TPTP Problem File: ITP154^2.p
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%------------------------------------------------------------------------------
% File : ITP154^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_205__6250810_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_205__6250810_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 352 ( 87 unt; 50 typ; 0 def)
% Number of atoms : 1011 ( 194 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 5267 ( 112 ~; 14 |; 90 &;4508 @)
% ( 0 <=>; 543 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 134 ( 134 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 49 usr; 5 con; 0-5 aty)
% Number of variables : 1125 ( 39 ^;1019 !; 17 ?;1125 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:01.254
%------------------------------------------------------------------------------
% 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 (47)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[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_Lattices_Olattice,type,
lattice:
!>[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_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[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_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic929149872er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic929674654er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic1704895705min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic2065218050nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic477160up_fin:
!>[A: $tType] : ( ( set @ A ) > 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_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_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
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_I3_J,axiom,
member @ a @ y @ carrier ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
member @ a @ x @ carrier ).
% assms(2)
thf(fact_2_rational__preference__axioms,axiom,
prefer1997167224erence @ a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_3_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_4_assms_I1_J,axiom,
finite_finite2 @ a @ carrier ).
% assms(1)
thf(fact_5_preference__axioms,axiom,
prefer199794634erence @ a @ carrier @ relation ).
% preference_axioms
thf(fact_6_trans__refl,axiom,
order_preorder_on @ a @ carrier @ relation ).
% trans_refl
thf(fact_7_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_8_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_9_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_10_total,axiom,
total_on @ a @ carrier @ relation ).
% total
thf(fact_11_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_12_reflexivity,axiom,
refl_on @ a @ carrier @ relation ).
% reflexivity
thf(fact_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_44_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% 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_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: B,C2: A,D: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
= ( ( ord_less_eq @ A @ A3 @ C2 )
& ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% Pair_le
thf(fact_50_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_51_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_52_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_53_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ord_less_eq @ A @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_54_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_55_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_56_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_57_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_58_subsetI,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% subsetI
thf(fact_59_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_60_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_61_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_62_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_63_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_64_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_65_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_66_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_67_in__mono,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_68_subsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_69_equalityE,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_70_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_71_equalityD1,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_72_equalityD2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_73_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_74_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_75_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_76_subset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_77_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_78_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_79_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% finite
thf(fact_80_finite__set__choice,axiom,
! [B: $tType,A: $tType,A2: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ? [X_1: B] : ( P @ X4 @ X_1 ) )
=> ? [F2: A > B] :
! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( P @ X2 @ ( F2 @ X2 ) ) ) ) ) ).
% finite_set_choice
thf(fact_81_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_82_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_83_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_84_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_85_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_86_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_87_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ( ord_less_eq @ A @ A3 @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_88_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ( ord_less_eq @ A @ X4 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ord_less_eq @ A @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_89_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_90_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_91_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_92_finite__subset,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_subset
thf(fact_93_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_94_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_95_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
= ( ( A3 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_96_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_97_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_98_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_99_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_100_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A4: set @ A,R: 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 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_101_total__onI,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A2 @ R2 ) ) ).
% total_onI
thf(fact_102_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_103_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_104_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_105_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_106_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_107_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_108_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: A,B7: A] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_109_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_110_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_111_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_112_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_113_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_114_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_115_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_116_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_117_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_118_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_119_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_120_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_121_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_122_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_123_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ 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 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_124_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( A3
= ( 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 @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_125_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ 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 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( 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 @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_127_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_128_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_129_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_130_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_131_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_132_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_133_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_134_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ~ ( ( A3 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_135_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A7: A,B7: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% prod_cases3
thf(fact_136_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A7: A,B7: B,C4: C,D3: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_137_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 ) ) )] :
~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A7 @ ( 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_138_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_139_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_140_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 )] :
( ! [A7: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_141_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 ) )] :
( ! [A7: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_142_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 ) ) )] :
( ! [A7: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A7 @ ( 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_143_prod__induct6,axiom,
! [F4: $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 @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_144_prod__induct7,axiom,
! [G3: $tType,F4: $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 @ F4 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_145_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A7: A,B7: B] :
( Y
!= ( product_Pair @ A @ B @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_146_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_147_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% subrelI
thf(fact_148_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_149_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_150_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_151_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_152_refl__onD,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A2 @ R2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_153_refl__onD1,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A2 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_154_refl__onD2,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A2 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_155_refl__on__domain,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A2 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ A @ A3 @ A2 )
& ( member @ A @ B2 @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_156_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_157_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_158_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_159_finite__transitivity__chain,axiom,
! [A: $tType,A2: set @ A,R3: A > A > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [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 @ A2 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A2 )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_160_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_161_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R3: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_162_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_163_rational__preference__axioms__def,axiom,
! [A: $tType] :
( ( prefer1801827867axioms @ A )
= ( total_on @ A ) ) ).
% rational_preference_axioms_def
thf(fact_164_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_165_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_166_less__by__empty,axiom,
! [A: $tType,A2: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( A2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B3 ) ) ).
% less_by_empty
thf(fact_167_finite__indexed__bound,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ B,P: B > A > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ? [X_1: A] : ( P @ X4 @ X_1 ) )
=> ? [M: A] :
! [X2: B] :
( ( member @ B @ X2 @ A2 )
=> ? [K: A] :
( ( ord_less_eq @ A @ K @ M )
& ( P @ X2 @ K ) ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_168_transitivity,axiom,
trans @ a @ relation ).
% transitivity
thf(fact_169_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_170_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C2 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_171_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_172_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_173_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_174_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_175_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_176_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [X3: A,Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_177_preorder__on__def,axiom,
! [A: $tType] :
( ( order_preorder_on @ A )
= ( ^ [A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R )
& ( trans @ A @ R ) ) ) ) ).
% preorder_on_def
thf(fact_178_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_179_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_180_under__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ B2 ) ) ) ) ).
% under_incr
thf(fact_181_Id__onI,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A2 ) ) ) ).
% Id_onI
thf(fact_182_trans__Id__on,axiom,
! [A: $tType,A2: set @ A] : ( trans @ A @ ( id_on @ A @ A2 ) ) ).
% trans_Id_on
thf(fact_183_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A2 ) )
= ( ( X = Y )
& ( member @ A @ X @ A2 ) ) ) ).
% Id_on_iff
thf(fact_184_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A2: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A2 ) ) ) ) ).
% Id_on_eqI
thf(fact_185_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A2 ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( C2
!= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_186_refl__on__Id__on,axiom,
! [A: $tType,A2: set @ A] : ( refl_on @ A @ A2 @ ( id_on @ A @ A2 ) ) ).
% refl_on_Id_on
thf(fact_187_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S )
=> ( ord_less_eq @ B @ ( F @ ( lattic1704895705min_on @ A @ B @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_188_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_189_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [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_190_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic1704895705min_on @ A @ B @ F @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_191_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ B3 ) @ ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_192_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ B3 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_193_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_194_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ord_less_eq @ A @ A3 @ ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_195_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ A3 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_196_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
=> ! [A8: A] :
( ( member @ A @ A8 @ A2 )
=> ( ord_less_eq @ A @ X @ A8 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_197_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A2 )
=> ( ord_less_eq @ A @ X @ A7 ) )
=> ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_198_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X )
=> ! [A8: A] :
( ( member @ A @ A8 @ A2 )
=> ( ord_less_eq @ A @ A8 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_199_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A2 )
=> ( ord_less_eq @ A @ A7 @ X ) )
=> ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_200_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_201_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_202_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ ( lattic929149872er_Max @ A @ B3 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_203_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: set @ A,N: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
=> ( ( M2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ M2 ) @ ( lattic929149872er_Max @ A @ N ) ) ) ) ) ) ).
% Max_mono
thf(fact_204_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_205_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ord_less_eq @ A @ A3 @ ( lattic929149872er_Max @ A @ A2 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_206_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B3 )
& ( ord_less_eq @ A @ X4 @ Xa ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B3 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A2 )
& ( ord_less_eq @ A @ X4 @ Xa ) ) )
=> ( ( lattic929149872er_Max @ A @ A2 )
= ( lattic929149872er_Max @ A @ B3 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_207_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A2 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A2 )
=> ( ( lattic929149872er_Max @ A @ A2 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_208_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( lattic929149872er_Max @ A @ A2 ) ) ) ) ) ).
% Max_ge
thf(fact_209_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic929149872er_Max @ A @ A2 ) @ A2 ) ) ) ) ).
% Max_in
thf(fact_210_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,M3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic929149872er_Max @ A @ A2 )
= M3 )
= ( ( member @ A @ M3 @ A2 )
& ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ M3 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_211_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic929149872er_Max @ A @ A2 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_212_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,M3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M3
= ( lattic929149872er_Max @ A @ A2 ) )
= ( ( member @ A @ M3 @ A2 )
& ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ M3 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_213_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X )
=> ! [A8: A] :
( ( member @ A @ A8 @ A2 )
=> ( ord_less_eq @ A @ A8 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_214_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A2 )
=> ( ord_less_eq @ A @ A7 @ X ) )
=> ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_215_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: set @ A,N: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
=> ( ( M2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ N ) @ ( lattic929674654er_Min @ A @ M2 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_216_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ B3 ) @ ( lattic929674654er_Min @ A @ A2 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_217_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_218_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic929674654er_Min @ A @ A2 ) @ A2 ) ) ) ) ).
% Min_in
thf(fact_219_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ X ) ) ) ) ).
% Min_le
thf(fact_220_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A2 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A2 )
=> ( ( lattic929674654er_Min @ A @ A2 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_221_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ A3 ) ) ) ) ).
% Min.coboundedI
thf(fact_222_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A2 )
=> ( ord_less_eq @ A @ X @ A7 ) )
=> ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_223_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) )
=> ! [A8: A] :
( ( member @ A @ A8 @ A2 )
=> ( ord_less_eq @ A @ X @ A8 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_224_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,M3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M3
= ( lattic929674654er_Min @ A @ A2 ) )
= ( ( member @ A @ M3 @ A2 )
& ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ M3 @ X3 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_225_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ X )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_226_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,M3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic929674654er_Min @ A @ A2 )
= M3 )
= ( ( member @ A @ M3 @ A2 )
& ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ M3 @ X3 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_227_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( ( inf_inf @ A @ ( lattic2065218050nf_fin @ A @ B3 ) @ ( lattic2065218050nf_fin @ A @ A2 ) )
= ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_228_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( ( sup_sup @ A @ ( lattic477160up_fin @ A @ B3 ) @ ( lattic477160up_fin @ A @ A2 ) )
= ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_229_Un__empty,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_230_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_231_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_232_finite__Int,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G5 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% finite_Int
thf(fact_233_finite__Un,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) )
= ( ( finite_finite2 @ A @ F5 )
& ( finite_finite2 @ A @ G5 ) ) ) ).
% finite_Un
thf(fact_234_inf__Pair__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( inf @ B )
& ( inf @ A ) )
=> ! [A3: A,B2: B,C2: A,D: B] :
( ( inf_inf @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
= ( product_Pair @ A @ B @ ( inf_inf @ A @ A3 @ C2 ) @ ( inf_inf @ B @ B2 @ D ) ) ) ) ).
% inf_Pair_Pair
thf(fact_235_sup__Pair__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( sup @ B )
& ( sup @ A ) )
=> ! [A3: A,B2: B,C2: A,D: B] :
( ( sup_sup @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
= ( product_Pair @ A @ B @ ( sup_sup @ A @ A3 @ C2 ) @ ( sup_sup @ B @ B2 @ D ) ) ) ) ).
% sup_Pair_Pair
thf(fact_236_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ A @ A3 @ ( lattic477160up_fin @ A @ A2 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_237_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( sup_sup @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ A3 )
= A3 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_238_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( sup_sup @ A @ X @ ( lattic477160up_fin @ A @ A2 ) )
= ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_239_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( inf_inf @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
= ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_240_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic477160up_fin @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
= ( sup_sup @ A @ ( lattic477160up_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_241_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic2065218050nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
= ( inf_inf @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ ( lattic2065218050nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_242_trans__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( trans @ A @ S2 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% trans_Int
thf(fact_243_refl__on__Un,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),B3: set @ A,S2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A2 @ R2 )
=> ( ( refl_on @ A @ B3 @ S2 )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_244_refl__on__Int,axiom,
! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),B3: set @ A,S2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A2 @ R2 )
=> ( ( refl_on @ A @ B3 @ S2 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_245_finite__UnI,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) ) ) ) ).
% finite_UnI
thf(fact_246_Un__infinite,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) ) ).
% Un_infinite
thf(fact_247_infinite__Un,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_Un
thf(fact_248_Un__Int__assoc__eq,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_249_Int__mono,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,B3: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_250_Int__lower1,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_251_Int__lower2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_252_Int__absorb1,axiom,
! [A: $tType,B3: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_253_Int__absorb2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_254_Int__greatest,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_255_Int__Collect__mono,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
% Type constructors (45)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( semilattice_sup @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_inf @ A10 )
=> ( semilattice_inf @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A9: $tType,A10: $tType] :
( ( order_bot @ A10 )
=> ( order_bot @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A9: $tType,A10: $tType] :
( ( lattice @ A10 )
=> ( lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 )
=> ( bot @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( sup @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Oinf,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_inf @ A10 )
=> ( inf @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_2,axiom,
! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_10,axiom,
! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Oinf_11,axiom,
! [A9: $tType] : ( inf @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_14,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_21,axiom,
sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oinf_22,axiom,
inf @ $o ).
thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_23,axiom,
! [A9: $tType,A10: $tType] :
( ( ( semilattice_sup @ A9 )
& ( semilattice_sup @ A10 ) )
=> ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__inf_24,axiom,
! [A9: $tType,A10: $tType] :
( ( ( semilattice_inf @ A9 )
& ( semilattice_inf @ A10 ) )
=> ( semilattice_inf @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_25,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order_bot @ A9 )
& ( order_bot @ A10 ) )
=> ( order_bot @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26,axiom,
! [A9: $tType,A10: $tType] :
( ( ( preorder @ A9 )
& ( preorder @ A10 ) )
=> ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Olattice_28,axiom,
! [A9: $tType,A10: $tType] :
( ( ( lattice @ A9 )
& ( lattice @ A10 ) )
=> ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_29,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( order @ A10 ) )
=> ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_30,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( ord @ A10 ) )
=> ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Obot_31,axiom,
! [A9: $tType,A10: $tType] :
( ( ( bot @ A9 )
& ( bot @ A10 ) )
=> ( bot @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osup_32,axiom,
! [A9: $tType,A10: $tType] :
( ( ( sup @ A9 )
& ( sup @ A10 ) )
=> ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Oinf_33,axiom,
! [A9: $tType,A10: $tType] :
( ( ( inf @ A9 )
& ( inf @ A10 ) )
=> ( inf @ ( product_prod @ A9 @ A10 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( 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 ) ) ) ).
%------------------------------------------------------------------------------