TPTP Problem File: ITP120^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP120^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Modular_Distrib_Lattice problem prob_203__3262370_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_203__3262370_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 322 ( 125 unt; 42 typ; 0 def)
% Number of atoms : 709 ( 195 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 2527 ( 22 ~; 2 |; 34 &;2169 @)
% ( 0 <=>; 300 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 213 ( 213 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 41 usr; 5 con; 0-6 aty)
% Number of variables : 824 ( 48 ^; 733 !; 6 ?; 824 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:28.434
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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_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_finite:
!>[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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic35693393ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux,type,
modula105138637_a_aux:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux,type,
modula477145868_b_aux:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux,type,
modula849153099_c_aux:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > 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_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oincomp,type,
modula1937309406incomp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_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_Oantisym,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ X )
=> ( X = Y ) ) ) ).
% local.antisym
thf(fact_1_local_Oantisym__conv,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv
thf(fact_2_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_3_local_Odual__order_Oeq__iff,axiom,
( ( ^ [Y2: a,Z: a] : Y2 = Z )
= ( ^ [A3: a,B3: a] :
( ( less_eq @ B3 @ A3 )
& ( less_eq @ A3 @ B3 ) ) ) ) ).
% local.dual_order.eq_iff
thf(fact_4_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_5_local_Oeq__iff,axiom,
( ( ^ [Y2: a,Z: a] : Y2 = Z )
= ( ^ [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
& ( less_eq @ Y3 @ X2 ) ) ) ) ).
% local.eq_iff
thf(fact_6_local_Oeq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( less_eq @ X @ Y ) ) ).
% local.eq_refl
thf(fact_7_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_8_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_9_local_Oorder_Oantisym,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% local.order.antisym
thf(fact_10_local_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: a,Z: a] : Y2 = Z )
= ( ^ [A3: a,B3: a] :
( ( less_eq @ A3 @ B3 )
& ( less_eq @ B3 @ A3 ) ) ) ) ).
% local.order.eq_iff
thf(fact_11_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_12_local_Oorder__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ Z2 )
=> ( less_eq @ X @ Z2 ) ) ) ).
% local.order_trans
thf(fact_13_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_14_local_Oinf_Ocommute,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ B2 )
= ( inf @ B2 @ A2 ) ) ).
% local.inf.commute
thf(fact_15_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_16_local_Oinf__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( inf @ ( inf @ X @ Y ) @ Z2 )
= ( inf @ X @ ( inf @ Y @ Z2 ) ) ) ).
% local.inf_assoc
thf(fact_17_local_Oinf__commute,axiom,
! [X: a,Y: a] :
( ( inf @ X @ Y )
= ( inf @ Y @ X ) ) ).
% local.inf_commute
thf(fact_18_local_Oinf__left__commute,axiom,
! [X: a,Y: a,Z2: a] :
( ( inf @ X @ ( inf @ Y @ Z2 ) )
= ( inf @ Y @ ( inf @ X @ Z2 ) ) ) ).
% local.inf_left_commute
thf(fact_19_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_20_local_Osup_Ocommute,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ B2 )
= ( sup @ B2 @ A2 ) ) ).
% local.sup.commute
thf(fact_21_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_22_local_Osup__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( sup @ ( sup @ X @ Y ) @ Z2 )
= ( sup @ X @ ( sup @ Y @ Z2 ) ) ) ).
% local.sup_assoc
thf(fact_23_local_Osup__commute,axiom,
! [X: a,Y: a] :
( ( sup @ X @ Y )
= ( sup @ Y @ X ) ) ).
% local.sup_commute
thf(fact_24_local_Osup__left__commute,axiom,
! [X: a,Y: a,Z2: a] :
( ( sup @ X @ ( sup @ Y @ Z2 ) )
= ( sup @ Y @ ( sup @ X @ Z2 ) ) ) ).
% local.sup_left_commute
thf(fact_25_local_Oinf_Oabsorb1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( inf @ A2 @ B2 )
= A2 ) ) ).
% local.inf.absorb1
thf(fact_26_local_Oinf_Oabsorb2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( inf @ A2 @ B2 )
= B2 ) ) ).
% local.inf.absorb2
thf(fact_27_local_Oinf_Oabsorb__iff1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( inf @ A2 @ B2 )
= A2 ) ) ).
% local.inf.absorb_iff1
thf(fact_28_local_Oinf_Oabsorb__iff2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( inf @ A2 @ B2 )
= B2 ) ) ).
% local.inf.absorb_iff2
thf(fact_29_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_30_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_31_local_Oinf_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% local.inf.cobounded1
thf(fact_32_local_Oinf_Ocobounded2,axiom,
! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ B2 ) ).
% local.inf.cobounded2
thf(fact_33_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_34_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_35_local_Oinf_OorderE,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( A2
= ( inf @ A2 @ B2 ) ) ) ).
% local.inf.orderE
thf(fact_36_local_Oinf_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( inf @ A2 @ B2 ) )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.inf.orderI
thf(fact_37_local_Oinf_Oorder__iff,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( A2
= ( inf @ A2 @ B2 ) ) ) ).
% local.inf.order_iff
thf(fact_38_local_Oinf__absorb1,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( inf @ X @ Y )
= X ) ) ).
% local.inf_absorb1
thf(fact_39_local_Oinf__absorb2,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( inf @ X @ Y )
= Y ) ) ).
% local.inf_absorb2
thf(fact_40_local_Oinf__greatest,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ X @ Z2 )
=> ( less_eq @ X @ ( inf @ Y @ Z2 ) ) ) ) ).
% local.inf_greatest
thf(fact_41_local_Oinf__le1,axiom,
! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% local.inf_le1
thf(fact_42_local_Oinf__le2,axiom,
! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% local.inf_le2
thf(fact_43_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_44_local_Oinf__unique,axiom,
! [F: a > a > a,X: a,Y: a] :
( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: a,Y4: a,Z3: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ( less_eq @ X3 @ Z3 )
=> ( less_eq @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% local.inf_unique
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_local_Ole__iff__inf,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( inf @ X @ Y )
= X ) ) ).
% local.le_iff_inf
thf(fact_50_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_51_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_52_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_53_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_54_local_Ole__iff__sup,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( sup @ X @ Y )
= Y ) ) ).
% local.le_iff_sup
thf(fact_55_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_56_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_57_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_58_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_59_local_Osup_Oabsorb1,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( sup @ A2 @ B2 )
= A2 ) ) ).
% local.sup.absorb1
thf(fact_60_local_Osup_Oabsorb2,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( sup @ A2 @ B2 )
= B2 ) ) ).
% local.sup.absorb2
thf(fact_61_local_Osup_Oabsorb__iff1,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( sup @ A2 @ B2 )
= A2 ) ) ).
% local.sup.absorb_iff1
thf(fact_62_local_Osup_Oabsorb__iff2,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( sup @ A2 @ B2 )
= B2 ) ) ).
% local.sup.absorb_iff2
thf(fact_63_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_64_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_65_local_Osup_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% local.sup.cobounded1
thf(fact_66_local_Osup_Ocobounded2,axiom,
! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% local.sup.cobounded2
thf(fact_67_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_68_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_69_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_70_local_Osup_OorderE,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( A2
= ( sup @ A2 @ B2 ) ) ) ).
% local.sup.orderE
thf(fact_71_local_Osup_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( sup @ A2 @ B2 ) )
=> ( less_eq @ B2 @ A2 ) ) ).
% local.sup.orderI
thf(fact_72_local_Osup_Oorder__iff,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( A2
= ( sup @ A2 @ B2 ) ) ) ).
% local.sup.order_iff
thf(fact_73_local_Osup__absorb1,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( sup @ X @ Y )
= X ) ) ).
% local.sup_absorb1
thf(fact_74_local_Osup__absorb2,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( sup @ X @ Y )
= Y ) ) ).
% local.sup_absorb2
thf(fact_75_local_Osup__ge1,axiom,
! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% local.sup_ge1
thf(fact_76_local_Osup__ge2,axiom,
! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% local.sup_ge2
thf(fact_77_local_Osup__least,axiom,
! [Y: a,X: a,Z2: a] :
( ( less_eq @ Y @ X )
=> ( ( less_eq @ Z2 @ X )
=> ( less_eq @ ( sup @ Y @ Z2 ) @ X ) ) ) ).
% local.sup_least
thf(fact_78_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_79_local_Osup__unique,axiom,
! [F: a > a > a,X: a,Y: a] :
( ! [X3: a,Y4: a] : ( less_eq @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: a,Y4: a] : ( less_eq @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: a,Y4: a,Z3: a] :
( ( less_eq @ Y4 @ X3 )
=> ( ( less_eq @ Z3 @ X3 )
=> ( less_eq @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% local.sup_unique
thf(fact_80_local_Odistrib__imp1,axiom,
! [X: a,Y: a,Z2: a] :
( ! [X3: a,Y4: a,Z3: a] :
( ( inf @ X3 @ ( sup @ Y4 @ Z3 ) )
= ( sup @ ( inf @ X3 @ Y4 ) @ ( inf @ X3 @ Z3 ) ) )
=> ( ( sup @ X @ ( inf @ Y @ Z2 ) )
= ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ) ).
% local.distrib_imp1
thf(fact_81_local_Odistrib__imp2,axiom,
! [X: a,Y: a,Z2: a] :
( ! [X3: a,Y4: a,Z3: a] :
( ( sup @ X3 @ ( inf @ Y4 @ Z3 ) )
= ( inf @ ( sup @ X3 @ Y4 ) @ ( sup @ X3 @ Z3 ) ) )
=> ( ( inf @ X @ ( sup @ Y @ Z2 ) )
= ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) ) ) ).
% local.distrib_imp2
thf(fact_82_local_Oa__join__d,axiom,
! [A2: a,B2: a,C: a] :
( ( sup @ A2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( sup @ A2 @ ( inf @ B2 @ C ) ) ) ).
% local.a_join_d
thf(fact_83_a__meet__d,axiom,
! [A2: a,B2: a,C: a] :
( ( inf @ A2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ C @ A2 ) ) ) ).
% a_meet_d
thf(fact_84_local_Ob__join__d,axiom,
! [B2: a,A2: a,C: a] :
( ( sup @ B2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( sup @ B2 @ ( inf @ C @ A2 ) ) ) ).
% local.b_join_d
thf(fact_85_b__meet__d,axiom,
! [B2: a,A2: a,C: a] :
( ( inf @ B2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( sup @ ( inf @ B2 @ C ) @ ( inf @ A2 @ B2 ) ) ) ).
% b_meet_d
thf(fact_86_c__meet__d,axiom,
! [C: a,A2: a,B2: a] :
( ( inf @ C @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( sup @ ( inf @ C @ A2 ) @ ( inf @ B2 @ C ) ) ) ).
% c_meet_d
thf(fact_87_local_Od__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 ) ) ) ).
% local.d_aux_def
thf(fact_88_local_Od__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 ) ) ).
% local.d_b_c_a
thf(fact_89_local_Od__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 ) ) ).
% local.d_c_a_b
thf(fact_90_local_Oa__meet__e,axiom,
! [A2: a,B2: a,C: a] :
( ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ).
% local.a_meet_e
thf(fact_91_local_Ob__meet__e,axiom,
! [B2: a,A2: a,C: a] :
( ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( inf @ B2 @ ( sup @ C @ A2 ) ) ) ).
% local.b_meet_e
thf(fact_92_local_Oc__meet__e,axiom,
! [C: a,A2: a,B2: a] :
( ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( inf @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.c_meet_e
thf(fact_93_local_Oe__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 ) ) ) ).
% local.e_aux_def
thf(fact_94_local_Oe__b__c__a,axiom,
! [B2: a,C: a,A2: a] :
( ( modula1593167561_e_aux @ a @ inf @ sup @ B2 @ C @ A2 )
= ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% local.e_b_c_a
thf(fact_95_local_Oe__c__a__b,axiom,
! [C: a,A2: a,B2: a] :
( ( modula1593167561_e_aux @ a @ inf @ sup @ C @ A2 @ B2 )
= ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% local.e_c_a_b
thf(fact_96_local_Odistrib__inf__le,axiom,
! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) @ ( inf @ X @ ( sup @ Y @ Z2 ) ) ) ).
% local.distrib_inf_le
thf(fact_97_local_Odistrib__sup__le,axiom,
! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z2 ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ).
% local.distrib_sup_le
thf(fact_98_local_Omodular,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ Y )
=> ( ( sup @ X @ ( inf @ Y @ Z2 ) )
= ( inf @ Y @ ( sup @ X @ Z2 ) ) ) ) ).
% local.modular
thf(fact_99_local_Oincomp__def,axiom,
! [X: a,Y: a] :
( ( modula1937309406incomp @ a @ less_eq @ X @ Y )
= ( ~ ( less_eq @ X @ Y )
& ~ ( less_eq @ Y @ X ) ) ) ).
% local.incomp_def
thf(fact_100_local_Oorder_Orefl,axiom,
! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% local.order.refl
thf(fact_101_local_Oorder__refl,axiom,
! [X: a] : ( less_eq @ X @ X ) ).
% local.order_refl
thf(fact_102_local_Oinf_Oidem,axiom,
! [A2: a] :
( ( inf @ A2 @ A2 )
= A2 ) ).
% local.inf.idem
thf(fact_103_local_Oinf_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.left_idem
thf(fact_104_local_Oinf_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.right_idem
thf(fact_105_local_Oinf__idem,axiom,
! [X: a] :
( ( inf @ X @ X )
= X ) ).
% local.inf_idem
thf(fact_106_local_Oinf__left__idem,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( inf @ X @ Y ) )
= ( inf @ X @ Y ) ) ).
% local.inf_left_idem
thf(fact_107_local_Oinf__right__idem,axiom,
! [X: a,Y: a] :
( ( inf @ ( inf @ X @ Y ) @ Y )
= ( inf @ X @ Y ) ) ).
% local.inf_right_idem
thf(fact_108_local_Osup_Oidem,axiom,
! [A2: a] :
( ( sup @ A2 @ A2 )
= A2 ) ).
% local.sup.idem
thf(fact_109_local_Osup_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.left_idem
thf(fact_110_local_Osup_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.right_idem
thf(fact_111_local_Osup__idem,axiom,
! [X: a] :
( ( sup @ X @ X )
= X ) ).
% local.sup_idem
thf(fact_112_local_Osup__left__idem,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( sup @ X @ Y ) )
= ( sup @ X @ Y ) ) ).
% local.sup_left_idem
thf(fact_113_local_Oc__a,axiom,
! [A2: a,B2: a,C: a] :
( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( modula105138637_a_aux @ a @ inf @ sup @ C @ A2 @ B2 ) ) ).
% local.c_a
thf(fact_114_local_OGreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ Y4 @ X ) )
=> ( ! [X3: a] :
( ( P @ X3 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( less_eq @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( greatest @ a @ less_eq @ P ) ) ) ) ) ).
% local.GreatestI2_order
thf(fact_115_local_OGreatest__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ Y4 @ 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_Oa__aux__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula105138637_a_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( sup @ ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% local.a_aux_def
thf(fact_119_local_Oc__aux__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( sup @ ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% local.c_aux_def
thf(fact_120_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_121_local_Ole__inf__iff,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ ( inf @ Y @ Z2 ) )
= ( ( less_eq @ X @ Y )
& ( less_eq @ X @ Z2 ) ) ) ).
% local.le_inf_iff
thf(fact_122_local_Ole__sup__iff,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ ( sup @ X @ Y ) @ Z2 )
= ( ( less_eq @ X @ Z2 )
& ( less_eq @ Y @ Z2 ) ) ) ).
% local.le_sup_iff
thf(fact_123_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_124_local_Oinf__sup__absorb,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( sup @ X @ Y ) )
= X ) ).
% local.inf_sup_absorb
thf(fact_125_local_Osup__inf__absorb,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( inf @ X @ Y ) )
= X ) ).
% local.sup_inf_absorb
thf(fact_126_local_Ob__a,axiom,
! [A2: a,B2: a,C: a] :
( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( modula105138637_a_aux @ a @ inf @ sup @ B2 @ C @ A2 ) ) ).
% local.b_a
thf(fact_127_local_OLeast1I,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ X4 @ Y4 ) )
& ! [Y4: a] :
( ( ( P @ Y4 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( less_eq @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( least @ a @ less_eq @ P ) ) ) ).
% local.Least1I
thf(fact_128_local_OLeast1__le,axiom,
! [P: a > $o,Z2: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ X4 @ Y4 ) )
& ! [Y4: a] :
( ( ( P @ Y4 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( less_eq @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z2 )
=> ( less_eq @ ( least @ a @ less_eq @ P ) @ Z2 ) ) ) ).
% local.Least1_le
thf(fact_129_local_OLeastI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ X @ Y4 ) )
=> ( ! [X3: a] :
( ( P @ X3 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( less_eq @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% local.LeastI2_order
thf(fact_130_local_OLeast__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ X @ Y4 ) )
=> ( ( least @ a @ less_eq @ P )
= X ) ) ) ).
% local.Least_equality
thf(fact_131_local_Ob__aux__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
= ( sup @ ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% local.b_aux_def
thf(fact_132_a__meet__b__eq__d,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
=> ( ( inf @ ( modula105138637_a_aux @ a @ inf @ sup @ A2 @ B2 @ C ) @ ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
= ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% a_meet_b_eq_d
thf(fact_133_lattice_Oa__aux_Ocong,axiom,
! [A: $tType] :
( ( modula105138637_a_aux @ A )
= ( modula105138637_a_aux @ A ) ) ).
% lattice.a_aux.cong
thf(fact_134_lattice_Oc__aux_Ocong,axiom,
! [A: $tType] :
( ( modula849153099_c_aux @ A )
= ( modula849153099_c_aux @ A ) ) ).
% lattice.c_aux.cong
thf(fact_135_lattice_Od__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1221160330_d_aux @ A )
= ( modula1221160330_d_aux @ A ) ) ).
% lattice.d_aux.cong
thf(fact_136_lattice_Oe__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1593167561_e_aux @ A )
= ( modula1593167561_e_aux @ A ) ) ).
% lattice.e_aux.cong
thf(fact_137_local_Ocomp__fun__idem__sup,axiom,
finite_comp_fun_idem @ a @ a @ sup ).
% local.comp_fun_idem_sup
thf(fact_138_local_Ocomp__fun__idem__inf,axiom,
finite_comp_fun_idem @ a @ a @ inf ).
% local.comp_fun_idem_inf
thf(fact_139_local_Osup_Osemigroup__axioms,axiom,
semigroup @ a @ sup ).
% local.sup.semigroup_axioms
thf(fact_140_local_Oinf_Osemigroup__axioms,axiom,
semigroup @ a @ inf ).
% local.inf.semigroup_axioms
thf(fact_141_local_Osup_Osemilattice__axioms,axiom,
semilattice @ a @ sup ).
% local.sup.semilattice_axioms
thf(fact_142_local_Oinf_Osemilattice__axioms,axiom,
semilattice @ a @ inf ).
% local.inf.semilattice_axioms
thf(fact_143_local_Osup_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ sup ).
% local.sup.abel_semigroup_axioms
thf(fact_144_local_Oinf_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ inf ).
% local.inf.abel_semigroup_axioms
thf(fact_145_local_OSup__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ sup ).
% local.Sup_fin.semilattice_set_axioms
thf(fact_146_local_OInf__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ inf ).
% local.Inf_fin.semilattice_set_axioms
thf(fact_147_local_Obdd__above__def,axiom,
! [A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ A4 )
= ( ? [M: a] :
! [X2: a] :
( ( member @ a @ X2 @ A4 )
=> ( less_eq @ X2 @ M ) ) ) ) ).
% local.bdd_above_def
thf(fact_148_local_Obdd__below__def,axiom,
! [A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ A4 )
= ( ? [M2: a] :
! [X2: a] :
( ( member @ a @ X2 @ A4 )
=> ( less_eq @ M2 @ X2 ) ) ) ) ).
% local.bdd_below_def
thf(fact_149_local_Obdd__belowI,axiom,
! [A4: set @ a,M3: a] :
( ! [X3: a] :
( ( member @ a @ X3 @ A4 )
=> ( less_eq @ M3 @ X3 ) )
=> ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% local.bdd_belowI
thf(fact_150_local_Obdd__aboveI,axiom,
! [A4: set @ a,M4: a] :
( ! [X3: a] :
( ( member @ a @ X3 @ A4 )
=> ( less_eq @ X3 @ M4 ) )
=> ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% local.bdd_aboveI
thf(fact_151_lattice_Oincomp_Ocong,axiom,
! [A: $tType] :
( ( modula1937309406incomp @ A )
= ( modula1937309406incomp @ A ) ) ).
% lattice.incomp.cong
thf(fact_152_lattice_Ob__aux_Ocong,axiom,
! [A: $tType] :
( ( modula477145868_b_aux @ A )
= ( modula477145868_b_aux @ A ) ) ).
% lattice.b_aux.cong
thf(fact_153_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_154_semilattice__set__def,axiom,
! [A: $tType] :
( ( lattic35693393ce_set @ A )
= ( semilattice @ A ) ) ).
% semilattice_set_def
thf(fact_155_semilattice__set_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( lattic35693393ce_set @ A @ F ) ) ).
% semilattice_set.intro
thf(fact_156_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_157_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_158_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_159_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_160_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F2: A > A > A] :
! [A3: A,B3: A,C3: A] :
( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
= ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_161_semilattice__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A] :
( ( lattic35693393ce_set @ A @ F )
=> ( semilattice @ A @ F ) ) ).
% semilattice_set.axioms
thf(fact_162_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ).
% semilattice.axioms(1)
thf(fact_163_local_Obdd__above__mono,axiom,
! [B5: set @ a,A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ B5 )
=> ( ( ord_less_eq @ ( set @ a ) @ A4 @ B5 )
=> ( condit2040224947_above @ a @ less_eq @ A4 ) ) ) ).
% local.bdd_above_mono
thf(fact_164_local_Obdd__below__mono,axiom,
! [B5: set @ a,A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ B5 )
=> ( ( ord_less_eq @ ( set @ a ) @ A4 @ B5 )
=> ( condit1201339847_below @ a @ less_eq @ A4 ) ) ) ).
% local.bdd_below_mono
thf(fact_165_local_Oantimono__def,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ( antimono @ a @ B @ less_eq @ F )
= ( ! [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ) ).
% local.antimono_def
thf(fact_166_local_OantimonoI,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ! [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% local.antimonoI
thf(fact_167_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_168_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_169_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_170_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_171_local_OmonoI,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ! [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( mono @ a @ B @ less_eq @ F ) ) ) ).
% local.monoI
thf(fact_172_local_Omono__def,axiom,
! [B: $tType] :
( ( order @ B )
=> ! [F: a > B] :
( ( mono @ a @ B @ less_eq @ F )
= ( ! [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% local.mono_def
thf(fact_173_semilattice_Oidem,axiom,
! [A: $tType,F: A > A > A,A2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A2 @ A2 )
= A2 ) ) ).
% semilattice.idem
thf(fact_174_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_175_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_176_subsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% subsetI
thf(fact_177_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_178_preorder__class_Oorder__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% preorder_class.order_refl
thf(fact_179_order_Oantimono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( antimono @ A @ B )
= ( antimono @ A @ B ) ) ) ).
% order.antimono.cong
thf(fact_180_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_181_order__class_Odual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.dual_order.antisym
thf(fact_182_order__class_Odual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_183_order__class_Odual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% order_class.dual_order.trans
thf(fact_184_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_185_order__class_Odual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% order_class.dual_order.refl
thf(fact_186_preorder__class_Oorder__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% preorder_class.order_trans
thf(fact_187_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_188_ord__class_Oord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_189_ord__class_Oord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_190_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_191_order__class_Oantisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_class.antisym_conv
thf(fact_192_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_193_order__class_Oorder_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order_class.order.trans
thf(fact_194_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_195_preorder__class_Oeq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% preorder_class.eq_refl
thf(fact_196_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_197_order__class_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_class.antisym
thf(fact_198_order__class_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% order_class.eq_iff
thf(fact_199_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_200_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_201_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_202_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_203_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_204_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_205_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_206_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_207_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_208_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_209_subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% subset_trans
thf(fact_210_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_211_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_212_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_213_equalityD2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_214_equalityD1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_215_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_216_equalityE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_217_subsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B5 ) ) ) ).
% subsetD
thf(fact_218_in__mono,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_219_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_220_order_OGreatest_Ocong,axiom,
! [A: $tType] :
( ( greatest @ A )
= ( greatest @ A ) ) ).
% order.Greatest.cong
thf(fact_221_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_222_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_223_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq: A > A > $o,A3: A,B3: A] : ( if @ A @ ( Less_eq @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% ord.min_def
thf(fact_224_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq: A > A > $o,A3: A,B3: A] : ( if @ A @ ( Less_eq @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% ord.max_def
thf(fact_225_local_Omono__inf,axiom,
! [B: $tType] :
( ( semilattice_inf @ B )
=> ! [F: a > B,A4: a,B5: a] :
( ( mono @ a @ B @ less_eq @ F )
=> ( ord_less_eq @ B @ ( F @ ( inf @ A4 @ B5 ) ) @ ( inf_inf @ B @ ( F @ A4 ) @ ( F @ B5 ) ) ) ) ) ).
% local.mono_inf
thf(fact_226_local_Omono__sup,axiom,
! [B: $tType] :
( ( semilattice_sup @ B )
=> ! [F: a > B,A4: a,B5: a] :
( ( mono @ a @ B @ less_eq @ F )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A4 ) @ ( F @ B5 ) ) @ ( F @ ( sup @ A4 @ B5 ) ) ) ) ) ).
% local.mono_sup
thf(fact_227_local_Obdd__above__finite,axiom,
! [A4: set @ a] :
( ( finite_finite @ a @ A4 )
=> ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% local.bdd_above_finite
thf(fact_228_local_Ofinite__has__minimal2,axiom,
! [A4: set @ a,A2: a] :
( ( finite_finite @ a @ A4 )
=> ( ( member @ a @ A2 @ A4 )
=> ? [X3: a] :
( ( member @ a @ X3 @ A4 )
& ( less_eq @ X3 @ A2 )
& ! [Xa: a] :
( ( member @ a @ Xa @ A4 )
=> ( ( less_eq @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% local.finite_has_minimal2
thf(fact_229_local_Ofinite__has__maximal2,axiom,
! [A4: set @ a,A2: a] :
( ( finite_finite @ a @ A4 )
=> ( ( member @ a @ A2 @ A4 )
=> ? [X3: a] :
( ( member @ a @ X3 @ A4 )
& ( less_eq @ A2 @ X3 )
& ! [Xa: a] :
( ( member @ a @ Xa @ A4 )
=> ( ( less_eq @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% local.finite_has_maximal2
thf(fact_230_local_Obdd__below__Int2,axiom,
! [B5: set @ a,A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ B5 )
=> ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B5 ) ) ) ).
% local.bdd_below_Int2
thf(fact_231_local_Obdd__below__Int1,axiom,
! [A4: set @ a,B5: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ A4 )
=> ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B5 ) ) ) ).
% local.bdd_below_Int1
thf(fact_232_local_Obdd__above__Int2,axiom,
! [B5: set @ a,A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ B5 )
=> ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B5 ) ) ) ).
% local.bdd_above_Int2
thf(fact_233_local_Obdd__above__Int1,axiom,
! [A4: set @ a,B5: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ A4 )
=> ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B5 ) ) ) ).
% local.bdd_above_Int1
thf(fact_234_local_Obdd__below__finite,axiom,
! [A4: set @ a] :
( ( finite_finite @ a @ A4 )
=> ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% local.bdd_below_finite
thf(fact_235_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C5 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_236_Int__subset__iff,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A4 @ B5 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C5 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C5 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_237_semilattice__inf__class_Oinf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% semilattice_inf_class.inf_right_idem
thf(fact_238_semilattice__inf__class_Oinf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% semilattice_inf_class.inf.right_idem
thf(fact_239_semilattice__inf__class_Oinf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% semilattice_inf_class.inf_left_idem
thf(fact_240_semilattice__inf__class_Oinf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% semilattice_inf_class.inf.left_idem
thf(fact_241_semilattice__inf__class_Oinf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% semilattice_inf_class.inf_idem
thf(fact_242_semilattice__inf__class_Oinf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% semilattice_inf_class.inf.idem
thf(fact_243_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_244_semilattice__sup__class_Osup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% semilattice_sup_class.sup.right_idem
thf(fact_245_semilattice__sup__class_Osup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% semilattice_sup_class.sup_left_idem
thf(fact_246_semilattice__sup__class_Osup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% semilattice_sup_class.sup.left_idem
thf(fact_247_semilattice__sup__class_Osup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% semilattice_sup_class.sup_idem
thf(fact_248_semilattice__sup__class_Osup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% semilattice_sup_class.sup.idem
thf(fact_249_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_250_semilattice__inf__class_Ole__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% semilattice_inf_class.le_inf_iff
thf(fact_251_semilattice__inf__class_Oinf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% semilattice_inf_class.inf.bounded_iff
thf(fact_252_semilattice__sup__class_Ole__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% semilattice_sup_class.le_sup_iff
thf(fact_253_semilattice__sup__class_Osup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% semilattice_sup_class.sup.bounded_iff
thf(fact_254_lattice__class_Osup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% lattice_class.sup_inf_absorb
thf(fact_255_lattice__class_Oinf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% lattice_class.inf_sup_absorb
% Type constructors (19)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 )
=> ( semilattice_sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_inf @ A8 )
=> ( semilattice_inf @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 )
=> ( lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_2,axiom,
! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_8,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_10,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
less_eq @ ( modula1221160330_d_aux @ a @ inf @ sup @ a2 @ b @ c ) @ ( modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c ) ).
thf(conj_1,conjecture,
( ( inf @ ( modula105138637_a_aux @ a @ inf @ sup @ b @ c @ a2 ) @ ( modula849153099_c_aux @ a @ inf @ sup @ a2 @ b @ c ) )
= ( modula1221160330_d_aux @ a @ inf @ sup @ a2 @ b @ c ) ) ).
%------------------------------------------------------------------------------