TPTP Problem File: ITP124^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP124^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Modular_Distrib_Lattice problem prob_46__3255746_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 : Modular_Distrib_Lattice/prob_46__3255746_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 303 ( 122 unt; 39 typ; 0 def)
% Number of atoms : 726 ( 226 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 2589 ( 52 ~; 7 |; 35 &;2208 @)
% ( 0 <=>; 287 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 175 ( 175 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 38 usr; 4 con; 0-6 aty)
% Number of variables : 737 ( 23 ^; 668 !; 12 ?; 737 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:28:37.161
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (37)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit2040224947_above:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit1201339847_below:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
finite_comp_fun_idem:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
lattic1263571978nf_fin:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic35693393ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
lattic1039401930up_fin:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
modula1221160330_d_aux:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
modula1593167561_e_aux:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder_OGreatest,type,
greatest:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder_Oantimono,type,
antimono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $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__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_c,type,
c: a ).
thf(sy_v_inf,type,
inf: a > a > a ).
thf(sy_v_less__eq,type,
less_eq: a > a > $o ).
thf(sy_v_sup,type,
sup: a > a > a ).
% Relevant facts (256)
thf(fact_0_local_Oinf_Oassoc,axiom,
! [A2: a,B2: a,C: a] :
( ( inf @ ( inf @ A2 @ B2 ) @ C )
= ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% local.inf.assoc
thf(fact_1_local_Oinf_Ocommute,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ B2 )
= ( inf @ B2 @ A2 ) ) ).
% local.inf.commute
thf(fact_2_local_Oinf_Oleft__commute,axiom,
! [B2: a,A2: a,C: a] :
( ( inf @ B2 @ ( inf @ A2 @ C ) )
= ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% local.inf.left_commute
thf(fact_3_local_Oinf__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( inf @ ( inf @ X @ Y ) @ Z )
= ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% local.inf_assoc
thf(fact_4_local_Oinf__commute,axiom,
! [X: a,Y: a] :
( ( inf @ X @ Y )
= ( inf @ Y @ X ) ) ).
% local.inf_commute
thf(fact_5_local_Oinf__left__commute,axiom,
! [X: a,Y: a,Z: a] :
( ( inf @ X @ ( inf @ Y @ Z ) )
= ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% local.inf_left_commute
thf(fact_6_local_Osup_Oassoc,axiom,
! [A2: a,B2: a,C: a] :
( ( sup @ ( sup @ A2 @ B2 ) @ C )
= ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% local.sup.assoc
thf(fact_7_local_Osup_Ocommute,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ B2 )
= ( sup @ B2 @ A2 ) ) ).
% local.sup.commute
thf(fact_8_local_Osup_Oleft__commute,axiom,
! [B2: a,A2: a,C: a] :
( ( sup @ B2 @ ( sup @ A2 @ C ) )
= ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% local.sup.left_commute
thf(fact_9_local_Osup__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( sup @ ( sup @ X @ Y ) @ Z )
= ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% local.sup_assoc
thf(fact_10_local_Osup__commute,axiom,
! [X: a,Y: a] :
( ( sup @ X @ Y )
= ( sup @ Y @ X ) ) ).
% local.sup_commute
thf(fact_11_local_Osup__left__commute,axiom,
! [X: a,Y: a,Z: a] :
( ( sup @ X @ ( sup @ Y @ Z ) )
= ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% local.sup_left_commute
thf(fact_12_local_Odistrib__imp1,axiom,
! [X: a,Y: a,Z: a] :
( ! [X2: a,Y2: a,Z2: a] :
( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
= ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
=> ( ( sup @ X @ ( inf @ Y @ Z ) )
= ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% local.distrib_imp1
thf(fact_13_local_Odistrib__imp2,axiom,
! [X: a,Y: a,Z: a] :
( ! [X2: a,Y2: a,Z2: a] :
( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
= ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
=> ( ( inf @ X @ ( sup @ Y @ Z ) )
= ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% local.distrib_imp2
thf(fact_14_e__aux__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% e_aux_def
thf(fact_15_local_Oinf_Oidem,axiom,
! [A2: a] :
( ( inf @ A2 @ A2 )
= A2 ) ).
% local.inf.idem
thf(fact_16_local_Oinf_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.left_idem
thf(fact_17_local_Oinf_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.right_idem
thf(fact_18_local_Oinf__idem,axiom,
! [X: a] :
( ( inf @ X @ X )
= X ) ).
% local.inf_idem
thf(fact_19_local_Oinf__left__idem,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( inf @ X @ Y ) )
= ( inf @ X @ Y ) ) ).
% local.inf_left_idem
thf(fact_20_local_Oinf__right__idem,axiom,
! [X: a,Y: a] :
( ( inf @ ( inf @ X @ Y ) @ Y )
= ( inf @ X @ Y ) ) ).
% local.inf_right_idem
thf(fact_21_local_Osup_Oidem,axiom,
! [A2: a] :
( ( sup @ A2 @ A2 )
= A2 ) ).
% local.sup.idem
thf(fact_22_local_Osup_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.left_idem
thf(fact_23_local_Osup_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.right_idem
thf(fact_24_local_Osup__idem,axiom,
! [X: a] :
( ( sup @ X @ X )
= X ) ).
% local.sup_idem
thf(fact_25_local_Osup__left__idem,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( sup @ X @ Y ) )
= ( sup @ X @ Y ) ) ).
% local.sup_left_idem
thf(fact_26_local_Oinf__sup__absorb,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( sup @ X @ Y ) )
= X ) ).
% local.inf_sup_absorb
thf(fact_27_local_Osup__inf__absorb,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( inf @ X @ Y ) )
= X ) ).
% local.sup_inf_absorb
thf(fact_28_d__aux__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% d_aux_def
thf(fact_29_d__b__c__a,axiom,
! [B2: a,C: a,A2: a] :
( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
= ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% d_b_c_a
thf(fact_30_d__c__a__b,axiom,
! [C: a,A2: a,B2: a] :
( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
= ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% d_c_a_b
thf(fact_31_lattice_Oe__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1593167561_e_aux @ A )
= ( modula1593167561_e_aux @ A ) ) ).
% lattice.e_aux.cong
thf(fact_32_local_Ocomp__fun__idem__sup,axiom,
finite_comp_fun_idem @ a @ a @ sup ).
% local.comp_fun_idem_sup
thf(fact_33_local_Ocomp__fun__idem__inf,axiom,
finite_comp_fun_idem @ a @ a @ inf ).
% local.comp_fun_idem_inf
thf(fact_34_local_Osup_Osemigroup__axioms,axiom,
semigroup @ a @ sup ).
% local.sup.semigroup_axioms
thf(fact_35_local_Oinf_Osemigroup__axioms,axiom,
semigroup @ a @ inf ).
% local.inf.semigroup_axioms
thf(fact_36_local_Osup_Osemilattice__axioms,axiom,
semilattice @ a @ sup ).
% local.sup.semilattice_axioms
thf(fact_37_local_Oinf_Osemilattice__axioms,axiom,
semilattice @ a @ inf ).
% local.inf.semilattice_axioms
thf(fact_38_local_Osup_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ sup ).
% local.sup.abel_semigroup_axioms
thf(fact_39_local_Oinf_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ inf ).
% local.inf.abel_semigroup_axioms
thf(fact_40_local_OSup__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ sup ).
% local.Sup_fin.semilattice_set_axioms
thf(fact_41_local_OInf__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ inf ).
% local.Inf_fin.semilattice_set_axioms
thf(fact_42_local_Odistrib__inf__le,axiom,
! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) @ ( inf @ X @ ( sup @ Y @ Z ) ) ) ).
% local.distrib_inf_le
thf(fact_43_local_Odistrib__sup__le,axiom,
! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ).
% local.distrib_sup_le
thf(fact_44_local_Oantisym,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ X )
=> ( X = Y ) ) ) ).
% local.antisym
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_local_Oantisym__conv,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv
thf(fact_50_local_Odual__order_Oantisym,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less_eq @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% local.dual_order.antisym
thf(fact_51_local_Odual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : Y3 = Z3 )
= ( ^ [A4: a,B3: a] :
( ( less_eq @ B3 @ A4 )
& ( less_eq @ A4 @ B3 ) ) ) ) ).
% local.dual_order.eq_iff
thf(fact_52_local_Odual__order_Otrans,axiom,
! [B2: a,A2: a,C: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less_eq @ C @ B2 )
=> ( less_eq @ C @ A2 ) ) ) ).
% local.dual_order.trans
thf(fact_53_local_Oeq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : Y3 = Z3 )
= ( ^ [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
& ( less_eq @ Y4 @ X3 ) ) ) ) ).
% local.eq_iff
thf(fact_54_local_Oeq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( less_eq @ X @ Y ) ) ).
% local.eq_refl
thf(fact_55_local_Oord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_eq_le_trans
thf(fact_56_local_Oord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_le_eq_trans
thf(fact_57_local_Oorder_Oantisym,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% local.order.antisym
thf(fact_58_local_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : Y3 = Z3 )
= ( ^ [A4: a,B3: a] :
( ( less_eq @ A4 @ B3 )
& ( less_eq @ B3 @ A4 ) ) ) ) ).
% local.order.eq_iff
thf(fact_59_local_Oorder_Otrans,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.order.trans
thf(fact_60_local_Oorder__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ Z )
=> ( less_eq @ X @ Z ) ) ) ).
% local.order_trans
thf(fact_61_local_Ole__infI2,axiom,
! [B2: a,X: a,A2: a] :
( ( less_eq @ B2 @ X )
=> ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% local.le_infI2
thf(fact_62_local_Ole__infI1,axiom,
! [A2: a,X: a,B2: a] :
( ( less_eq @ A2 @ X )
=> ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% local.le_infI1
thf(fact_63_local_Ole__infI,axiom,
! [X: a,A2: a,B2: a] :
( ( less_eq @ X @ A2 )
=> ( ( less_eq @ X @ B2 )
=> ( less_eq @ X @ ( inf @ A2 @ B2 ) ) ) ) ).
% local.le_infI
thf(fact_64_local_Ole__infE,axiom,
! [X: a,A2: a,B2: a] :
( ( less_eq @ X @ ( inf @ A2 @ B2 ) )
=> ~ ( ( less_eq @ X @ A2 )
=> ~ ( less_eq @ X @ B2 ) ) ) ).
% local.le_infE
thf(fact_65_local_Ole__iff__inf,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( inf @ X @ Y )
= X ) ) ).
% local.le_iff_inf
thf(fact_66_local_Oinf__unique,axiom,
! [F: a > a > a,X: a,Y: a] :
( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( less_eq @ X2 @ Y2 )
=> ( ( less_eq @ X2 @ Z2 )
=> ( less_eq @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% local.inf_unique
thf(fact_67_local_Oinf__mono,axiom,
! [A2: a,C: a,B2: a,D: a] :
( ( less_eq @ A2 @ C )
=> ( ( less_eq @ B2 @ D )
=> ( less_eq @ ( inf @ A2 @ B2 ) @ ( inf @ C @ D ) ) ) ) ).
% local.inf_mono
thf(fact_68_local_Oinf__le2,axiom,
! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% local.inf_le2
thf(fact_69_local_Oinf__le1,axiom,
! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% local.inf_le1
thf(fact_70_local_Oinf__greatest,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ X @ Z )
=> ( less_eq @ X @ ( inf @ Y @ Z ) ) ) ) ).
% local.inf_greatest
thf(fact_71_local_Oinf__absorb2,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( inf @ X @ Y )
= Y ) ) ).
% local.inf_absorb2
thf(fact_72_local_Oinf__absorb1,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( inf @ X @ Y )
= X ) ) ).
% local.inf_absorb1
thf(fact_73_local_Oinf_Oorder__iff,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( A2
= ( inf @ A2 @ B2 ) ) ) ).
% local.inf.order_iff
thf(fact_74_local_Oinf_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( inf @ A2 @ B2 ) )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.inf.orderI
thf(fact_75_local_Oinf_OorderE,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( A2
= ( inf @ A2 @ B2 ) ) ) ).
% local.inf.orderE
thf(fact_76_local_Oinf_OcoboundedI2,axiom,
! [B2: a,C: a,A2: a] :
( ( less_eq @ B2 @ C )
=> ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.coboundedI2
thf(fact_77_local_Oinf_OcoboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( less_eq @ A2 @ C )
=> ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.coboundedI1
thf(fact_78_local_Oinf_Ocobounded2,axiom,
! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ B2 ) ).
% local.inf.cobounded2
thf(fact_79_local_Oinf_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% local.inf.cobounded1
thf(fact_80_local_Oinf_OboundedI,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ A2 @ C )
=> ( less_eq @ A2 @ ( inf @ B2 @ C ) ) ) ) ).
% local.inf.boundedI
thf(fact_81_local_Oinf_OboundedE,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
=> ~ ( ( less_eq @ A2 @ B2 )
=> ~ ( less_eq @ A2 @ C ) ) ) ).
% local.inf.boundedE
thf(fact_82_local_Oinf_Oabsorb__iff2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( inf @ A2 @ B2 )
= B2 ) ) ).
% local.inf.absorb_iff2
thf(fact_83_local_Oinf_Oabsorb__iff1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( inf @ A2 @ B2 )
= A2 ) ) ).
% local.inf.absorb_iff1
thf(fact_84_local_Oinf_Oabsorb2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( inf @ A2 @ B2 )
= B2 ) ) ).
% local.inf.absorb2
thf(fact_85_local_Oinf_Oabsorb1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( inf @ A2 @ B2 )
= A2 ) ) ).
% local.inf.absorb1
thf(fact_86_local_Osup__unique,axiom,
! [F: a > a > a,X: a,Y: a] :
( ! [X2: a,Y2: a] : ( less_eq @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: a,Y2: a] : ( less_eq @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( less_eq @ Y2 @ X2 )
=> ( ( less_eq @ Z2 @ X2 )
=> ( less_eq @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% local.sup_unique
thf(fact_87_local_Osup__mono,axiom,
! [A2: a,C: a,B2: a,D: a] :
( ( less_eq @ A2 @ C )
=> ( ( less_eq @ B2 @ D )
=> ( less_eq @ ( sup @ A2 @ B2 ) @ ( sup @ C @ D ) ) ) ) ).
% local.sup_mono
thf(fact_88_local_Osup__least,axiom,
! [Y: a,X: a,Z: a] :
( ( less_eq @ Y @ X )
=> ( ( less_eq @ Z @ X )
=> ( less_eq @ ( sup @ Y @ Z ) @ X ) ) ) ).
% local.sup_least
thf(fact_89_local_Osup__ge2,axiom,
! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% local.sup_ge2
thf(fact_90_local_Osup__ge1,axiom,
! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% local.sup_ge1
thf(fact_91_local_Osup__absorb2,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( sup @ X @ Y )
= Y ) ) ).
% local.sup_absorb2
thf(fact_92_local_Osup__absorb1,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( sup @ X @ Y )
= X ) ) ).
% local.sup_absorb1
thf(fact_93_local_Osup_Oorder__iff,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( A2
= ( sup @ A2 @ B2 ) ) ) ).
% local.sup.order_iff
thf(fact_94_local_Osup_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( sup @ A2 @ B2 ) )
=> ( less_eq @ B2 @ A2 ) ) ).
% local.sup.orderI
thf(fact_95_local_Osup_OorderE,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( A2
= ( sup @ A2 @ B2 ) ) ) ).
% local.sup.orderE
thf(fact_96_local_Osup_Omono,axiom,
! [C: a,A2: a,D: a,B2: a] :
( ( less_eq @ C @ A2 )
=> ( ( less_eq @ D @ B2 )
=> ( less_eq @ ( sup @ C @ D ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% local.sup.mono
thf(fact_97_local_Osup_OcoboundedI2,axiom,
! [C: a,B2: a,A2: a] :
( ( less_eq @ C @ B2 )
=> ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.sup.coboundedI2
thf(fact_98_local_Osup_OcoboundedI1,axiom,
! [C: a,A2: a,B2: a] :
( ( less_eq @ C @ A2 )
=> ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.sup.coboundedI1
thf(fact_99_local_Osup_Ocobounded2,axiom,
! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% local.sup.cobounded2
thf(fact_100_local_Osup_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% local.sup.cobounded1
thf(fact_101_local_Osup_OboundedI,axiom,
! [B2: a,A2: a,C: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less_eq @ C @ A2 )
=> ( less_eq @ ( sup @ B2 @ C ) @ A2 ) ) ) ).
% local.sup.boundedI
thf(fact_102_local_Osup_OboundedE,axiom,
! [B2: a,C: a,A2: a] :
( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
=> ~ ( ( less_eq @ B2 @ A2 )
=> ~ ( less_eq @ C @ A2 ) ) ) ).
% local.sup.boundedE
thf(fact_103_local_Osup_Oabsorb__iff2,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( sup @ A2 @ B2 )
= B2 ) ) ).
% local.sup.absorb_iff2
thf(fact_104_local_Osup_Oabsorb__iff1,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( sup @ A2 @ B2 )
= A2 ) ) ).
% local.sup.absorb_iff1
thf(fact_105_local_Osup_Oabsorb2,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( sup @ A2 @ B2 )
= B2 ) ) ).
% local.sup.absorb2
thf(fact_106_local_Osup_Oabsorb1,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( sup @ A2 @ B2 )
= A2 ) ) ).
% local.sup.absorb1
thf(fact_107_local_Ole__supI2,axiom,
! [X: a,B2: a,A2: a] :
( ( less_eq @ X @ B2 )
=> ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% local.le_supI2
thf(fact_108_local_Ole__supI1,axiom,
! [X: a,A2: a,B2: a] :
( ( less_eq @ X @ A2 )
=> ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% local.le_supI1
thf(fact_109_local_Ole__supI,axiom,
! [A2: a,X: a,B2: a] :
( ( less_eq @ A2 @ X )
=> ( ( less_eq @ B2 @ X )
=> ( less_eq @ ( sup @ A2 @ B2 ) @ X ) ) ) ).
% local.le_supI
thf(fact_110_local_Ole__supE,axiom,
! [A2: a,B2: a,X: a] :
( ( less_eq @ ( sup @ A2 @ B2 ) @ X )
=> ~ ( ( less_eq @ A2 @ X )
=> ~ ( less_eq @ B2 @ X ) ) ) ).
% local.le_supE
thf(fact_111_local_Ole__iff__sup,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( sup @ X @ Y )
= Y ) ) ).
% local.le_iff_sup
thf(fact_112_local_Oorder_Orefl,axiom,
! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% local.order.refl
thf(fact_113_local_Oorder__refl,axiom,
! [X: a] : ( less_eq @ X @ X ) ).
% local.order_refl
thf(fact_114_local_OGreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( less_eq @ Y2 @ X ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( less_eq @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( greatest @ a @ less_eq @ P ) ) ) ) ) ).
% local.GreatestI2_order
thf(fact_115_local_OGreatest__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( less_eq @ Y2 @ X ) )
=> ( ( greatest @ a @ less_eq @ P )
= X ) ) ) ).
% local.Greatest_equality
thf(fact_116_local_Omax__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( max @ a @ less_eq @ A2 @ B2 )
= B2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( max @ a @ less_eq @ A2 @ B2 )
= A2 ) ) ) ).
% local.max_def
thf(fact_117_local_Omin__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= A2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= B2 ) ) ) ).
% local.min_def
thf(fact_118_local_Ole__inf__iff,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ X @ ( inf @ Y @ Z ) )
= ( ( less_eq @ X @ Y )
& ( less_eq @ X @ Z ) ) ) ).
% local.le_inf_iff
thf(fact_119_local_Oinf_Obounded__iff,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
= ( ( less_eq @ A2 @ B2 )
& ( less_eq @ A2 @ C ) ) ) ).
% local.inf.bounded_iff
thf(fact_120_local_Osup_Obounded__iff,axiom,
! [B2: a,C: a,A2: a] :
( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
= ( ( less_eq @ B2 @ A2 )
& ( less_eq @ C @ A2 ) ) ) ).
% local.sup.bounded_iff
thf(fact_121_local_Ole__sup__iff,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ ( sup @ X @ Y ) @ Z )
= ( ( less_eq @ X @ Z )
& ( less_eq @ Y @ Z ) ) ) ).
% local.le_sup_iff
thf(fact_122_local_OLeast1I,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y2: a] :
( ( P @ Y2 )
=> ( less_eq @ X4 @ Y2 ) )
& ! [Y2: a] :
( ( ( P @ Y2 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( less_eq @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( P @ ( least @ a @ less_eq @ P ) ) ) ).
% local.Least1I
thf(fact_123_local_OLeast1__le,axiom,
! [P: a > $o,Z: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y2: a] :
( ( P @ Y2 )
=> ( less_eq @ X4 @ Y2 ) )
& ! [Y2: a] :
( ( ( P @ Y2 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( less_eq @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( ( P @ Z )
=> ( less_eq @ ( least @ a @ less_eq @ P ) @ Z ) ) ) ).
% local.Least1_le
thf(fact_124_local_OLeastI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( less_eq @ X @ Y2 ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( less_eq @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% local.LeastI2_order
thf(fact_125_local_OLeast__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( less_eq @ X @ Y2 ) )
=> ( ( least @ a @ less_eq @ P )
= X ) ) ) ).
% local.Least_equality
thf(fact_126_lattice_Od__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1221160330_d_aux @ A )
= ( modula1221160330_d_aux @ A ) ) ).
% lattice.d_aux.cong
thf(fact_127_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( semigroup @ A @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_128_semilattice__set__def,axiom,
! [A: $tType] :
( ( lattic35693393ce_set @ A )
= ( semilattice @ A ) ) ).
% semilattice_set_def
thf(fact_129_semilattice__set_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( lattic35693393ce_set @ A @ F ) ) ).
% semilattice_set.intro
thf(fact_130_semilattice__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A] :
( ( lattic35693393ce_set @ A @ F )
=> ( semilattice @ A @ F ) ) ).
% semilattice_set.axioms
thf(fact_131_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ).
% semilattice.axioms(1)
thf(fact_132_local_Obdd__above__def,axiom,
! [A3: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ A3 )
= ( ? [M: a] :
! [X3: a] :
( ( member @ a @ X3 @ A3 )
=> ( less_eq @ X3 @ M ) ) ) ) ).
% local.bdd_above_def
thf(fact_133_local_Obdd__below__def,axiom,
! [A3: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ A3 )
= ( ? [M2: a] :
! [X3: a] :
( ( member @ a @ X3 @ A3 )
=> ( less_eq @ M2 @ X3 ) ) ) ) ).
% local.bdd_below_def
thf(fact_134_local_Otransp__le,axiom,
transp @ a @ less_eq ).
% local.transp_le
thf(fact_135_local_Obdd__belowI,axiom,
! [A3: set @ a,M3: a] :
( ! [X2: a] :
( ( member @ a @ X2 @ A3 )
=> ( less_eq @ M3 @ X2 ) )
=> ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% local.bdd_belowI
thf(fact_136_local_Obdd__aboveI,axiom,
! [A3: set @ a,M4: a] :
( ! [X2: a] :
( ( member @ a @ X2 @ A3 )
=> ( less_eq @ X2 @ M4 ) )
=> ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% local.bdd_aboveI
thf(fact_137_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ B2 @ ( F @ A2 @ C ) )
= ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_138_semilattice_Oright__idem,axiom,
! [A: $tType,F: A > A > A,A2: A,B2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ ( F @ A2 @ B2 ) @ B2 )
= ( F @ A2 @ B2 ) ) ) ).
% semilattice.right_idem
thf(fact_139_semilattice_Oleft__idem,axiom,
! [A: $tType,F: A > A > A,A2: A,B2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A2 @ ( F @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ) ).
% semilattice.left_idem
thf(fact_140_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: A > A > A,A2: A,B2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ A2 @ B2 )
= ( F @ B2 @ A2 ) ) ) ).
% abel_semigroup.commute
thf(fact_141_semilattice_Oidem,axiom,
! [A: $tType,F: A > A > A,A2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A2 @ A2 )
= A2 ) ) ).
% semilattice.idem
thf(fact_142_semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A5: A,B4: A,C2: A] :
( ( F @ ( F @ A5 @ B4 ) @ C2 )
= ( F @ A5 @ ( F @ B4 @ C2 ) ) )
=> ( semigroup @ A @ F ) ) ).
% semigroup.intro
thf(fact_143_semigroup_Oassoc,axiom,
! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
( ( semigroup @ A @ F )
=> ( ( F @ ( F @ A2 @ B2 ) @ C )
= ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% semigroup.assoc
thf(fact_144_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F2: A > A > A] :
! [A4: A,B3: A,C3: A] :
( ( F2 @ ( F2 @ A4 @ B3 ) @ C3 )
= ( F2 @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_145_local_Obdd__below__mono,axiom,
! [B5: set @ a,A3: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ B5 )
=> ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
=> ( condit1201339847_below @ a @ less_eq @ A3 ) ) ) ).
% local.bdd_below_mono
thf(fact_146_local_Obdd__above__mono,axiom,
! [B5: set @ a,A3: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ B5 )
=> ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
=> ( condit2040224947_above @ a @ less_eq @ A3 ) ) ) ).
% local.bdd_above_mono
thf(fact_147_local_Obdd__below__finite,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% local.bdd_below_finite
thf(fact_148_local_Obdd__above__finite,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% local.bdd_above_finite
thf(fact_149_local_Ofinite__has__maximal2,axiom,
! [A3: set @ a,A2: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ A2 @ A3 )
=> ? [X2: a] :
( ( member @ a @ X2 @ A3 )
& ( less_eq @ A2 @ X2 )
& ! [Xa: a] :
( ( member @ a @ Xa @ A3 )
=> ( ( less_eq @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% local.finite_has_maximal2
thf(fact_150_local_Ofinite__has__minimal2,axiom,
! [A3: set @ a,A2: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ A2 @ A3 )
=> ? [X2: a] :
( ( member @ a @ X2 @ A3 )
& ( less_eq @ X2 @ A2 )
& ! [Xa: a] :
( ( member @ a @ Xa @ A3 )
=> ( ( less_eq @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% local.finite_has_minimal2
thf(fact_151_local_OantimonoD,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B,X: a,Y: a] :
( ( antimono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% local.antimonoD
thf(fact_152_local_OantimonoE,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B,X: a,Y: a] :
( ( antimono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% local.antimonoE
thf(fact_153_local_OantimonoI,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ! [X2: a,Y2: a] :
( ( less_eq @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% local.antimonoI
thf(fact_154_local_Oantimono__def,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ( antimono @ a @ B @ less_eq @ F )
= ( ! [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% local.antimono_def
thf(fact_155_local_OmonoD,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B,X: a,Y: a] :
( ( mono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% local.monoD
thf(fact_156_local_OmonoE,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B,X: a,Y: a] :
( ( mono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% local.monoE
thf(fact_157_local_OmonoI,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ! [X2: a,Y2: a] :
( ( less_eq @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( mono @ a @ B @ less_eq @ F ) ) ) ).
% local.monoI
thf(fact_158_local_Omono__def,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ( mono @ a @ B @ less_eq @ F )
= ( ! [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% local.mono_def
thf(fact_159_local_OSup__fin_OcoboundedI,axiom,
! [A3: set @ a,A2: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ A2 @ A3 )
=> ( less_eq @ A2 @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% local.Sup_fin.coboundedI
thf(fact_160_local_OInf__fin_OcoboundedI,axiom,
! [A3: set @ a,A2: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ A2 @ A3 )
=> ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A2 ) ) ) ).
% local.Inf_fin.coboundedI
thf(fact_161_local_OSup__fin_Oin__idem,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ X @ A3 )
=> ( ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
= ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% local.Sup_fin.in_idem
thf(fact_162_local_OInf__fin_Oin__idem,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ X @ A3 )
=> ( ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
= ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ).
% local.Inf_fin.in_idem
thf(fact_163_local_Oinf__Sup__absorb,axiom,
! [A3: set @ a,A2: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ A2 @ A3 )
=> ( ( inf @ A2 @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
= A2 ) ) ) ).
% local.inf_Sup_absorb
thf(fact_164_local_Osup__Inf__absorb,axiom,
! [A3: set @ a,A2: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( member @ a @ A2 @ A3 )
=> ( ( sup @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A2 )
= A2 ) ) ) ).
% local.sup_Inf_absorb
thf(fact_165_semilattice__inf_OInf__fin_Ocong,axiom,
! [A: $tType] :
( ( lattic1263571978nf_fin @ A )
= ( lattic1263571978nf_fin @ A ) ) ).
% semilattice_inf.Inf_fin.cong
thf(fact_166_semilattice__sup_OSup__fin_Ocong,axiom,
! [A: $tType] :
( ( lattic1039401930up_fin @ A )
= ( lattic1039401930up_fin @ A ) ) ).
% semilattice_sup.Sup_fin.cong
thf(fact_167_local_OInf__fin__le__Sup__fin,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% local.Inf_fin_le_Sup_fin
thf(fact_168_local_OSup__fin_Osubset__imp,axiom,
! [A3: set @ a,B5: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( finite_finite2 @ a @ B5 )
=> ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ ( lattic1039401930up_fin @ a @ sup @ B5 ) ) ) ) ) ).
% local.Sup_fin.subset_imp
thf(fact_169_local_OInf__fin_Osubset__imp,axiom,
! [A3: set @ a,B5: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( finite_finite2 @ a @ B5 )
=> ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ B5 ) @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% local.Inf_fin.subset_imp
thf(fact_170_local_Ofinite__has__minimal,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ? [X2: a] :
( ( member @ a @ X2 @ A3 )
& ! [Xa: a] :
( ( member @ a @ Xa @ A3 )
=> ( ( less_eq @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% local.finite_has_minimal
thf(fact_171_local_Ofinite__has__maximal,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ? [X2: a] :
( ( member @ a @ X2 @ A3 )
& ! [Xa: a] :
( ( member @ a @ Xa @ A3 )
=> ( ( less_eq @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% local.finite_has_maximal
thf(fact_172_local_OInf__fin_Osubset,axiom,
! [A3: set @ a,B5: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B5 @ A3 )
=> ( ( inf @ ( lattic1263571978nf_fin @ a @ inf @ B5 ) @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
= ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% local.Inf_fin.subset
thf(fact_173_local_OSup__fin_Osubset,axiom,
! [A3: set @ a,B5: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B5 @ A3 )
=> ( ( sup @ ( lattic1039401930up_fin @ a @ sup @ B5 ) @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
= ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ).
% local.Sup_fin.subset
thf(fact_174_local_OInf__fin_Obounded__iff,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
= ( ! [X3: a] :
( ( member @ a @ X3 @ A3 )
=> ( less_eq @ X @ X3 ) ) ) ) ) ) ).
% local.Inf_fin.bounded_iff
thf(fact_175_local_OInf__fin_OboundedI,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ! [A5: a] :
( ( member @ a @ A5 @ A3 )
=> ( less_eq @ X @ A5 ) )
=> ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% local.Inf_fin.boundedI
thf(fact_176_local_OInf__fin_OboundedE,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
=> ! [A6: a] :
( ( member @ a @ A6 @ A3 )
=> ( less_eq @ X @ A6 ) ) ) ) ) ).
% local.Inf_fin.boundedE
thf(fact_177_local_OSup__fin_Obounded__iff,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X )
= ( ! [X3: a] :
( ( member @ a @ X3 @ A3 )
=> ( less_eq @ X3 @ X ) ) ) ) ) ) ).
% local.Sup_fin.bounded_iff
thf(fact_178_local_OSup__fin_OboundedI,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ! [A5: a] :
( ( member @ a @ A5 @ A3 )
=> ( less_eq @ A5 @ X ) )
=> ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X ) ) ) ) ).
% local.Sup_fin.boundedI
thf(fact_179_local_OSup__fin_OboundedE,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X )
=> ! [A6: a] :
( ( member @ a @ A6 @ A3 )
=> ( less_eq @ A6 @ X ) ) ) ) ) ).
% local.Sup_fin.boundedE
thf(fact_180_local_Obdd__above__empty,axiom,
condit2040224947_above @ a @ less_eq @ ( bot_bot @ ( set @ a ) ) ).
% local.bdd_above_empty
thf(fact_181_local_Obdd__below__empty,axiom,
condit1201339847_below @ a @ less_eq @ ( bot_bot @ ( set @ a ) ) ).
% local.bdd_below_empty
thf(fact_182_local_Onot__empty__eq__Iic__eq__empty,axiom,
! [H: a] :
( ( bot_bot @ ( set @ a ) )
!= ( set_atMost @ a @ less_eq @ H ) ) ).
% local.not_empty_eq_Iic_eq_empty
thf(fact_183_local_Onot__empty__eq__Ici__eq__empty,axiom,
! [L: a] :
( ( bot_bot @ ( set @ a ) )
!= ( set_atLeast @ a @ less_eq @ L ) ) ).
% local.not_empty_eq_Ici_eq_empty
thf(fact_184_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_185_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_186_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_187_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_188_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_189_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_190_subsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% subsetI
thf(fact_191_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_192_local_OatLeast__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
= ( less_eq @ K @ I ) ) ).
% local.atLeast_iff
thf(fact_193_local_OatMost__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
= ( less_eq @ I @ K ) ) ).
% local.atMost_iff
thf(fact_194_local_Obdd__below__Ici,axiom,
! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% local.bdd_below_Ici
thf(fact_195_local_Obdd__above__Iic,axiom,
! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atMost @ a @ less_eq @ B2 ) ) ).
% local.bdd_above_Iic
thf(fact_196_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_197_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_198_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_199_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_200_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_201_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z3: set @ A] : Y3 = Z3 )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_202_subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_203_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_204_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_205_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [T: A] :
( ( member @ A @ T @ A7 )
=> ( member @ A @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_206_equalityD2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_207_equalityD1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_208_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A7 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_209_equalityE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_210_subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B5 ) ) ) ).
% subsetD
thf(fact_211_in__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_212_local_OSup__fin_Oclosed,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ! [X2: a,Y2: a] : ( member @ a @ ( sup @ X2 @ Y2 ) @ ( insert @ a @ X2 @ ( insert @ a @ Y2 @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( member @ a @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ A3 ) ) ) ) ).
% local.Sup_fin.closed
thf(fact_213_local_OSup__fin_Oinsert__not__elem,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ~ ( member @ a @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ A3 ) )
= ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ) ).
% local.Sup_fin.insert_not_elem
thf(fact_214_local_OInf__fin_Oclosed,axiom,
! [A3: set @ a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ! [X2: a,Y2: a] : ( member @ a @ ( inf @ X2 @ Y2 ) @ ( insert @ a @ X2 @ ( insert @ a @ Y2 @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( member @ a @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A3 ) ) ) ) ).
% local.Inf_fin.closed
thf(fact_215_local_OInf__fin_Oinsert__not__elem,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ~ ( member @ a @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ A3 ) )
= ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ) ).
% local.Inf_fin.insert_not_elem
thf(fact_216_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_217_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_218_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_219_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_220_local_Obdd__above__insert,axiom,
! [A2: a,A3: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ ( insert @ a @ A2 @ A3 ) )
= ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% local.bdd_above_insert
thf(fact_221_local_Obdd__below__insert,axiom,
! [A2: a,A3: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ ( insert @ a @ A2 @ A3 ) )
= ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% local.bdd_below_insert
thf(fact_222_local_OInf__fin_Osingleton,axiom,
! [X: a] :
( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ ( bot_bot @ ( set @ a ) ) ) )
= X ) ).
% local.Inf_fin.singleton
thf(fact_223_local_OSup__fin_Osingleton,axiom,
! [X: a] :
( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ ( bot_bot @ ( set @ a ) ) ) )
= X ) ).
% local.Sup_fin.singleton
thf(fact_224_local_OInf__fin_Oinsert,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ A3 ) )
= ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% local.Inf_fin.insert
thf(fact_225_local_OSup__fin_Oinsert,axiom,
! [A3: set @ a,X: a] :
( ( finite_finite2 @ a @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ A3 ) )
= ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ).
% local.Sup_fin.insert
thf(fact_226_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_227_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_228_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_229_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_230_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_231_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_232_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_233_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_234_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_235_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_236_insert__mono,axiom,
! [A: $tType,C4: set @ A,D2: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_237_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B,S2: set @ B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S2 )
=> ( ord_less_eq @ A @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert @ B @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_238_local_OatLeastAtMost__singleton_H,axiom,
! [A2: a,B2: a] :
( ( A2 = B2 )
=> ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
= ( insert @ a @ A2 @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% local.atLeastAtMost_singleton'
thf(fact_239_finite__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
thf(fact_240_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_241_local_OIcc__eq__Icc,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_atLeastAtMost @ a @ less_eq @ L @ H )
= ( set_atLeastAtMost @ a @ less_eq @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( less_eq @ L @ H )
& ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% local.Icc_eq_Icc
thf(fact_242_local_OatLeastAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.atLeastAtMost_iff
thf(fact_243_local_OatLeastatMost__empty__iff,axiom,
! [A2: a,B2: a] :
( ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
= ( bot_bot @ ( set @ a ) ) )
= ( ~ ( less_eq @ A2 @ B2 ) ) ) ).
% local.atLeastatMost_empty_iff
thf(fact_244_local_OatLeastatMost__empty__iff2,axiom,
! [A2: a,B2: a] :
( ( ( bot_bot @ ( set @ a ) )
= ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) )
= ( ~ ( less_eq @ A2 @ B2 ) ) ) ).
% local.atLeastatMost_empty_iff2
thf(fact_245_local_Obdd__below__Icc,axiom,
! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% local.bdd_below_Icc
thf(fact_246_local_Obdd__above__Icc,axiom,
! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% local.bdd_above_Icc
thf(fact_247_local_OatLeastatMost__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
= ( ~ ( less_eq @ A2 @ B2 )
| ( ( less_eq @ C @ A2 )
& ( less_eq @ B2 @ D ) ) ) ) ).
% local.atLeastatMost_subset_iff
thf(fact_248_local_OatLeastAtMost__singleton,axiom,
! [A2: a] :
( ( set_atLeastAtMost @ a @ less_eq @ A2 @ A2 )
= ( insert @ a @ A2 @ ( bot_bot @ ( set @ a ) ) ) ) ).
% local.atLeastAtMost_singleton
thf(fact_249_local_OatLeastAtMost__singleton__iff,axiom,
! [A2: a,B2: a,C: a] :
( ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
= ( insert @ a @ C @ ( bot_bot @ ( set @ a ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C ) ) ) ).
% local.atLeastAtMost_singleton_iff
thf(fact_250_local_OIcc__subset__Ici__iff,axiom,
! [L: a,H: a,L2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atLeast @ a @ less_eq @ L2 ) )
= ( ~ ( less_eq @ L @ H )
| ( less_eq @ L2 @ L ) ) ) ).
% local.Icc_subset_Ici_iff
thf(fact_251_local_OIcc__subset__Iic__iff,axiom,
! [L: a,H: a,H2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atMost @ a @ less_eq @ H2 ) )
= ( ~ ( less_eq @ L @ H )
| ( less_eq @ H @ H2 ) ) ) ).
% local.Icc_subset_Iic_iff
thf(fact_252_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_253_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [X_1: B] : ( P @ X2 @ X_1 ) )
=> ? [F3: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( P @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_254_comp__fun__idem_Ofun__left__idem,axiom,
! [A: $tType,B: $tType,F: A > B > B,X: A,Z: B] :
( ( finite_comp_fun_idem @ A @ B @ F )
=> ( ( F @ X @ ( F @ X @ Z ) )
= ( F @ X @ Z ) ) ) ).
% comp_fun_idem.fun_left_idem
thf(fact_255_order__class_Ofinite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% order_class.finite_has_minimal2
% Type constructors (7)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
order @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2 )
= ( modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c ) ) ).
%------------------------------------------------------------------------------