P1 ) ) ) ) ).
% x.poly_add_comm
thf(fact_728_x_Opoly__mult__in__carrier,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_mult_in_carrier
thf(fact_729_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P6: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P6 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P6 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_730_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_731_x_Opoly__add__zero_I1_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= P ) ) ) ).
% x.poly_add_zero(1)
thf(fact_732_x_Opoly__add__zero_I2_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= P ) ) ) ).
% x.poly_add_zero(2)
thf(fact_733_x_Opoly__mult__l__distr,axiom,
! [K2: set_list_a,P1: list_list_a,P22: list_list_a,P32: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P1 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P22 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P32 )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) @ P32 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P32 ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P32 ) ) ) ) ) ) ) ).
% x.poly_mult_l_distr
thf(fact_734_x_Opoly__add__normalize_I3_J,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.poly_add_normalize(3)
thf(fact_735_x_Opoly__add__normalize_I2_J,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.poly_add_normalize(2)
thf(fact_736_x_Opoly__add__normalize__aux,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_add_normalize_aux
thf(fact_737_x_Opoly__mult__zero_I2_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= nil_list_a ) ) ).
% x.poly_mult_zero(2)
thf(fact_738_x_Opoly__mult__zero_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= nil_list_a ) ) ).
% x.poly_mult_zero(1)
thf(fact_739_x_Opoly__add__coeff__aux,axiom,
! [P22: list_list_a,P1: list_list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P22 ) @ ( size_s349497388124573686list_a @ P1 ) )
=> ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) )
= ( ^ [I2: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ I2 ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ I2 ) ) ) ) ) ).
% x.poly_add_coeff_aux
thf(fact_740_x_Opoly__mult__l__distr_H,axiom,
! [P1: list_list_a,P22: list_list_a,P32: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P32 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) @ P32 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P32 ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P32 ) ) ) ) ) ) ).
% x.poly_mult_l_distr'
thf(fact_741_x_Opoly__mult__normalize,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_mult_normalize
thf(fact_742_x_Opoly__add__is__polynomial,axiom,
! [K2: set_list_a,P1: list_list_a,P22: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ K2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ K2 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) ) ) ) ).
% x.poly_add_is_polynomial
thf(fact_743_x_Opoly__mult__is__polynomial,axiom,
! [K2: set_list_a,P1: list_list_a,P22: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ K2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ K2 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) ) ) ) ).
% x.poly_mult_is_polynomial
thf(fact_744_x_Opoly__add__replicate__zero_I1_J,axiom,
! [K2: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= P ) ) ) ).
% x.poly_add_replicate_zero(1)
thf(fact_745_x_Opoly__add__replicate__zero_I2_J,axiom,
! [K2: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P )
= P ) ) ) ).
% x.poly_add_replicate_zero(2)
thf(fact_746_x_Opoly__add__coeff,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) )
= ( ^ [I2: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ I2 ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ I2 ) ) ) ) ) ) ).
% x.poly_add_coeff
thf(fact_747_x_Opoly__add__zero_H_I2_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_zero'(2)
thf(fact_748_x_Opoly__add__zero_H_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_zero'(1)
thf(fact_749_x_Oconst__term__simprules_I3_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(3)
thf(fact_750_x_Ocombine__append,axiom,
! [Ks: list_list_a,Us3: list_list_a,Ks2: list_list_a,Vs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us3 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us3 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Vs2 ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks @ Ks2 ) @ ( append_list_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ) ).
% x.combine_append
thf(fact_751_x_Ocombine__replicate,axiom,
! [Us3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( replicate_list_a @ ( size_s349497388124573686list_a @ Us3 ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Us3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.combine_replicate
thf(fact_752_x_Ocombine__append__replicate,axiom,
! [Us3: list_list_a,Ks: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) @ Us3 )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us3 ) ) ) ).
% x.combine_append_replicate
thf(fact_753_x_Opoly__add__replicate__zero_H_I1_J,axiom,
! [P: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_replicate_zero'(1)
thf(fact_754_x_Opoly__add__replicate__zero_H_I2_J,axiom,
! [P: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_replicate_zero'(2)
thf(fact_755_x_Opoly__mult__prepend__replicate__zero,axiom,
! [P1: list_list_a,P22: list_list_a,N: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_mult_prepend_replicate_zero
thf(fact_756_x_Ocombine__append__zero,axiom,
! [Us3: list_list_a,Ks: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ Us3 )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us3 ) ) ) ).
% x.combine_append_zero
thf(fact_757_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_758_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_759_polynomial__in__carrier,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_760_x_Ocombine__in__carrier,axiom,
! [Ks: list_list_a,Us3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.combine_in_carrier
thf(fact_761_poly__mult__var,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_762_pdivides__imp__degree__le,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_763_x_Oconst__term__simprules_I2_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(2)
thf(fact_764_poly__mult_Osimps_I1_J,axiom,
! [P22: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P22 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_765_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_766_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_767_poly__mult__r__distr,axiom,
! [K2: set_a,P1: list_a,P22: list_a,P32: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( ( polynomial_a_b @ r @ K2 @ P32 )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P22 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P22 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_r_distr
thf(fact_768_poly__mult__l__distr,axiom,
! [K2: set_a,P1: list_a,P22: list_a,P32: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( ( polynomial_a_b @ r @ K2 @ P32 )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P22 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_l_distr
thf(fact_769_poly__add__closed,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_add_closed
thf(fact_770_poly__mult__closed,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_771_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_772_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_773_poly__mult__l__distr_H,axiom,
! [P1: list_a,P22: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P22 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_774_poly__mult__r__distr_H,axiom,
! [P1: list_a,P22: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P22 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P22 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ).
% poly_mult_r_distr'
thf(fact_775_poly__add__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_776_poly__add__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_add_comm
thf(fact_777_poly__mult__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_778_poly__mult__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_mult_comm
thf(fact_779_poly__add__zero_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= P ) ) ) ).
% poly_add_zero(1)
thf(fact_780_poly__add__zero_I2_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= P ) ) ) ).
% poly_add_zero(2)
thf(fact_781_poly__mult__integral,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( ( ( poly_mult_a_b @ r @ P1 @ P22 )
= nil_a )
=> ( ( P1 = nil_a )
| ( P22 = nil_a ) ) ) ) ) ) ).
% poly_mult_integral
thf(fact_782_x_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z2 ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_783_x_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.m_comm
thf(fact_784_x_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z2 ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_785_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_786_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_787_poly__add__normalize__aux,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_788_poly__add__normalize_I2_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_789_poly__add__normalize_I3_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_790_poly__mult__normalize,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_normalize
thf(fact_791_poly__add__coeff__aux,axiom,
! [P22: list_a,P1: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P22 ) @ ( size_size_list_a @ P1 ) )
=> ( ( coeff_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) )
= ( ^ [I2: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I2 ) @ ( coeff_a_b @ r @ P22 @ I2 ) ) ) ) ) ).
% poly_add_coeff_aux
thf(fact_792_poly__add__is__polynomial,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_793_poly__mult__is__polynomial,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ K2 )
=> ( polynomial_a_b @ r @ K2 @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_794_poly__mult__monom__assoc,axiom,
! [P: list_a,Q: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_795_poly__add__replicate__zero_I1_J,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= P ) ) ) ).
% poly_add_replicate_zero(1)
thf(fact_796_poly__add__replicate__zero_I2_J,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= P ) ) ) ).
% poly_add_replicate_zero(2)
thf(fact_797_x_Or__distr,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z2 @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z2 @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_798_x_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z2 ) ) ) ) ) ) ).
% x.l_distr
thf(fact_799_poly__mult__semiassoc,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_800_poly__add__coeff,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( coeff_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) )
= ( ^ [I2: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I2 ) @ ( coeff_a_b @ r @ P22 @ I2 ) ) ) ) ) ) ).
% poly_add_coeff
thf(fact_801_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_802_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_803_const__term__simprules_I3_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_804_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K2: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K2 )
& ? [Y4: list_a] :
( ( member_list_a @ Y4 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_805_pdivides__zero,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_806_combine__append,axiom,
! [Ks: list_a,Us3: list_a,Ks2: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) @ ( embedded_combine_a_b @ r @ Ks2 @ Vs2 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks2 ) @ ( append_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_807_x_Ocombine_Osimps_I1_J,axiom,
! [K: list_a,Ks: list_list_a,U: list_a,Us3: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K @ Ks ) @ ( cons_list_a @ U @ Us3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ U ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us3 ) ) ) ).
% x.combine.simps(1)
thf(fact_808_combine__replicate,axiom,
! [Us3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( replicate_a @ ( size_size_list_a @ Us3 ) @ ( zero_a_b @ r ) ) @ Us3 )
= ( zero_a_b @ r ) ) ) ).
% combine_replicate
thf(fact_809_poly__mult__replicate__zero_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= nil_a ) ) ).
% poly_mult_replicate_zero(2)
thf(fact_810_poly__mult__replicate__zero_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= nil_a ) ) ).
% poly_mult_replicate_zero(1)
thf(fact_811_combine__append__replicate,axiom,
! [Us3: list_a,Ks: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ Us3 ) ) ) ).
% combine_append_replicate
thf(fact_812_poly__mult__prepend__replicate__zero,axiom,
! [P1: list_a,P22: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_813_poly__add__replicate__zero_H_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_814_poly__add__replicate__zero_H_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_815_poly__mult__append__zero__rcancel,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( polynomial_a_b @ r @ K2 @ Q )
=> ( ( ( poly_mult_a_b @ r @ P @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( append_a @ R3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P @ Q )
= R3 ) ) ) ) ) ).
% poly_mult_append_zero_rcancel
thf(fact_816_poly__mult__append__zero__lcancel,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( polynomial_a_b @ r @ K2 @ Q )
=> ( ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( append_a @ R3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P @ Q )
= R3 ) ) ) ) ) ).
% poly_mult_append_zero_lcancel
thf(fact_817_x_Ocombine__r__distr,axiom,
! [Ks: list_list_a,Us3: list_list_a,K: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us3 ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Ks ) @ Us3 ) ) ) ) ) ).
% x.combine_r_distr
thf(fact_818_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_819_combine__prepend__replicate,axiom,
! [Ks: list_a,Us3: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ Ks ) @ Us3 )
= ( embedded_combine_a_b @ r @ Ks @ ( drop_a @ N @ Us3 ) ) ) ) ) ).
% combine_prepend_replicate
thf(fact_820_x_Ocombine_Oelims,axiom,
! [X: list_list_a,Xa: list_list_a,Y: list_a] :
( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Xa )
= Y )
=> ( ! [K5: list_a,Ks3: list_list_a] :
( ( X
= ( cons_list_a @ K5 @ Ks3 ) )
=> ! [U2: list_a,Us4: list_list_a] :
( ( Xa
= ( cons_list_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K5 @ U2 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ~ ( ( Xa = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.combine.elims
thf(fact_821_poly__add__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_822_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_823_poly__add__append__replicate,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_824_poly__mult__var_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_825_poly__mult__var_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_826_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_827_x_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_828_x_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_829_x_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_830_x_Omonoid__cancelI,axiom,
( ! [A3: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B3 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B3 ) ) ) ) )
=> ( ! [A3: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ C2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B3 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_831_x_Ofactors__mult,axiom,
! [Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fa @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% x.factors_mult
thf(fact_832_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_833_m__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_834_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_835_m__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_836_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_837_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_838_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_839_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_840_r__distr,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z2 @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z2 @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z2 @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_841_l__distr,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% l_distr
thf(fact_842_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_843_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_844_combine__r__distr,axiom,
! [Ks: list_a,Us3: list_a,K: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K @ ( embedded_combine_a_b @ r @ Ks @ Us3 ) )
= ( embedded_combine_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ K ) @ Ks ) @ Us3 ) ) ) ) ) ).
% combine_r_distr
thf(fact_845_combine_Oelims,axiom,
! [X: list_a,Xa: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X @ Xa )
= Y )
=> ( ! [K5: a,Ks3: list_a] :
( ( X
= ( cons_a @ K5 @ Ks3 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K5 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_846_const__term__simprules__shell_I2_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_847_const__term__simprules_I2_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_848_x_Ofactors__closed,axiom,
! [Fs: list_list_a,A: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.factors_closed
thf(fact_849_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_850_poly__mult__monom_H,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% poly_mult_monom'
thf(fact_851_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_852_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_853_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_854_poly__mult__degree__le__1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ).
% poly_mult_degree_le_1
thf(fact_855_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_856_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_857_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_858_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_859_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_860_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_861_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_862_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_863_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_864_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_865_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_866_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_867_poly__mult__degree__le,axiom,
! [X: list_a,Y: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_868_poly__mult__degree__eq,axiom,
! [K2: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P1 )
=> ( ( polynomial_a_b @ r @ K2 @ P22 )
=> ( ( ( ( P1 = nil_a )
| ( P22 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ( P1 = nil_a )
| ( P22 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% poly_mult_degree_eq
thf(fact_869_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_870_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_871_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_872_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_873_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_874_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_875_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_876_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_877_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K5: nat] :
( ( ord_less_eq_nat @ K5 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K5 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K5 ) ) ) ) ).
% ex_least_nat_le
thf(fact_886_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_887_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_888_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_889_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_890_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_891_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_892_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_893_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_894_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_895_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_904_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_905_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_906_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_907_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_908_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_909_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_910_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_911_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_912_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_913_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K5: nat] :
( ( ord_less_nat @ zero_zero_nat @ K5 )
& ( ( plus_plus_nat @ I @ K5 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_914_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P2 @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_915_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P2 @ D ) ) ) ) ).
% nat_diff_split
thf(fact_916_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_917_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_918_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_919_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_920_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_921_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_922_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_923_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_924_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_925_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_926_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_927_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_928_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_929_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_930_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_931_x_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.order_gt_0_iff_finite
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat ) @ one_one_nat ).
%------------------------------------------------------------------------------