TPTP Problem File: ITP109^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP109^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_1098__6259092_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 : Lower_Semicontinuous/prob_1098__6259092_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 417 ( 131 unt; 65 typ; 0 def)
% Number of atoms : 812 ( 325 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 4226 ( 74 ~; 10 |; 45 &;3738 @)
% ( 0 <=>; 359 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 129 ( 129 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 60 usr; 6 con; 0-4 aty)
% Number of variables : 1058 ( 87 ^; 903 !; 16 ?;1058 :)
% ( 52 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:28:51.464
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Extended__Real_Oereal,type,
extended_ereal: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (59)
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_Oplus,type,
plus:
!>[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_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[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_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__bits,type,
semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_OscaleR,type,
real_Vector_scaleR:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s1003936772cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V1076094709vector:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
real_V148923926lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V1015715145lgebra:
!>[A: $tType] : $o ).
thf(sy_c_Convex_Oconvex,type,
convex:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups1340683514dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on,type,
lower_673667120vex_on:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V1908273582scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_C,type,
c: set @ b ).
thf(sy_v_a,type,
a2: a > real ).
thf(sy_v_aa____,type,
aa: a > real ).
thf(sy_v_f,type,
f: b > extended_ereal ).
thf(sy_v_i____,type,
i: a ).
thf(sy_v_s,type,
s: set @ a ).
thf(sy_v_sa____,type,
sa: set @ a ).
thf(sy_v_y,type,
y: a > b ).
% Relevant facts (255)
thf(fact_0_insert_Ohyps_I3_J,axiom,
~ ( member @ a @ i @ sa ) ).
% insert.hyps(3)
thf(fact_1_asm,axiom,
( ( aa @ i )
!= ( one_one @ real ) ) ).
% asm
thf(fact_2__092_060open_062_I_092_060Sum_062x_092_060in_062s_O_Aa_Ax_A_K_092_060_094sub_062R_Ay_Ax_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_J_A_061_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_092_060close_062,axiom,
( ( groups1340683514dd_sum @ a @ b
@ ^ [X: a] : ( real_V1908273582scaleR @ b @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) @ ( real_V1908273582scaleR @ b @ ( aa @ X ) @ ( y @ X ) ) )
@ sa )
= ( real_V1908273582scaleR @ b @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) ) ) ).
% \<open>(\<Sum>x\<in>s. a x *\<^sub>R y x /\<^sub>R (1 - a i)) = (\<Sum>j\<in>s. a j *\<^sub>R y j) /\<^sub>R (1 - a i)\<close>
thf(fact_3_insert_Ohyps_I2_J,axiom,
( sa
!= ( bot_bot @ ( set @ a ) ) ) ).
% insert.hyps(2)
thf(fact_4_insert_Ohyps_I1_J,axiom,
finite_finite2 @ a @ sa ).
% insert.hyps(1)
thf(fact_5__092_060open_062f_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_A_061_Af_A_I_I_I1_A_N_Aa_Ai_J_A_K_Ainverse_A_I1_A_N_Aa_Ai_J_J_A_K_092_060_094sub_062R_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
( ( f
@ ( plus_plus @ b
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ sa )
@ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) )
= ( f
@ ( plus_plus @ b
@ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% \<open>f ((\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i) = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i)\<close>
thf(fact_6_calculation,axiom,
( ( f
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ ( insert @ a @ i @ sa ) ) )
= ( f
@ ( plus_plus @ b
@ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% calculation
thf(fact_7__092_060open_062f_A_I_092_060Sum_062j_092_060in_062insert_Ai_As_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_061_Af_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
( ( f
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ ( insert @ a @ i @ sa ) ) )
= ( f
@ ( plus_plus @ b
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ sa )
@ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% \<open>f (\<Sum>j\<in>insert i s. a j *\<^sub>R y j) = f ((\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i)\<close>
thf(fact_8_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [U: real,A2: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
= A2 ) ) ).
% scaleR_collapse
thf(fact_9_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [B2: A,U: real,A2: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
= ( plus_plus @ A @ A2 @ ( real_V1908273582scaleR @ A @ U @ B2 ) ) )
= ( ( A2 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_10_yai_I1_J,axiom,
member @ b @ ( y @ i ) @ c ).
% yai(1)
thf(fact_11_scaleR__scaleR,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) )
= ( real_V1908273582scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X2 ) ) ) ).
% scaleR_scaleR
thf(fact_12_scaleR__one,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X2: A] :
( ( real_V1908273582scaleR @ A @ ( one_one @ real ) @ X2 )
= X2 ) ) ).
% scaleR_one
thf(fact_13__092_060open_062sum_Aa_As_A_061_A1_A_N_Aa_Ai_092_060close_062,axiom,
( ( groups1340683514dd_sum @ a @ real @ aa @ sa )
= ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) ).
% \<open>sum a s = 1 - a i\<close>
thf(fact_14_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( ( inverse_inverse @ A @ X2 )
= ( one_one @ A ) )
= ( X2
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_15_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_16_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_17_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V148923926lgebra @ A )
=> ! [A2: real,X2: A,Y: A] :
( ( times_times @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ Y )
= ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y ) ) ) ) ).
% mult_scaleR_left
thf(fact_18_assms_I1_J,axiom,
finite_finite2 @ a @ s ).
% assms(1)
thf(fact_19_assms_I2_J,axiom,
( s
!= ( bot_bot @ ( set @ a ) ) ) ).
% assms(2)
thf(fact_20_assms_I7_J,axiom,
! [I: a] :
( ( member @ a @ I @ s )
=> ( member @ b @ ( y @ I ) @ c ) ) ).
% assms(7)
thf(fact_21_insert_Oprems_I3_J,axiom,
! [I: a] :
( ( member @ a @ I @ ( insert @ a @ i @ sa ) )
=> ( member @ b @ ( y @ I ) @ c ) ) ).
% insert.prems(3)
thf(fact_22_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_23_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_24_insert_Oprems_I1_J,axiom,
( ( groups1340683514dd_sum @ a @ real @ aa @ ( insert @ a @ i @ sa ) )
= ( one_one @ real ) ) ).
% insert.prems(1)
thf(fact_25_assms_I3_J,axiom,
lower_673667120vex_on @ b @ c @ f ).
% assms(3)
thf(fact_26_fis,axiom,
finite_finite2 @ a @ ( insert @ a @ i @ sa ) ).
% fis
thf(fact_27_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V148923926lgebra @ A )
=> ! [X2: A,A2: real,Y: A] :
( ( times_times @ A @ X2 @ ( real_V1908273582scaleR @ A @ A2 @ Y ) )
= ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y ) ) ) ) ).
% mult_scaleR_right
thf(fact_28_assms_I4_J,axiom,
convex @ b @ c ).
% assms(4)
thf(fact_29_assms_I5_J,axiom,
( ( groups1340683514dd_sum @ a @ real @ a2 @ s )
= ( one_one @ real ) ) ).
% assms(5)
thf(fact_30_real__scaleR__def,axiom,
( ( real_V1908273582scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_31_scale__left__commute,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) )
= ( real_V1908273582scaleR @ A @ B2 @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) ) ) ) ).
% scale_left_commute
thf(fact_32_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_33_scaleR__add__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X2: A,Y: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( plus_plus @ A @ X2 @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_add_right
thf(fact_34_scaleR__add__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X2 )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) ) ) ) ).
% scaleR_add_left
thf(fact_35_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X2: real,Y: real,Xa: A] :
( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ X2 @ Y ) @ Xa )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ X2 @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_36_scale__right__diff__distrib,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X2: A,Y: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( minus_minus @ A @ X2 @ Y ) )
= ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% scale_right_diff_distrib
thf(fact_37_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X2: A] :
( ( ( times_times @ A @ Y @ X2 )
= ( times_times @ A @ X2 @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_38_scale__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,F: C > A,A3: set @ C] :
( ( real_V1908273582scaleR @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) )
= ( groups1340683514dd_sum @ C @ A
@ ^ [X: C] : ( real_V1908273582scaleR @ A @ A2 @ ( F @ X ) )
@ A3 ) ) ) ).
% scale_sum_right
thf(fact_39_scale__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( real_V1076094709vector @ A )
=> ! [F: C > real,A3: set @ C,X2: A] :
( ( real_V1908273582scaleR @ A @ ( groups1340683514dd_sum @ C @ real @ F @ A3 ) @ X2 )
= ( groups1340683514dd_sum @ C @ A
@ ^ [A4: C] : ( real_V1908273582scaleR @ A @ ( F @ A4 ) @ X2 )
@ A3 ) ) ) ).
% scale_sum_left
thf(fact_40_scaleR__right_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,G: C > A,A3: set @ C] :
( ( real_V1908273582scaleR @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ G @ A3 ) )
= ( groups1340683514dd_sum @ C @ A
@ ^ [X: C] : ( real_V1908273582scaleR @ A @ A2 @ ( G @ X ) )
@ A3 ) ) ) ).
% scaleR_right.sum
thf(fact_41_scaleR__left_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V1076094709vector @ A )
=> ! [G: C > real,A3: set @ C,X2: A] :
( ( real_V1908273582scaleR @ A @ ( groups1340683514dd_sum @ C @ real @ G @ A3 ) @ X2 )
= ( groups1340683514dd_sum @ C @ A
@ ^ [X: C] : ( real_V1908273582scaleR @ A @ ( G @ X ) @ X2 )
@ A3 ) ) ) ).
% scaleR_left.sum
thf(fact_42_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_43_scale__left__diff__distrib,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X2 )
= ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) ) ) ) ).
% scale_left_diff_distrib
thf(fact_44_scaleR__left_Odiff,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X2: real,Y: real,Xa: A] :
( ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ X2 @ Y ) @ Xa )
= ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ X2 @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% scaleR_left.diff
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V1015715145lgebra @ A )
=> ! [A2: real,X2: A] :
( ( inverse_inverse @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) )
= ( real_V1908273582scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X2 ) ) ) ) ).
% inverse_scaleR_distrib
thf(fact_50_asum,axiom,
( member @ b
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( divide_divide @ real @ ( aa @ J ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) @ ( y @ J ) )
@ sa )
@ c ) ).
% asum
thf(fact_51_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X2 @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_52_affine__hull__finite__step,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [S: set @ A,A2: A,W: real,Y: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ? [U2: A > real] :
( ( ( groups1340683514dd_sum @ A @ real @ U2 @ ( insert @ A @ A2 @ S ) )
= W )
& ( ( groups1340683514dd_sum @ A @ A
@ ^ [X: A] : ( real_V1908273582scaleR @ A @ ( U2 @ X ) @ X )
@ ( insert @ A @ A2 @ S ) )
= Y ) ) )
= ( ? [V: real,U2: A > real] :
( ( ( groups1340683514dd_sum @ A @ real @ U2 @ S )
= ( minus_minus @ real @ W @ V ) )
& ( ( groups1340683514dd_sum @ A @ A
@ ^ [X: A] : ( real_V1908273582scaleR @ A @ ( U2 @ X ) @ X )
@ S )
= ( minus_minus @ A @ Y @ ( real_V1908273582scaleR @ A @ V @ A2 ) ) ) ) ) ) ) ) ).
% affine_hull_finite_step
thf(fact_53_a1,axiom,
( ( groups1340683514dd_sum @ a @ real
@ ^ [J: a] : ( divide_divide @ real @ ( aa @ J ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
@ sa )
= ( one_one @ real ) ) ).
% a1
thf(fact_54__092_060open_062sum_Aa_As_A_P_A_I1_A_N_Aa_Ai_J_A_061_A1_092_060close_062,axiom,
( ( divide_divide @ real @ ( groups1340683514dd_sum @ a @ real @ aa @ sa ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
= ( one_one @ real ) ) ).
% \<open>sum a s / (1 - a i) = 1\<close>
thf(fact_55_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X: A] : X = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_56_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y2: A,Z: A] : Y2 = Z
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_57_finite__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
thf(fact_58_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A2: B,F: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( F @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups1340683514dd_sum @ B @ A @ F @ A3 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_59_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_60_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_61_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_62_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_63_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_64_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_65_insert__absorb2,axiom,
! [A: $tType,X2: A,A3: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
= ( insert @ A @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_66_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_67_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_68_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_69_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_70_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_71_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_72_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_73_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_74_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_75_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_76_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_77_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_78_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_79_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_80_finite__Diff2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_Diff2
thf(fact_81_finite__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% finite_Diff
thf(fact_82_insert__Diff1,axiom,
! [A: $tType,X2: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ X2 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_83_Diff__insert0,axiom,
! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X2 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_84_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% inverse_divide
thf(fact_85_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_86_finite__Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_87_sum__diff1__nat,axiom,
! [A: $tType,A2: A,A3: set @ A,F: A > nat] :
( ( ( member @ A @ A2 @ A3 )
=> ( ( groups1340683514dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
& ( ~ ( member @ A @ A2 @ A3 )
=> ( ( groups1340683514dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) ) ) ) ).
% sum_diff1_nat
thf(fact_88_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_89_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ~ ( member @ A @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_90_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( member @ A @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_91_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_92_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_93_Diff__infinite__finite,axiom,
! [A: $tType,T: set @ A,S: set @ A] :
( ( finite_finite2 @ A @ T )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_94_insert__Diff__if,axiom,
! [A: $tType,X2: A,B3: set @ A,A3: set @ A] :
( ( ( member @ A @ X2 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
& ( ~ ( member @ A @ X2 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
= ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_95_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,A3: set @ B,R: A] :
( ( divide_divide @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
= ( groups1340683514dd_sum @ B @ A
@ ^ [N: B] : ( divide_divide @ A @ ( F @ N ) @ R )
@ A3 ) ) ) ).
% sum_divide_distrib
thf(fact_96_convex__ereal__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [S2: set @ A,F: A > extended_ereal,G: A > extended_ereal] :
( ( lower_673667120vex_on @ A @ S2 @ F )
=> ( ( lower_673667120vex_on @ A @ S2 @ G )
=> ( lower_673667120vex_on @ A @ S2
@ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% convex_ereal_add
thf(fact_97_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y: A,Z2: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_98_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y: A,Z2: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_99_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_100_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_101_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_102_Diff__insert__absorb,axiom,
! [A: $tType,X2: A,A3: set @ A] :
( ~ ( member @ A @ X2 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_103_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_104_insert__Diff,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_105_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_106_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: A,A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ( member @ A @ A6 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_107_infinite__coinduct,axiom,
! [A: $tType,X4: ( set @ A ) > $o,A3: set @ A] :
( ( X4 @ A3 )
=> ( ! [A7: set @ A] :
( ( X4 @ A7 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A7 )
& ( ( X4 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_108_infinite__remove,axiom,
! [A: $tType,S: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_109_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_110_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% divide_inverse
thf(fact_111_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A4 ) ) ) ) ).
% divide_inverse_commute
thf(fact_112_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_113_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_114_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_115_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_116_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_117_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_118_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [X_1: B] : ( P @ X3 @ X_1 ) )
=> ? [F2: A > B] :
! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( P @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% finite_set_choice
thf(fact_119_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B6: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B6 ) )
& ~ ( member @ A @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_120_insert__commute,axiom,
! [A: $tType,X2: A,Y: A,A3: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_121_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B3 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C3 ) )
& ~ ( member @ A @ B2 @ C3 )
& ( B3
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_122_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_123_insert__ident,axiom,
! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X2 @ A3 )
=> ( ~ ( member @ A @ X2 @ B3 )
=> ( ( ( insert @ A @ X2 @ A3 )
= ( insert @ A @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_124_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A3: set @ A] :
( ( member @ A @ X2 @ A3 )
=> ~ ! [B6: set @ A] :
( ( A3
= ( insert @ A @ X2 @ B6 ) )
=> ( member @ A @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_125_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_126_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_127_insertE,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_128_sum_Ocong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A,H: B > A] :
( ( A3 = B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
= ( groups1340683514dd_sum @ B @ A @ H @ B3 ) ) ) ) ) ).
% sum.cong
thf(fact_129_sum_Oeq__general,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ B,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
( ! [Y3: B] :
( ( member @ B @ Y3 @ B3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A3 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: C] :
( ( ( member @ C @ Ya @ A3 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( member @ B @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
= ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% sum.eq_general
thf(fact_130_sum_Oeq__general__inverses,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ B,K: B > C,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
( ! [Y3: B] :
( ( member @ B @ Y3 @ B3 )
=> ( ( member @ C @ ( K @ Y3 ) @ A3 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( member @ B @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
= ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_131_sum_Oreindex__bij__witness,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,I2: C > B,J2: B > C,T: set @ C,H: C > A,G: B > A] :
( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( member @ C @ ( J2 @ A6 ) @ T ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T )
=> ( ( J2 @ ( I2 @ B7 ) )
= B7 ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T )
=> ( member @ B @ ( I2 @ B7 ) @ S ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ S )
= ( groups1340683514dd_sum @ C @ A @ H @ T ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_132_if__smult,axiom,
! [A: $tType] :
( ( real_Vector_scaleR @ A )
=> ! [P: $o,X2: real,Y: real,V2: A] :
( ( P
=> ( ( real_V1908273582scaleR @ A @ ( if @ real @ P @ X2 @ Y ) @ V2 )
= ( real_V1908273582scaleR @ A @ X2 @ V2 ) ) )
& ( ~ P
=> ( ( real_V1908273582scaleR @ A @ ( if @ real @ P @ X2 @ Y ) @ V2 )
= ( real_V1908273582scaleR @ A @ Y @ V2 ) ) ) ) ) ).
% if_smult
thf(fact_133_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_134_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X2 @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_135_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups1340683514dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups1340683514dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups1340683514dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S )
= ( groups1340683514dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_136_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% empty_def
thf(fact_137_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R2: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B3 )
& ( R2 @ X3 @ Xa2 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_138_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_139_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_140_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A4: A,B4: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( X = A4 )
| ( member @ A @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_141_sum__delta__notmem_I4_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
( ~ ( member @ A @ X2 @ S2 )
=> ( ( groups1340683514dd_sum @ A @ B
@ ^ [Y4: A] : ( if @ B @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% sum_delta_notmem(4)
thf(fact_142_sum__delta__notmem_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
( ~ ( member @ A @ X2 @ S2 )
=> ( ( groups1340683514dd_sum @ A @ B
@ ^ [Y4: A] : ( if @ B @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% sum_delta_notmem(3)
thf(fact_143_sum__delta__notmem_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
( ~ ( member @ A @ X2 @ S2 )
=> ( ( groups1340683514dd_sum @ A @ B
@ ^ [Y4: A] : ( if @ B @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% sum_delta_notmem(2)
thf(fact_144_sum__delta__notmem_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
( ~ ( member @ A @ X2 @ S2 )
=> ( ( groups1340683514dd_sum @ A @ B
@ ^ [Y4: A] : ( if @ B @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% sum_delta_notmem(1)
thf(fact_145_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B3: set @ C,A3: set @ B] :
( ( groups1340683514dd_sum @ B @ A
@ ^ [I3: B] : ( groups1340683514dd_sum @ C @ A @ ( G @ I3 ) @ B3 )
@ A3 )
= ( groups1340683514dd_sum @ C @ A
@ ^ [J: C] :
( groups1340683514dd_sum @ B @ A
@ ^ [I3: B] : ( G @ I3 @ J )
@ A3 )
@ B3 ) ) ) ).
% sum.swap
thf(fact_146_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_147_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_148_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_149_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_150_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_151_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_152_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_153_finite_OinsertI,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_154_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,A3: set @ A,G: C > B,B3: set @ C] :
( ( times_times @ B @ ( groups1340683514dd_sum @ A @ B @ F @ A3 ) @ ( groups1340683514dd_sum @ C @ B @ G @ B3 ) )
= ( groups1340683514dd_sum @ A @ B
@ ^ [I3: A] :
( groups1340683514dd_sum @ C @ B
@ ^ [J: C] : ( times_times @ B @ ( F @ I3 ) @ ( G @ J ) )
@ B3 )
@ A3 ) ) ) ).
% sum_product
thf(fact_155_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R: A,F: B > A,A3: set @ B] :
( ( times_times @ A @ R @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) )
= ( groups1340683514dd_sum @ B @ A
@ ^ [N: B] : ( times_times @ A @ R @ ( F @ N ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_156_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,A3: set @ B,R: A] :
( ( times_times @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
= ( groups1340683514dd_sum @ B @ A
@ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_157_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H: B > A,A3: set @ B] :
( ( groups1340683514dd_sum @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H @ X ) )
@ A3 )
= ( plus_plus @ A @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ H @ A3 ) ) ) ) ).
% sum.distrib
thf(fact_158_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_159_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_160_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,A3: set @ B] :
( ( groups1340683514dd_sum @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
@ A3 )
= ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_161_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ C,G: B > C > A,R2: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups1340683514dd_sum @ B @ A
@ ^ [X: B] :
( groups1340683514dd_sum @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B3 )
& ( R2 @ X @ Y4 ) ) ) )
@ A3 )
= ( groups1340683514dd_sum @ C @ A
@ ^ [Y4: C] :
( groups1340683514dd_sum @ B @ A
@ ^ [X: B] : ( G @ X @ Y4 )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( R2 @ X @ Y4 ) ) ) )
@ B3 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_162_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ! [A7: set @ A] :
( ~ ( finite_finite2 @ A @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_163_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( F3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_164_finite_Oinducts,axiom,
! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: set @ A,A6: A] :
( ( finite_finite2 @ A @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( insert @ A @ A6 @ A7 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_165_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_166_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ? [A5: set @ A,B5: A] :
( ( A4
= ( insert @ A @ B5 @ A5 ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_167_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A7: set @ A] :
( ? [A6: A] :
( A2
= ( insert @ A @ A6 @ A7 ) )
=> ~ ( finite_finite2 @ A @ A7 ) ) ) ) ).
% finite.cases
thf(fact_168_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,X2: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( X2 @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( times_times @ A @ ( X2 @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_169_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X2 @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X2 @ A3 )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_170_convex__singleton,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: A] : ( convex @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% convex_singleton
thf(fact_171_real__divide__square__eq,axiom,
! [R: real,A2: real] :
( ( divide_divide @ real @ ( times_times @ real @ R @ A2 ) @ ( times_times @ real @ R @ R ) )
= ( divide_divide @ real @ A2 @ R ) ) ).
% real_divide_square_eq
thf(fact_172_convex__empty,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( convex @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% convex_empty
thf(fact_173_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_174_bits__div__by__1,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% bits_div_by_1
thf(fact_175_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_176_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_177_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_178_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_179_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_180_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_181_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_182_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_183_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_184_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_185_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_186_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ B5 @ A4 ) ) ) ) ).
% mult.commute
thf(fact_187_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_188_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_189_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_190_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_191_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_192_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% add.commute
thf(fact_193_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_194_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_195_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_196_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_197_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_198_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus @ A @ I2 @ K )
= ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_200_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_201_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A2 = B2 )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_202_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_203_convex__set__plus,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [S: set @ A,T: set @ A] :
( ( convex @ A @ S )
=> ( ( convex @ A @ T )
=> ( convex @ A @ ( plus_plus @ ( set @ A ) @ S @ T ) ) ) ) ) ).
% convex_set_plus
thf(fact_204_finite__set__sum,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ X3 ) ) )
=> ( finite_finite2 @ B @ ( groups1340683514dd_sum @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% finite_set_sum
thf(fact_205_convex__set__sum,axiom,
! [B: $tType,A: $tType] :
( ( real_V1076094709vector @ B )
=> ! [A3: set @ A,B3: A > ( set @ B )] :
( ! [I5: A] :
( ( member @ A @ I5 @ A3 )
=> ( convex @ B @ ( B3 @ I5 ) ) )
=> ( convex @ B @ ( groups1340683514dd_sum @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% convex_set_sum
thf(fact_206_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_207_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_208_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_209_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_210_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_211_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_212_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s1003936772cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_213_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s1003936772cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% left_diff_distrib'
thf(fact_214_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_215_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_216_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_217_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_218_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_219_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_add
thf(fact_220_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_221_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_222_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_223_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_224_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_225_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_226_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_227_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X: A] : X )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_228_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X2: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( minus_minus @ A @ X2 @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_229_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% eq_add_iff2
thf(fact_230_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 )
= D ) ) ) ).
% eq_add_iff1
thf(fact_231_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_232_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% minus_apply
thf(fact_233_vector__space__over__itself_Oscale__one,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( times_times @ A @ ( one_one @ A ) @ X2 )
= X2 ) ) ).
% vector_space_over_itself.scale_one
thf(fact_234_segment__degen__1,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [U: real,A2: A,B2: A] :
( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V1908273582scaleR @ A @ U @ B2 ) )
= B2 )
= ( ( A2 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% segment_degen_1
thf(fact_235_vector__space__over__itself_Oscale__scale,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
= ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_236_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
= ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ X2 ) ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_237_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ X2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ B2 @ X2 ) ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_238_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,X2: A,Y: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ X2 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ A2 @ Y ) ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_239_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ X2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ B2 @ X2 ) ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_240_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,X2: A,Y: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ X2 @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ A2 @ Y ) ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_241_add__scaleR__degen,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [U: real,B2: A,V2: real,A2: A] :
( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ B2 ) @ ( real_V1908273582scaleR @ A @ V2 @ A2 ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ A2 ) @ ( real_V1908273582scaleR @ A @ V2 @ B2 ) ) )
=> ( ( U != V2 )
=> ( A2 = B2 ) ) ) ) ).
% add_scaleR_degen
thf(fact_242_vector__space__over__itself_Oscale__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( field @ A )
=> ! [F: C > A,A3: set @ C,X2: A] :
( ( times_times @ A @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) @ X2 )
= ( groups1340683514dd_sum @ C @ A
@ ^ [A4: C] : ( times_times @ A @ ( F @ A4 ) @ X2 )
@ A3 ) ) ) ).
% vector_space_over_itself.scale_sum_left
thf(fact_243_vector__space__over__itself_Oscale__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( field @ A )
=> ! [A2: A,F: C > A,A3: set @ C] :
( ( times_times @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) )
= ( groups1340683514dd_sum @ C @ A
@ ^ [X: C] : ( times_times @ A @ A2 @ ( F @ X ) )
@ A3 ) ) ) ).
% vector_space_over_itself.scale_sum_right
thf(fact_244_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% fun_diff_def
thf(fact_245_vector__space__over__itself_Ofinite__Basis,axiom,
! [A: $tType] :
( ( field @ A )
=> ( finite_finite2 @ A @ ( insert @ A @ ( one_one @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vector_space_over_itself.finite_Basis
thf(fact_246_set__plus__intro,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [A2: A,C4: set @ A,B2: A,D2: set @ A] :
( ( member @ A @ A2 @ C4 )
=> ( ( member @ A @ B2 @ D2 )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ ( set @ A ) @ C4 @ D2 ) ) ) ) ) ).
% set_plus_intro
thf(fact_247_sum__clauses_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,X2: B,F: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ X2 @ S )
=> ( ( groups1340683514dd_sum @ B @ A @ F @ ( insert @ B @ X2 @ S ) )
= ( groups1340683514dd_sum @ B @ A @ F @ S ) ) )
& ( ~ ( member @ B @ X2 @ S )
=> ( ( groups1340683514dd_sum @ B @ A @ F @ ( insert @ B @ X2 @ S ) )
= ( plus_plus @ A @ ( F @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ F @ S ) ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_248_set__times__intro,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [A2: A,C4: set @ A,B2: A,D2: set @ A] :
( ( member @ A @ A2 @ C4 )
=> ( ( member @ A @ B2 @ D2 )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ ( set @ A ) @ C4 @ D2 ) ) ) ) ) ).
% set_times_intro
thf(fact_249_set__times__elim,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [X2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ X2 @ ( times_times @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ! [A6: A,B7: A] :
( ( X2
= ( times_times @ A @ A6 @ B7 ) )
=> ( ( member @ A @ A6 @ A3 )
=> ~ ( member @ A @ B7 @ B3 ) ) ) ) ) ).
% set_times_elim
thf(fact_250_finite__set__times,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [S2: set @ A,T2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ A @ T2 )
=> ( finite_finite2 @ A @ ( times_times @ ( set @ A ) @ S2 @ T2 ) ) ) ) ) ).
% finite_set_times
thf(fact_251_set__plus__elim,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ X2 @ ( plus_plus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ! [A6: A,B7: A] :
( ( X2
= ( plus_plus @ A @ A6 @ B7 ) )
=> ( ( member @ A @ A6 @ A3 )
=> ~ ( member @ A @ B7 @ B3 ) ) ) ) ) ).
% set_plus_elim
thf(fact_252_finite__set__plus,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [S2: set @ A,T2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ A @ T2 )
=> ( finite_finite2 @ A @ ( plus_plus @ ( set @ A ) @ S2 @ T2 ) ) ) ) ) ).
% finite_set_plus
thf(fact_253_set__one,axiom,
! [A: $tType] :
( ( one @ A )
=> ( ( one_one @ ( set @ A ) )
= ( insert @ A @ ( one_one @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_one
thf(fact_254_ai1,axiom,
ord_less_eq @ real @ ( aa @ i ) @ ( one_one @ real ) ).
% ai1
% Subclasses (12)
thf(subcl_Real__Vector__Spaces_Oreal__vector___HOL_Otype,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( type @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oplus,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( plus @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ominus,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( minus @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__group__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( ab_group_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Osemigroup__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( semigroup_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( comm_monoid_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__semigroup__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( ab_semigroup_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Real__Vector__Spaces_OscaleR,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( real_Vector_scaleR @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__semigroup__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( cancel_semigroup_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( cancel1352612707id_add @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__ab__semigroup__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( cancel146912293up_add @ A ) ) ).
% Type constructors (80)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 )
=> ( minus @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s1003936772cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__bits,axiom,
semiring_bits @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_1,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__mult_2,axiom,
! [A8: $tType] :
( ( ab_semigroup_mult @ A8 )
=> ( ab_semigroup_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__mult_3,axiom,
! [A8: $tType] :
( ( comm_monoid_mult @ A8 )
=> ( comm_monoid_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add_4,axiom,
! [A8: $tType] :
( ( ab_semigroup_add @ A8 )
=> ( ab_semigroup_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_5,axiom,
! [A8: $tType] :
( ( comm_monoid_add @ A8 )
=> ( comm_monoid_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__mult_6,axiom,
! [A8: $tType] :
( ( semigroup_mult @ A8 )
=> ( semigroup_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add_7,axiom,
! [A8: $tType] :
( ( semigroup_add @ A8 )
=> ( semigroup_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__mult_8,axiom,
! [A8: $tType] :
( ( monoid_mult @ A8 )
=> ( monoid_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_9,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Otimes_10,axiom,
! [A8: $tType] :
( ( times @ A8 )
=> ( times @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_11,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Oplus_12,axiom,
! [A8: $tType] :
( ( plus @ A8 )
=> ( plus @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oone_13,axiom,
! [A8: $tType] :
( ( one @ A8 )
=> ( one @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_14,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_15,axiom,
minus @ $o ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
real_V1015715145lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
real_V148923926lgebra @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_16,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V1076094709vector @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_17,axiom,
cancel146912293up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_18,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_19,axiom,
comm_s1003936772cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_20,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_OscaleR,axiom,
real_Vector_scaleR @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_21,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_22,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_23,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_24,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_25,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_26,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_27,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_28,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_29,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_30,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_31,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Otimes_32,axiom,
times @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_33,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_34,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Groups_Oone_35,axiom,
one @ real ).
thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_36,axiom,
ordere779506340up_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__mult_37,axiom,
ab_semigroup_mult @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__mult_38,axiom,
comm_monoid_mult @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_39,axiom,
ab_semigroup_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_40,axiom,
comm_monoid_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__mult_41,axiom,
semigroup_mult @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_42,axiom,
semigroup_add @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Omonoid__mult_43,axiom,
monoid_mult @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Otimes_44,axiom,
times @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ominus_45,axiom,
minus @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oplus_46,axiom,
plus @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oone_47,axiom,
one @ extended_ereal ).
% 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,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Free types (1)
thf(tfree_0,hypothesis,
real_V1076094709vector @ b ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( f
@ ( plus_plus @ b
@ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) )
= ( f
@ ( plus_plus @ b
@ ( real_V1908273582scaleR @ b @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) )
@ ( groups1340683514dd_sum @ a @ b
@ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( aa @ J ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
%------------------------------------------------------------------------------