TPTP Problem File: ITP155^2.p
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%------------------------------------------------------------------------------
% File : ITP155^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_260__6251520_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_260__6251520_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 336 ( 87 unt; 43 typ; 0 def)
% Number of atoms : 932 ( 162 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 4853 ( 92 ~; 21 |; 97 &;4183 @)
% ( 0 <=>; 460 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 118 ( 118 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 41 usr; 5 con; 0-5 aty)
% Number of variables : 1060 ( 50 ^; 949 !; 20 ?;1060 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:29.951
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (39)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[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_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[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_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[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_Oas__good__as,type,
prefer951318096ood_as:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
prefer310429814s_good:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
prefer1532642881r_than:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
prefer199794634erence:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
prefer1997167224erence:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
prefer1801827867axioms:
!>[A: $tType] : ( ( set @ A ) > ( 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_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_carrier,type,
carrier: set @ a ).
thf(sy_v_relation,type,
relation: set @ ( product_prod @ a @ a ) ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (256)
thf(fact_0_assms_I2_J,axiom,
member @ a @ y @ carrier ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
member @ a @ x @ carrier ).
% assms(1)
thf(fact_2_xy__in__eachothers__nbt,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( member @ a @ X @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) )
| ( member @ a @ Y @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ) ) ).
% xy_in_eachothers_nbt
thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062,axiom,
ord_less_eq @ nat @ ( finite_card @ a @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) ) @ ( finite_card @ a @ ( prefer1532642881r_than @ a @ y @ carrier @ relation ) ) ).
% \<open>card (no_better_than x carrier relation) \<le> card (no_better_than y carrier relation)\<close>
thf(fact_4_nbt__nest,axiom,
! [Y: a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
| ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% nbt_nest
thf(fact_5_nbt__subset__carrier,axiom,
! [X: a] :
( ( member @ a @ X @ carrier )
=> ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ carrier ) ) ).
% nbt_subset_carrier
thf(fact_6_rational__preference__axioms,axiom,
prefer1997167224erence @ a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_7__092_060open_062finite_Acarrier_092_060close_062,axiom,
finite_finite2 @ a @ carrier ).
% \<open>finite carrier\<close>
thf(fact_8_preference__axioms,axiom,
prefer199794634erence @ a @ carrier @ relation ).
% preference_axioms
thf(fact_9_trans__refl,axiom,
order_preorder_on @ a @ carrier @ relation ).
% trans_refl
thf(fact_10_nbt__subset,axiom,
! [X: a,Y: a] :
( ( finite_finite2 @ a @ carrier )
=> ( ( member @ a @ X @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
| ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ) ) ) ).
% nbt_subset
thf(fact_11_no__better__subset__pref,axiom,
! [X: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
=> ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ).
% no_better_subset_pref
thf(fact_12_no__better__thansubset__rel,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ).
% no_better_thansubset_rel
thf(fact_13_no__better__than__nonepty,axiom,
! [X: a] :
( ( carrier
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( member @ a @ X @ carrier )
=> ( ( prefer1532642881r_than @ a @ X @ carrier @ relation )
!= ( bot_bot @ ( set @ a ) ) ) ) ) ).
% no_better_than_nonepty
thf(fact_14_total,axiom,
total_on @ a @ carrier @ relation ).
% total
thf(fact_15_fnt__carrier__fnt__nbt,axiom,
! [X2: a] :
( ( member @ a @ X2 @ carrier )
=> ( finite_finite2 @ a @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) ) ) ).
% fnt_carrier_fnt_nbt
thf(fact_16_same__nbt__same__pref,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( ( member @ a @ X @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) )
& ( member @ a @ Y @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) )
= ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% same_nbt_same_pref
thf(fact_17_worse__in__no__better,axiom,
! [X: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
=> ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% worse_in_no_better
thf(fact_18_reflexivity,axiom,
refl_on @ a @ carrier @ relation ).
% reflexivity
thf(fact_19_compl,axiom,
! [X2: a] :
( ( member @ a @ X2 @ carrier )
=> ! [Xa: a] :
( ( member @ a @ Xa @ carrier )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Xa ) @ relation )
| ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X2 ) @ relation ) ) ) ) ).
% compl
thf(fact_20_completeD,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
| ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% completeD
thf(fact_21_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_22_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_23_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_24_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_25_weak__is__transitive,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member @ a @ X @ carrier )
& ( member @ a @ Y @ carrier )
& ( member @ a @ Z @ carrier ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation ) ) ) ) ).
% weak_is_transitive
thf(fact_26_strict__is__neg__transitive,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member @ a @ X @ carrier )
& ( member @ a @ Y @ carrier )
& ( member @ a @ Z @ carrier ) )
=> ( ( ( 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 @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) )
| ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% strict_is_neg_transitive
thf(fact_27_not__outside,axiom,
! [X: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
=> ( member @ a @ X @ carrier ) ) ).
% not_outside
thf(fact_28_strict__not__refl__weak,axiom,
! [X: a,Y: a] :
( ( ( member @ a @ X @ carrier )
& ( member @ a @ Y @ carrier ) )
=> ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
= ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ).
% strict_not_refl_weak
thf(fact_29_rational__preference_Oaxioms_I1_J,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% rational_preference.axioms(1)
thf(fact_30_rational__preference_Oindifferent__imp__weak__pref_I2_J,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( 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 ) ) ) ).
% rational_preference.indifferent_imp_weak_pref(2)
thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( 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 ) ) ) ).
% rational_preference.indifferent_imp_weak_pref(1)
thf(fact_32_preference__def,axiom,
! [A: $tType] :
( ( prefer199794634erence @ A )
= ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
=> ( member @ A @ X3 @ Carrier2 ) )
& ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
=> ( member @ A @ Y2 @ Carrier2 ) )
& ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_33_preference_Ointro,axiom,
! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
=> ( member @ A @ X4 @ Carrier ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
=> ( member @ A @ Y3 @ Carrier ) )
=> ( ( order_preorder_on @ A @ Carrier @ Relation )
=> ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_34_preference_Otrans__refl,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% preference.trans_refl
thf(fact_35_preference_Onot__outside,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
=> ( member @ A @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_36_preference_Oreflexivity,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( refl_on @ A @ Carrier @ Relation ) ) ).
% preference.reflexivity
thf(fact_37_preference_Oindiff__trans,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( ( 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 ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_38_rational__preference_Ocompl,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ! [X2: A] :
( ( member @ A @ X2 @ Carrier )
=> ! [Xa: A] :
( ( member @ A @ Xa @ Carrier )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa ) @ Relation )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_39_rational__preference_Ototal,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( total_on @ A @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_40_rational__preference_OcompleteD,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_41_rational__preference_Ostrict__trans,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( 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 ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_42_rational__preference_Oweak__is__transitive,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X @ Carrier )
& ( member @ A @ Y @ Carrier )
& ( member @ A @ Z @ Carrier ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_43_rational__preference_Ofnt__carrier__fnt__rel,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( finite_finite2 @ A @ Carrier )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ Relation ) ) ) ).
% rational_preference.fnt_carrier_fnt_rel
thf(fact_44_rational__preference_Ostrict__not__refl__weak,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X @ Carrier )
& ( member @ A @ Y @ Carrier ) )
=> ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
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
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_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_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_rational__preference_Ostrict__is__neg__transitive,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X @ Carrier )
& ( member @ A @ Y @ Carrier )
& ( member @ A @ Z @ Carrier ) )
=> ( ( ( 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 @ Z ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) )
| ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% rational_preference.strict_is_neg_transitive
thf(fact_50_rational__preference_Oworse__in__no__better,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
=> ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_51_rational__preference_Osame__nbt__same__pref,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( ( member @ A @ X @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) )
& ( member @ A @ Y @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.same_nbt_same_pref
thf(fact_52_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( finite_finite2 @ A @ Carrier )
=> ! [X2: A] :
( ( member @ A @ X2 @ Carrier )
=> ( finite_finite2 @ A @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% rational_preference.fnt_carrier_fnt_nbt
thf(fact_53_rational__preference_Ono__better__than__nonepty,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( Carrier
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ X @ Carrier )
=> ( ( prefer1532642881r_than @ A @ X @ Carrier @ Relation )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_54_rational__preference_Ono__better__thansubset__rel,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_55_rational__preference_Ono__better__subset__pref,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
=> ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_56_rational__preference_Onbt__subset,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( finite_finite2 @ A @ Carrier )
=> ( ( member @ A @ X @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
| ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% rational_preference.nbt_subset
thf(fact_57_rational__preference_Oxy__in__eachothers__nbt,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( member @ A @ X @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) )
| ( member @ A @ Y @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% rational_preference.xy_in_eachothers_nbt
thf(fact_58_rational__preference_Onbt__subset__carrier,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X @ Carrier )
=> ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% rational_preference.nbt_subset_carrier
thf(fact_59_rational__preference_Onbt__nest,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
| ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_60_at__lst__asgd__not__ge,axiom,
! [X: a,Y: a] :
( ( carrier
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( member @ a @ X @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ~ ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
=> ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ) ).
% at_lst_asgd_not_ge
thf(fact_61_same__at__least__as__equal,axiom,
! [Z: a,Y: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) )
=> ( ( prefer310429814s_good @ a @ Z @ carrier @ relation )
= ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% same_at_least_as_equal
thf(fact_62_pref__in__at__least__as,axiom,
! [X: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
=> ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% pref_in_at_least_as
thf(fact_63_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_64_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_65_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C2: A,D: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
= ( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% Pair_le
thf(fact_66_as__good__as__sameIff,axiom,
! [Z: a,Y: a] :
( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
= ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% as_good_as_sameIff
thf(fact_67_as__good__asIff,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
= ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ).
% as_good_asIff
thf(fact_68_fnt__carrier__fnt__rel,axiom,
( ( finite_finite2 @ a @ carrier )
=> ( finite_finite2 @ ( product_prod @ a @ a ) @ relation ) ) ).
% fnt_carrier_fnt_rel
thf(fact_69_card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% card_mono
thf(fact_70_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_71_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_72_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_73_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_74_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_75_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_76_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_77_at__least__as__goodD,axiom,
! [A: $tType,Z: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_78_at__lst__asgd__ge,axiom,
! [A: $tType,X: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_79_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_80_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_81_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_82_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_83_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_84_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_85_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_86_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_87_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_88_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_89_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A4 )
=> ( member @ A @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_90_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_91_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_92_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_93_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2 )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_94_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_95_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_96_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ? [X_1: B] : ( P @ X4 @ X_1 ) )
=> ? [F2: A > B] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 @ ( F2 @ X2 ) ) ) ) ) ).
% finite_set_choice
thf(fact_97_rational__preference_Osame__at__least__as__equal,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) )
=> ( ( prefer310429814s_good @ A @ Z @ Carrier @ Relation )
= ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.same_at_least_as_equal
thf(fact_98_rational__preference_Opref__in__at__least__as,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
=> ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_99_rational__preference_Oas__good__as__sameIff,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% rational_preference.as_good_as_sameIff
thf(fact_100_rational__preference_Oas__good__asIff,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ).
% rational_preference.as_good_asIff
thf(fact_101_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( Carrier
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ X @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ~ ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_102_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X: A,X5: A,Y: B,Y5: B] :
( ( ord_less_eq @ A @ X @ X5 )
=> ( ( ord_less_eq @ B @ Y @ Y5 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% Pair_mono
thf(fact_103_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less_eq @ A @ A2 @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_104_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less_eq @ A @ X4 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_105_bot__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( bot @ B )
& ( bot @ A ) )
=> ( ( bot_bot @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% bot_prod_def
thf(fact_106_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_107_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_108_finite__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_109_infinite__super,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_super
thf(fact_110_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_111_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_112_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_113_card__subset__eq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_114_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B5: set @ A] :
( ( finite_finite2 @ A @ B5 )
& ( ( finite_card @ A @ B5 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_115_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F3: set @ A,C3: nat] :
( ! [G2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G2 @ F3 )
=> ( ( finite_finite2 @ A @ G2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G2 ) @ C3 ) ) )
=> ( ( finite_finite2 @ A @ F3 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_116_card__seteq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_seteq
thf(fact_117_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
=> ( ( ( finite_card @ A @ T4 )
= N )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_118_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R: B > A > $o] :
( ( finite_finite2 @ A @ B3 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ? [B6: A] :
( ( member @ A @ B6 @ B3 )
& ( R @ A5 @ B6 ) ) )
=> ( ! [A1: B,A22: B,B7: A] :
( ( member @ B @ A1 @ A3 )
=> ( ( member @ B @ A22 @ A3 )
=> ( ( member @ A @ B7 @ B3 )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_119_ex__card,axiom,
! [A: $tType,N: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ A3 ) )
=> ? [S2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ A3 )
& ( ( finite_card @ A @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_120_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B8: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B8 ) )
= ( ( A2 = A6 )
& ( B2 = B8 ) ) ) ).
% old.prod.inject
thf(fact_121_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_122_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_123_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_124_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_125_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_126_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A7: A,B9: A] :
( ( ord_less_eq @ A @ B9 @ A7 )
& ( ord_less_eq @ A @ A7 @ B9 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_127_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_128_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B7: A] :
( ( ord_less_eq @ A @ A5 @ B7 )
=> ( P @ A5 @ B7 ) )
=> ( ! [A5: A,B7: A] :
( ( P @ B7 @ A5 )
=> ( P @ A5 @ B7 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_129_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_130_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_131_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_132_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_133_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_134_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A7: A,B9: A] :
( ( ord_less_eq @ A @ A7 @ B9 )
& ( ord_less_eq @ A @ B9 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_135_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_136_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_137_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_138_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_139_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_140_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_141_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_142_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_143_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_144_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_145_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_146_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_147_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G3 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_148_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_149_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_150_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_151_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_152_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_153_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_154_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B8: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B8 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B8 ) ) ) ).
% Pair_inject
thf(fact_155_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B7: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% prod_cases3
thf(fact_156_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_157_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_158_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_159_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5,G5: G4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F2 @ G5 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_160_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_161_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_162_prod__induct5,axiom,
! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_163_prod__induct6,axiom,
! [F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_164_prod__induct7,axiom,
! [G4: $tType,F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F2 @ G5 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_165_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B7: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_166_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_167_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_168_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N2: set @ nat] :
? [M3: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N2 )
=> ( ord_less_eq @ nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_169_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_170_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_171_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_172_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_173_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_174_total__onI,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( X4 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_on @ A @ A3 @ R ) ) ).
% total_onI
thf(fact_175_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_176_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_177_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% subrelI
thf(fact_178_refl__on__empty,axiom,
! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% refl_on_empty
thf(fact_179_refl__on__domain,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ A3 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( ( member @ A @ A2 @ A3 )
& ( member @ A @ B2 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_180_refl__onD2,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A3 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ A @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_181_refl__onD1,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A3 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ A @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_182_refl__onD,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ A3 @ R )
=> ( ( member @ A @ A2 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_183_total__on__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% total_on_empty
thf(fact_184_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_185_infinite__nat__iff__unbounded__le,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( member @ nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_186_finite__transitivity__chain,axiom,
! [A: $tType,A3: set @ A,R3: A > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X4: A] :
~ ( R3 @ X4 @ X4 )
=> ( ! [X4: A,Y3: A,Z3: A] :
( ( R3 @ X4 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X4 @ Z3 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A3 )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_187_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% less_by_empty
thf(fact_188_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_189_rational__preference__def,axiom,
! [A: $tType] :
( ( prefer1997167224erence @ A )
= ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
& ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% rational_preference_def
thf(fact_190_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S3: B,R3: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R3 )
=> ( ( S4 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_191_rational__preference__axioms_Ointro,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ Carrier @ Relation )
=> ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% rational_preference_axioms.intro
thf(fact_192_rational__preference__axioms__def,axiom,
! [A: $tType] :
( ( prefer1801827867axioms @ A )
= ( total_on @ A ) ) ).
% rational_preference_axioms_def
thf(fact_193_rational__preference_Oaxioms_I2_J,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% rational_preference.axioms(2)
thf(fact_194_rational__preference_Ointro,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
=> ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% rational_preference.intro
thf(fact_195_finite__indexed__bound,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,P: B > A > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ? [X_1: A] : ( P @ X4 @ X_1 ) )
=> ? [M2: A] :
! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ? [K: A] :
( ( ord_less_eq @ A @ K @ M2 )
& ( P @ X2 @ K ) ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_196_transitivity,axiom,
trans @ a @ relation ).
% transitivity
thf(fact_197_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_198_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_199_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_200_transI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R ) ) )
=> ( trans @ A @ R ) ) ).
% transI
thf(fact_201_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X3: A,Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% trans_def
thf(fact_202_preference_Otransitivity,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( trans @ A @ Relation ) ) ).
% preference.transitivity
thf(fact_203_preorder__on__def,axiom,
! [A: $tType] :
( ( order_preorder_on @ A )
= ( ^ [A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R2 )
& ( trans @ A @ R2 ) ) ) ) ).
% preorder_on_def
thf(fact_204_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_205_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_206_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_207_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_208_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_209_card_Oinfinite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_210_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_211_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_212_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_213_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_214_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
| ( ord_less_eq @ nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_215_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( ( ord_less_eq @ nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_216_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq @ nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_217_le__trans,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ J2 @ K2 )
=> ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_218_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_219_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_220_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_221_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_222_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_223_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_224_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_225_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_226_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_227_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_228_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_229_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_230_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_231_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_232_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_233_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_234_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_235_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_236_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_237_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_238_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_239_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_240_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_241_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_242_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_243_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_244_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_245_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_246_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_247_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_248_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_249_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_250_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_251_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_252_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_253_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B9: A] :
( ( ord_less @ A @ A7 @ B9 )
| ( A7 = B9 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_254_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B9: A] :
( ( ord_less_eq @ A @ A7 @ B9 )
& ( A7 != B9 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_255_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
% Type constructors (36)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( 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_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_12,axiom,
! [A8: $tType] :
( ( zero @ A8 )
=> ( zero @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_19,axiom,
bot @ $o ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_20,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order_bot @ A8 )
& ( order_bot @ A9 ) )
=> ( order_bot @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_21,axiom,
! [A8: $tType,A9: $tType] :
( ( ( preorder @ A8 )
& ( preorder @ A9 ) )
=> ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_22,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_23,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order @ A8 )
& ( order @ A9 ) )
=> ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_24,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ord @ A8 )
& ( ord @ A9 ) )
=> ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Obot_25,axiom,
! [A8: $tType,A9: $tType] :
( ( ( bot @ A8 )
& ( bot @ A9 ) )
=> ( bot @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_26,axiom,
! [A8: $tType,A9: $tType] :
( ( ( zero @ A8 )
& ( zero @ A9 ) )
=> ( zero @ ( product_prod @ A8 @ A9 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ y @ carrier @ relation ) ).
%------------------------------------------------------------------------------