TPTP Problem File: ITP153^2.p
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%------------------------------------------------------------------------------
% File : ITP153^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_180__6250282_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_180__6250282_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 324 ( 113 unt; 46 typ; 0 def)
% Number of atoms : 777 ( 217 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4707 ( 88 ~; 15 |; 79 &;4193 @)
% ( 0 <=>; 332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 124 ( 124 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 4 con; 0-5 aty)
% Number of variables : 1093 ( 47 ^; 985 !; 11 ?;1093 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:28:41.875
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (43)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_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_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_1409979114der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
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_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_ORange,type,
range:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
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_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
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_y,type,
y: a ).
thf(sy_v_z,type,
z: a ).
% Relevant facts (256)
thf(fact_0_assms,axiom,
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ y @ z ) @ relation ) ) ).
% assms
thf(fact_1_rational__preference__axioms,axiom,
prefer1997167224erence @ a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_2_preference__axioms,axiom,
prefer199794634erence @ a @ carrier @ relation ).
% preference_axioms
thf(fact_3_trans__refl,axiom,
order_preorder_on @ a @ carrier @ relation ).
% trans_refl
thf(fact_4_total,axiom,
total_on @ a @ carrier @ relation ).
% total
thf(fact_5_reflexivity,axiom,
refl_on @ a @ carrier @ relation ).
% reflexivity
thf(fact_6_compl,axiom,
! [X: a] :
( ( member @ a @ X @ carrier )
=> ! [Xa: a] :
( ( member @ a @ Xa @ carrier )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Xa ) @ relation )
| ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X ) @ relation ) ) ) ) ).
% compl
thf(fact_7_completeD,axiom,
! [X2: a,Y: a] :
( ( member @ a @ X2 @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( X2 != Y )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
| ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) ) ) ) ) ).
% completeD
thf(fact_8_not__outside,axiom,
! [X2: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
=> ( member @ a @ X2 @ carrier ) ) ).
% not_outside
thf(fact_9_strict__is__neg__transitive,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member @ a @ X2 @ carrier )
& ( member @ a @ Y @ carrier )
& ( member @ a @ Z @ carrier ) )
=> ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
=> ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ 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_10_weak__is__transitive,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member @ a @ X2 @ carrier )
& ( member @ a @ Y @ carrier )
& ( member @ a @ Z @ carrier ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation ) ) ) ) ).
% weak_is_transitive
thf(fact_11_strict__not__refl__weak,axiom,
! [X2: a,Y: a] :
( ( ( member @ a @ X2 @ carrier )
& ( member @ a @ Y @ carrier ) )
=> ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
= ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) ) ) ) ).
% strict_not_refl_weak
thf(fact_12_indiff__trans,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ 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 @ X2 @ Z ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_13_strict__trans,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ 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 @ X2 @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_14_pref__in__at__least__as,axiom,
! [X2: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
=> ( member @ a @ X2 @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% pref_in_at_least_as
thf(fact_15_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_16_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_17_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_18_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_19_preference_Onot__outside,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
=> ( member @ A @ X2 @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_20_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_21_preference_Oindiff__trans,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ 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 @ X2 @ Z ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_22_rational__preference_Ocompl,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ! [X: A] :
( ( member @ A @ X @ Carrier )
=> ! [Xa: A] :
( ( member @ A @ Xa @ Carrier )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa ) @ Relation )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_23_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_24_rational__preference_OcompleteD,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X2 @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( X2 != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_25_rational__preference_Ostrict__trans,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ 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 @ X2 @ Z ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_26_rational__preference_Oweak__is__transitive,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X2 @ Carrier )
& ( member @ A @ Y @ Carrier )
& ( member @ A @ Z @ Carrier ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_27_rational__preference_Ostrict__not__refl__weak,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X2 @ Carrier )
& ( member @ A @ Y @ Carrier ) )
=> ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_28_rational__preference_Ostrict__is__neg__transitive,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X2 @ Carrier )
& ( member @ A @ Y @ Carrier )
& ( member @ A @ Z @ Carrier ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ 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_29_worse__in__no__better,axiom,
! [X2: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
=> ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% worse_in_no_better
thf(fact_30_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_31_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_32_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_33_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_34_total__onI,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( 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 @ A4 @ R ) ) ).
% total_onI
thf(fact_35_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A5 )
=> ( ( 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_36_refl__onD,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ A @ A2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_37_refl__onD1,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ( member @ A @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_38_refl__onD2,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ( member @ A @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_39_refl__on__domain,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( ( member @ A @ A2 @ A4 )
& ( member @ A @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_40_at__least__as__goodD,axiom,
! [A: $tType,Z: A,Y: A,B4: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B4 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_41_at__lst__asgd__ge,axiom,
! [A: $tType,X2: A,Y: A,B4: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ B4 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_42_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_43_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_44_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
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,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% 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,F3: A > B,G3: A > B] :
( ! [X4: A] :
( ( F3 @ X4 )
= ( G3 @ X4 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_50_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B5: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_51_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_52_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_53_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_54_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_55_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B5: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_56_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_57_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_58_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_59_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_60_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_61_rational__preference_Oworse__in__no__better,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
=> ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_62_rational__preference_Opref__in__at__least__as,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
=> ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_63_at__lst__asgd__not__ge,axiom,
! [X2: a,Y: a] :
( ( carrier
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( member @ a @ X2 @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ~ ( member @ a @ X2 @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
=> ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation ) ) ) ) ) ).
% at_lst_asgd_not_ge
thf(fact_64_no__better__subset__pref,axiom,
! [X2: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
=> ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) ) ) ).
% no_better_subset_pref
thf(fact_65_no__better__thansubset__rel,axiom,
! [X2: a,Y: a] :
( ( member @ a @ X2 @ carrier )
=> ( ( member @ a @ Y @ carrier )
=> ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) )
=> ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation ) ) ) ) ).
% no_better_thansubset_rel
thf(fact_66_no__better__than__nonepty,axiom,
! [X2: a] :
( ( carrier
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( member @ a @ X2 @ carrier )
=> ( ( prefer1532642881r_than @ a @ X2 @ carrier @ relation )
!= ( bot_bot @ ( set @ a ) ) ) ) ) ).
% no_better_than_nonepty
thf(fact_67_nbt__nest,axiom,
! [Y: a,X2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) )
| ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% nbt_nest
thf(fact_68_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_69_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_70_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_71_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_72_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_73_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_74_rational__preference_Onbt__nest,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X2: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) )
| ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_75_rational__preference_Ono__better__than__nonepty,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( Carrier
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ X2 @ Carrier )
=> ( ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_76_rational__preference__axioms__def,axiom,
! [A: $tType] :
( ( prefer1801827867axioms @ A )
= ( total_on @ A ) ) ).
% rational_preference_axioms_def
thf(fact_77_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_78_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_79_rational__preference_Ono__better__subset__pref,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
=> ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_80_rational__preference_Ono__better__thansubset__rel,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( member @ A @ X2 @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_81_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( Carrier
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ X2 @ Carrier )
=> ( ( member @ A @ Y @ Carrier )
=> ( ~ ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_82_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_83_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_84_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C3: A,D3: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% Pair_le
thf(fact_85_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ A @ X4 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_86_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_87_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_88_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_89_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_90_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_91_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: 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 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_92_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_93_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_94_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_95_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_96_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_97_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2 )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_98_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_99_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_100_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_101_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_102_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_103_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_104_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_105_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_106_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_107_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_108_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X2: A,X5: A,Y: B,Y5: B] :
( ( ord_less_eq @ A @ X2 @ X5 )
=> ( ( ord_less_eq @ B @ Y @ Y5 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% Pair_mono
thf(fact_109_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_110_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_111_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_112_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_113_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_114_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_115_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_116_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_117_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_118_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% le_funD
thf(fact_119_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% le_funE
thf(fact_120_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_121_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_122_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_124_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_125_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_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_127_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_128_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_129_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_130_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_131_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_132_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_133_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_134_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
& ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_135_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_136_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_137_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_138_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_139_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_140_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_141_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_142_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A7: A,B7: A] :
( ( ord_less_eq @ A @ B7 @ A7 )
& ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_143_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_144_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_145_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_146_less__by__empty,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
( ( A4
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B4 ) ) ).
% less_by_empty
thf(fact_147_transitivity,axiom,
trans @ a @ relation ).
% transitivity
thf(fact_148_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_149_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_150_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_151_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_152_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_153_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_154_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_155_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_156_preorder__on__def,axiom,
! [A: $tType] :
( ( order_preorder_on @ A )
= ( ^ [A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R2 )
& ( trans @ A @ R2 ) ) ) ) ).
% preorder_on_def
thf(fact_157_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_158_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_159_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_160_Id__onI,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A4 ) ) ) ).
% Id_onI
thf(fact_161_Id__on__iff,axiom,
! [A: $tType,X2: A,Y: A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( id_on @ A @ A4 ) )
= ( ( X2 = Y )
& ( member @ A @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_162_Id__on__eqI,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( A2 = B2 )
=> ( ( member @ A @ A2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_163_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A4 ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( C3
!= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_164_trans__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( trans @ A @ ( id_on @ A @ A4 ) ) ).
% trans_Id_on
thf(fact_165_refl__on__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( refl_on @ A @ A4 @ ( id_on @ A @ A4 ) ) ).
% refl_on_Id_on
thf(fact_166_under__incr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A2 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% under_incr
thf(fact_167_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_168_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_169_fst__bot,axiom,
! [B: $tType,A: $tType] :
( ( ( bot @ A )
& ( bot @ B ) )
=> ( ( product_fst @ A @ B @ ( bot_bot @ ( product_prod @ A @ B ) ) )
= ( bot_bot @ A ) ) ) ).
% fst_bot
thf(fact_170_snd__bot,axiom,
! [B: $tType,A: $tType] :
( ( ( bot @ A )
& ( bot @ B ) )
=> ( ( product_snd @ B @ A @ ( bot_bot @ ( product_prod @ B @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% snd_bot
thf(fact_171_trans__singleton,axiom,
! [A: $tType,A2: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_172_insert__absorb2,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
= ( insert @ A @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_173_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_174_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_175_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_176_insert__subset,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A4 ) @ B4 )
= ( ( member @ A @ X2 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_177_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_178_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_179_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_180_insert__subsetI,axiom,
! [A: $tType,X2: A,A4: set @ A,X6: set @ A] :
( ( member @ A @ X2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_181_less__eq__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( ord_less_eq @ ( product_prod @ A @ B ) )
= ( ^ [X3: product_prod @ A @ B,Y2: product_prod @ A @ B] :
( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y2 ) )
& ( ord_less_eq @ B @ ( product_snd @ A @ B @ X3 ) @ ( product_snd @ A @ B @ Y2 ) ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_182_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_183_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_184_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_185_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_186_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_187_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_188_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_189_subset__insert,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_190_insert__mono,axiom,
! [A: $tType,C4: set @ A,D4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% insert_mono
thf(fact_191_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B8: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B8 ) )
& ~ ( member @ A @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_192_insert__commute,axiom,
! [A: $tType,X2: A,Y: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_193_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B2 @ B4 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A4 = B4 ) )
& ( ( A2 != B2 )
=> ? [C5: set @ A] :
( ( A4
= ( insert @ A @ B2 @ C5 ) )
& ~ ( member @ A @ B2 @ C5 )
& ( B4
= ( insert @ A @ A2 @ C5 ) )
& ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_194_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_195_insert__ident,axiom,
! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A4 )
=> ( ~ ( member @ A @ X2 @ B4 )
=> ( ( ( insert @ A @ X2 @ A4 )
= ( insert @ A @ X2 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_196_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( member @ A @ X2 @ A4 )
=> ~ ! [B8: set @ A] :
( ( A4
= ( insert @ A @ X2 @ B8 ) )
=> ( member @ A @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_197_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_198_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_199_insertE,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_200_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y4: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y4 = Z2 )
= ( ^ [S3: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S3 )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S3 )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_201_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_202_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_203_surjective__pairing,axiom,
! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
( T3
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% surjective_pairing
thf(fact_204_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_205_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_206_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_207_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_208_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_209_refl__on__singleton,axiom,
! [A: $tType,X2: A] : ( refl_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_210_total__on__singleton,axiom,
! [A: $tType,X2: A] : ( total_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_211_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_212_subset__singleton__iff,axiom,
! [A: $tType,X6: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
| ( X6
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_213_snd__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
=> ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% snd_mono
thf(fact_214_fst__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
=> ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_mono
thf(fact_215_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X2 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_216_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_217_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
( ( P @ Y @ X2 )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_218_the__elem__eq,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ).
% the_elem_eq
thf(fact_219_is__singletonI,axiom,
! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_220_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_221_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_222_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X4: A] :
( A4
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_223_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X3: A] :
( A5
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_224_linear__order__on__singleton,axiom,
! [A: $tType,X2: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_225_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z ) ) ).
% sndI
thf(fact_226_lnear__order__on__empty,axiom,
! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% lnear_order_on_empty
thf(fact_227_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y ) ) ).
% fstI
thf(fact_228_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A7: B] :
( P2
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_229_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B7: B] :
( P2
= ( product_Pair @ A @ B @ A2 @ B7 ) ) ) ) ).
% eq_fst_iff
thf(fact_230_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_231_Range__insert,axiom,
! [A: $tType,B: $tType,A2: B,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( range @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B2 ) @ R ) )
= ( insert @ A @ B2 @ ( range @ B @ A @ R ) ) ) ).
% Range_insert
thf(fact_232_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_233_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% swap_simp
thf(fact_234_Range__Id__on,axiom,
! [A: $tType,A4: set @ A] :
( ( range @ A @ A @ ( id_on @ A @ A4 ) )
= A4 ) ).
% Range_Id_on
thf(fact_235_Range__empty,axiom,
! [B: $tType,A: $tType] :
( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Range_empty
thf(fact_236_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_237_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
thf(fact_238_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B2 @ ( range @ B @ A @ R ) )
=> ~ ! [A6: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_239_Range__iff,axiom,
! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A2 @ ( range @ B @ A @ R ) )
= ( ? [Y2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y2 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_240_Range_Ocases,axiom,
! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
=> ~ ! [A6: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_241_Range_Osimps,axiom,
! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
= ( ? [A7: A,B7: B] :
( ( A2 = B7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_242_Range_Ointros,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
=> ( member @ B @ B2 @ ( range @ A @ B @ R ) ) ) ).
% Range.intros
thf(fact_243_Range_Oinducts,axiom,
! [A: $tType,B: $tType,X2: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
( ( member @ B @ X2 @ ( range @ A @ B @ R ) )
=> ( ! [A6: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R )
=> ( P @ B5 ) )
=> ( P @ X2 ) ) ) ).
% Range.inducts
thf(fact_244_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
( ( ( range @ B @ A @ R )
= ( bot_bot @ ( set @ A ) ) )
= ( R
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Range_empty_iff
thf(fact_245_Range__mono,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
=> ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R ) @ ( range @ A @ B @ S ) ) ) ).
% Range_mono
thf(fact_246_Domain__insert,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R ) )
= ( insert @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% Domain_insert
thf(fact_247_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_248_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A4 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_249_Compl__iff,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% Compl_iff
thf(fact_250_ComplI,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ~ ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_251_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_252_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_253_Domain__Id__on,axiom,
! [A: $tType,A4: set @ A] :
( ( domain @ A @ A @ ( id_on @ A @ A4 ) )
= A4 ) ).
% Domain_Id_on
thf(fact_254_Domain__empty,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Domain_empty
thf(fact_255_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
% Type constructors (21)
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___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_Set_Oset___Orderings_Oorder__bot_1,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A8: $tType] : ( preorder @ ( 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___Orderings_Obot_5,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
bot @ $o ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_11,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order_bot @ A8 )
& ( order_bot @ A9 ) )
=> ( order_bot @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_12,axiom,
! [A8: $tType,A9: $tType] :
( ( ( preorder @ A8 )
& ( preorder @ A9 ) )
=> ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_13,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order @ A8 )
& ( order @ A9 ) )
=> ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_14,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ord @ A8 )
& ( ord @ A9 ) )
=> ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Obot_15,axiom,
! [A8: $tType,A9: $tType] :
( ( ( bot @ A8 )
& ( bot @ A9 ) )
=> ( bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( member @ a @ z @ carrier )
& ( member @ a @ y @ carrier ) ) ).
%------------------------------------------------------------------------------