TPTP Problem File: ITP118^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP118^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Minkowskis_Theorem problem prob_62__6246984_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Minkowskis_Theorem/prob_62__6246984_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 435 ( 100 unt; 68 typ; 0 def)
% Number of atoms : 1027 ( 200 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3681 ( 54 ~; 13 |; 75 &;3055 @)
% ( 0 <=>; 484 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 94 ( 94 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 64 usr; 7 con; 0-6 aty)
% Number of variables : 1013 ( 65 ^; 864 !; 18 ?;1013 :)
% ( 66 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:28:18.191
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec: $tType > $tType > $tType ).
thf(ty_t_Numeral__Type_Onum1,type,
numeral_num1: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_n,type,
n: $tType ).
% Explicit typings (62)
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Odist__norm,type,
real_V1639655670t_norm:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V1229719638ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim2085082626_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Finite__Cartesian__Product_Oaxis,type,
finite1218308419n_axis:
!>[B: $tType,A: $tType] : ( B > A > ( finite_Cartesian_vec @ A @ B ) ) ).
thf(sy_c_Finite__Cartesian__Product_Oaxis__index,type,
finite1115014511_index:
!>[A: $tType,N: $tType] : ( ( finite_Cartesian_vec @ A @ N ) > N ) ).
thf(sy_c_Finite__Cartesian__Product_Omap__matrix,type,
finite1614961371matrix:
!>[A: $tType,B: $tType,I: $tType,J: $tType] : ( ( A > B ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ B @ I ) @ J ) ) ).
thf(sy_c_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec2:
!>[A: $tType,B: $tType] : ( A > ( finite_Cartesian_vec @ A @ B ) ) ).
thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth,type,
finite1433825200ec_nth:
!>[A: $tType,B: $tType] : ( ( finite_Cartesian_vec @ A @ B ) > B > A ) ).
thf(sy_c_Finite__Cartesian__Product_Ovector__scalar__mult,type,
finite431509953r_mult:
!>[A: $tType,N: $tType] : ( A > ( finite_Cartesian_vec @ A @ N ) > ( finite_Cartesian_vec @ A @ N ) ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__jpyvdnlcjd_Oof__int__vec,type,
minkow1937162798nt_vec:
!>[B: $tType,A: $tType] : ( ( finite_Cartesian_vec @ int @ B ) > ( finite_Cartesian_vec @ A @ B ) ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V2000881966t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V1127708846m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a: set @ ( finite_Cartesian_vec @ real @ n ) ).
thf(sy_v_i____,type,
i: n ).
thf(sy_v_m____,type,
m: int ).
thf(sy_v_x____,type,
x: finite_Cartesian_vec @ real @ n ).
thf(sy_v_y____,type,
y: finite_Cartesian_vec @ real @ n ).
% Relevant facts (255)
thf(fact_0__C1_C_I2_J,axiom,
member @ ( finite_Cartesian_vec @ real @ n ) @ y @ a ).
% "1"(2)
thf(fact_1__C1_C_I1_J,axiom,
member @ ( finite_Cartesian_vec @ real @ n ) @ x @ a ).
% "1"(1)
thf(fact_2_m,axiom,
( ( ring_1_of_int @ real @ m )
= ( minus_minus @ real @ ( finite1433825200ec_nth @ real @ n @ x @ i ) @ ( finite1433825200ec_nth @ real @ n @ y @ i ) ) ) ).
% m
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_Areal__of__int_Am_A_061_Ax_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: int] :
( ( ring_1_of_int @ real @ M )
!= ( minus_minus @ real @ ( finite1433825200ec_nth @ real @ n @ x @ i ) @ ( finite1433825200ec_nth @ real @ n @ y @ i ) ) ) ).
% \<open>\<And>thesis. (\<And>m. real_of_int m = x $ i - y $ i \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_vector__minus__component,axiom,
! [A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( minus @ A ) )
=> ! [X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B,I2: B] :
( ( finite1433825200ec_nth @ A @ B @ ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ X @ Y ) @ I2 )
= ( minus_minus @ A @ ( finite1433825200ec_nth @ A @ B @ X @ I2 ) @ ( finite1433825200ec_nth @ A @ B @ Y @ I2 ) ) ) ) ).
% vector_minus_component
thf(fact_5_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% of_int_abs
thf(fact_6_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_7__092_060open_062x_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060in_062_A_092_060int_062_092_060close_062,axiom,
member @ real @ ( minus_minus @ real @ ( finite1433825200ec_nth @ real @ n @ x @ i ) @ ( finite1433825200ec_nth @ real @ n @ y @ i ) ) @ ( ring_1_Ints @ real ) ).
% \<open>x $ i - y $ i \<in> \<int>\<close>
thf(fact_8_of__int__vec__nth,axiom,
! [A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( ring_1 @ A ) )
=> ! [V: finite_Cartesian_vec @ int @ B,N2: B] :
( ( finite1433825200ec_nth @ A @ B @ ( minkow1937162798nt_vec @ B @ A @ V ) @ N2 )
= ( ring_1_of_int @ A @ ( finite1433825200ec_nth @ int @ B @ V @ N2 ) ) ) ) ).
% of_int_vec_nth
thf(fact_9_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: int,Z: int] :
( ( ( ring_1_of_int @ A @ W )
= ( ring_1_of_int @ A @ Z ) )
= ( W = Z ) ) ) ).
% of_int_eq_iff
thf(fact_10_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_abs
thf(fact_11_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_12_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A3: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A3 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_13_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B3: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B3 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A2 ) ) ) ) ).
% abs_minus_commute
thf(fact_14_lambda__skolem,axiom,
! [N: $tType,A: $tType] :
( ( finite_finite @ N )
=> ! [P: N > A > $o] :
( ( ! [I3: N] :
? [X3: A] : ( P @ I3 @ X3 ) )
= ( ? [X2: finite_Cartesian_vec @ A @ N] :
! [I3: N] : ( P @ I3 @ ( finite1433825200ec_nth @ A @ N @ X2 @ I3 ) ) ) ) ) ).
% lambda_skolem
thf(fact_15_vec__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ( ( ^ [Y2: finite_Cartesian_vec @ A @ B,Z2: finite_Cartesian_vec @ A @ B] : Y2 = Z2 )
= ( ^ [X2: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
! [I3: B] :
( ( finite1433825200ec_nth @ A @ B @ X2 @ I3 )
= ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% vec_eq_iff
thf(fact_16_assms,axiom,
! [V: finite_Cartesian_vec @ real @ n,I2: n] :
( ( member @ ( finite_Cartesian_vec @ real @ n ) @ V @ a )
=> ( member @ real @ ( finite1433825200ec_nth @ real @ n @ V @ I2 ) @ ( ring_1_Ints @ real ) ) ) ).
% assms
thf(fact_17_of__int__vec__eq__iff,axiom,
! [A: $tType,N: $tType] :
( ( ( finite_finite @ N )
& ( ring_char_0 @ A ) )
=> ! [A2: finite_Cartesian_vec @ int @ N,B3: finite_Cartesian_vec @ int @ N] :
( ( ( minkow1937162798nt_vec @ N @ A @ A2 )
= ( minkow1937162798nt_vec @ N @ A @ B3 ) )
= ( A2 = B3 ) ) ) ).
% of_int_vec_eq_iff
thf(fact_18_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B3: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_19_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_20_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_21_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q: A,P: A > $o] :
( ( member @ A @ Q @ ( ring_1_Ints @ A ) )
=> ( ! [Z3: int] : ( P @ ( ring_1_of_int @ A @ Z3 ) )
=> ( P @ Q ) ) ) ) ).
% Ints_induct
thf(fact_22_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q: A] :
( ( member @ A @ Q @ ( ring_1_Ints @ A ) )
=> ~ ! [Z3: int] :
( Q
!= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% Ints_cases
thf(fact_23_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_24_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A2 = B3 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_25_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A3: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A3 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_26_vec__nth__inject,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B] :
( ( ( finite1433825200ec_nth @ A @ B @ X )
= ( finite1433825200ec_nth @ A @ B @ Y ) )
= ( X = Y ) ) ) ).
% vec_nth_inject
thf(fact_27_cond__component,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ B )
=> ! [B3: $o,X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B,I2: B] :
( ( B3
=> ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B3 @ X @ Y ) @ I2 )
= ( finite1433825200ec_nth @ A @ B @ X @ I2 ) ) )
& ( ~ B3
=> ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B3 @ X @ Y ) @ I2 )
= ( finite1433825200ec_nth @ A @ B @ Y @ I2 ) ) ) ) ) ).
% cond_component
thf(fact_28_nth__map__matrix,axiom,
! [A: $tType,C2: $tType,D2: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C2 ) )
=> ! [F: D2 > A,X: finite_Cartesian_vec @ ( finite_Cartesian_vec @ D2 @ B ) @ C2,I2: C2,J2: B] :
( ( finite1433825200ec_nth @ A @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ B ) @ C2 @ ( finite1614961371matrix @ D2 @ A @ B @ C2 @ F @ X ) @ I2 ) @ J2 )
= ( F @ ( finite1433825200ec_nth @ D2 @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ D2 @ B ) @ C2 @ X @ I2 ) @ J2 ) ) ) ) ).
% nth_map_matrix
thf(fact_29_dist__of__int,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A )
=> ! [M2: int,N2: int] :
( ( real_V2000881966t_dist @ A @ ( ring_1_of_int @ A @ M2 ) @ ( ring_1_of_int @ A @ N2 ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ M2 @ N2 ) ) ) ) ) ).
% dist_of_int
thf(fact_30_vec__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( minus @ A )
& ( finite_finite @ B ) )
=> ! [X: A,Y: A] :
( ( finite_Cartesian_vec2 @ A @ B @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ ( finite_Cartesian_vec2 @ A @ B @ X ) @ ( finite_Cartesian_vec2 @ A @ B @ Y ) ) ) ) ).
% vec_sub
thf(fact_31_vector__sub__rdistrib,axiom,
! [B: $tType,A: $tType] :
( ( ( ring @ A )
& ( finite_finite @ B ) )
=> ! [A2: A,B3: A,X: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ ( minus_minus @ A @ A2 @ B3 ) @ X )
= ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ ( finite431509953r_mult @ A @ B @ A2 @ X ) @ ( finite431509953r_mult @ A @ B @ B3 @ X ) ) ) ) ).
% vector_sub_rdistrib
thf(fact_32__C1_C_I3_J,axiom,
ord_less @ real @ ( real_V2000881966t_dist @ ( finite_Cartesian_vec @ real @ n ) @ y @ x ) @ ( one_one @ real ) ).
% "1"(3)
thf(fact_33_norm__of__int,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A )
=> ! [Z: int] :
( ( real_V1127708846m_norm @ A @ ( ring_1_of_int @ A @ Z ) )
= ( abs_abs @ real @ ( ring_1_of_int @ real @ Z ) ) ) ) ).
% norm_of_int
thf(fact_34_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_diff_of_int
thf(fact_35_axis__nth,axiom,
! [B: $tType,A: $tType] :
( ( ( zero @ A )
& ( finite_finite @ B ) )
=> ! [I2: B,X: A] :
( ( finite1433825200ec_nth @ A @ B @ ( finite1218308419n_axis @ B @ A @ I2 @ X ) @ I2 )
= X ) ) ).
% axis_nth
thf(fact_36_vec_Oscale__left__diff__distrib,axiom,
! [B: $tType,A: $tType] :
( ( ( field @ A )
& ( finite_finite @ B ) )
=> ! [A2: A,B3: A,X: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ ( minus_minus @ A @ A2 @ B3 ) @ X )
= ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ ( finite431509953r_mult @ A @ B @ A2 @ X ) @ ( finite431509953r_mult @ A @ B @ B3 @ X ) ) ) ) ).
% vec.scale_left_diff_distrib
thf(fact_37_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Z: int] :
( ( archim2085082626_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archim2085082626_floor @ A @ X ) @ Z ) ) ) ).
% floor_diff_of_int
thf(fact_38_norm__axis__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [M2: A] :
( ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ real @ A ) @ ( finite1218308419n_axis @ A @ real @ M2 @ ( one_one @ real ) ) )
= ( one_one @ real ) ) ) ).
% norm_axis_1
thf(fact_39_vec__eq,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [M2: A,N2: A] :
( ( ( finite_Cartesian_vec2 @ A @ B @ M2 )
= ( finite_Cartesian_vec2 @ A @ B @ N2 ) )
= ( M2 = N2 ) ) ) ).
% vec_eq
thf(fact_40_vec__inj,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [X: A,Y: A] :
( ( ( finite_Cartesian_vec2 @ A @ B @ X )
= ( finite_Cartesian_vec2 @ A @ B @ Y ) )
= ( X = Y ) ) ) ).
% vec_inj
thf(fact_41_vec_Oscale__one,axiom,
! [B: $tType,A: $tType] :
( ( ( field @ A )
& ( finite_finite @ B ) )
=> ! [X: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ ( one_one @ A ) @ X )
= X ) ) ).
% vec.scale_one
thf(fact_42_vector__smult__lid,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_mult @ A )
& ( finite_finite @ B ) )
=> ! [X: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ ( one_one @ A ) @ X )
= X ) ) ).
% vector_smult_lid
thf(fact_43_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% of_int_less_iff
thf(fact_49_one__index,axiom,
! [N: $tType,A: $tType] :
( ( ( one @ A )
& ( finite_finite @ N ) )
=> ! [I2: N] :
( ( finite1433825200ec_nth @ A @ N @ ( one_one @ ( finite_Cartesian_vec @ A @ N ) ) @ I2 )
= ( one_one @ A ) ) ) ).
% one_index
thf(fact_50_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( one_one @ A ) )
= ( Z
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_51_of__int__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( one_one @ int ) )
= ( one_one @ A ) ) ) ).
% of_int_1
thf(fact_52_abs__norm__cancel,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [A2: A] :
( ( abs_abs @ real @ ( real_V1127708846m_norm @ A @ A2 ) )
= ( real_V1127708846m_norm @ A @ A2 ) ) ) ).
% abs_norm_cancel
thf(fact_53_floor__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ( ( archim2085082626_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_54_ceiling__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_55_vec__1,axiom,
! [B: $tType,A: $tType] :
( ( ( one @ A )
& ( finite_finite @ B ) )
=> ( ( finite_Cartesian_vec2 @ A @ B @ ( one_one @ A ) )
= ( one_one @ ( finite_Cartesian_vec @ A @ B ) ) ) ) ).
% vec_1
thf(fact_56_floor__of__int,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Z: int] :
( ( archim2085082626_floor @ A @ ( ring_1_of_int @ A @ Z ) )
= Z ) ) ).
% floor_of_int
thf(fact_57_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Z: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z ) )
= Z ) ) ).
% ceiling_of_int
thf(fact_58_vec__component,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [X: A,I2: B] :
( ( finite1433825200ec_nth @ A @ B @ ( finite_Cartesian_vec2 @ A @ B @ X ) @ I2 )
= X ) ) ).
% vec_component
thf(fact_59_vector__ssub__ldistrib,axiom,
! [B: $tType,A: $tType] :
( ( ( ring @ A )
& ( finite_finite @ B ) )
=> ! [C: A,X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ C @ ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ X @ Y ) )
= ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ ( finite431509953r_mult @ A @ B @ C @ X ) @ ( finite431509953r_mult @ A @ B @ C @ Y ) ) ) ) ).
% vector_ssub_ldistrib
thf(fact_60_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_less_iff
thf(fact_61_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_62_norm__one,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A )
=> ( ( real_V1127708846m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_63_dist__diff_I2_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [A2: A,B3: A] :
( ( real_V2000881966t_dist @ A @ ( minus_minus @ A @ A2 @ B3 ) @ A2 )
= ( real_V1127708846m_norm @ A @ B3 ) ) ) ).
% dist_diff(2)
thf(fact_64_dist__diff_I1_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [A2: A,B3: A] :
( ( real_V2000881966t_dist @ A @ A2 @ ( minus_minus @ A @ A2 @ B3 ) )
= ( real_V1127708846m_norm @ A @ B3 ) ) ) ).
% dist_diff(1)
thf(fact_65_floor__less__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim2085082626_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_66_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_67_floor__diff__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( archim2085082626_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim2085082626_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_68_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_69_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_70_floor__less__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archim2085082626_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_71_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_72_vec_Oscale__left__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( field @ A )
& ( finite_finite @ B ) )
=> ! [A2: A,B3: A,X: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ A2 @ ( finite431509953r_mult @ A @ B @ B3 @ X ) )
= ( finite431509953r_mult @ A @ B @ B3 @ ( finite431509953r_mult @ A @ B @ A2 @ X ) ) ) ) ).
% vec.scale_left_commute
thf(fact_73_dist__norm,axiom,
! [A: $tType] :
( ( real_V1639655670t_norm @ A )
=> ( ( real_V2000881966t_dist @ A )
= ( ^ [X2: A,Y3: A] : ( real_V1127708846m_norm @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) ) ) ) ).
% dist_norm
thf(fact_74_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_75_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [Y: A,X: A,E: real] :
( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X ) @ E )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% dist_commute_lessI
thf(fact_76_dist__commute,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( ( real_V2000881966t_dist @ A )
= ( ^ [X2: A,Y3: A] : ( real_V2000881966t_dist @ A @ Y3 @ X2 ) ) ) ) ).
% dist_commute
thf(fact_77_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_78_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim2085082626_floor @ A @ X ) @ ( archim2085082626_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_79_norm__minus__commute,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [A2: A,B3: A] :
( ( real_V1127708846m_norm @ A @ ( minus_minus @ A @ A2 @ B3 ) )
= ( real_V1127708846m_norm @ A @ ( minus_minus @ A @ B3 @ A2 ) ) ) ) ).
% norm_minus_commute
thf(fact_80_real__norm__def,axiom,
( ( real_V1127708846m_norm @ real )
= ( abs_abs @ real ) ) ).
% real_norm_def
thf(fact_81_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_82_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_83_norm__bound__component__lt__cart,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: finite_Cartesian_vec @ real @ A,E: real,I2: A] :
( ( ord_less @ real @ ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ real @ A ) @ X ) @ E )
=> ( ord_less @ real @ ( abs_abs @ real @ ( finite1433825200ec_nth @ real @ A @ X @ I2 ) ) @ E ) ) ) ).
% norm_bound_component_lt_cart
thf(fact_84_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,D: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_85_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A2 @ B3 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_86_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_87_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_88_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_89_dist__real__def,axiom,
( ( real_V2000881966t_dist @ real )
= ( ^ [X2: real,Y3: real] : ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) ) ) ).
% dist_real_def
thf(fact_90_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_91_vec_Oscale__right__diff__distrib,axiom,
! [B: $tType,A: $tType] :
( ( ( field @ A )
& ( finite_finite @ B ) )
=> ! [A2: A,X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B] :
( ( finite431509953r_mult @ A @ B @ A2 @ ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ X @ Y ) )
= ( minus_minus @ ( finite_Cartesian_vec @ A @ B ) @ ( finite431509953r_mult @ A @ B @ A2 @ X ) @ ( finite431509953r_mult @ A @ B @ A2 @ Y ) ) ) ) ).
% vec.scale_right_diff_distrib
thf(fact_92_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_93_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_94_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim2085082626_floor @ A @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_95_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : ( ord_less @ real @ ( minus_minus @ real @ R @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim2085082626_floor @ real @ R ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_96_real__abs__dist,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] :
( ( abs_abs @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) )
= ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ).
% real_abs_dist
thf(fact_97_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_98_axis__index__axis,axiom,
! [N: $tType] :
( ( finite_finite @ N )
=> ! [U: N] :
( ( finite1115014511_index @ real @ N @ ( finite1218308419n_axis @ N @ real @ U @ ( one_one @ real ) ) )
= U ) ) ).
% axis_index_axis
thf(fact_99_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_100_ceiling__split,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [P: int > $o,T: A] :
( ( P @ ( archimedean_ceiling @ A @ T ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T )
& ( ord_less_eq @ A @ T @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_101_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_102_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_103_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ W @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_104_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_105_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_le_iff
thf(fact_106_one__le__floor,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim2085082626_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_107_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_108_ceiling__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_109_floor__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archim2085082626_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_110_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y3: real] :
( ( ord_less @ real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_111_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_112_less__eq__vec__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( finite_finite @ B ) )
=> ( ( ord_less_eq @ ( finite_Cartesian_vec @ A @ B ) )
= ( ^ [X2: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
! [I3: B] : ( ord_less_eq @ A @ ( finite1433825200ec_nth @ A @ B @ X2 @ I3 ) @ ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_113_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_114_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_115_ceiling__le,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_116_le__floor__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archim2085082626_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% le_floor_iff
thf(fact_117_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_118_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_119_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_120_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_121_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,D: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_122_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_123_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B3 )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% abs_le_D1
thf(fact_124_norm__vector__1,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ( ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ A @ numeral_num1 ) )
= ( ^ [X2: finite_Cartesian_vec @ A @ numeral_num1] : ( real_V1127708846m_norm @ A @ ( finite1433825200ec_nth @ A @ numeral_num1 @ X2 @ ( one_one @ numeral_num1 ) ) ) ) ) ) ).
% norm_vector_1
thf(fact_125_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_126_dist__vector__1,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ( ( real_V2000881966t_dist @ ( finite_Cartesian_vec @ A @ numeral_num1 ) )
= ( ^ [X2: finite_Cartesian_vec @ A @ numeral_num1,Y3: finite_Cartesian_vec @ A @ numeral_num1] : ( real_V2000881966t_dist @ A @ ( finite1433825200ec_nth @ A @ numeral_num1 @ X2 @ ( one_one @ numeral_num1 ) ) @ ( finite1433825200ec_nth @ A @ numeral_num1 @ Y3 @ ( one_one @ numeral_num1 ) ) ) ) ) ) ).
% dist_vector_1
thf(fact_127_norm__real,axiom,
( ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ real @ numeral_num1 ) )
= ( ^ [X2: finite_Cartesian_vec @ real @ numeral_num1] : ( abs_abs @ real @ ( finite1433825200ec_nth @ real @ numeral_num1 @ X2 @ ( one_one @ numeral_num1 ) ) ) ) ) ).
% norm_real
thf(fact_128_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim2085082626_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_129_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_130_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A2 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_131_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B3 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B3 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_132_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B3 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_133_norm__le__componentwise__cart,axiom,
! [A: $tType,N: $tType,B: $tType] :
( ( ( real_V55928688vector @ B )
& ( finite_finite @ N )
& ( real_V55928688vector @ A ) )
=> ! [X: finite_Cartesian_vec @ A @ N,Y: finite_Cartesian_vec @ B @ N] :
( ! [I4: N] : ( ord_less_eq @ real @ ( real_V1127708846m_norm @ A @ ( finite1433825200ec_nth @ A @ N @ X @ I4 ) ) @ ( real_V1127708846m_norm @ B @ ( finite1433825200ec_nth @ B @ N @ Y @ I4 ) ) )
=> ( ord_less_eq @ real @ ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ A @ N ) @ X ) @ ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ B @ N ) @ Y ) ) ) ) ).
% norm_le_componentwise_cart
thf(fact_134_Finite__Cartesian__Product_Onorm__nth__le,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( finite_finite @ B ) )
=> ! [X: finite_Cartesian_vec @ A @ B,I2: B] : ( ord_less_eq @ real @ ( real_V1127708846m_norm @ A @ ( finite1433825200ec_nth @ A @ B @ X @ I2 ) ) @ ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ A @ B ) @ X ) ) ) ).
% Finite_Cartesian_Product.norm_nth_le
thf(fact_135_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim2085082626_floor @ real @ R ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_136_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim2085082626_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_137_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_138_norm__bound__component__le__cart,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: finite_Cartesian_vec @ real @ A,E: real,I2: A] :
( ( ord_less_eq @ real @ ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ real @ A ) @ X ) @ E )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( finite1433825200ec_nth @ real @ A @ X @ I2 ) ) @ E ) ) ) ).
% norm_bound_component_le_cart
thf(fact_139_component__le__norm__cart,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: finite_Cartesian_vec @ real @ A,I2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( finite1433825200ec_nth @ real @ A @ X @ I2 ) ) @ ( real_V1127708846m_norm @ ( finite_Cartesian_vec @ real @ A ) @ X ) ) ) ).
% component_le_norm_cart
thf(fact_140_dist__real,axiom,
( ( real_V2000881966t_dist @ ( finite_Cartesian_vec @ real @ numeral_num1 ) )
= ( ^ [X2: finite_Cartesian_vec @ real @ numeral_num1,Y3: finite_Cartesian_vec @ real @ numeral_num1] : ( abs_abs @ real @ ( minus_minus @ real @ ( finite1433825200ec_nth @ real @ numeral_num1 @ X2 @ ( one_one @ numeral_num1 ) ) @ ( finite1433825200ec_nth @ real @ numeral_num1 @ Y3 @ ( one_one @ numeral_num1 ) ) ) ) ) ) ).
% dist_real
thf(fact_141_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_142_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [R: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R ) ) @ ( one_one @ A ) ) @ R ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_143_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [A2: A,B3: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V1127708846m_norm @ A @ A2 ) @ ( real_V1127708846m_norm @ A @ B3 ) ) @ ( real_V1127708846m_norm @ A @ ( minus_minus @ A @ A2 @ B3 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_144_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [A2: A,B3: A,C: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V2000881966t_dist @ A @ A2 @ B3 ) @ ( real_V2000881966t_dist @ A @ B3 @ C ) ) ) @ ( real_V2000881966t_dist @ A @ A2 @ C ) ) ) ).
% abs_dist_diff_le
thf(fact_145_metric__eq__thm,axiom,
! [A5: $tType] :
( ( real_V2090557954_space @ A5 )
=> ! [X5: A5,S: set @ A5,Y4: A5] :
( ( member @ A5 @ X5 @ S )
=> ( ( member @ A5 @ Y4 @ S )
=> ( ( X5 = Y4 )
= ( ! [X2: A5] :
( ( member @ A5 @ X2 @ S )
=> ( ( real_V2000881966t_dist @ A5 @ X5 @ X2 )
= ( real_V2000881966t_dist @ A5 @ Y4 @ X2 ) ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_146_dist__vec__nth__le,axiom,
! [D2: $tType,C2: $tType] :
( ( ( real_V2090557954_space @ C2 )
& ( finite_finite @ D2 ) )
=> ! [X: finite_Cartesian_vec @ C2 @ D2,I2: D2,Y: finite_Cartesian_vec @ C2 @ D2] : ( ord_less_eq @ real @ ( real_V2000881966t_dist @ C2 @ ( finite1433825200ec_nth @ C2 @ D2 @ X @ I2 ) @ ( finite1433825200ec_nth @ C2 @ D2 @ Y @ I2 ) ) @ ( real_V2000881966t_dist @ ( finite_Cartesian_vec @ C2 @ D2 ) @ X @ Y ) ) ) ).
% dist_vec_nth_le
thf(fact_147_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [A2: A,B3: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V1127708846m_norm @ A @ A2 ) @ ( real_V1127708846m_norm @ A @ B3 ) ) ) @ ( real_V1127708846m_norm @ A @ ( minus_minus @ A @ A2 @ B3 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_148_ceiling__correct,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_149_ceiling__unique,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_150_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_151_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A2: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B3 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D3: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_152_eucl__less__le__not__le,axiom,
! [A: $tType] :
( ( ordere890947078_space @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_153_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( A2 != B3 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_154_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_155_vector__one__nth,axiom,
! [A: $tType,X: finite_Cartesian_vec @ A @ numeral_num1] :
( ( finite_Cartesian_vec2 @ A @ numeral_num1 @ ( finite1433825200ec_nth @ A @ numeral_num1 @ X @ ( one_one @ numeral_num1 ) ) )
= X ) ).
% vector_one_nth
thf(fact_156_less__vec__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( finite_finite @ B ) )
=> ( ( ord_less @ ( finite_Cartesian_vec @ A @ B ) )
= ( ^ [X2: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
( ( ord_less_eq @ ( finite_Cartesian_vec @ A @ B ) @ X2 @ Y3 )
& ~ ( ord_less_eq @ ( finite_Cartesian_vec @ A @ B ) @ Y3 @ X2 ) ) ) ) ) ).
% less_vec_def
thf(fact_157_complete__real,axiom,
! [S2: set @ real] :
( ? [X6: real] : ( member @ real @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member @ real @ X4 @ S2 )
=> ( ord_less_eq @ real @ X4 @ Z4 ) )
=> ? [Y5: real] :
( ! [X6: real] :
( ( member @ real @ X6 @ S2 )
=> ( ord_less_eq @ real @ X6 @ Y5 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ S2 )
=> ( ord_less_eq @ real @ X4 @ Z4 ) )
=> ( ord_less_eq @ real @ Y5 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_158_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_159_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A6: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A6 )
& ( ord_less_eq @ A @ A6 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_160_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_161_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
=> ( P @ A7 @ B5 ) )
=> ( ! [A7: A,B5: A] :
( ( P @ B5 @ A7 )
=> ( P @ A7 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_162_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_163_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_164_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_165_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_166_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_167_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A6: A,B4: A] :
( ( ord_less_eq @ A @ A6 @ B4 )
& ( ord_less_eq @ A @ B4 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_168_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_169_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_170_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_171_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_172_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_173_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_174_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_175_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_176_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_177_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_178_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_179_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_180_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_181_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_182_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_183_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_184_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B5: A] :
( ( ord_less @ A @ A2 @ B5 )
| ( ord_less @ A @ B5 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_185_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( A2 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_186_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_187_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_188_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_189_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A7: A,B5: A] :
( ( ord_less @ A @ A7 @ B5 )
=> ( P @ A7 @ B5 ) )
=> ( ! [A7: A] : ( P @ A7 @ A7 )
=> ( ! [A7: A,B5: A] :
( ( P @ B5 @ A7 )
=> ( P @ A7 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_190_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X3: A] : ( P2 @ X3 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_191_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_192_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_193_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_194_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_195_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_196_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_197_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_198_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X4 )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_199_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_200_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_201_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_202_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_203_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C: A] :
( ( A2 = B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_204_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_205_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_206_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_207_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% less_asym'
thf(fact_208_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_209_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_210_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_211_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% order.asym
thf(fact_212_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_213_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_214_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_215_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% lt_ex
thf(fact_216_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_217_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_218_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_219_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_220_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_221_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_222_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_223_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_224_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_225_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_226_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_227_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_228_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_229_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_230_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_231_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_232_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_233_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_234_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_235_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_236_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_237_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_238_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_239_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_240_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_241_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_242_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_243_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_244_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B4: A] :
( ( ord_less @ A @ A6 @ B4 )
| ( A6 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_245_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B4: A] :
( ( ord_less_eq @ A @ A6 @ B4 )
& ( A6 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_246_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_247_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_248_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z @ W2 )
=> ( ( ord_less @ A @ W2 @ X )
=> ( ord_less_eq @ A @ Y @ W2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_249_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W2: A] :
( ( ord_less @ A @ X @ W2 )
=> ( ( ord_less @ A @ W2 @ Y )
=> ( ord_less_eq @ A @ W2 @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_250_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_251_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A6: A] :
( ( ord_less @ A @ B4 @ A6 )
| ( A6 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_252_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A6: A] :
( ( ord_less_eq @ A @ B4 @ A6 )
& ( A6 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_253_forall__1,axiom,
( ( ^ [P2: numeral_num1 > $o] :
! [X3: numeral_num1] : ( P2 @ X3 ) )
= ( ^ [P3: numeral_num1 > $o] : ( P3 @ ( one_one @ numeral_num1 ) ) ) ) ).
% forall_1
thf(fact_254_ex__1,axiom,
( ( ^ [P2: numeral_num1 > $o] :
? [X3: numeral_num1] : ( P2 @ X3 ) )
= ( ^ [P3: numeral_num1 > $o] : ( P3 @ ( one_one @ numeral_num1 ) ) ) ) ).
% ex_1
% Subclasses (1)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
% Type constructors (106)
thf(tcon_Numeral__Type_Onum1___Cardinality_OCARD__1,axiom,
cARD_1 @ numeral_num1 ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A5: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A5 > A8 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A5: $tType,A8: $tType] :
( ( ( finite_finite @ A5 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( A5 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A5 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A5: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A5 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A5: $tType,A8: $tType] :
( ( minus @ A8 )
=> ( minus @ ( A5 > A8 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Ominus_4,axiom,
minus @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A5: $tType] : ( preorder @ ( set @ A5 ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__mult_6,axiom,
! [A5: $tType] :
( ( monoid_mult @ A5 )
=> ( monoid_mult @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_7,axiom,
! [A5: $tType] :
( ( finite_finite @ A5 )
=> ( finite_finite @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A5: $tType] : ( order @ ( set @ A5 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A5: $tType] : ( ord @ ( set @ A5 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_10,axiom,
! [A5: $tType] : ( minus @ ( set @ A5 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_11,axiom,
! [A5: $tType] :
( ( zero @ A5 )
=> ( zero @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oone_12,axiom,
! [A5: $tType] :
( ( one @ A5 )
=> ( one @ ( set @ A5 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_18,axiom,
minus @ $o ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
condit1037483654norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit1656338222tinuum @ real ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
ordere890947078_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
real_V1229719638ebra_1 @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_20,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_21,axiom,
ordere142940540dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
archim1727834104eiling @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_22,axiom,
cancel146912293up_add @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Odist__norm,axiom,
real_V1639655670t_norm @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_23,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_24,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_25,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_26,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_27,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_28,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_29,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_30,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_31,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_32,axiom,
order @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_33,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_34,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_35,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_36,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_37,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Rings_Oring_38,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Groups_Oone_39,axiom,
one @ real ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_40,axiom,
cancel146912293up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_41,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_42,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_43,axiom,
monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_44,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_45,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_46,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_47,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ominus_48,axiom,
minus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_49,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Oring_50,axiom,
ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_51,axiom,
one @ numeral_num1 ).
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space_52,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A5 )
& ( finite_finite @ A8 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_53,axiom,
! [A5: $tType,A8: $tType] :
( ( ( real_V55928688vector @ A5 )
& ( finite_finite @ A8 ) )
=> ( real_V55928688vector @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Ometric__space_54,axiom,
! [A5: $tType,A8: $tType] :
( ( ( real_V2090557954_space @ A5 )
& ( finite_finite @ A8 ) )
=> ( real_V2090557954_space @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add__abs_55,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A5 )
& ( finite_finite @ A8 ) )
=> ( ordere142940540dd_abs @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ocancel__ab__semigroup__add_56,axiom,
! [A5: $tType,A8: $tType] :
( ( ( cancel146912293up_add @ A5 )
& ( finite_finite @ A8 ) )
=> ( cancel146912293up_add @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Odist__norm_57,axiom,
! [A5: $tType,A8: $tType] :
( ( ( real_V55928688vector @ A5 )
& ( finite_finite @ A8 ) )
=> ( real_V1639655670t_norm @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_58,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ordere890947078_space @ A5 )
& ( finite_finite @ A8 ) )
=> ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_59,axiom,
! [A5: $tType,A8: $tType] :
( ( ( order @ A5 )
& ( finite_finite @ A8 ) )
=> ( preorder @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_60,axiom,
! [A5: $tType,A8: $tType] :
( ( ( linorder @ A5 )
& ( cARD_1 @ A8 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Omonoid__mult_61,axiom,
! [A5: $tType,A8: $tType] :
( ( ( monoid_mult @ A5 )
& ( finite_finite @ A8 ) )
=> ( monoid_mult @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_62,axiom,
! [A5: $tType,A8: $tType] :
( ( ( finite_finite @ A5 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_63,axiom,
! [A5: $tType,A8: $tType] :
( ( ( group_add @ A5 )
& ( finite_finite @ A8 ) )
=> ( group_add @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_64,axiom,
! [A5: $tType,A8: $tType] :
( ( ( order @ A5 )
& ( finite_finite @ A8 ) )
=> ( order @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Nat_Oring__char__0_65,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ring_char_0 @ A5 )
& ( finite_finite @ A8 ) )
=> ( ring_char_0 @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_66,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ord @ A5 )
& ( finite_finite @ A8 ) )
=> ( ord @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Rings_Oring__1_67,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ring_1 @ A5 )
& ( finite_finite @ A8 ) )
=> ( ring_1 @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ominus_68,axiom,
! [A5: $tType,A8: $tType] :
( ( ( minus @ A5 )
& ( finite_finite @ A8 ) )
=> ( minus @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_69,axiom,
! [A5: $tType,A8: $tType] :
( ( ( zero @ A5 )
& ( finite_finite @ A8 ) )
=> ( zero @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Rings_Oring_70,axiom,
! [A5: $tType,A8: $tType] :
( ( ( ring @ A5 )
& ( finite_finite @ A8 ) )
=> ( ring @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oone_71,axiom,
! [A5: $tType,A8: $tType] :
( ( ( one @ A5 )
& ( finite_finite @ A8 ) )
=> ( one @ ( finite_Cartesian_vec @ A5 @ A8 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Free types (1)
thf(tfree_0,hypothesis,
finite_finite @ n ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ring_1_of_int @ real @ ( abs_abs @ int @ m ) )
= ( abs_abs @ real @ ( finite1433825200ec_nth @ real @ n @ ( minus_minus @ ( finite_Cartesian_vec @ real @ n ) @ x @ y ) @ i ) ) ) ).
%------------------------------------------------------------------------------