TPTP Problem File: SLH0044^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00109_004112__17042822_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1456 ( 303 unt; 177 typ; 0 def)
% Number of atoms : 4336 (1299 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17157 ( 321 ~; 45 |; 185 &;13920 @)
% ( 0 <=>;2686 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 561 ( 561 >; 0 *; 0 +; 0 <<)
% Number of symbols : 163 ( 160 usr; 12 con; 0-4 aty)
% Number of variables : 4013 ( 138 ^;3716 !; 159 ?;4013 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:35:52.207
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (160)
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thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
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thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001tf__a,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_Itf__a_J,type,
ord_max_set_a: set_a > set_a > set_a ).
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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno6951661231331188332t_unit: partia2670972154091845814t_unit > list_list_a > list_a > $o ).
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_001tf__a_001tf__b,type,
polynomial_roots_a_b: partia2175431115845679010xt_a_b > list_a > multiset_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_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_Oeval_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
eval_l34571156754992824t_unit: partia2670972154091845814t_unit > list_list_a > list_a > list_a ).
thf(sy_c_Polynomials_Oring_Oeval_001tf__a_001tf__b,type,
eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).
thf(sy_c_Polynomials_Oring_Omonom_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_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__dense_001tf__a_001tf__b,type,
poly_of_dense_a_b: partia2175431115845679010xt_a_b > list_P3592885314253461005_a_nat > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_Polynomials_Ovar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
var_li8453953174693405341t_unit: partia2670972154091845814t_unit > list_list_a ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oabelian__group_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_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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_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_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_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_f,type,
f: nat > a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1269)
thf(fact_0_length__build__poly,axiom,
! [F: nat > a,N: nat] : ( ord_less_eq_nat @ ( size_size_list_a @ ( bounde1002222742488328185ly_a_b @ r @ F @ N ) ) @ N ) ).
% length_build_poly
thf(fact_1_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_2_ring_Obuild__poly_Ocong,axiom,
bounde1002222742488328185ly_a_b = bounde1002222742488328185ly_a_b ).
% ring.build_poly.cong
thf(fact_3_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_4_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_5_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_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_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_27_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_28_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_63_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_64_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_65_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_66_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_67_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_68_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_69_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_70_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_71_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_72_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_73_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_74_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_75_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_76_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_77_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_78_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_79_ring_Olength__build__poly,axiom,
! [R: partia2175431115845679010xt_a_b,F: nat > a,N: nat] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( bounde1002222742488328185ly_a_b @ R @ F @ N ) ) @ N ) ) ).
% ring.length_build_poly
thf(fact_80_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_81_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_82_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_83_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_84_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_85_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_86_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_87_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_88_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_89_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_90_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_91_normalize__coeff,axiom,
! [P: list_a] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_coeff
thf(fact_92_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_93_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_94_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_95_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_96_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_97_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_98_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_99_ring_Odense__repr_Ocong,axiom,
dense_repr_a_b = dense_repr_a_b ).
% ring.dense_repr.cong
thf(fact_100_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_dense__repr_Osimps_I1_J,axiom,
( ( dense_repr_a_b @ r @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ).
% dense_repr.simps(1)
thf(fact_125_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_126_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_127_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_128_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_129_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_130_append_Oright__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ A @ nil_Pr7402525243500994295_a_nat )
= A ) ).
% append.right_neutral
thf(fact_131_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_132_append__Nil2,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ Xs @ nil_Pr7402525243500994295_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_133_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_134_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_135_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_136_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_137_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_138_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_139_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_140_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_141_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_142_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_143_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_144_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_145_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_146_append__Nil,axiom,
! [Ys: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_147_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_148_append_Oleft__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_149_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_150_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_151_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_152_drop__Nil,axiom,
! [N: nat] :
( ( drop_P2883665741211355575_a_nat @ N @ nil_Pr7402525243500994295_a_nat )
= nil_Pr7402525243500994295_a_nat ) ).
% drop_Nil
thf(fact_153_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_154_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_155_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_156_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_157_diff__size__le__size__Diff,axiom,
! [M2: multiset_a,M3: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M2 ) @ ( size_size_multiset_a @ M3 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ M3 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_158_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_159_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_160_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_161_splitted__def,axiom,
! [P: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_162_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ G )
=> ( abelian_monoid_a_b @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_163_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_164_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_165_subset__code_I1_J,axiom,
! [Xs: list_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B4 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_166_subset__code_I1_J,axiom,
! [Xs: list_nat_a,B4: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( set_nat_a2 @ Xs ) @ B4 )
= ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ ( set_nat_a2 @ Xs ) )
=> ( member_nat_a @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_167_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_168_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_169_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_170_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_171_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_172_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_173_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_174_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_175_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_176_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_177_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_178_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_179_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_180_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_181_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_182_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_183_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_184_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P4 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_185_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_186_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_187_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_188_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_189_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_190_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_191_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_192_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_193_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_194_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_195_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_196_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_197_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_198_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_199_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_200_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P2 @ M5 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_201_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P2 @ M5 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_202_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_203_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_204_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_205_order__less__asym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_206_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_207_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_208_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_209_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_210_order__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_211_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_212_ord__eq__less__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( 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 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_213_ord__eq__less__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_214_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_215_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_216_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( 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 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_217_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_218_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_219_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_220_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_221_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_222_order__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_223_order__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_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_less_subst1
thf(fact_224_order__less__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_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_225_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_226_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_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_subst2
thf(fact_227_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_228_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_229_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_230_order__less__not__sym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_231_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_232_order__less__imp__triv,axiom,
! [X: set_a,Y: set_a,P2: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_233_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_234_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_235_order__less__imp__not__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_236_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_237_order__less__imp__not__eq2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_238_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_239_order__less__imp__not__less,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_240_ring_Osplitted_Ocong,axiom,
polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).
% ring.splitted.cong
thf(fact_241_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_242_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_243_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_244_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_245_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_246_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_247_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_248_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_249_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_279_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_280_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_281_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_282_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_283_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_order__le__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_296_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_297_order__le__less__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_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_308_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_309_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_310_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_311_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_312_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_313_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_314_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_315_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_316_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_317_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_318_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_319_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_320_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_321_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_322_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_323_set__drop__subset,axiom,
! [N: nat,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ N @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_drop_subset
thf(fact_324_in__set__dropD,axiom,
! [X: list_a,N: nat,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( drop_list_a @ N @ Xs ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_325_in__set__dropD,axiom,
! [X: nat > a,N: nat,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ ( drop_nat_a @ N @ Xs ) ) )
=> ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_326_in__set__dropD,axiom,
! [X: a,N: nat,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( drop_a @ N @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_327_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_328_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_329_ring_Opolynomial__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 ) ) ) ).
% ring.polynomial_incl
thf(fact_330_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ M @ Xs ) ) @ ( set_a2 @ ( drop_a @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_331_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_332_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_333_ring_Osplitted__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polyno8329700637149614481ed_a_b @ R @ P )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ R @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% ring.splitted_def
thf(fact_334_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_335_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_336_ring_Onormalize__gives__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 )
=> ( polynomial_a_b @ R @ K2 @ ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.normalize_gives_polynomial
thf(fact_337_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_338_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_339_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_340_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_341_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_342_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_343_ring_Ois__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( abelian_group_a_b @ R ) ) ).
% ring.is_abelian_group
thf(fact_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_length__replicate,axiom,
! [N: nat,X: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_357_nth__replicate,axiom,
! [I: nat,N: nat,X: a] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_a @ ( replicate_a @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_358_drop__replicate,axiom,
! [I: nat,K: nat,X: a] :
( ( drop_a @ I @ ( replicate_a @ K @ X ) )
= ( replicate_a @ ( minus_minus_nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_359_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_360_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_361_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_362_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_363_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_364_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_365_append__replicate__commute,axiom,
! [N: nat,X: a,K: nat] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K @ X ) )
= ( append_a @ ( replicate_a @ K @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_366_ring_Omonom__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ R @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_367_ring_Omonom__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ R @ A @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_368_replicate__eqI,axiom,
! [Xs: list_list_a,N: nat,X: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_list_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_369_replicate__eqI,axiom,
! [Xs: list_nat_a,N: nat,X: nat > a] :
( ( ( size_size_list_nat_a @ Xs )
= N )
=> ( ! [Y2: nat > a] :
( ( member_nat_a @ Y2 @ ( set_nat_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_nat_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_370_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_371_replicate__length__same,axiom,
! [Xs: list_a,X: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_372_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_373_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_374_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_375_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_376_ring_Oring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_377_abelian__groupE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_378_abelian__groupE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_379_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_380_abelian__monoid_Ozero__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_381_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_382_abelian__monoidE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_383_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_384_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_385_ring_Oreplicate__zero__coeff,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.replicate_zero_coeff
thf(fact_386_ring_Oreplicate__zero__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( replicate_a @ N @ ( zero_a_b @ R ) ) )
= ( ^ [Uu: nat] : ( zero_a_b @ R ) ) ) ) ).
% ring.replicate_zero_coeff
thf(fact_387_nth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_388_nth__mem,axiom,
! [N: nat,Xs: list_nat_a] :
( ( ord_less_nat @ N @ ( size_size_list_nat_a @ Xs ) )
=> ( member_nat_a @ ( nth_nat_a @ Xs @ N ) @ ( set_nat_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_389_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_390_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P2: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P2 @ X2 ) )
=> ( P2 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_391_in__set__conv__nth,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_392_in__set__conv__nth,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat_a @ Xs ) )
& ( ( nth_nat_a @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_393_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_394_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P2: list_a > $o,X: list_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P2 @ ( nth_list_a @ Xs @ I3 ) ) )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_395_all__nth__imp__all__set,axiom,
! [Xs: list_nat_a,P2: ( nat > a ) > $o,X: nat > a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat_a @ Xs ) )
=> ( P2 @ ( nth_nat_a @ Xs @ I3 ) ) )
=> ( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_396_all__nth__imp__all__set,axiom,
! [Xs: list_a,P2: a > $o,X: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I3 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_397_all__set__conv__all__nth,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P2 @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_398_ring_Onormalize__replicate__zero,axiom,
! [R: partia2670972154091845814t_unit,N: nat,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P ) )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ).
% ring.normalize_replicate_zero
thf(fact_399_ring_Onormalize__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,P: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P ) )
= ( normalize_a_b @ R @ P ) ) ) ).
% ring.normalize_replicate_zero
thf(fact_400_ring_Oprefix__replicate__zero__coeff,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ P )
= ( coeff_6360649920519955023t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P ) ) ) ) ).
% ring.prefix_replicate_zero_coeff
thf(fact_401_ring_Oprefix__replicate__zero__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ P )
= ( coeff_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P ) ) ) ) ).
% ring.prefix_replicate_zero_coeff
thf(fact_402_ring_Odense__repr__replicate__zero,axiom,
! [R: partia2670972154091845814t_unit,N: nat,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( dense_5814815041220002634t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P ) )
= ( dense_5814815041220002634t_unit @ R @ P ) ) ) ).
% ring.dense_repr_replicate_zero
thf(fact_403_ring_Odense__repr__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,P: list_a] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P ) )
= ( dense_repr_a_b @ R @ P ) ) ) ).
% ring.dense_repr_replicate_zero
thf(fact_404_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_405_ring_Oconst__term__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( P != nil_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_406_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P )
!= ( zero_a_b @ R ) )
=> ( P != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_407_ring_Onormalize__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ R @ P ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_408_ring_Onormalize__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_409_Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( coeff_a_b @ R @ P @ I ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% Polynomials.ring.coeff_in_carrier
thf(fact_410_Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P @ I ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% Polynomials.ring.coeff_in_carrier
thf(fact_411_ring_Oexp__base__closed,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ R @ X @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_412_ring_Oexp__base__closed,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ R @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_413_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_414_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_415_ring_Onormalize__trick,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ R ) ) @ ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.normalize_trick
thf(fact_416_ring_Onormalize__trick,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) ) @ ( zero_a_b @ R ) ) @ ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.normalize_trick
thf(fact_417_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_418_ring_Onormalize__def_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( P
= ( append_list_a @ ( replicate_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ ( zero_l4142658623432671053t_unit @ R ) ) @ ( drop_list_a @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) ) @ P ) ) ) ) ).
% ring.normalize_def'(1)
thf(fact_419_ring_Onormalize__def_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) ) @ ( zero_a_b @ R ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) ) @ P ) ) ) ) ).
% ring.normalize_def'(1)
thf(fact_420_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_421_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_422_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P5 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_423_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_424_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_425_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_426_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_427_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_428_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_429_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_430_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_431_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_432_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_433_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_434_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_435_eval__in__carrier,axiom,
! [P: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_436_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_437_combine__eq__eval,axiom,
! [Ks: list_a,X: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X ) ) ).
% combine_eq_eval
thf(fact_438_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_439_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_440_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_441_eval__normalize,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_455_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_456_ring_Opoly__add_Ocong,axiom,
poly_add_a_b = poly_add_a_b ).
% ring.poly_add.cong
thf(fact_457_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_458_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_459_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_460_list_Oset__intros_I2_J,axiom,
! [Y: nat > a,X22: list_nat_a,X21: nat > a] :
( ( member_nat_a @ Y @ ( set_nat_a2 @ X22 ) )
=> ( member_nat_a @ Y @ ( set_nat_a2 @ ( cons_nat_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_461_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_462_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_463_list_Oset__intros_I1_J,axiom,
! [X21: nat > a,X22: list_nat_a] : ( member_nat_a @ X21 @ ( set_nat_a2 @ ( cons_nat_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_464_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_465_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z22: list_list_a] :
( A
!= ( cons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_466_list_Oset__cases,axiom,
! [E: nat > a,A: list_nat_a] :
( ( member_nat_a @ E @ ( set_nat_a2 @ A ) )
=> ( ! [Z22: list_nat_a] :
( A
!= ( cons_nat_a @ E @ Z22 ) )
=> ~ ! [Z1: nat > a,Z22: list_nat_a] :
( ( A
= ( cons_nat_a @ Z1 @ Z22 ) )
=> ~ ( member_nat_a @ E @ ( set_nat_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_467_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_468_set__ConsD,axiom,
! [Y: list_a,X: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_469_set__ConsD,axiom,
! [Y: nat > a,X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ Y @ ( set_nat_a2 @ ( cons_nat_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat_a @ Y @ ( set_nat_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_470_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_471_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_472_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_473_list__induct2_H,axiom,
! [P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o,Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] : ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] : ( P2 @ nil_Pr7402525243500994295_a_nat @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_483_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_484_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_485_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_486_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_487_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_488_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_489_ring_Ocombine__eq__eval,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks @ ( polyno2922411391617481336se_a_b @ R @ X @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ R @ Ks @ X ) ) ) ).
% ring.combine_eq_eval
thf(fact_490_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_491_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_492_list__induct4,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_P3592885314253461005_a_nat > list_a > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ 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_Pr7402525243500994295_a_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ 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_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_493_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_494_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_495_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_496_list__induct4,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_a,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ 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_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ 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_P5205166803686508359_a_nat @ 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_497_list__induct4,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P2: list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ 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_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ 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_P5205166803686508359_a_nat @ 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_498_list__induct4,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ 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_Pr7402525243500994295_a_nat @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a,Z3: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ 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_P5205166803686508359_a_nat @ 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_499_list__induct4,axiom,
! [Xs: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P2: list_a > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ 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_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_500_list__induct4,axiom,
! [Xs: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P2: list_a > list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( 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_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_501_list__induct3,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_502_list__induct3,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_a,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: a,Zs2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_503_list__induct3,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_a,Zs: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: a,Ys4: list_a,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_504_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_505_list__induct3,axiom,
! [Xs: list_a,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,P2: list_a > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P2 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z3: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_506_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_507_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_508_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_509_list__induct2,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ( P2 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y2 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_510_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_511_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_512_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_513_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_514_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_515_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_516_in__set__conv__decomp__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_517_in__set__conv__decomp__first,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
= ( ? [Ys3: list_nat_a,Zs3: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys3 @ ( cons_nat_a @ X @ Zs3 ) ) )
& ~ ( member_nat_a @ X @ ( set_nat_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_518_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_519_in__set__conv__decomp__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_520_in__set__conv__decomp__last,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
= ( ? [Ys3: list_nat_a,Zs3: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys3 @ ( cons_nat_a @ X @ Zs3 ) ) )
& ~ ( member_nat_a @ X @ ( set_nat_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_521_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_522_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys4: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_523_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys4: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_524_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys4: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_525_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys4: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_526_in__set__conv__decomp,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_527_in__set__conv__decomp,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
= ( ? [Ys3: list_nat_a,Zs3: list_nat_a] :
( Xs
= ( append_nat_a @ Ys3 @ ( cons_nat_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_528_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_529_append__Cons__eq__iff,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs4: list_list_a,Ys5: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) )
= ( append_list_a @ Xs4 @ ( cons_list_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_530_append__Cons__eq__iff,axiom,
! [X: nat > a,Xs: list_nat_a,Ys: list_nat_a,Xs4: list_nat_a,Ys5: list_nat_a] :
( ~ ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ( ~ ( member_nat_a @ X @ ( set_nat_a2 @ Ys ) )
=> ( ( ( append_nat_a @ Xs @ ( cons_nat_a @ X @ Ys ) )
= ( append_nat_a @ Xs4 @ ( cons_nat_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_531_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_532_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys4: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ~ ( P2 @ X2 ) ) ) ).
% split_list_propE
thf(fact_533_split__list__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys4: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_534_split__list__first,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ? [Ys4: list_nat_a,Zs2: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X @ Zs2 ) ) )
& ~ ( member_nat_a @ X @ ( set_nat_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_535_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_536_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys4: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 ) ) ) ).
% split_list_prop
thf(fact_537_split__list__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys4: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_538_split__list__last,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ? [Ys4: list_nat_a,Zs2: list_nat_a] :
( ( Xs
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X @ Zs2 ) ) )
& ~ ( member_nat_a @ X @ ( set_nat_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_539_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_540_split__list,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys4: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_541_split__list,axiom,
! [X: nat > a,Xs: list_nat_a] :
( ( member_nat_a @ X @ ( set_nat_a2 @ Xs ) )
=> ? [Ys4: list_nat_a,Zs2: list_nat_a] :
( Xs
= ( append_nat_a @ Ys4 @ ( cons_nat_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_542_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_543_rev__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 @ ( append7679239579558125090_a_nat @ Xs2 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_544_rev__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 @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_545_append__eq__Cons__conv,axiom,
! [Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Ys @ Zs )
= ( cons_P5205166803686508359_a_nat @ X @ Xs ) )
= ( ( ( Ys = nil_Pr7402525243500994295_a_nat )
& ( Zs
= ( cons_P5205166803686508359_a_nat @ X @ Xs ) ) )
| ? [Ys6: list_P3592885314253461005_a_nat] :
( ( Ys
= ( cons_P5205166803686508359_a_nat @ X @ Ys6 ) )
& ( ( append7679239579558125090_a_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_546_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_547_Cons__eq__append__conv,axiom,
! [X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X @ Xs )
= ( append7679239579558125090_a_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr7402525243500994295_a_nat )
& ( ( cons_P5205166803686508359_a_nat @ X @ Xs )
= Zs ) )
| ? [Ys6: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X @ Ys6 )
= Ys )
& ( Xs
= ( append7679239579558125090_a_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_548_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_549_rev__exhaust,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( Xs != nil_Pr7402525243500994295_a_nat )
=> ~ ! [Ys4: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat] :
( Xs
!= ( append7679239579558125090_a_nat @ Ys4 @ ( cons_P5205166803686508359_a_nat @ Y2 @ nil_Pr7402525243500994295_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_550_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys4: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys4 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_551_rev__induct,axiom,
! [P2: list_P3592885314253461005_a_nat > $o,Xs: list_P3592885314253461005_a_nat] :
( ( P2 @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X2: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append7679239579558125090_a_nat @ Xs2 @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_552_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_553_ring_Odense__repr_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a] :
( ( ring_a_b @ R )
=> ( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_554_replicate__app__Cons__same,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( append_a @ ( replicate_a @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_555_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_556_same__length__different,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( Xs != Ys )
=> ( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ? [Pre: list_P3592885314253461005_a_nat,X2: product_prod_a_nat,Xs5: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Ys7: list_P3592885314253461005_a_nat] :
( ( X2 != Y2 )
& ( Xs
= ( append7679239579558125090_a_nat @ Pre @ ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) @ Xs5 ) ) )
& ( Ys
= ( append7679239579558125090_a_nat @ Pre @ ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ Y2 @ nil_Pr7402525243500994295_a_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_557_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X2: a,Xs5: list_a,Y2: a,Ys7: list_a] :
( ( X2 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs5 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_558_replicate__append__same,axiom,
! [I: nat,X: product_prod_a_nat] :
( ( append7679239579558125090_a_nat @ ( replic5595554873386817213_a_nat @ I @ X ) @ ( cons_P5205166803686508359_a_nat @ X @ nil_Pr7402525243500994295_a_nat ) )
= ( cons_P5205166803686508359_a_nat @ X @ ( replic5595554873386817213_a_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_559_replicate__append__same,axiom,
! [I: nat,X: a] :
( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_560_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( eval_l34571156754992824t_unit @ R @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_561_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( eval_a_b @ R @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_562_ring_Oconst__term__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( const_6738166269504826821t_unit @ R @ P )
= ( eval_l34571156754992824t_unit @ R @ P @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_563_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P )
= ( eval_a_b @ R @ P @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_564_ring_Opoly__add__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ 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 ) ) ) ) ) ) ) ).
% ring.poly_add_append_zero
thf(fact_565_ring_Opoly__add__append__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) @ ( append_list_a @ Q @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( poly_a7601779127272115787t_unit @ R @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ).
% ring.poly_add_append_zero
thf(fact_566_ring_Opoly__add__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.poly_add_in_carrier
thf(fact_567_ring_Opoly__add__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.poly_add_in_carrier
thf(fact_568_ring_Opoly__add__comm,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.poly_add_comm
thf(fact_569_ring_Opoly__add__comm,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ R @ P22 @ P1 ) ) ) ) ) ).
% ring.poly_add_comm
thf(fact_570_ring_Oeval__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_571_ring_Oeval__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_572_ring_Omonom__def,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( monom_7446464087056152608t_unit @ R @ A @ N )
= ( cons_list_a @ A @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.monom_def
thf(fact_573_ring_Omonom__def,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( monom_a_b @ R @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_def
thf(fact_574_ring_Opoly__add__normalize__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.poly_add_normalize_aux
thf(fact_575_ring_Opoly__add__normalize__aux,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ R @ ( normal637505603836502915t_unit @ R @ P1 ) @ P22 ) ) ) ) ) ).
% ring.poly_add_normalize_aux
thf(fact_576_ring_Opoly__add__normalize_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_add_normalize(2)
thf(fact_577_ring_Opoly__add__normalize_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ R @ P1 @ ( normal637505603836502915t_unit @ R @ P22 ) ) ) ) ) ) ).
% ring.poly_add_normalize(2)
thf(fact_578_ring_Opoly__add__normalize_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_add_normalize(3)
thf(fact_579_ring_Opoly__add__normalize_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ R @ ( normal637505603836502915t_unit @ R @ P1 ) @ ( normal637505603836502915t_unit @ R @ P22 ) ) ) ) ) ) ).
% ring.poly_add_normalize(3)
thf(fact_580_ring_Oeval__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( normalize_a_b @ R @ P ) @ A )
= ( eval_a_b @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_581_ring_Oeval__normalize,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) @ A )
= ( eval_l34571156754992824t_unit @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_582_ring_Opoly__add__zero_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.poly_add_zero'(2)
thf(fact_583_ring_Opoly__add__zero_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ nil_list_a @ P )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(2)
thf(fact_584_ring_Opoly__add__zero_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.poly_add_zero'(1)
thf(fact_585_ring_Opoly__add__zero_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P @ nil_list_a )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_zero'(1)
thf(fact_586_ring_Oconst__term__eq__last,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.const_term_eq_last
thf(fact_587_ring_Oconst__term__eq__last,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ) ).
% ring.const_term_eq_last
thf(fact_588_ring_Oconst__term__explicit,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ R @ P )
= A )
=> ~ ! [P5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P5 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ) ).
% ring.const_term_explicit
thf(fact_589_ring_Oconst__term__explicit,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
= A )
=> ~ ! [P5: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P5 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( P
!= ( append_list_a @ P5 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ) ).
% ring.const_term_explicit
thf(fact_590_ring_Opoly__add__replicate__zero_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.poly_add_replicate_zero'(1)
thf(fact_591_ring_Opoly__add__replicate__zero_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(1)
thf(fact_592_ring_Opoly__add__replicate__zero_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.poly_add_replicate_zero'(2)
thf(fact_593_ring_Opoly__add__replicate__zero_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.poly_add_replicate_zero'(2)
thf(fact_594_ring_Oeval__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P ) @ A )
= ( eval_a_b @ R @ P @ A ) ) ) ) ) ).
% ring.eval_replicate
thf(fact_595_ring_Oeval__replicate,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P ) @ A )
= ( eval_l34571156754992824t_unit @ R @ P @ A ) ) ) ) ) ).
% ring.eval_replicate
thf(fact_596_ring_Opoly__add__append__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ 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 ) ) ) ) ) ) ).
% ring.poly_add_append_replicate
thf(fact_597_ring_Opoly__add__append__replicate,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_a7601779127272115787t_unit @ R @ ( append_list_a @ P @ ( replicate_list_a @ ( size_s349497388124573686list_a @ Q ) @ ( zero_l4142658623432671053t_unit @ R ) ) ) @ Q )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P @ Q ) ) ) ) ) ) ).
% ring.poly_add_append_replicate
thf(fact_598_is__root__def,axiom,
! [P: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_599_ring_Ocombine__prepend__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us3: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ Ks ) @ Us3 )
= ( embedded_combine_a_b @ R @ Ks @ ( drop_a @ N @ Us3 ) ) ) ) ) ) ).
% ring.combine_prepend_replicate
thf(fact_600_ring_Ocombine__prepend__replicate,axiom,
! [R: partia2670972154091845814t_unit,Ks: list_list_a,Us3: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd2435972518007585703t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ Ks ) @ Us3 )
= ( embedd2435972518007585703t_unit @ R @ Ks @ ( drop_list_a @ N @ Us3 ) ) ) ) ) ) ).
% ring.combine_prepend_replicate
thf(fact_601_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_602_ring_Ocombine__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ Us3 )
= ( embedded_combine_a_b @ R @ Ks @ Us3 ) ) ) ) ).
% ring.combine_append_zero
thf(fact_603_ring_Ocombine__append__zero,axiom,
! [R: partia2670972154091845814t_unit,Us3: list_list_a,Ks: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd2435972518007585703t_unit @ R @ ( append_list_a @ Ks @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) @ Us3 )
= ( embedd2435972518007585703t_unit @ R @ Ks @ Us3 ) ) ) ) ).
% ring.combine_append_zero
thf(fact_604_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K3: a] : ( normalize_a_b @ r @ ( cons_a @ K3 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_605_ring_Ocombine__append__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a,Ks: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( append_a @ Ks @ ( replicate_a @ N @ ( zero_a_b @ R ) ) ) @ Us3 )
= ( embedded_combine_a_b @ R @ Ks @ Us3 ) ) ) ) ).
% ring.combine_append_replicate
thf(fact_606_ring_Ocombine__append__replicate,axiom,
! [R: partia2670972154091845814t_unit,Us3: list_list_a,Ks: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd2435972518007585703t_unit @ R @ ( append_list_a @ Ks @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) ) @ Us3 )
= ( embedd2435972518007585703t_unit @ R @ Ks @ Us3 ) ) ) ) ).
% ring.combine_append_replicate
thf(fact_607_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_608_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_609_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X )
= X ) ) ) ).
% ring.eval_var
thf(fact_610_ring_Oeval__var,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ X )
= X ) ) ) ).
% ring.eval_var
thf(fact_611_ring_Opoly__of__const__def,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( poly_of_const_a_b @ R )
= ( ^ [K3: a] : ( normalize_a_b @ R @ ( cons_a @ K3 @ nil_a ) ) ) ) ) ).
% ring.poly_of_const_def
thf(fact_612_ring_Ois__root__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X )
= ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
& ( ( eval_l34571156754992824t_unit @ R @ P @ X )
= ( zero_l4142658623432671053t_unit @ R ) )
& ( P != nil_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_613_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P @ X )
= ( zero_a_b @ R ) )
& ( P != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_614_ring_Ocombine_Osimps_I2_J,axiom,
! [R: partia2670972154091845814t_unit,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd2435972518007585703t_unit @ R @ nil_list_a @ Us3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.combine.simps(2)
thf(fact_615_ring_Ocombine_Osimps_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ nil_a @ Us3 )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(2)
thf(fact_616_ring_Ocombine_Osimps_I3_J,axiom,
! [R: partia2670972154091845814t_unit,Ks: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd2435972518007585703t_unit @ R @ Ks @ nil_list_a )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.combine.simps(3)
thf(fact_617_ring_Ocombine_Osimps_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks @ nil_a )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(3)
thf(fact_618_ring_Ocombine__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( embedded_combine_a_b @ R @ Ks @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.combine_in_carrier
thf(fact_619_ring_Ocombine__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,Ks: list_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( embedd2435972518007585703t_unit @ R @ Ks @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.combine_in_carrier
thf(fact_620_ring_Ocombine__replicate,axiom,
! [R: partia2175431115845679010xt_a_b,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( replicate_a @ ( size_size_list_a @ Us3 ) @ ( zero_a_b @ R ) ) @ Us3 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.combine_replicate
thf(fact_621_ring_Ocombine__replicate,axiom,
! [R: partia2670972154091845814t_unit,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd2435972518007585703t_unit @ R @ ( replicate_list_a @ ( size_s349497388124573686list_a @ Us3 ) @ ( zero_l4142658623432671053t_unit @ R ) ) @ Us3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.combine_replicate
thf(fact_622_poly__degree__bound__from__coeff,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ N @ K4 )
=> ( ( coeff_a_b @ r @ X @ K4 )
= ( zero_a_b @ r ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
| ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% poly_degree_bound_from_coeff
thf(fact_623_eval__poly__add__aux,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_624_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_625_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_626_bound_Ointro,axiom,
! [N: nat,F: nat > a,Z2: a] :
( ! [M6: nat] :
( ( ord_less_nat @ N @ M6 )
=> ( ( F @ M6 )
= Z2 ) )
=> ( bound_a @ Z2 @ N @ F ) ) ).
% bound.intro
thf(fact_627_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_628_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_629_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_630_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_631_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_632_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_633_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_634_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_635_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_636_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_637_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_638_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_639_coeff__in__carrier,axiom,
! [P: list_a,I: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% coeff_in_carrier
thf(fact_640_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_641_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_642_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_643_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_644_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_645_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_646_eval__poly__add,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_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_655_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_656_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_657_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_658_poly__degree__bound__from__coeff__1,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ N @ K4 )
=> ( ( coeff_a_b @ r @ X @ K4 )
= ( zero_a_b @ r ) ) )
=> ( member_list_a @ X @ ( bounde2262800523058855161ls_a_b @ r @ N ) ) ) ) ).
% poly_degree_bound_from_coeff_1
thf(fact_659_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_660_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,K5: set_a,P6: list_a] : ( member_list_a @ P6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K5 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_661_ring_Opoly__mult_Ocong,axiom,
poly_mult_a_b = poly_mult_a_b ).
% ring.poly_mult.cong
thf(fact_662_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_663_ring_Oring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_664_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_665_ring_Oring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_666_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_667_ring_Oring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_668_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_669_ring_Oring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_670_ring_Opoly__mult_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( poly_mult_a_b @ R @ nil_a @ P22 )
= nil_a ) ) ).
% ring.poly_mult.simps(1)
thf(fact_671_abelian__groupE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_672_abelian__groupE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_673_abelian__groupE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( abelian_group_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_674_abelian__groupE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_675_abelian__groupE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_676_abelian__groupE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_677_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_678_abelian__monoid_Oa__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_679_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z2 ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_680_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z2 ) )
= ( add_li7652885771158616974t_unit @ G @ Y @ ( add_li7652885771158616974t_unit @ G @ X @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_681_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X @ Y ) @ Z2 )
= ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_682_abelian__monoid_Oa__assoc,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_683_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ Y )
= ( add_a_b @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_684_abelian__monoid_Oa__comm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ Y )
= ( add_li7652885771158616974t_unit @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_685_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_686_abelian__monoidE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_687_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_688_abelian__monoidE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_689_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_690_abelian__monoidE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_691_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_692_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_693_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_694_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_695_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_696_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_697_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_698_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_699_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_700_ring_Oring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_701_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_702_ring_Oring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_703_abelian__groupE_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_704_abelian__groupE_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( add_li7652885771158616974t_unit @ R @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_705_abelian__groupE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_706_abelian__groupE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_707_abelian__groupI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Z3: a] :
( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y2 ) @ Z3 )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y2 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y2 )
= ( add_a_b @ R @ Y2 @ X2 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ Xa @ X2 )
= ( zero_a_b @ R ) ) ) )
=> ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_708_abelian__groupI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Z3: list_a] :
( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y2 ) @ Z3 )
= ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z3 ) ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ X2 ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ R ) )
& ( ( add_li7652885771158616974t_unit @ R @ Xa @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) )
=> ( abelia3891852623213500406t_unit @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_709_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P @ I ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_710_Bounded__Degree__Polynomials_Oring_Ocoeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( member_a @ ( coeff_a_b @ R @ P @ I ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% Bounded_Degree_Polynomials.ring.coeff_in_carrier
thf(fact_711_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_712_abelian__monoidE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_713_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_714_abelian__monoid_Ol__zero,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_715_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( zero_a_b @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_716_abelian__monoid_Or__zero,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( zero_l4142658623432671053t_unit @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_717_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y6: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X @ Y6 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y6 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_718_abelian__monoid_Ominus__unique,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y6: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X @ Y6 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y = Y6 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_719_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X2: a,Y2: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y2 ) @ Z3 )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y2 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y2 )
= ( add_a_b @ R @ Y2 @ X2 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_720_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X2: list_a,Y2: list_a,Z3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y2 ) @ Z3 )
= ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z3 ) ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ X2 ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_721_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_722_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_723_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_724_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_725_ring_Opoly__mult__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.poly_mult_in_carrier
thf(fact_726_ring_Opoly__mult__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_m7087347720095500472t_unit @ R @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.poly_mult_in_carrier
thf(fact_727_ring_Opoly__mult__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P @ nil_a )
= nil_a ) ) ) ).
% ring.poly_mult_zero(2)
thf(fact_728_ring_Opoly__mult__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P @ nil_list_a )
= nil_list_a ) ) ) ).
% ring.poly_mult_zero(2)
thf(fact_729_ring_Opoly__mult__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ nil_a @ P )
= nil_a ) ) ) ).
% ring.poly_mult_zero(1)
thf(fact_730_ring_Opoly__mult__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ nil_list_a @ P )
= nil_list_a ) ) ) ).
% ring.poly_mult_zero(1)
thf(fact_731_ring_Opoly__mult__l__distr_H,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a,P32: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.poly_mult_l_distr'
thf(fact_732_ring_Opoly__mult__l__distr_H,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a,P32: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P32 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 ) @ P32 )
= ( poly_a7601779127272115787t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ P1 @ P32 ) @ ( poly_m7087347720095500472t_unit @ R @ P22 @ P32 ) ) ) ) ) ) ) ).
% ring.poly_mult_l_distr'
thf(fact_733_ring_Opoly__mult__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.poly_mult_normalize
thf(fact_734_ring_Opoly__mult__normalize,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ R @ ( normal637505603836502915t_unit @ R @ P1 ) @ P22 ) ) ) ) ) ).
% ring.poly_mult_normalize
thf(fact_735_ring_Opoly__add__coeff__aux,axiom,
! [R: partia2670972154091845814t_unit,P22: list_list_a,P1: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P22 ) @ ( size_s349497388124573686list_a @ P1 ) )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 ) )
= ( ^ [I2: nat] : ( add_li7652885771158616974t_unit @ R @ ( coeff_6360649920519955023t_unit @ R @ P1 @ I2 ) @ ( coeff_6360649920519955023t_unit @ R @ P22 @ I2 ) ) ) ) ) ) ).
% ring.poly_add_coeff_aux
thf(fact_736_ring_Opoly__add__coeff__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P22: list_a,P1: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_add_coeff_aux
thf(fact_737_ring_Opoly__add__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.poly_add_coeff
thf(fact_738_ring_Opoly__add__coeff,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P1 @ P22 ) )
= ( ^ [I2: nat] : ( add_li7652885771158616974t_unit @ R @ ( coeff_6360649920519955023t_unit @ R @ P1 @ I2 ) @ ( coeff_6360649920519955023t_unit @ R @ P22 @ I2 ) ) ) ) ) ) ) ).
% ring.poly_add_coeff
thf(fact_739_ring_Oeval__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_add_a_b @ R @ P @ Q ) @ A )
= ( add_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( eval_a_b @ R @ Q @ A ) ) ) ) ) ) ) ).
% ring.eval_poly_add
thf(fact_740_ring_Oeval__poly__add,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( eval_l34571156754992824t_unit @ R @ Q @ A ) ) ) ) ) ) ) ).
% ring.eval_poly_add
thf(fact_741_ring_Opoly__mult__prepend__replicate__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.poly_mult_prepend_replicate_zero
thf(fact_742_ring_Opoly__mult__prepend__replicate__zero,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ R @ ( append_list_a @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P1 ) @ P22 ) ) ) ) ) ).
% ring.poly_mult_prepend_replicate_zero
thf(fact_743_ring_Ocombine__append,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us3: list_a,Ks2: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ 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 ) ) ) ) ) ) ) ) ) ).
% ring.combine_append
thf(fact_744_ring_Ocombine__append,axiom,
! [R: partia2670972154091845814t_unit,Ks: list_list_a,Us3: list_list_a,Ks2: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us3 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( embedd2435972518007585703t_unit @ R @ Ks @ Us3 ) @ ( embedd2435972518007585703t_unit @ R @ Ks2 @ Vs2 ) )
= ( embedd2435972518007585703t_unit @ R @ ( append_list_a @ Ks @ Ks2 ) @ ( append_list_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ) ) ).
% ring.combine_append
thf(fact_745_ring_Odegree__oneE,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ 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 @ R ) )
=> ! [B3: list_a] :
( ( member_list_a @ B3 @ K2 )
=> ( P
!= ( cons_list_a @ A3 @ ( cons_list_a @ B3 @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% ring.degree_oneE
thf(fact_746_ring_Odegree__oneE,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ) ) ).
% ring.degree_oneE
thf(fact_747_ring_Oeval__poly__add__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_add_a_b @ R @ P @ Q ) @ A )
= ( add_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( eval_a_b @ R @ Q @ A ) ) ) ) ) ) ) ) ).
% ring.eval_poly_add_aux
thf(fact_748_ring_Oeval__poly__add__aux,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( eval_l34571156754992824t_unit @ R @ Q @ A ) ) ) ) ) ) ) ) ).
% ring.eval_poly_add_aux
thf(fact_749_bound__def,axiom,
( bound_a
= ( ^ [Z4: a,N2: nat,F2: nat > a] :
! [M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( ( F2 @ M4 )
= Z4 ) ) ) ) ).
% bound_def
thf(fact_750_bound_Obound,axiom,
! [Z2: a,N: nat,F: nat > a,M: nat] :
( ( bound_a @ Z2 @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z2 ) ) ) ).
% bound.bound
thf(fact_751_bound__below,axiom,
! [Z2: a,M: nat,F: nat > a,N: nat] :
( ( bound_a @ Z2 @ M @ F )
=> ( ( ( F @ N )
!= Z2 )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_752_ring_Opoly__mult__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ 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 ) ) ) ) ) ) ) ).
% ring.poly_mult_append_zero
thf(fact_753_ring_Opoly__mult__append__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) @ Q )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ).
% ring.poly_mult_append_zero
thf(fact_754_ring_Opoly__degree__bound__from__coeff,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ N @ K4 )
=> ( ( coeff_6360649920519955023t_unit @ R @ X @ K4 )
= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat ) @ N )
| ( X
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff
thf(fact_755_ring_Opoly__degree__bound__from__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ N @ K4 )
=> ( ( coeff_a_b @ R @ X @ K4 )
= ( zero_a_b @ R ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
| ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff
thf(fact_756_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2175431115845679010xt_a_b,N: nat,F: nat > a,M: nat] :
( ( abelian_monoid_a_b @ G )
=> ( ( bound_a @ ( zero_a_b @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_757_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2670972154091845814t_unit,N: nat,F: nat > list_a,M: nat] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_758_eval__append__aux,axiom,
! [P: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_759_const__term__zero,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P5: list_a] :
( ( polynomial_a_b @ r @ K2 @ P5 )
=> ( ( P5 != nil_a )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_760_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_761_a__lcos__m__assoc,axiom,
! [M2: set_a,G2: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G2 @ ( a_l_coset_a_b @ r @ H @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G2 @ H ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_762_combine_Oelims,axiom,
! [X: list_a,Xa2: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X @ Xa2 )
= Y )
=> ( ! [K4: a,Ks3: list_a] :
( ( X
= ( cons_a @ K4 @ Ks3 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K4 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_763_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_764_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_765_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_766_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_767_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_768_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_769_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_770_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_771_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_772_eval__poly__in__carrier,axiom,
! [K2: set_a,P: list_a,X: a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_789_ring_Oring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_790_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_791_ring_Oring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_792_univ__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= ( poly_add_a_b @ R ) ) ).
% univ_poly_add
thf(fact_793_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_794_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_795_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_796_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_797_ring_Ocarrier__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( polynomial_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_798_ring_Ocarrier__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( polyno1315193887021588240t_unit @ R @ ( partia5361259788508890537t_unit @ R ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_799_ring_Opoly__add__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_add_closed
thf(fact_800_ring_Opoly__mult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_mult_closed
thf(fact_801_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_802_ring_Oring__simprules_I25_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_803_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_804_ring_Oring__simprules_I24_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_805_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_806_ring_Oring__simprules_I23_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_807_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_808_ring_Oring__simprules_I13_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_809_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_810_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_811_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_812_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_813_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_814_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_815_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_816_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_817_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_818_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_819_ring_Opoly__coeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( member_a @ ( coeff_a_b @ R @ P @ I ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_820_ring_Opoly__coeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P @ I ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_821_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_822_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_823_ring_Opoly__add__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ( poly_add_a_b @ R @ P @ nil_a )
= P ) ) ) ) ).
% ring.poly_add_zero(1)
thf(fact_824_ring_Opoly__add__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ( poly_add_a_b @ R @ nil_a @ P )
= P ) ) ) ) ).
% ring.poly_add_zero(2)
thf(fact_825_ring_Opoly__mult__l__distr,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a,P32: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ) ).
% ring.poly_mult_l_distr
thf(fact_826_ring_Ocombine_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: list_a,Ks: list_list_a,U: list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd2435972518007585703t_unit @ R @ ( cons_list_a @ K @ Ks ) @ ( cons_list_a @ U @ Us3 ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ K @ U ) @ ( embedd2435972518007585703t_unit @ R @ Ks @ Us3 ) ) ) ) ).
% ring.combine.simps(1)
thf(fact_827_ring_Ocombine_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: a,Ks: list_a,U: a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ ( cons_a @ K @ Ks ) @ ( cons_a @ U @ Us3 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedded_combine_a_b @ R @ Ks @ Us3 ) ) ) ) ).
% ring.combine.simps(1)
thf(fact_828_ring_Opolynomial__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.polynomial_in_carrier
thf(fact_829_ring_Opolynomial__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.polynomial_in_carrier
thf(fact_830_ring_Opoly__add__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_add_is_polynomial
thf(fact_831_ring_Opoly__mult__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% ring.poly_mult_is_polynomial
thf(fact_832_ring_Opoly__add__replicate__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ( poly_a7601779127272115787t_unit @ R @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) )
= P ) ) ) ) ).
% ring.poly_add_replicate_zero(1)
thf(fact_833_ring_Opoly__add__replicate__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.poly_add_replicate_zero(1)
thf(fact_834_ring_Opoly__add__replicate__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ( poly_a7601779127272115787t_unit @ R @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P )
= P ) ) ) ) ).
% ring.poly_add_replicate_zero(2)
thf(fact_835_ring_Opoly__add__replicate__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% ring.poly_add_replicate_zero(2)
thf(fact_836_ring_Oappend__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ( P != nil_list_a )
=> ( polyno1315193887021588240t_unit @ R @ K2 @ ( append_list_a @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% ring.append_is_polynomial
thf(fact_837_ring_Oappend__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ) ).
% ring.append_is_polynomial
thf(fact_838_ring_Ocombine_Oelims,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,Xa2: list_list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( embedd2435972518007585703t_unit @ R @ X @ Xa2 )
= Y )
=> ( ! [K4: list_a,Ks3: list_list_a] :
( ( X
= ( cons_list_a @ K4 @ Ks3 ) )
=> ! [U2: list_a,Us4: list_list_a] :
( ( Xa2
= ( cons_list_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ K4 @ U2 ) @ ( embedd2435972518007585703t_unit @ R @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ~ ( ( Xa2 = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% ring.combine.elims
thf(fact_839_ring_Ocombine_Oelims,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,Xa2: list_a,Y: a] :
( ( ring_a_b @ R )
=> ( ( ( embedded_combine_a_b @ R @ X @ Xa2 )
= Y )
=> ( ! [K4: a,Ks3: list_a] :
( ( X
= ( cons_a @ K4 @ Ks3 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K4 @ U2 ) @ ( embedded_combine_a_b @ R @ Ks3 @ Us4 ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% ring.combine.elims
thf(fact_840_ring_Oconst__term__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ~ ! [P5: list_list_a] :
( ( polyno1315193887021588240t_unit @ R @ K2 @ P5 )
=> ( ( P5 != nil_list_a )
=> ( P
!= ( append_list_a @ P5 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_841_ring_Oconst__term__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ R @ P )
= ( zero_a_b @ R ) )
=> ~ ! [P5: list_a] :
( ( polynomial_a_b @ R @ K2 @ P5 )
=> ( ( P5 != nil_a )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_842_ring_Oeval__append__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,B: a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ A ) @ B ) ) ) ) ) ) ).
% ring.eval_append_aux
thf(fact_843_ring_Oeval__append__aux,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,B: list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ B @ nil_list_a ) ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ A ) @ B ) ) ) ) ) ) ).
% ring.eval_append_aux
thf(fact_844_ring_Opoly__degree__bound__from__coeff__1,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ N @ K4 )
=> ( ( coeff_6360649920519955023t_unit @ R @ X @ K4 )
= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ( member_list_list_a @ X @ ( bounde872414301498847361t_unit @ R @ N ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff_1
thf(fact_845_ring_Opoly__degree__bound__from__coeff__1,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ N @ K4 )
=> ( ( coeff_a_b @ R @ X @ K4 )
= ( zero_a_b @ R ) ) )
=> ( member_list_a @ X @ ( bounde2262800523058855161ls_a_b @ R @ N ) ) ) ) ) ).
% ring.poly_degree_bound_from_coeff_1
thf(fact_846_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_847_eval__append,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_848_monoid__cancelI,axiom,
( ! [A3: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B3 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B3 ) ) ) ) )
=> ( ! [A3: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B3 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B3 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_849_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a,G2: a,H: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ G2 @ ( a_l_coset_a_b @ G @ H @ M2 ) )
= ( a_l_coset_a_b @ G @ ( add_a_b @ G @ G2 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_850_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2670972154091845814t_unit,M2: set_list_a,G2: list_a,H: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ G2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_l_co7008843373686234386t_unit @ G @ G2 @ ( a_l_co7008843373686234386t_unit @ G @ H @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ G2 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_851_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_852_nat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_853_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_854_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_855_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_856_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_857_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_858_ring_Opoly__of__dense_Ocong,axiom,
poly_of_dense_a_b = poly_of_dense_a_b ).
% ring.poly_of_dense.cong
thf(fact_859_univ__poly__mult,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= ( poly_mult_a_b @ R ) ) ).
% univ_poly_mult
thf(fact_860_ring_Oeval__monom,axiom,
! [R: partia2175431115845679010xt_a_b,B: a,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( monom_a_b @ R @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ R @ B @ ( pow_a_1026414303147256608_b_nat @ R @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_861_ring_Oeval__monom,axiom,
! [R: partia2670972154091845814t_unit,B: list_a,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ B @ N ) @ A )
= ( mult_l7073676228092353617t_unit @ R @ B @ ( pow_li1142815632869257134it_nat @ R @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_862_ring_Oeval__append,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P @ Q ) @ A )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ R @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ R @ Q @ A ) ) ) ) ) ) ) ).
% ring.eval_append
thf(fact_863_ring_Oeval__append,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( append_list_a @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( pow_li1142815632869257134it_nat @ R @ A @ ( size_s349497388124573686list_a @ Q ) ) ) @ ( eval_l34571156754992824t_unit @ R @ Q @ A ) ) ) ) ) ) ) ).
% ring.eval_append
thf(fact_864_ring_Omonom__decomp,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P )
=> ( P
= ( poly_of_dense_a_b @ R @ ( dense_repr_a_b @ R @ P ) ) ) ) ) ) ).
% ring.monom_decomp
thf(fact_865_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ G @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_866_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2670972154091845814t_unit,H2: set_list_a,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ G @ X @ H2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_867_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ ( zero_a_b @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_868_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2670972154091845814t_unit,M2: set_list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_l_co7008843373686234386t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_869_subfield__long__division__theorem__shell,axiom,
! [K2: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ? [Q2: list_a,R3: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ Q2 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_870_factors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_871_poly__add__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 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat )
!= ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) ) ) ) ) ) ) ).
% poly_add_degree_eq
thf(fact_872_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_873_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_874_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_875_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_876_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_877_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_878_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_879_poly__add__length__le,axiom,
! [P1: list_a,P22: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ ( ord_max_nat @ ( size_size_list_a @ P1 ) @ ( size_size_list_a @ P22 ) ) ) ).
% poly_add_length_le
thf(fact_880_poly__add__length__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 )
=> ( ( ( size_size_list_a @ P1 )
!= ( size_size_list_a @ P22 ) )
=> ( ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P22 ) )
= ( ord_max_nat @ ( size_size_list_a @ P1 ) @ ( size_size_list_a @ P22 ) ) ) ) ) ) ) ).
% poly_add_length_eq
thf(fact_881_poly__add__degree,axiom,
! [P1: list_a,P22: list_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ one_one_nat ) @ ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) ) ) ).
% poly_add_degree
thf(fact_882_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_883_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_884_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_885_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_886_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_887_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
& ( ord_less_nat @ Y @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_888_max_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_889_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_890_max_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.assoc
thf(fact_891_less__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z2 @ X )
| ( ord_less_nat @ Z2 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_892_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_893_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( ord_max_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_894_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_895_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_896_max__def,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_897_max__def,axiom,
( ord_max_set_a
= ( ^ [A2: set_a,B2: set_a] : ( if_set_a @ ( ord_less_eq_set_a @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_898_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_899_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_900_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_901_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_902_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_903_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_904_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_905_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_906_le__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z2 @ X )
| ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_907_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_908_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_909_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_910_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_911_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_912_max__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_max_set_a @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_913_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_914_max__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_max_set_a @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_915_ring_Osubring__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_916_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_917_ring_Osubring__props_I7_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,H1: list_a,H22: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_918_ring_Osubring__props_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_919_ring_Osubring__props_I6_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,H1: list_a,H22: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_920_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_921_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_922_ring_Osubring__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_923_ring_Opoly__add__length__le,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( poly_add_a_b @ R @ P1 @ P22 ) ) @ ( ord_max_nat @ ( size_size_list_a @ P1 ) @ ( size_size_list_a @ P22 ) ) ) ) ).
% ring.poly_add_length_le
thf(fact_924_ring_Opoly__add__length__eq,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P1 )
=> ( ( polynomial_a_b @ R @ K2 @ P22 )
=> ( ( ( size_size_list_a @ P1 )
!= ( size_size_list_a @ P22 ) )
=> ( ( size_size_list_a @ ( poly_add_a_b @ R @ P1 @ P22 ) )
= ( ord_max_nat @ ( size_size_list_a @ P1 ) @ ( size_size_list_a @ P22 ) ) ) ) ) ) ) ) ).
% ring.poly_add_length_eq
thf(fact_925_ring_Opoly__add__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_add_a_b @ R @ P1 @ P22 ) ) @ one_one_nat ) @ ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) ) ) ) ).
% ring.poly_add_degree
thf(fact_926_ring_Opoly__add__degree__eq,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P1: list_a,P22: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P1 )
=> ( ( polynomial_a_b @ R @ K2 @ P22 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat )
!= ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_add_a_b @ R @ P1 @ P22 ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% ring.poly_add_degree_eq
thf(fact_927_ring_Osubfield__long__division__theorem__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,B: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ? [Q2: list_a,R3: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K2 ) @ B @ Q2 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% ring.subfield_long_division_theorem_shell
thf(fact_928_pirreducible__degree,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_929_line__extension__smult__closed,axiom,
! [K2: set_a,E2: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_930_Span__mem__iff__length__version,axiom,
! [K2: set_a,Us3: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
= ( ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ r @ Ks4 @ Us3 ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_931_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_932_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E2: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E2 )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_933_Span__in__carrier,axiom,
! [K2: set_a,Us3: list_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_934_mono__Span__subset,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% mono_Span_subset
thf(fact_935_mono__Span__sublist,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( set_a2 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_936_Span__same__set,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs2 ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% Span_same_set
thf(fact_937_Span__base__incl,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_base_incl
thf(fact_938_Span__subgroup__props_I1_J,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_939_Span__strict__incl,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs2 ) )
& ~ ( member_a @ X2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ) ).
% Span_strict_incl
thf(fact_940_Span__subgroup__props_I3_J,axiom,
! [K2: set_a,Us3: list_a,V1: a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( ( member_a @ V2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( add_a_b @ r @ V1 @ V2 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_subgroup_props(3)
thf(fact_941_Span__subgroup__props_I2_J,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_subgroup_props(2)
thf(fact_942_mono__Span,axiom,
! [K2: set_a,Us3: list_a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ) ).
% mono_Span
thf(fact_943_Span__smult__closed,axiom,
! [K2: set_a,Us3: list_a,K: a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V3 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_944_mono__Span__append_I2_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Vs2 @ Us3 ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_945_mono__Span__append_I1_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_946_ring_OSpan__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.Span_in_carrier
thf(fact_947_ring_OSpan__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.Span_in_carrier
thf(fact_948_ring_Oline__extension__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,A: list_a,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_949_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_950_ring_Oline__extension__mem__iff,axiom,
! [R: partia2670972154091845814t_unit,U: list_a,K2: set_list_a,A: list_a,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K2 )
& ? [Y4: list_a] :
( ( member_list_a @ Y4 @ E2 )
& ( U
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_951_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,A: a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E2 )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_952_ring_Omono__Span__sublist,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( set_a2 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.mono_Span_sublist
thf(fact_953_ring_Omono__Span__sublist,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( set_list_a2 @ Vs2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.mono_Span_sublist
thf(fact_954_ring_Omono__Span__subset,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.mono_Span_subset
thf(fact_955_ring_Omono__Span__subset,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.mono_Span_subset
thf(fact_956_ring_OSpan__base__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_base_incl
thf(fact_957_ring_OSpan__base__incl,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_base_incl
thf(fact_958_ring_OSpan__same__set,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs2 ) )
=> ( ( embedded_Span_a_b @ R @ K2 @ Us3 )
= ( embedded_Span_a_b @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.Span_same_set
thf(fact_959_ring_OSpan__same__set,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( set_list_a2 @ Us3 )
= ( set_list_a2 @ Vs2 ) )
=> ( ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 )
= ( embedd4402942584324845940t_unit @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.Span_same_set
thf(fact_960_ring_OSpan__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.Span_subgroup_props(1)
thf(fact_961_ring_OSpan__subgroup__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.Span_subgroup_props(1)
thf(fact_962_ring_OSpan__strict__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs2 ) )
& ~ ( member_a @ X2 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ) ) ) ).
% ring.Span_strict_incl
thf(fact_963_ring_OSpan__strict__incl,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_less_set_list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs2 ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Vs2 ) )
& ~ ( member_list_a @ X2 @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ) ) ) ).
% ring.Span_strict_incl
thf(fact_964_ring_OSpan__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( zero_a_b @ R ) @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_subgroup_props(2)
thf(fact_965_ring_OSpan__subgroup__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_subgroup_props(2)
thf(fact_966_ring_OSpan__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,V1: a,V2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) )
=> ( ( member_a @ V2 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) )
=> ( member_a @ ( add_a_b @ R @ V1 @ V2 ) @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.Span_subgroup_props(3)
thf(fact_967_ring_OSpan__subgroup__props_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,V1: list_a,V2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ V1 @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) )
=> ( ( member_list_a @ V2 @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ V1 @ V2 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.Span_subgroup_props(3)
thf(fact_968_ring_Omono__Span,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ) ) ).
% ring.mono_Span
thf(fact_969_ring_Omono__Span,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,U: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ ( cons_list_a @ U @ Us3 ) ) ) ) ) ) ) ).
% ring.mono_Span
thf(fact_970_ring_OSpan__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,K: a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.Span_smult_closed
thf(fact_971_ring_OSpan__smult__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,K: list_a,V3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ V3 @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ V3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.Span_smult_closed
thf(fact_972_ring_Omono__Span__append_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ ( append_a @ Vs2 @ Us3 ) ) ) ) ) ) ) ).
% ring.mono_Span_append(2)
thf(fact_973_ring_Omono__Span__append_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ ( append_list_a @ Vs2 @ Us3 ) ) ) ) ) ) ) ).
% ring.mono_Span_append(2)
thf(fact_974_ring_Omono__Span__append_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ ( append_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ).
% ring.mono_Span_append(1)
thf(fact_975_ring_Omono__Span__append_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ ( append_list_a @ Us3 @ Vs2 ) ) ) ) ) ) ) ).
% ring.mono_Span_append(1)
thf(fact_976_ring_OSpan__mem__iff__length__version,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) )
= ( ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 )
& ( ( size_size_list_a @ Ks4 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ R @ Ks4 @ Us3 ) ) ) ) ) ) ) ) ).
% ring.Span_mem_iff_length_version
thf(fact_977_ring_OSpan__mem__iff__length__version,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) )
= ( ? [Ks4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks4 ) @ K2 )
& ( ( size_s349497388124573686list_a @ Ks4 )
= ( size_s349497388124573686list_a @ Us3 ) )
& ( A
= ( embedd2435972518007585703t_unit @ R @ Ks4 @ Us3 ) ) ) ) ) ) ) ) ).
% ring.Span_mem_iff_length_version
thf(fact_978_ring_Oline__extension__smult__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,E2: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ! [K4: list_a,V: list_a] :
( ( member_list_a @ K4 @ K2 )
=> ( ( member_list_a @ V @ E2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ U ) @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_979_ring_Oline__extension__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a,A: a,K: a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K4 @ V ) @ E2 ) ) )
=> ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E2 ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_980_ring_Opirreducible__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K2 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ) ).
% ring.pirreducible_degree
thf(fact_981_Span__is__subalgebra,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ r ) ) ) ).
% Span_is_subalgebra
thf(fact_982_Span__subalgebraI,axiom,
! [K2: set_a,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ! [V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V4 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V4 )
=> ( ord_less_eq_set_a @ E2 @ V4 ) ) )
=> ( E2
= ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_983_subalgebra__Span__incl,axiom,
! [K2: set_a,V5: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V5 )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ V5 ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_984_subalgebra__in__carrier,axiom,
! [K2: set_a,V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ord_less_eq_set_a @ V5 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_985_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_986_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_987_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_988_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V5: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ R )
=> ( ord_less_eq_set_a @ V5 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_989_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,V5: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd1768981623711841426t_unit @ K2 @ V5 @ R )
=> ( ord_le8861187494160871172list_a @ V5 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_990_ring_OSpan__subalgebraI,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ! [V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V4 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V4 )
=> ( ord_less_eq_set_a @ E2 @ V4 ) ) )
=> ( E2
= ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.Span_subalgebraI
thf(fact_991_ring_Osubalgebra__Span__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V5: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ V5 )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ V5 ) ) ) ) ) ).
% ring.subalgebra_Span_incl
thf(fact_992_ring_OSpan__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ R ) ) ) ) ).
% ring.Span_is_subalgebra
thf(fact_993_ring_OSpan__is__subalgebra,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ R ) ) ) ) ).
% ring.Span_is_subalgebra
thf(fact_994_Span__finite__dimension,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_finite_dimension
thf(fact_995_Span__append__eq__set__add,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_996_psubsetI,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_set_a @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_997_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F3: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F3 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F3 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_998_subset__antisym,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_999_subsetI,axiom,
! [A4: set_list_a,B4: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A4 )
=> ( member_list_a @ X2 @ B4 ) )
=> ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_1000_subsetI,axiom,
! [A4: set_nat_a,B4: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A4 )
=> ( member_nat_a @ X2 @ B4 ) )
=> ( ord_le871467723717165285_nat_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_1001_subsetI,axiom,
! [A4: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_1002_set__add__closed,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A4 @ B4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1003_set__add__comm,axiom,
! [I4: set_a,J3: set_a] :
( ( ord_less_eq_set_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I4 @ J3 )
= ( set_add_a_b @ r @ J3 @ I4 ) ) ) ) ).
% set_add_comm
thf(fact_1004_setadd__subset__G,axiom,
! [H2: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1005_sum__space__dim_I1_J,axiom,
! [K2: set_a,E2: set_a,F3: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F3 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( set_add_a_b @ r @ E2 @ F3 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1006_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_1007_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E2: set_a,V5: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ( ord_less_eq_set_a @ V5 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V5 ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_1008_ring_Osum__space__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a,F3: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ F3 )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ ( set_add_a_b @ R @ E2 @ F3 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_1009_ring_Otelescopic__base__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F3: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F3 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ F3 )
=> ( ( embedd8708762675212832759on_a_b @ R @ F3 @ E2 )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_1010_Collect__mono__iff,axiom,
! [P2: a > $o,Q3: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q3 ) )
= ( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1011_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_1012_subset__trans,axiom,
! [A4: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ord_less_eq_set_a @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_1013_Collect__mono,axiom,
! [P2: a > $o,Q3: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q3 @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q3 ) ) ) ).
% Collect_mono
thf(fact_1014_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_1015_double__diff,axiom,
! [A4: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ( minus_minus_set_a @ B4 @ ( minus_minus_set_a @ C3 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_1016_Diff__subset,axiom,
! [A4: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_1017_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A5 )
=> ( member_list_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1018_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A5 )
=> ( member_nat_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1019_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1020_equalityD2,axiom,
! [A4: set_a,B4: set_a] :
( ( A4 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_1021_equalityD1,axiom,
! [A4: set_a,B4: set_a] :
( ( A4 = B4 )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_1022_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
=> ( member_list_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1023_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A5 )
=> ( member_nat_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1024_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1025_equalityE,axiom,
! [A4: set_a,B4: set_a] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_1026_Diff__mono,axiom,
! [A4: set_a,C3: set_a,D2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ B4 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1027_subsetD,axiom,
! [A4: set_list_a,B4: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B4 )
=> ( ( member_list_a @ C @ A4 )
=> ( member_list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1028_subsetD,axiom,
! [A4: set_nat_a,B4: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A4 @ B4 )
=> ( ( member_nat_a @ C @ A4 )
=> ( member_nat_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1029_subsetD,axiom,
! [A4: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1030_in__mono,axiom,
! [A4: set_list_a,B4: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B4 )
=> ( ( member_list_a @ X @ A4 )
=> ( member_list_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1031_in__mono,axiom,
! [A4: set_nat_a,B4: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A4 @ B4 )
=> ( ( member_nat_a @ X @ A4 )
=> ( member_nat_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1032_in__mono,axiom,
! [A4: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( member_a @ X @ A4 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1033_psubsetD,axiom,
! [A4: set_list_a,B4: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A4 @ B4 )
=> ( ( member_list_a @ C @ A4 )
=> ( member_list_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1034_psubsetD,axiom,
! [A4: set_nat_a,B4: set_nat_a,C: nat > a] :
( ( ord_less_set_nat_a @ A4 @ B4 )
=> ( ( member_nat_a @ C @ A4 )
=> ( member_nat_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1035_psubsetD,axiom,
! [A4: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1036_psubset__trans,axiom,
! [A4: set_a,B4: set_a,C3: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C3 )
=> ( ord_less_set_a @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_1037_psubset__imp__ex__mem,axiom,
! [A4: set_list_a,B4: set_list_a] :
( ( ord_less_set_list_a @ A4 @ B4 )
=> ? [B3: list_a] : ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1038_psubset__imp__ex__mem,axiom,
! [A4: set_nat_a,B4: set_nat_a] :
( ( ord_less_set_nat_a @ A4 @ B4 )
=> ? [B3: nat > a] : ( member_nat_a @ B3 @ ( minus_490503922182417452_nat_a @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1039_psubset__imp__ex__mem,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1040_abelian__group_Osetadd__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,K2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ G @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_1041_abelian__group_Osetadd__subset__G,axiom,
! [G: partia2670972154091845814t_unit,H2: set_list_a,K2: set_list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ G @ H2 @ K2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_1042_abelian__monoid_Oset__add__closed,axiom,
! [G: partia2175431115845679010xt_a_b,A4: set_a,B4: set_a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ B4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ G @ A4 @ B4 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_1043_abelian__monoid_Oset__add__closed,axiom,
! [G: partia2670972154091845814t_unit,A4: set_list_a,B4: set_list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ A4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ord_le8861187494160871172list_a @ B4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ G @ A4 @ B4 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_1044_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E2 @ R ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_1045_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a,V5: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ R )
=> ( ( ord_less_eq_set_a @ V5 @ E2 )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ V5 ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_1046_psubsetE,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_1047_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_1048_psubset__imp__subset,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_1049_psubset__subset__trans,axiom,
! [A4: set_a,B4: set_a,C3: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ord_less_set_a @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_1050_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1051_subset__psubset__trans,axiom,
! [A4: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C3 )
=> ( ord_less_set_a @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_1052_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1053_ring_OSpan__finite__dimension,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( embedd1345800358437254783t_unit @ R @ K2 @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_finite_dimension
thf(fact_1054_ring_OSpan__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_finite_dimension
thf(fact_1055_ring_OSpan__append__eq__set__add,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_Span_a_b @ R @ K2 @ ( append_a @ Us3 @ Vs2 ) )
= ( set_add_a_b @ R @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) ) ) ) ) ) ) ).
% ring.Span_append_eq_set_add
thf(fact_1056_ring_OSpan__append__eq__set__add,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a,Vs2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd4402942584324845940t_unit @ R @ K2 @ ( append_list_a @ Us3 @ Vs2 ) )
= ( set_ad92425877771022410t_unit @ R @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs2 ) ) ) ) ) ) ) ).
% ring.Span_append_eq_set_add
thf(fact_1057_Span__incl,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ r @ K2 @ ( set_a2 @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ).
% Span_incl
thf(fact_1058_unique__decomposition,axiom,
! [K2: set_a,Us3: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ? [X2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X2 ) @ K2 )
& ( ( size_size_list_a @ X2 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ r @ X2 @ Us3 ) )
& ! [Y5: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ Y5 ) @ K2 )
& ( ( size_size_list_a @ Y5 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ r @ Y5 @ Us3 ) ) )
=> ( Y5 = X2 ) ) ) ) ) ) ).
% unique_decomposition
thf(fact_1059_replacement__theorem,axiom,
! [K2: set_a,Us5: list_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us5 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us5 @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ? [Vs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( set_a2 @ Vs2 ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Us5 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Vs3 @ Us3 ) ) ) ) ) ) ) ).
% replacement_theorem
thf(fact_1060_independent__backwards_I2_J,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ).
% independent_backwards(2)
thf(fact_1061_li__Nil,axiom,
! [K2: set_a] : ( embedd5208550302661555450nt_a_b @ r @ K2 @ nil_a ) ).
% li_Nil
thf(fact_1062_independent__backwards_I3_J,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_backwards(3)
thf(fact_1063_independent__backwards_I1_J,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ).
% independent_backwards(1)
thf(fact_1064_independent__split_I2_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 ) ) ) ).
% independent_split(2)
thf(fact_1065_independent__split_I1_J,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 ) ) ) ).
% independent_split(1)
thf(fact_1066_set__mult__closed,axiom,
! [H2: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ r @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_mult_closed
thf(fact_1067_independent__in__carrier,axiom,
! [K2: set_a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_in_carrier
thf(fact_1068_li__Cons,axiom,
! [U: a,K2: set_a,Us3: list_a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ).
% li_Cons
thf(fact_1069_independent__same__set,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs2 ) )
=> ( ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Vs2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 ) ) ) ) ) ).
% independent_same_set
thf(fact_1070_independent_Ocases,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
=> ( ( A22 != nil_a )
=> ~ ! [U2: a,Us4: list_a] :
( ( A22
= ( cons_a @ U2 @ Us4 ) )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U2 @ ( embedded_Span_a_b @ r @ A1 @ Us4 ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ A1 @ Us4 ) ) ) ) ) ) ).
% independent.cases
thf(fact_1071_independent_Osimps,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
= ( ? [K5: set_a] :
( ( A1 = K5 )
& ( A22 = nil_a ) )
| ? [U3: a,K5: set_a,Us6: list_a] :
( ( A1 = K5 )
& ( A22
= ( cons_a @ U3 @ Us6 ) )
& ( member_a @ U3 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ U3 @ ( embedded_Span_a_b @ r @ K5 @ Us6 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K5 @ Us6 ) ) ) ) ).
% independent.simps
thf(fact_1072_independent__rotate1__aux,axiom,
! [K2: set_a,U: a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ ( append_a @ Us3 @ Vs2 ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ ( append_a @ Us3 @ ( cons_a @ U @ nil_a ) ) @ Vs2 ) ) ) ) ).
% independent_rotate1_aux
thf(fact_1073_independent__strict__incl,axiom,
! [K2: set_a,U: a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ).
% independent_strict_incl
thf(fact_1074_filter__base,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ~ ! [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs4 )
=> ( ( embedded_Span_a_b @ r @ K2 @ Vs4 )
!= ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ).
% filter_base
thf(fact_1075_independent__replacement,axiom,
! [K2: set_a,U: a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us3 ) ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs2 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( cons_a @ X2 @ Us3 ) ) ) ) ) ) ) ).
% independent_replacement
thf(fact_1076_independent__length__le,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ ( size_size_list_a @ Vs2 ) ) ) ) ) ) ).
% independent_length_le
thf(fact_1077_ring_Oindependent__backwards_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,U: a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 ) ) ) ).
% ring.independent_backwards(2)
thf(fact_1078_ring_Oli__Nil,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ nil_a ) ) ).
% ring.li_Nil
thf(fact_1079_ring_Oindependent__backwards_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,U: list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3875673156127067906t_unit @ R @ K2 @ ( cons_list_a @ U @ Us3 ) )
=> ( member_list_a @ U @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.independent_backwards(3)
thf(fact_1080_ring_Oindependent__backwards_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,U: a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( member_a @ U @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.independent_backwards(3)
thf(fact_1081_ring_Oindependent__backwards_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,U: a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) )
=> ~ ( member_a @ U @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ).
% ring.independent_backwards(1)
thf(fact_1082_ring_Oindependent__split_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs2 ) ) ) ) ).
% ring.independent_split(1)
thf(fact_1083_ring_Oindependent__split_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 ) ) ) ) ).
% ring.independent_split(2)
thf(fact_1084_ring_Oindependent__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3875673156127067906t_unit @ R @ K2 @ Us3 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.independent_in_carrier
thf(fact_1085_ring_Oindependent__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.independent_in_carrier
thf(fact_1086_ring_Oli__Cons,axiom,
! [R: partia2670972154091845814t_unit,U: list_a,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ R ) )
=> ( ~ ( member_list_a @ U @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) )
=> ( ( embedd3875673156127067906t_unit @ R @ K2 @ Us3 )
=> ( embedd3875673156127067906t_unit @ R @ K2 @ ( cons_list_a @ U @ Us3 ) ) ) ) ) ) ).
% ring.li_Cons
thf(fact_1087_ring_Oli__Cons,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ U @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ) ).
% ring.li_Cons
thf(fact_1088_ring_Oindependent__same__set,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ( set_a2 @ Us3 )
= ( set_a2 @ Vs2 ) )
=> ( ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Vs2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs2 ) ) ) ) ) ) ).
% ring.independent_same_set
thf(fact_1089_ring_Oindependent_Osimps,axiom,
! [R: partia2670972154091845814t_unit,A1: set_list_a,A22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3875673156127067906t_unit @ R @ A1 @ A22 )
= ( ? [K5: set_list_a] :
( ( A1 = K5 )
& ( A22 = nil_list_a ) )
| ? [U3: list_a,K5: set_list_a,Us6: list_list_a] :
( ( A1 = K5 )
& ( A22
= ( cons_list_a @ U3 @ Us6 ) )
& ( member_list_a @ U3 @ ( partia5361259788508890537t_unit @ R ) )
& ~ ( member_list_a @ U3 @ ( embedd4402942584324845940t_unit @ R @ K5 @ Us6 ) )
& ( embedd3875673156127067906t_unit @ R @ K5 @ Us6 ) ) ) ) ) ).
% ring.independent.simps
thf(fact_1090_ring_Oindependent_Osimps,axiom,
! [R: partia2175431115845679010xt_a_b,A1: set_a,A22: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ A1 @ A22 )
= ( ? [K5: set_a] :
( ( A1 = K5 )
& ( A22 = nil_a ) )
| ? [U3: a,K5: set_a,Us6: list_a] :
( ( A1 = K5 )
& ( A22
= ( cons_a @ U3 @ Us6 ) )
& ( member_a @ U3 @ ( partia707051561876973205xt_a_b @ R ) )
& ~ ( member_a @ U3 @ ( embedded_Span_a_b @ R @ K5 @ Us6 ) )
& ( embedd5208550302661555450nt_a_b @ R @ K5 @ Us6 ) ) ) ) ) ).
% ring.independent.simps
thf(fact_1091_ring_Oindependent_Ocases,axiom,
! [R: partia2670972154091845814t_unit,A1: set_list_a,A22: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3875673156127067906t_unit @ R @ A1 @ A22 )
=> ( ( A22 != nil_list_a )
=> ~ ! [U2: list_a,Us4: list_list_a] :
( ( A22
= ( cons_list_a @ U2 @ Us4 ) )
=> ( ( member_list_a @ U2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ~ ( member_list_a @ U2 @ ( embedd4402942584324845940t_unit @ R @ A1 @ Us4 ) )
=> ~ ( embedd3875673156127067906t_unit @ R @ A1 @ Us4 ) ) ) ) ) ) ) ).
% ring.independent.cases
thf(fact_1092_ring_Oindependent_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,A1: set_a,A22: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ A1 @ A22 )
=> ( ( A22 != nil_a )
=> ~ ! [U2: a,Us4: list_a] :
( ( A22
= ( cons_a @ U2 @ Us4 ) )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ U2 @ ( embedded_Span_a_b @ R @ A1 @ Us4 ) )
=> ~ ( embedd5208550302661555450nt_a_b @ R @ A1 @ Us4 ) ) ) ) ) ) ) ).
% ring.independent.cases
thf(fact_1093_ring_Oindependent__rotate1__aux,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,U: a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ ( append_a @ Us3 @ Vs2 ) ) )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ ( append_a @ Us3 @ ( cons_a @ U @ nil_a ) ) @ Vs2 ) ) ) ) ) ).
% ring.independent_rotate1_aux
thf(fact_1094_ring_Oindependent__strict__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,U: a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ord_less_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) ) ) ) ) ) ).
% ring.independent_strict_incl
thf(fact_1095_ring_OSpan__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ R @ K2 @ ( set_a2 @ Us3 ) ) @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_incl
thf(fact_1096_ring_OSpan__incl,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_mu3586181839180059898t_unit @ R @ K2 @ ( set_list_a2 @ Us3 ) ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ).
% ring.Span_incl
thf(fact_1097_ring_Ofilter__base,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ~ ! [Vs4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs4 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd3875673156127067906t_unit @ R @ K2 @ Vs4 )
=> ( ( embedd4402942584324845940t_unit @ R @ K2 @ Vs4 )
!= ( embedd4402942584324845940t_unit @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.filter_base
thf(fact_1098_ring_Ofilter__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ~ ! [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs4 )
=> ( ( embedded_Span_a_b @ R @ K2 @ Vs4 )
!= ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ) ) ) ).
% ring.filter_base
thf(fact_1099_ring_Oindependent__replacement,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,U: a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ ( cons_a @ U @ Us3 ) ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs2 ) )
& ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( cons_a @ X2 @ Us3 ) ) ) ) ) ) ) ) ).
% ring.independent_replacement
thf(fact_1100_ring_Oindependent__length__le,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ ( size_size_list_a @ Vs2 ) ) ) ) ) ) ) ).
% ring.independent_length_le
thf(fact_1101_ring_Oreplacement__theorem,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us5: list_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ Us5 @ Us3 ) )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs2 )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ ( append_a @ Us5 @ Us3 ) ) @ ( embedded_Span_a_b @ R @ K2 @ Vs2 ) )
=> ? [Vs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( set_a2 @ Vs2 ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Us5 ) )
& ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ Vs3 @ Us3 ) ) ) ) ) ) ) ) ).
% ring.replacement_theorem
thf(fact_1102_ring_Ounique__decomposition,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) )
=> ? [X2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X2 ) @ K2 )
& ( ( size_size_list_a @ X2 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ R @ X2 @ Us3 ) )
& ! [Y5: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ Y5 ) @ K2 )
& ( ( size_size_list_a @ Y5 )
= ( size_size_list_a @ Us3 ) )
& ( A
= ( embedded_combine_a_b @ R @ Y5 @ Us3 ) ) )
=> ( Y5 = X2 ) ) ) ) ) ) ) ).
% ring.unique_decomposition
thf(fact_1103_independent__rotate1,axiom,
! [K2: set_a,Us3: list_a,Vs2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ ( rotate1_a @ Us3 ) @ Vs2 ) ) ) ) ).
% independent_rotate1
thf(fact_1104_complete__base,axiom,
! [K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ? [Vs4: list_a] :
( ( ( size_size_list_a @ ( append_a @ Vs4 @ Us3 ) )
= N )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ ( append_a @ Vs4 @ Us3 ) )
& ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Vs4 @ Us3 ) )
= E2 ) ) ) ) ) ) ).
% complete_base
thf(fact_1105_dimension__is__inj,axiom,
! [K2: set_a,N: nat,E2: set_a,M: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ E2 )
=> ( N = M ) ) ) ) ).
% dimension_is_inj
thf(fact_1106_finite__dimension__def,axiom,
! [K2: set_a,E2: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
= ( ? [N2: nat] : ( embedd2795209813406577254on_a_b @ r @ N2 @ K2 @ E2 ) ) ) ).
% finite_dimension_def
thf(fact_1107_finite__dimensionI,axiom,
! [N: nat,K2: set_a,E2: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 ) ) ).
% finite_dimensionI
thf(fact_1108_finite__dimensionE_H,axiom,
! [K2: set_a,E2: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N3 @ K2 @ E2 ) ) ).
% finite_dimensionE'
thf(fact_1109_space__subgroup__props_I3_J,axiom,
! [K2: set_a,N: nat,E2: set_a,V1: a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ V1 @ E2 )
=> ( ( member_a @ V2 @ E2 )
=> ( member_a @ ( add_a_b @ r @ V1 @ V2 ) @ E2 ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_1110_space__subgroup__props_I2_J,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( member_a @ ( zero_a_b @ r ) @ E2 ) ) ) ).
% space_subgroup_props(2)
thf(fact_1111_telescopic__base__aux,axiom,
! [K2: set_a,F3: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F3 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F3 )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F3 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1112_space__subgroup__props_I5_J,axiom,
! [K2: set_a,N: nat,E2: set_a,K: a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V3 ) @ E2 ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_1113_unique__dimension,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E2 )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X2 @ K2 @ E2 )
& ! [Y5: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y5 @ K2 @ E2 )
=> ( Y5 = X2 ) ) ) ) ) ).
% unique_dimension
thf(fact_1114_space__subgroup__props_I1_J,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_1115_dimensionI,axiom,
! [K2: set_a,Us3: list_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= E2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( size_size_list_a @ Us3 ) @ K2 @ E2 ) ) ) ) ).
% dimensionI
thf(fact_1116_independent__length__le__dimension,axiom,
! [K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ N ) ) ) ) ) ).
% independent_length_le_dimension
thf(fact_1117_independent__length__eq__dimension,axiom,
! [K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ( ( size_size_list_a @ Us3 )
= N )
= ( ( embedded_Span_a_b @ r @ K2 @ Us3 )
= E2 ) ) ) ) ) ) ).
% independent_length_eq_dimension
thf(fact_1118_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_1119_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_1120_rotate1__is__Nil__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat] :
( ( ( rotate4887004114156536554_a_nat @ Xs )
= nil_Pr7402525243500994295_a_nat )
= ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_1121_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_1122_rotate1__replicate,axiom,
! [N: nat,A: a] :
( ( rotate1_a @ ( replicate_a @ N @ A ) )
= ( replicate_a @ N @ A ) ) ).
% rotate1_replicate
thf(fact_1123_exists__base,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ? [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( embedd5208550302661555450nt_a_b @ r @ K2 @ Vs4 )
& ( ( size_size_list_a @ Vs4 )
= N )
& ( ( embedded_Span_a_b @ r @ K2 @ Vs4 )
= E2 ) ) ) ) ).
% exists_base
thf(fact_1124_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1125_dimension__one,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K2 @ K2 ) ) ).
% dimension_one
thf(fact_1126_dimension__independent,axiom,
! [K2: set_a,Us3: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ( embedd2795209813406577254on_a_b @ r @ ( size_size_list_a @ Us3 ) @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ).
% dimension_independent
thf(fact_1127_ring_Odimension__is__inj,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,M: nat] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ E2 )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_1128_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_1129_rotate1_Osimps_I1_J,axiom,
( ( rotate4887004114156536554_a_nat @ nil_Pr7402525243500994295_a_nat )
= nil_Pr7402525243500994295_a_nat ) ).
% rotate1.simps(1)
thf(fact_1130_ring_Ofinite__dimensionI,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,K2: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 ) ) ) ).
% ring.finite_dimensionI
thf(fact_1131_ring_Ofinite__dimensionE_H,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ R @ N3 @ K2 @ E2 ) ) ) ).
% ring.finite_dimensionE'
thf(fact_1132_ring_Ofinite__dimension__def,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 )
= ( ? [N2: nat] : ( embedd2795209813406577254on_a_b @ R @ N2 @ K2 @ E2 ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_1133_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ E2 ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_1134_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( member_a @ ( zero_a_b @ R ) @ E2 ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_1135_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a,V1: list_a,V2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ( member_list_a @ V1 @ E2 )
=> ( ( member_list_a @ V2 @ E2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ V1 @ V2 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_1136_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,V1: a,V2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( member_a @ V1 @ E2 )
=> ( ( member_a @ V2 @ E2 )
=> ( member_a @ ( add_a_b @ R @ V1 @ V2 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_1137_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a,K: list_a,V3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ V3 @ E2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ V3 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_1138_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,K: a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ E2 ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_1139_ring_Otelescopic__base__aux,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F3: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F3 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F3 )
=> ( ( embedd2795209813406577254on_a_b @ R @ one_one_nat @ F3 @ E2 )
=> ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 ) ) ) ) ) ) ).
% ring.telescopic_base_aux
thf(fact_1140_ring_Odimension__one,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( embedd2795209813406577254on_a_b @ R @ one_one_nat @ K2 @ K2 ) ) ) ).
% ring.dimension_one
thf(fact_1141_ring_Ounique__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E2 )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ R @ X2 @ K2 @ E2 )
& ! [Y5: nat] :
( ( embedd2795209813406577254on_a_b @ R @ Y5 @ K2 @ E2 )
=> ( Y5 = X2 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_1142_rotate1_Osimps_I2_J,axiom,
! [X: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
( ( rotate4887004114156536554_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) )
= ( append7679239579558125090_a_nat @ Xs @ ( cons_P5205166803686508359_a_nat @ X @ nil_Pr7402525243500994295_a_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_1143_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_1144_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_1145_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_1146_ring_Odimension__independent,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( embedd2795209813406577254on_a_b @ R @ ( size_size_list_a @ Us3 ) @ K2 @ ( embedded_Span_a_b @ R @ K2 @ Us3 ) ) ) ) ).
% ring.dimension_independent
thf(fact_1147_ring_OdimensionI,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ( ( embedded_Span_a_b @ R @ K2 @ Us3 )
= E2 )
=> ( embedd2795209813406577254on_a_b @ R @ ( size_size_list_a @ Us3 ) @ K2 @ E2 ) ) ) ) ) ).
% ring.dimensionI
thf(fact_1148_ring_Oindependent__rotate1,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us3: list_a,Vs2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ Us3 @ Vs2 ) )
=> ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ ( rotate1_a @ Us3 ) @ Vs2 ) ) ) ) ) ).
% ring.independent_rotate1
thf(fact_1149_ring_Oindependent__length__le__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us3 ) @ N ) ) ) ) ) ) ).
% ring.independent_length_le_dimension
thf(fact_1150_ring_Oindependent__length__eq__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ( ( ( size_size_list_a @ Us3 )
= N )
= ( ( embedded_Span_a_b @ R @ K2 @ Us3 )
= E2 ) ) ) ) ) ) ) ).
% ring.independent_length_eq_dimension
thf(fact_1151_ring_Oexists__base,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E2 )
=> ? [Vs4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs4 ) @ ( partia5361259788508890537t_unit @ R ) )
& ( embedd3875673156127067906t_unit @ R @ K2 @ Vs4 )
& ( ( size_s349497388124573686list_a @ Vs4 )
= N )
& ( ( embedd4402942584324845940t_unit @ R @ K2 @ Vs4 )
= E2 ) ) ) ) ) ).
% ring.exists_base
thf(fact_1152_ring_Oexists__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ? [Vs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs4 ) @ ( partia707051561876973205xt_a_b @ R ) )
& ( embedd5208550302661555450nt_a_b @ R @ K2 @ Vs4 )
& ( ( size_size_list_a @ Vs4 )
= N )
& ( ( embedded_Span_a_b @ R @ K2 @ Vs4 )
= E2 ) ) ) ) ) ).
% ring.exists_base
thf(fact_1153_ring_Ocomplete__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E2: set_a,Us3: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E2 )
=> ( ( embedd5208550302661555450nt_a_b @ R @ K2 @ Us3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ E2 )
=> ? [Vs4: list_a] :
( ( ( size_size_list_a @ ( append_a @ Vs4 @ Us3 ) )
= N )
& ( embedd5208550302661555450nt_a_b @ R @ K2 @ ( append_a @ Vs4 @ Us3 ) )
& ( ( embedded_Span_a_b @ R @ K2 @ ( append_a @ Vs4 @ Us3 ) )
= E2 ) ) ) ) ) ) ) ).
% ring.complete_base
thf(fact_1154_dimension__backwards,axiom,
! [K2: set_a,N: nat,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ E2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E3 )
& ~ ( member_a @ X2 @ E3 )
& ( E2
= ( embedd971793762689825387on_a_b @ r @ K2 @ X2 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_1155_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1156_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_1157_nat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_Suc2
thf(fact_1158_Suc__dim,axiom,
! [V3: a,E2: set_a,N: nat,K2: set_a] :
( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V3 @ E2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ r @ K2 @ V3 @ E2 ) ) ) ) ) ).
% Suc_dim
thf(fact_1159_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1160_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1161_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1162_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1163_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1164_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1165_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_1166_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1167_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1168_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1169_local_Onat__pow__Suc,axiom,
! [X: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ X ) ) ).
% local.nat_pow_Suc
thf(fact_1170_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1171_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1172_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1173_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1174_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1175_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1176_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1177_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1178_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1179_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M6: nat] :
( M7
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_1180_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1181_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1182_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1183_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P2 @ M5 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1184_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1185_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1186_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1187_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1188_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P2 @ I3 @ J2 )
=> ( ( P2 @ J2 @ K4 )
=> ( P2 @ I3 @ K4 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1189_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1190_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1191_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1192_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M8: nat] :
( ( M
= ( suc @ M8 ) )
& ( ord_less_nat @ N @ M8 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1193_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1194_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1195_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1196_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P2 @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1197_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1198_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1199_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1200_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1201_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1202_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1203_replicate__Suc,axiom,
! [N: nat,X: a] :
( ( replicate_a @ ( suc @ N ) @ X )
= ( cons_a @ X @ ( replicate_a @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1204_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1205_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1206_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1207_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1208_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1209_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1210_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1211_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1212_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1213_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1214_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1215_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1216_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1217_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y4: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1218_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1219_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_1220_subcringI,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H2 @ r ) ) ) ).
% subcringI
thf(fact_1221_Span__subgroup__props_I4_J,axiom,
! [K2: set_a,Us3: list_a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V3 @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ).
% Span_subgroup_props(4)
thf(fact_1222_subring__props_I5_J,axiom,
! [K2: set_a,H: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_1223_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_1224_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_1225_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_1226_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_1227_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_1228_a__transpose__inv,axiom,
! [X: a,Y: a,Z2: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z2 )
=> ( ( 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 @ ( a_inv_a_b @ r @ X ) @ Z2 )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_1229_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_1230_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_1231_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_1232_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_1233_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_1234_space__subgroup__props_I4_J,axiom,
! [K2: set_a,N: nat,E2: set_a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ V3 @ E2 )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ E2 ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_1235_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_1236_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_1237_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_1238_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_1239_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_1240_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_1241_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_1242_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_1243_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ! [Xa3: a] :
( ( member_a @ Xa3 @ H2 )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa3 ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_1244_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1245_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1246_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_1247_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_1248_Span__mem__imp__non__trivial__combine,axiom,
! [K2: set_a,Us3: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ~ ! [K4: a] :
( ( member_a @ K4 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ! [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K2 )
=> ( ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us3 ) )
=> ( ( embedded_combine_a_b @ r @ ( cons_a @ K4 @ Ks3 ) @ ( cons_a @ A @ Us3 ) )
!= ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% Span_mem_imp_non_trivial_combine
thf(fact_1249_poly__add__monom,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P ) ) @ P )
= ( cons_a @ A @ P ) ) ) ) ).
% poly_add_monom
thf(fact_1250_non__empty__bounded__degree__polynomials,axiom,
! [K: nat] :
( ( bounde2262800523058855161ls_a_b @ r @ K )
!= bot_bot_set_list_a ) ).
% non_empty_bounded_degree_polynomials
thf(fact_1251_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1252_lead__coeff__not__zero,axiom,
! [K2: set_a,A: a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ P ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_1253_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1254_space__subgroup__props_I6_J,axiom,
! [K2: set_a,N: nat,E2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E2 )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ E2 )
=> ( member_a @ A @ E2 ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_1255_lead__coeff__in__carrier,axiom,
! [K2: set_a,A: a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ P ) )
=> ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% lead_coeff_in_carrier
thf(fact_1256_dependent__imp__non__trivial__combine,axiom,
! [K2: set_a,Us3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( embedd5208550302661555450nt_a_b @ r @ K2 @ Us3 )
=> ~ ! [Ks3: list_a] :
( ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us3 ) )
=> ( ( ( embedded_combine_a_b @ r @ Ks3 @ Us3 )
= ( zero_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K2 )
=> ( ( set_a2 @ Ks3 )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).
% dependent_imp_non_trivial_combine
thf(fact_1257_Span__m__inv__simprule,axiom,
! [K2: set_a,Us3: list_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
=> ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) ) ) ) ) ) ) ).
% Span_m_inv_simprule
thf(fact_1258_Span__mem__iff,axiom,
! [K2: set_a,Us3: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us3 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ? [Ks4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K2 )
& ( ( embedded_combine_a_b @ r @ ( cons_a @ X3 @ Ks4 ) @ ( cons_a @ A @ Us3 ) )
= ( zero_a_b @ r ) ) ) ) ) ) ) ) ) ).
% Span_mem_iff
thf(fact_1259_const__is__polynomial,axiom,
! [A: a,K2: set_a] :
( ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_1260_monom__is__polynomial,axiom,
! [K2: set_a,A: a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K2 @ ( monom_a_b @ r @ A @ N ) ) ) ) ).
% monom_is_polynomial
thf(fact_1261_dimension_Osimps,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
= ( ? [K5: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V6: a,E4: set_a,N2: nat,K5: set_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K5 @ V6 @ E4 ) )
& ( member_a @ V6 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V6 @ E4 )
& ( embedd2795209813406577254on_a_b @ r @ N2 @ K5 @ E4 ) ) ) ) ).
% dimension.simps
thf(fact_1262_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_1263_zero__dim,axiom,
! [K2: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_1264_dimension__zero,axiom,
! [K2: set_a,E2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ E2 )
=> ( E2
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_1265_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_1266_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1267_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1268_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
% Helper facts (9)
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_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_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 @ ( bounde1002222742488328185ly_a_b @ r @ f @ n ) ) @ one_one_nat ) @ ( minus_minus_nat @ n @ one_one_nat ) ).
%------------------------------------------------------------------------------