TPTP Problem File: ITP156^2.p
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%------------------------------------------------------------------------------
% File : ITP156^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_474__6256716_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 : Preferences/prob_474__6256716_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 399 ( 70 unt; 58 typ; 0 def)
% Number of atoms : 1015 ( 235 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4501 ( 54 ~; 13 |; 81 &;3881 @)
% ( 0 <=>; 472 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 61 ( 61 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 56 usr; 5 con; 0-5 aty)
% Number of variables : 1126 ( 33 ^;1034 !; 7 ?;1126 :)
% ( 52 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:31:20.776
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (54)
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[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_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[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_Groups_Omonoid__add,type,
monoid_add:
!>[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_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_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_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere1818651114id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V1587248058vector:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1598680935umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique455577585es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
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_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
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_Preferences__Mirabelle__stygcjuplb_Oweak__convex__pref,type,
prefer1289595409x_pref:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V1908273582scaleR:
!>[A: $tType] : ( real > A > A ) ).
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_relation,type,
relation: set @ ( product_prod @ a @ a ) ).
thf(sy_v_u,type,
u: real ).
thf(sy_v_v,type,
v: real ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (254)
thf(fact_0_assms_I1_J,axiom,
prefer1289595409x_pref @ a @ relation ).
% assms(1)
thf(fact_1_indifferent__imp__weak__pref_I2_J,axiom,
! [X: a,Y: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ).
% indifferent_imp_weak_pref(2)
thf(fact_2_indifferent__imp__weak__pref_I1_J,axiom,
! [X: a,Y: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ).
% indifferent_imp_weak_pref(1)
thf(fact_3_indiff__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
=> ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_4_strict__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
=> ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_5_assms_I2_J,axiom,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ x @ y ) @ relation ).
% assms(2)
thf(fact_6__092_060open_062u_A_092_060noteq_062_A0_A_092_060and_062_Au_A_092_060noteq_062_A1_A_092_060longrightarrow_062_Au_A_K_092_060_094sub_062R_Ax_A_L_Av_A_K_092_060_094sub_062R_Ay_A_092_060succeq_062_Ay_092_060close_062,axiom,
( ( ( u
!= ( zero_zero @ real ) )
& ( u
!= ( one_one @ real ) ) )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( plus_plus @ a @ ( real_V1908273582scaleR @ a @ u @ x ) @ ( real_V1908273582scaleR @ a @ v @ y ) ) @ y ) @ relation ) ) ).
% \<open>u \<noteq> 0 \<and> u \<noteq> 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<succeq> y\<close>
thf(fact_7_u__0,axiom,
( ( u
= ( zero_zero @ real ) )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( plus_plus @ a @ ( real_V1908273582scaleR @ a @ u @ x ) @ ( real_V1908273582scaleR @ a @ v @ y ) ) @ y ) @ relation ) ) ).
% u_0
thf(fact_8_assms_I4_J,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ v ).
% assms(4)
thf(fact_9_assms_I5_J,axiom,
( ( plus_plus @ real @ u @ v )
= ( one_one @ real ) ) ).
% assms(5)
thf(fact_10_assms_I3_J,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ u ).
% assms(3)
thf(fact_11_scaleR__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V1076094709vector @ B )
& ( real_V1076094709vector @ A ) )
=> ! [R: real,A2: A,B2: B] :
( ( real_V1908273582scaleR @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( real_V1908273582scaleR @ A @ R @ A2 ) @ ( real_V1908273582scaleR @ B @ R @ B2 ) ) ) ) ).
% scaleR_Pair
thf(fact_12_scale__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V1076094709vector @ B )
& ( real_V1076094709vector @ A ) )
=> ! [X: real,A2: A,B2: B] :
( ( real_V1908273582scaleR @ ( product_prod @ A @ B ) @ X @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( real_V1908273582scaleR @ A @ X @ A2 ) @ ( real_V1908273582scaleR @ B @ X @ B2 ) ) ) ) ).
% scale_prod
thf(fact_13_add__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( plus @ B )
& ( plus @ A ) )
=> ! [A2: A,B2: B,C: A,D: B] :
( ( plus_plus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C @ D ) )
= ( product_Pair @ A @ B @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ B @ B2 @ D ) ) ) ) ).
% add_Pair
thf(fact_14_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_16_set__plus__intro,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [A2: A,C2: set @ A,B2: A,D2: set @ A] :
( ( member @ A @ A2 @ C2 )
=> ( ( member @ A @ B2 @ D2 )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ ( set @ A ) @ C2 @ D2 ) ) ) ) ) ).
% set_plus_intro
thf(fact_17_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_18_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_19_pth__9_I3_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,W: A,D: real,Z: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ W ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ X ) @ Z ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ C @ D ) @ X ) @ ( plus_plus @ A @ W @ Z ) ) ) ) ).
% pth_9(3)
thf(fact_20_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_21_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_22_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_23_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_24_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_25_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_26_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_27_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_28_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_29_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_30_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_31_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_32_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_33_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_34_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_35_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_36_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_37_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_38_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_39_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_40_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_41_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_42_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_47_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_48_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_49_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_50_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_51_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_52_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_53_zero__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( zero @ B )
& ( zero @ A ) )
=> ( ( zero_zero @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% zero_prod_def
thf(fact_54_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_55_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_56_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C3: A] :
( B4
= ( plus_plus @ A @ A5 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_57_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_right_mono
thf(fact_58_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_59_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_60_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_61_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_62_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_63_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_64_pth__4_I1_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X: A] :
( ( real_V1908273582scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% pth_4(1)
thf(fact_65_convex__ge__imp__conved,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [Pr: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Pr )
=> ! [Alpha: real,Beta: real] :
( ( ( ( plus_plus @ real @ Alpha @ Beta )
= ( one_one @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Alpha )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Beta ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ Alpha @ X4 ) @ ( real_V1908273582scaleR @ A @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
=> ( prefer1289595409x_pref @ A @ Pr ) ) ) ).
% convex_ge_imp_conved
thf(fact_66_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_67_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_68_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_69_pth__d,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X: A] :
( ( plus_plus @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% pth_d
thf(fact_70_pth__7_I1_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ X )
= X ) ) ).
% pth_7(1)
thf(fact_71_pth__1,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X: A] :
( X
= ( real_V1908273582scaleR @ A @ ( one_one @ real ) @ X ) ) ) ).
% pth_1
thf(fact_72_pth__4_I2_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real] :
( ( real_V1908273582scaleR @ A @ C @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% pth_4(2)
thf(fact_73_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_74_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_75_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_76_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_77_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_78_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_79_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_80_set__plus__elim,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ A4 @ B5 ) )
=> ~ ! [A6: A,B6: A] :
( ( X
= ( plus_plus @ A @ A6 @ B6 ) )
=> ( ( member @ A @ A6 @ A4 )
=> ~ ( member @ A @ B6 @ B5 ) ) ) ) ) ).
% set_plus_elim
thf(fact_81_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_82_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_83_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_84_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_85_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_86_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B6: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_87_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D3: $tType,C5: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A6: A,B6: B,C4: C5,D4: D3,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B6 @ ( product_Pair @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_88_prod__induct6,axiom,
! [F: $tType,E: $tType,D3: $tType,C5: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) )] :
( ! [A6: A,B6: B,C4: C5,D4: D3,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) @ B6 @ ( product_Pair @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F ) @ D4 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_89_prod__induct5,axiom,
! [E: $tType,D3: $tType,C5: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) )] :
( ! [A6: A,B6: B,C4: C5,D4: D3,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) @ B6 @ ( product_Pair @ C5 @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_90_prod__induct4,axiom,
! [D3: $tType,C5: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ D3 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ D3 ) )] :
( ! [A6: A,B6: B,C4: C5,D4: D3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ D3 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ D3 ) @ B6 @ ( product_Pair @ C5 @ D3 @ C4 @ D4 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_91_prod__induct3,axiom,
! [C5: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C5 ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C5 )] :
( ! [A6: A,B6: B,C4: C5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C5 ) @ A6 @ ( product_Pair @ B @ C5 @ B6 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_92_prod__cases7,axiom,
! [A: $tType,B: $tType,C5: $tType,D3: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A6: A,B6: B,C4: C5,D4: D3,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B6 @ ( product_Pair @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_93_prod__cases6,axiom,
! [A: $tType,B: $tType,C5: $tType,D3: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A6: A,B6: B,C4: C5,D4: D3,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) ) @ B6 @ ( product_Pair @ C5 @ ( product_prod @ D3 @ ( product_prod @ E @ F ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F ) @ D4 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_94_prod__cases5,axiom,
! [A: $tType,B: $tType,C5: $tType,D3: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) )] :
~ ! [A6: A,B6: B,C4: C5,D4: D3,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ ( product_prod @ D3 @ E ) ) @ B6 @ ( product_Pair @ C5 @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_95_prod__cases4,axiom,
! [A: $tType,B: $tType,C5: $tType,D3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C5 @ D3 ) )] :
~ ! [A6: A,B6: B,C4: C5,D4: D3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C5 @ D3 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C5 @ D3 ) @ B6 @ ( product_Pair @ C5 @ D3 @ C4 @ D4 ) ) ) ) ).
% prod_cases4
thf(fact_96_prod__cases3,axiom,
! [A: $tType,B: $tType,C5: $tType,Y: product_prod @ A @ ( product_prod @ B @ C5 )] :
~ ! [A6: A,B6: B,C4: C5] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C5 ) @ A6 @ ( product_Pair @ B @ C5 @ B6 @ C4 ) ) ) ).
% prod_cases3
thf(fact_97_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_98_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_99_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X4: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_100_pth__a,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X: A,Y: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( zero_zero @ real ) @ X ) @ Y )
= Y ) ) ).
% pth_a
thf(fact_101_pth__8,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,D: real] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( real_V1908273582scaleR @ A @ D @ X ) )
= ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ C @ D ) @ X ) ) ) ).
% pth_8
thf(fact_102_pth__6,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,Y: A] :
( ( real_V1908273582scaleR @ A @ C @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( real_V1908273582scaleR @ A @ C @ Y ) ) ) ) ).
% pth_6
thf(fact_103_pth__c_I1_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,D: real,Y: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( real_V1908273582scaleR @ A @ D @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ ( real_V1908273582scaleR @ A @ C @ X ) ) ) ) ).
% pth_c(1)
thf(fact_104_pth__c_I2_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,Z: A,D: real,Y: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ Z ) @ ( real_V1908273582scaleR @ A @ D @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ Z ) ) ) ) ).
% pth_c(2)
thf(fact_105_pth__c_I3_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,D: real,Y: A,Z: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ Z ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ Z ) ) ) ) ).
% pth_c(3)
thf(fact_106_pth__c_I4_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,W: A,D: real,Y: A,Z: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ W ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ Z ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ W ) @ Z ) ) ) ) ).
% pth_c(4)
thf(fact_107_pth__b_I1_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,D: real,Y: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( real_V1908273582scaleR @ A @ D @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( real_V1908273582scaleR @ A @ D @ Y ) ) ) ) ).
% pth_b(1)
thf(fact_108_pth__b_I2_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,Z: A,D: real,Y: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ Z ) @ ( real_V1908273582scaleR @ A @ D @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( plus_plus @ A @ Z @ ( real_V1908273582scaleR @ A @ D @ Y ) ) ) ) ) ).
% pth_b(2)
thf(fact_109_pth__b_I3_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,D: real,Y: A,Z: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ Z ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ Z ) ) ) ) ).
% pth_b(3)
thf(fact_110_pth__b_I4_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,W: A,D: real,Y: A,Z: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ W ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ Z ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( plus_plus @ A @ W @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ Y ) @ Z ) ) ) ) ) ).
% pth_b(4)
thf(fact_111_pth__9_I1_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,Z: A,D: real] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ Z ) @ ( real_V1908273582scaleR @ A @ D @ X ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ C @ D ) @ X ) @ Z ) ) ) ).
% pth_9(1)
thf(fact_112_pth__9_I2_J,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C: real,X: A,D: real,Z: A] :
( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ C @ X ) @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ D @ X ) @ Z ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ C @ D ) @ X ) @ Z ) ) ) ).
% pth_9(2)
thf(fact_113_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_114_scale__zero__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X: A] :
( ( real_V1908273582scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scale_zero_left
thf(fact_115_scale__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X: A] :
( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scale_eq_0_iff
thf(fact_116_scaleR__one,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X: A] :
( ( real_V1908273582scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_117_scale__cancel__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ A2 @ Y ) )
= ( ( X = Y )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scale_cancel_left
thf(fact_118_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [X: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_119_scale__cancel__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X: A,B2: real] :
( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scale_cancel_right
thf(fact_120_scale__zero__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real] :
( ( real_V1908273582scaleR @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scale_zero_right
thf(fact_121_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_122_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V1908273582scaleR @ A @ A2 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_123_set__plus__mono2,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [C2: set @ A,D2: set @ A,E3: set @ A,F3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D2 )
=> ( ( ord_less_eq @ ( set @ A ) @ E3 @ F3 )
=> ( ord_less_eq @ ( set @ A ) @ ( plus_plus @ ( set @ A ) @ C2 @ E3 ) @ ( plus_plus @ ( set @ A ) @ D2 @ F3 ) ) ) ) ) ).
% set_plus_mono2
thf(fact_124_set__plus__mono2__b,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [C2: set @ A,D2: set @ A,E3: set @ A,F3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D2 )
=> ( ( ord_less_eq @ ( set @ A ) @ E3 @ F3 )
=> ( ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ C2 @ E3 ) )
=> ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ D2 @ F3 ) ) ) ) ) ) ).
% set_plus_mono2_b
thf(fact_125_set__zero__plus2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ A,B5: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( plus_plus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% set_zero_plus2
thf(fact_126_scale__left__commute,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X ) )
= ( real_V1908273582scaleR @ A @ B2 @ ( real_V1908273582scaleR @ A @ A2 @ X ) ) ) ) ).
% scale_left_commute
thf(fact_127_scale__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X: A,A2: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ) ).
% scale_right_imp_eq
thf(fact_128_scaleR__add__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_add_right
thf(fact_129_scale__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ A2 @ Y ) )
=> ( X = Y ) ) ) ) ).
% scale_left_imp_eq
thf(fact_130_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [X: real,Y: real,Xa: A] :
( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ X @ Y ) @ Xa )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ X @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_131_scaleR__add__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_add_left
thf(fact_132_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_133_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [B2: real,A2: real,C: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ C ) @ ( real_V1908273582scaleR @ A @ B2 @ C ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_134_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_135_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [B2: A,A2: A,C: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ C @ A2 ) @ ( real_V1908273582scaleR @ A @ C @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_136_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: real,C: A,D: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ C ) @ ( real_V1908273582scaleR @ A @ B2 @ D ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_137_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_138_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V1908273582scaleR @ A @ A2 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_139_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_140_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_141_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V1908273582scaleR @ A @ A2 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_142_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C: A,D: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C @ D ) )
= ( ( ord_less_eq @ A @ A2 @ C )
& ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% Pair_le
thf(fact_143_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_144_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_145_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_146_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_147_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X: A,X5: A,Y: B,Y4: B] :
( ( ord_less_eq @ A @ X @ X5 )
=> ( ( ord_less_eq @ B @ Y @ Y4 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) ) ) ) ) ).
% Pair_mono
thf(fact_148_weak__convexI,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [Pr: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A,Alpha: real,Beta: real] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Pr )
=> ( ( ( plus_plus @ real @ Alpha @ Beta )
= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Alpha )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Beta )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ Alpha @ X4 ) @ ( real_V1908273582scaleR @ A @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
=> ( prefer1289595409x_pref @ A @ Pr ) ) ) ).
% weak_convexI
thf(fact_149_subsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ A @ X4 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% subsetI
thf(fact_150_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_151_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_152_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_153_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_154_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_155_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_156_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_157_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_158_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_159_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_160_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_161_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_162_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_163_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_164_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_165_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_166_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_167_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_168_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_169_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_170_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_171_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_172_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_173_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_174_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_175_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_176_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_177_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_178_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_179_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_180_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_181_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_182_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_183_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_184_add__mono1,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_185_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_186_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_187_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing
thf(fact_188_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_189_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_190_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_191_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_192_zero__less__two,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_193_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [C: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ( ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ C @ A2 ) @ ( real_V1908273582scaleR @ A @ C @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_194_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [C: real,A2: A,B2: A] :
( ( ord_less @ real @ C @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ C @ A2 ) @ ( real_V1908273582scaleR @ A @ C @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_195_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [C: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ C @ A2 ) @ ( real_V1908273582scaleR @ A @ C @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_196_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_197_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_198_subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_199_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_200_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_201_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A7 )
=> ( member @ A @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_202_equalityD2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_203_equalityD1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_204_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A7 )
=> ( member @ A @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_205_equalityE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_206_subsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B5 ) ) ) ).
% subsetD
thf(fact_207_in__mono,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_208_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V1908273582scaleR @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_209_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V1587248058vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V1908273582scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_210_weak__convexD,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [Pr: set @ ( product_prod @ A @ A ),X: A,Y: A,U: real,V: real] :
( ( prefer1289595409x_pref @ A @ Pr )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ U )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ V )
=> ( ( ( plus_plus @ real @ U @ V )
= ( one_one @ real ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ X ) @ ( real_V1908273582scaleR @ A @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ) ).
% weak_convexD
thf(fact_211_weak__convex__pref__def,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( ( prefer1289595409x_pref @ A )
= ( ^ [Pr2: set @ ( product_prod @ A @ A )] :
! [X3: A,Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y6 ) @ Pr2 )
=> ! [Alpha2: real,Beta2: real] :
( ( ( ( plus_plus @ real @ Alpha2 @ Beta2 )
= ( one_one @ real ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Alpha2 )
& ( ord_less @ real @ ( zero_zero @ real ) @ Beta2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ Alpha2 @ X3 ) @ ( real_V1908273582scaleR @ A @ Beta2 @ Y6 ) ) @ Y6 ) @ Pr2 ) ) ) ) ) ) ).
% weak_convex_pref_def
thf(fact_212_discrete,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_213_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_214_approachable__lt__le2,axiom,
! [A: $tType,Q: A > $o,F4: A > real,P: A > $o] :
( ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( Q @ X3 )
=> ( ( ord_less @ real @ ( F4 @ X3 ) @ D5 )
=> ( P @ X3 ) ) ) ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ord_less_eq @ real @ ( F4 @ X3 ) @ D5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) ) ) ) ) ).
% approachable_lt_le2
thf(fact_215_psubsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_216_psubsetE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_217_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_218_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_219_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C2 )
=> ( ord_less @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_220_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_221_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C2 )
=> ( ord_less @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_222_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_223_less__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( ord_less @ ( product_prod @ A @ B ) )
= ( ^ [X3: product_prod @ A @ B,Y6: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X3 @ Y6 )
& ~ ( ord_less_eq @ ( product_prod @ A @ B ) @ Y6 @ X3 ) ) ) ) ) ).
% less_prod_def
thf(fact_224_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_225_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [X_1: A] : ( ord_less @ A @ X6 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_226_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [Q2: A,R: A] :
( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_227_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_228_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C2 )
=> ( ord_less @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% psubset_trans
thf(fact_229_psubsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_230_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_231_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_232_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_233_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_234_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_235_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_236_add__scaleR__degen,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [U: real,B2: A,V: real,A2: A] :
( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ B2 ) @ ( real_V1908273582scaleR @ A @ V @ A2 ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ A2 ) @ ( real_V1908273582scaleR @ A @ V @ B2 ) ) )
=> ( ( U != V )
=> ( A2 = B2 ) ) ) ) ).
% add_scaleR_degen
thf(fact_237_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F4: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
=> ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_238_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F4: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
=> ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_239_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F4: A > B,G3: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G3 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F4 @ G3 ) ) ) ).
% le_funI
thf(fact_240_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F5: A > B,G4: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F5 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_241_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F4: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F4 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F4 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst2,axiom,
! [A: $tType,C5: $tType] :
( ( ( order @ C5 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F4: A > C5,C: C5] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C5 @ ( F4 @ B2 ) @ C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C5 @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) )
=> ( ord_less_eq @ C5 @ ( F4 @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_243_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F4: B > A,B2: B,C: B] :
( ( A2
= ( F4 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F4 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_244_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F4: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F4 @ B2 )
= C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F4 @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X3: A,Y6: A] :
( ( ord_less_eq @ A @ X3 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_246_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_247_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_248_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_249_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_250_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_251_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_252_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_253_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
% Subclasses (10)
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_Ozero,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( zero @ 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_Omonoid__add,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( monoid_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___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 ) ) ).
% Type constructors (75)
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere890947078_space @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
ordere890947078_space @ real ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
! [A8: $tType] :
( ( ab_semigroup_add @ A8 )
=> ( ab_semigroup_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
! [A8: $tType] :
( ( comm_monoid_add @ A8 )
=> ( comm_monoid_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
! [A8: $tType] :
( ( semigroup_add @ A8 )
=> ( semigroup_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__add,axiom,
! [A8: $tType] :
( ( monoid_add @ A8 )
=> ( monoid_add @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A8: $tType] :
( ( zero @ A8 )
=> ( zero @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oplus,axiom,
! [A8: $tType] :
( ( plus @ A8 )
=> ( plus @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oone,axiom,
! [A8: $tType] :
( ( one @ A8 )
=> ( one @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
real_V1587248058vector @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere1818651114id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V1076094709vector @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_8,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_9,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_10,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_11,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_12,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_13,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_14,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_15,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_16,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_17,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Groups_Oone_18,axiom,
one @ real ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__semigroup__monoid__add__imp__le_19,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere516151231imp_le @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oordered__real__vector_20,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( real_V1587248058vector @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ostrict__ordered__ab__semigroup__add_21,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( strict2144017051up_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__cancel__ab__semigroup__add_22,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere223160158up_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__semigroup__add__imp__le_23,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere236663937imp_le @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_24,axiom,
! [A8: $tType,A9: $tType] :
( ( ( real_V55928688vector @ A8 )
& ( real_V55928688vector @ A9 ) )
=> ( real_V55928688vector @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ostrict__ordered__comm__monoid__add_25,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( strict797366125id_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__cancel__comm__monoid__add_26,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere1818651114id_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__semigroup__add_27,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere779506340up_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__vector_28,axiom,
! [A8: $tType,A9: $tType] :
( ( ( real_V1076094709vector @ A8 )
& ( real_V1076094709vector @ A9 ) )
=> ( real_V1076094709vector @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__comm__monoid__add_29,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ordere890947078_space @ A8 )
& ( ordere890947078_space @ A9 ) )
=> ( ordere216010020id_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ocancel__comm__monoid__add_30,axiom,
! [A8: $tType,A9: $tType] :
( ( ( cancel1352612707id_add @ A8 )
& ( cancel1352612707id_add @ A9 ) )
=> ( cancel1352612707id_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ocancel__semigroup__add_31,axiom,
! [A8: $tType,A9: $tType] :
( ( ( cancel_semigroup_add @ A8 )
& ( cancel_semigroup_add @ A9 ) )
=> ( cancel_semigroup_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oab__semigroup__add_32,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ab_semigroup_add @ A8 )
& ( ab_semigroup_add @ A9 ) )
=> ( ab_semigroup_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ocomm__monoid__add_33,axiom,
! [A8: $tType,A9: $tType] :
( ( ( comm_monoid_add @ A8 )
& ( comm_monoid_add @ A9 ) )
=> ( comm_monoid_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Osemigroup__add_34,axiom,
! [A8: $tType,A9: $tType] :
( ( ( semigroup_add @ A8 )
& ( semigroup_add @ A9 ) )
=> ( semigroup_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_35,axiom,
! [A8: $tType,A9: $tType] :
( ( ( preorder @ A8 )
& ( preorder @ A9 ) )
=> ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Omonoid__add_36,axiom,
! [A8: $tType,A9: $tType] :
( ( ( monoid_add @ A8 )
& ( monoid_add @ A9 ) )
=> ( monoid_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_37,axiom,
! [A8: $tType,A9: $tType] :
( ( ( group_add @ A8 )
& ( group_add @ A9 ) )
=> ( group_add @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_38,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order @ A8 )
& ( order @ A9 ) )
=> ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_39,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ord @ A8 )
& ( ord @ A9 ) )
=> ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_40,axiom,
! [A8: $tType,A9: $tType] :
( ( ( zero @ A8 )
& ( zero @ A9 ) )
=> ( zero @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oplus_41,axiom,
! [A8: $tType,A9: $tType] :
( ( ( plus @ A8 )
& ( plus @ A9 ) )
=> ( plus @ ( product_prod @ A8 @ A9 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
real_V1076094709vector @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( plus_plus @ a @ ( real_V1908273582scaleR @ a @ u @ x ) @ ( real_V1908273582scaleR @ a @ v @ y ) ) @ y ) @ relation ).
%------------------------------------------------------------------------------