TPTP Problem File: ITP123^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP123^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Modular_Distrib_Lattice problem prob_319__3264956_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_319__3264956_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.2.0, 0.33 v8.1.0, 0.00 v7.5.0
% Syntax : Number of formulae : 299 ( 129 unt; 38 typ; 0 def)
% Number of atoms : 599 ( 194 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2079 ( 51 ~; 4 |; 39 &;1756 @)
% ( 0 <=>; 229 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 396 ( 396 >; 0 *; 0 +; 0 <<)
% Number of symbols : 38 ( 37 usr; 3 con; 0-7 aty)
% Number of variables : 774 ( 25 ^; 709 !; 8 ?; 774 :)
% ( 32 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:30:15.471
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (36)
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_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_Oclass_Olattice,type,
lattice:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Oord_Ois__arg__max,type,
lattices_is_arg_max:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Oord_Ois__arg__min,type,
lattices_is_arg_min:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic1693879045er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
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_OM5__lattice,type,
modula1618706180attice:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice,type,
modula1192512645attice:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > 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_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib,type,
modula141148890istrib:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > 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_Oorder_OGreatest,type,
greatest:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr1161524159ssThan:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_inf,type,
inf: a > a > a ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_less__eq,type,
less_eq: a > a > $o ).
thf(sy_v_sup,type,
sup: a > a > a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_z,type,
z: 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_Odual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ~ ( less @ A2 @ B2 ) ) ).
% local.dual_order.asym
thf(fact_14_local_Odual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% local.dual_order.strict_implies_not_eq
thf(fact_15_local_Odual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( less @ B2 @ A2 )
=> ( ( less @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans
thf(fact_16_local_Oless__asym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_asym
thf(fact_17_local_Oless__asym_H,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.less_asym'
thf(fact_18_local_Oless__imp__neq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_neq
thf(fact_19_local_Oless__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_not_eq
thf(fact_20_local_Oless__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( Y != X ) ) ).
% local.less_imp_not_eq2
thf(fact_21_local_Oless__imp__not__less,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_imp_not_less
thf(fact_22_local_Oless__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( less @ X @ Y )
=> ( ( less @ Y @ X )
=> P ) ) ).
% local.less_imp_triv
thf(fact_23_local_Oless__irrefl,axiom,
! [X: a] :
~ ( less @ X @ X ) ).
% local.less_irrefl
thf(fact_24_local_Oless__not__sym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_not_sym
thf(fact_25_local_Oless__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less @ X @ Y )
=> ( ( less @ Y @ Z2 )
=> ( less @ X @ Z2 ) ) ) ).
% local.less_trans
thf(fact_26_local_Oord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_27_local_Oord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_28_local_Oorder_Oasym,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.order.asym
thf(fact_29_local_Oorder_Oirrefl,axiom,
! [A2: a] :
~ ( less @ A2 @ A2 ) ).
% local.order.irrefl
thf(fact_30_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% local.order.strict_implies_not_eq
thf(fact_31_local_Oorder_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans
thf(fact_32_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_33_local_Oinf_Ocommute,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ B2 )
= ( inf @ B2 @ A2 ) ) ).
% local.inf.commute
thf(fact_34_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_35_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_36_local_Oinf__commute,axiom,
! [X: a,Y: a] :
( ( inf @ X @ Y )
= ( inf @ Y @ X ) ) ).
% local.inf_commute
thf(fact_37_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_38_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_39_local_Osup_Ocommute,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ B2 )
= ( sup @ B2 @ A2 ) ) ).
% local.sup.commute
thf(fact_40_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_41_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_42_local_Osup__commute,axiom,
! [X: a,Y: a] :
( ( sup @ X @ Y )
= ( sup @ Y @ X ) ) ).
% local.sup_commute
thf(fact_43_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_44_local_Oantisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv1
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_Oantisym__conv2,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv2
thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( less_eq @ B2 @ A2 )
=> ( less @ B2 @ A2 ) ) ) ).
% local.dual_order.not_eq_order_implies_strict
thf(fact_51_local_Odual__order_Oorder__iff__strict,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( less @ B2 @ A2 )
| ( A2 = B2 ) ) ) ).
% local.dual_order.order_iff_strict
thf(fact_52_local_Odual__order_Ostrict__iff__order,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
= ( ( less_eq @ B2 @ A2 )
& ( A2 != B2 ) ) ) ).
% local.dual_order.strict_iff_order
thf(fact_53_local_Odual__order_Ostrict__implies__order,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ( less_eq @ B2 @ A2 ) ) ).
% local.dual_order.strict_implies_order
thf(fact_54_local_Odual__order_Ostrict__trans1,axiom,
! [B2: a,A2: a,C: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans1
thf(fact_55_local_Odual__order_Ostrict__trans2,axiom,
! [B2: a,A2: a,C: a] :
( ( less @ B2 @ A2 )
=> ( ( less_eq @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans2
thf(fact_56_local_OleD,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ~ ( less @ X @ Y ) ) ).
% local.leD
thf(fact_57_local_Ole__imp__less__or__eq,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less @ X @ Y )
| ( X = Y ) ) ) ).
% local.le_imp_less_or_eq
thf(fact_58_local_Ole__less,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( less @ X @ Y )
| ( X = Y ) ) ) ).
% local.le_less
thf(fact_59_local_Ole__less__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ Y )
=> ( ( less @ Y @ Z2 )
=> ( less @ X @ Z2 ) ) ) ).
% local.le_less_trans
thf(fact_60_local_Ole__neq__trans,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( less @ A2 @ B2 ) ) ) ).
% local.le_neq_trans
thf(fact_61_local_Oless__imp__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( less_eq @ X @ Y ) ) ).
% local.less_imp_le
thf(fact_62_local_Oless__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
= ( ( less_eq @ X @ Y )
& ( X != Y ) ) ) ).
% local.less_le
thf(fact_63_local_Oless__le__not__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
= ( ( less_eq @ X @ Y )
& ~ ( less_eq @ Y @ X ) ) ) ).
% local.less_le_not_le
thf(fact_64_local_Oless__le__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less @ X @ Y )
=> ( ( less_eq @ Y @ Z2 )
=> ( less @ X @ Z2 ) ) ) ).
% local.less_le_trans
thf(fact_65_local_Oorder_Onot__eq__order__implies__strict,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( less_eq @ A2 @ B2 )
=> ( less @ A2 @ B2 ) ) ) ).
% local.order.not_eq_order_implies_strict
thf(fact_66_local_Oorder_Oorder__iff__strict,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( less @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% local.order.order_iff_strict
thf(fact_67_local_Oorder_Ostrict__iff__order,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
= ( ( less_eq @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% local.order.strict_iff_order
thf(fact_68_local_Oorder_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.order.strict_implies_order
thf(fact_69_local_Oorder_Ostrict__trans1,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans1
thf(fact_70_local_Oorder_Ostrict__trans2,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans2
thf(fact_71_local_Oinf_Oabsorb1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( inf @ A2 @ B2 )
= A2 ) ) ).
% local.inf.absorb1
thf(fact_72_local_Oinf_Oabsorb2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( inf @ A2 @ B2 )
= B2 ) ) ).
% local.inf.absorb2
thf(fact_73_local_Oinf_Oabsorb__iff1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( inf @ A2 @ B2 )
= A2 ) ) ).
% local.inf.absorb_iff1
thf(fact_74_local_Oinf_Oabsorb__iff2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( inf @ A2 @ B2 )
= B2 ) ) ).
% local.inf.absorb_iff2
thf(fact_75_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_76_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_77_local_Oinf_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% local.inf.cobounded1
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_OcoboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( less_eq @ A2 @ C )
=> ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.coboundedI1
thf(fact_80_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_81_local_Oinf_OorderE,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( A2
= ( inf @ A2 @ B2 ) ) ) ).
% local.inf.orderE
thf(fact_82_local_Oinf_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( inf @ A2 @ B2 ) )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.inf.orderI
thf(fact_83_local_Oinf_Oorder__iff,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( A2
= ( inf @ A2 @ B2 ) ) ) ).
% local.inf.order_iff
thf(fact_84_local_Oinf__absorb1,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( inf @ X @ Y )
= X ) ) ).
% local.inf_absorb1
thf(fact_85_local_Oinf__absorb2,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( inf @ X @ Y )
= Y ) ) ).
% local.inf_absorb2
thf(fact_86_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_87_local_Oinf__le1,axiom,
! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% local.inf_le1
thf(fact_88_local_Oinf__le2,axiom,
! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% local.inf_le2
thf(fact_89_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_90_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_91_local_Ole__iff__inf,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( inf @ X @ Y )
= X ) ) ).
% local.le_iff_inf
thf(fact_92_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_93_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_94_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_95_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_96_local_Ole__iff__sup,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( sup @ X @ Y )
= Y ) ) ).
% local.le_iff_sup
thf(fact_97_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_98_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_99_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_100_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_101_local_Osup_Oabsorb1,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( sup @ A2 @ B2 )
= A2 ) ) ).
% local.sup.absorb1
thf(fact_102_local_Osup_Oabsorb2,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( sup @ A2 @ B2 )
= B2 ) ) ).
% local.sup.absorb2
thf(fact_103_local_Osup_Oabsorb__iff1,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( sup @ A2 @ B2 )
= A2 ) ) ).
% local.sup.absorb_iff1
thf(fact_104_local_Osup_Oabsorb__iff2,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( sup @ A2 @ B2 )
= B2 ) ) ).
% local.sup.absorb_iff2
thf(fact_105_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_106_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_107_local_Osup_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% local.sup.cobounded1
thf(fact_108_local_Osup_Ocobounded2,axiom,
! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% local.sup.cobounded2
thf(fact_109_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_110_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_111_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_112_local_Osup_OorderE,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( A2
= ( sup @ A2 @ B2 ) ) ) ).
% local.sup.orderE
thf(fact_113_local_Osup_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( sup @ A2 @ B2 ) )
=> ( less_eq @ B2 @ A2 ) ) ).
% local.sup.orderI
thf(fact_114_local_Osup_Oorder__iff,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( A2
= ( sup @ A2 @ B2 ) ) ) ).
% local.sup.order_iff
thf(fact_115_local_Osup__absorb1,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( sup @ X @ Y )
= X ) ) ).
% local.sup_absorb1
thf(fact_116_local_Osup__absorb2,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( sup @ X @ Y )
= Y ) ) ).
% local.sup_absorb2
thf(fact_117_local_Osup__ge1,axiom,
! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% local.sup_ge1
thf(fact_118_local_Osup__ge2,axiom,
! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% local.sup_ge2
thf(fact_119_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_120_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_121_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_122_local_Oinf_Ostrict__boundedE,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ ( inf @ B2 @ C ) )
=> ~ ( ( less @ A2 @ B2 )
=> ~ ( less @ A2 @ C ) ) ) ).
% local.inf.strict_boundedE
thf(fact_123_local_Oinf_Ostrict__coboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( less @ A2 @ C )
=> ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.strict_coboundedI1
thf(fact_124_local_Oinf_Ostrict__coboundedI2,axiom,
! [B2: a,C: a,A2: a] :
( ( less @ B2 @ C )
=> ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.strict_coboundedI2
thf(fact_125_local_Oinf_Ostrict__order__iff,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
= ( ( A2
= ( inf @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% local.inf.strict_order_iff
thf(fact_126_local_Oless__infI1,axiom,
! [A2: a,X: a,B2: a] :
( ( less @ A2 @ X )
=> ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% local.less_infI1
thf(fact_127_local_Oless__infI2,axiom,
! [B2: a,X: a,A2: a] :
( ( less @ B2 @ X )
=> ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% local.less_infI2
thf(fact_128_local_Oless__supI1,axiom,
! [X: a,A2: a,B2: a] :
( ( less @ X @ A2 )
=> ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% local.less_supI1
thf(fact_129_local_Oless__supI2,axiom,
! [X: a,B2: a,A2: a] :
( ( less @ X @ B2 )
=> ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% local.less_supI2
thf(fact_130_local_Osup_Ostrict__boundedE,axiom,
! [B2: a,C: a,A2: a] :
( ( less @ ( sup @ B2 @ C ) @ A2 )
=> ~ ( ( less @ B2 @ A2 )
=> ~ ( less @ C @ A2 ) ) ) ).
% local.sup.strict_boundedE
thf(fact_131_local_Osup_Ostrict__coboundedI1,axiom,
! [C: a,A2: a,B2: a] :
( ( less @ C @ A2 )
=> ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.sup.strict_coboundedI1
thf(fact_132_local_Osup_Ostrict__coboundedI2,axiom,
! [C: a,B2: a,A2: a] :
( ( less @ C @ B2 )
=> ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.sup.strict_coboundedI2
thf(fact_133_local_Osup_Ostrict__order__iff,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
= ( ( A2
= ( sup @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% local.sup.strict_order_iff
thf(fact_134_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_135_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_136_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_137_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_138_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_139_local_Oorder_Orefl,axiom,
! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% local.order.refl
thf(fact_140_local_Oorder__refl,axiom,
! [X: a] : ( less_eq @ X @ X ) ).
% local.order_refl
thf(fact_141_local_Oinf_Oidem,axiom,
! [A2: a] :
( ( inf @ A2 @ A2 )
= A2 ) ).
% local.inf.idem
thf(fact_142_local_Oinf_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.left_idem
thf(fact_143_local_Oinf_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.right_idem
thf(fact_144_local_Oinf__idem,axiom,
! [X: a] :
( ( inf @ X @ X )
= X ) ).
% local.inf_idem
thf(fact_145_local_Oinf__left__idem,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( inf @ X @ Y ) )
= ( inf @ X @ Y ) ) ).
% local.inf_left_idem
thf(fact_146_local_Oinf__right__idem,axiom,
! [X: a,Y: a] :
( ( inf @ ( inf @ X @ Y ) @ Y )
= ( inf @ X @ Y ) ) ).
% local.inf_right_idem
thf(fact_147_local_Osup_Oidem,axiom,
! [A2: a] :
( ( sup @ A2 @ A2 )
= A2 ) ).
% local.sup.idem
thf(fact_148_local_Osup_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.left_idem
thf(fact_149_local_Osup_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.right_idem
thf(fact_150_local_Osup__idem,axiom,
! [X: a] :
( ( sup @ X @ X )
= X ) ).
% local.sup_idem
thf(fact_151_local_Osup__left__idem,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( sup @ X @ Y ) )
= ( sup @ X @ Y ) ) ).
% local.sup_left_idem
thf(fact_152_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_153_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_154_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_155_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_156_local_Ois__arg__max__def,axiom,
! [B: $tType,F: B > a,P: B > $o,X: B] :
( ( lattices_is_arg_max @ a @ B @ less @ F @ P @ X )
= ( ( P @ X )
& ~ ? [Y3: B] :
( ( P @ Y3 )
& ( less @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% local.is_arg_max_def
thf(fact_157_local_Ois__arg__min__def,axiom,
! [B: $tType,F: B > a,P: B > $o,X: B] :
( ( lattices_is_arg_min @ a @ B @ less @ F @ P @ X )
= ( ( P @ X )
& ~ ? [Y3: B] :
( ( P @ Y3 )
& ( less @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% local.is_arg_min_def
thf(fact_158_local_ON5__lattice__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula1192512645attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
= ( ( ( inf @ A2 @ C )
= ( inf @ B2 @ C ) )
& ( less @ A2 @ B2 )
& ( ( sup @ A2 @ C )
= ( sup @ B2 @ C ) ) ) ) ).
% local.N5_lattice_def
thf(fact_159_local_Ono__distrib__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula141148890istrib @ a @ inf @ less @ sup @ A2 @ B2 @ C )
= ( less @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ C @ A2 ) ) @ ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ) ).
% local.no_distrib_def
thf(fact_160_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_161_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_162_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_163_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_164_local_Oinf__sup__absorb,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( sup @ X @ Y ) )
= X ) ).
% local.inf_sup_absorb
thf(fact_165_local_Osup__inf__absorb,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( inf @ X @ Y ) )
= X ) ).
% local.sup_inf_absorb
thf(fact_166_local_OM5__lattice__def,axiom,
! [A2: a,B2: a,C: a] :
( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
= ( ( ( inf @ A2 @ B2 )
= ( inf @ B2 @ C ) )
& ( ( inf @ C @ A2 )
= ( inf @ B2 @ C ) )
& ( ( sup @ A2 @ B2 )
= ( sup @ B2 @ C ) )
& ( ( sup @ C @ A2 )
= ( sup @ B2 @ C ) )
& ( less @ ( inf @ A2 @ B2 ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% local.M5_lattice_def
thf(fact_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_local_OInf__fin_Osemilattice__order__set__axioms,axiom,
lattic1693879045er_set @ a @ inf @ less_eq @ less ).
% local.Inf_fin.semilattice_order_set_axioms
thf(fact_183_local_OM5__lattice__incomp,axiom,
! [A2: a,B2: a,C: a] :
( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
=> ( modula1937309406incomp @ a @ less_eq @ A2 @ B2 ) ) ).
% local.M5_lattice_incomp
thf(fact_184_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_185_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_186_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_187_lattice_Ono__distrib_Ocong,axiom,
! [A: $tType] :
( ( modula141148890istrib @ A )
= ( modula141148890istrib @ A ) ) ).
% lattice.no_distrib.cong
thf(fact_188_lattice_ON5__lattice_Ocong,axiom,
! [A: $tType] :
( ( modula1192512645attice @ A )
= ( modula1192512645attice @ A ) ) ).
% lattice.N5_lattice.cong
thf(fact_189_lattice_OM5__lattice_Ocong,axiom,
! [A: $tType] :
( ( modula1618706180attice @ A )
= ( modula1618706180attice @ A ) ) ).
% lattice.M5_lattice.cong
thf(fact_190_lattice_Oincomp_Ocong,axiom,
! [A: $tType] :
( ( modula1937309406incomp @ A )
= ( modula1937309406incomp @ A ) ) ).
% lattice.incomp.cong
thf(fact_191_lattice_Oe__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1593167561_e_aux @ A )
= ( modula1593167561_e_aux @ A ) ) ).
% lattice.e_aux.cong
thf(fact_192_lattice_Od__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1221160330_d_aux @ A )
= ( modula1221160330_d_aux @ A ) ) ).
% lattice.d_aux.cong
thf(fact_193_local_Oinf_Osemilattice__order__axioms,axiom,
semilattice_order @ a @ inf @ less_eq @ less ).
% local.inf.semilattice_order_axioms
thf(fact_194_local_Oorder_Oordering__axioms,axiom,
ordering @ a @ less_eq @ less ).
% local.order.ordering_axioms
thf(fact_195_local_Ocomp__fun__idem__sup,axiom,
finite_comp_fun_idem @ a @ a @ sup ).
% local.comp_fun_idem_sup
thf(fact_196_local_Ocomp__fun__idem__inf,axiom,
finite_comp_fun_idem @ a @ a @ inf ).
% local.comp_fun_idem_inf
thf(fact_197_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_198_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_199_lattice_Oc__aux_Ocong,axiom,
! [A: $tType] :
( ( modula849153099_c_aux @ A )
= ( modula849153099_c_aux @ A ) ) ).
% lattice.c_aux.cong
thf(fact_200_lattice_Ob__aux_Ocong,axiom,
! [A: $tType] :
( ( modula477145868_b_aux @ A )
= ( modula477145868_b_aux @ A ) ) ).
% lattice.b_aux.cong
thf(fact_201_lattice_Oa__aux_Ocong,axiom,
! [A: $tType] :
( ( modula105138637_a_aux @ A )
= ( modula105138637_a_aux @ A ) ) ).
% lattice.a_aux.cong
thf(fact_202_local_Osup_Osemigroup__axioms,axiom,
semigroup @ a @ sup ).
% local.sup.semigroup_axioms
thf(fact_203_local_Oinf_Osemigroup__axioms,axiom,
semigroup @ a @ inf ).
% local.inf.semigroup_axioms
thf(fact_204_local_Osup_Osemilattice__axioms,axiom,
semilattice @ a @ sup ).
% local.sup.semilattice_axioms
thf(fact_205_local_Oinf_Osemilattice__axioms,axiom,
semilattice @ a @ inf ).
% local.inf.semilattice_axioms
thf(fact_206_local_Osup_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ sup ).
% local.sup.abel_semigroup_axioms
thf(fact_207_local_Oinf_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ inf ).
% local.inf.abel_semigroup_axioms
thf(fact_208_semilattice__order__set_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic1693879045er_set @ A @ F @ Less_eq @ Less )
=> ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% semilattice_order_set.axioms(1)
thf(fact_209_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_210_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_211_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_212_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_213_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_214_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_215_ord_Ois__arg__min_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( lattices_is_arg_min @ A @ B )
= ( lattices_is_arg_min @ A @ B ) ) ).
% ord.is_arg_min.cong
thf(fact_216_ord_Ois__arg__max_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( lattices_is_arg_max @ A @ B )
= ( lattices_is_arg_max @ A @ B ) ) ).
% ord.is_arg_max.cong
thf(fact_217_ord_Ois__arg__min__def,axiom,
! [B: $tType,A: $tType] :
( ( lattices_is_arg_min @ A @ B )
= ( ^ [Less2: A > A > $o,F2: B > A,P2: B > $o,X2: B] :
( ( P2 @ X2 )
& ~ ? [Y3: B] :
( ( P2 @ Y3 )
& ( Less2 @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% ord.is_arg_min_def
thf(fact_218_ord_Ois__arg__max__def,axiom,
! [B: $tType,A: $tType] :
( ( lattices_is_arg_max @ A @ B )
= ( ^ [Less2: A > A > $o,F2: B > A,P2: B > $o,X2: B] :
( ( P2 @ X2 )
& ~ ? [Y3: B] :
( ( P2 @ Y3 )
& ( Less2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% ord.is_arg_max_def
thf(fact_219_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ).
% semilattice.axioms(1)
thf(fact_220_local_OSup__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ sup ).
% local.Sup_fin.semilattice_set_axioms
thf(fact_221_local_OInf__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ inf ).
% local.Inf_fin.semilattice_set_axioms
thf(fact_222_semilattice__order__set_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic1693879045er_set @ A @ F @ Less_eq @ Less )
=> ( lattic35693393ce_set @ A @ F ) ) ).
% semilattice_order_set.axioms(2)
thf(fact_223_semilattice__set__def,axiom,
! [A: $tType] :
( ( lattic35693393ce_set @ A )
= ( semilattice @ A ) ) ).
% semilattice_set_def
thf(fact_224_semilattice__set_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( lattic35693393ce_set @ A @ F ) ) ).
% semilattice_set.intro
thf(fact_225_semilattice__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A] :
( ( lattic35693393ce_set @ A @ F )
=> ( semilattice @ A @ F ) ) ).
% semilattice_set.axioms
thf(fact_226_semilattice__order__set__def,axiom,
! [A: $tType] :
( ( lattic1693879045er_set @ A )
= ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 )
& ( lattic35693393ce_set @ A @ F2 ) ) ) ) ).
% semilattice_order_set_def
thf(fact_227_semilattice__order__set_Ointro,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( lattic35693393ce_set @ A @ F )
=> ( lattic1693879045er_set @ A @ F @ Less_eq @ Less ) ) ) ).
% semilattice_order_set.intro
thf(fact_228_semilattice__order_Omono,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A,D: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ C )
=> ( ( Less_eq @ B2 @ D )
=> ( Less_eq @ ( F @ A2 @ B2 ) @ ( F @ C @ D ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_229_semilattice__order_OorderE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( A2
= ( F @ A2 @ B2 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_230_semilattice__order_OorderI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( A2
= ( F @ A2 @ B2 ) )
=> ( Less_eq @ A2 @ B2 ) ) ) ).
% semilattice_order.orderI
thf(fact_231_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( F @ A2 @ B2 )
= A2 ) ) ) ).
% semilattice_order.absorb1
thf(fact_232_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A2 )
=> ( ( F @ A2 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb2
thf(fact_233_semilattice__order_OboundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ ( F @ B2 @ C ) )
=> ~ ( ( Less_eq @ A2 @ B2 )
=> ~ ( Less_eq @ A2 @ C ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_234_semilattice__order_OboundedI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ A2 @ C )
=> ( Less_eq @ A2 @ ( F @ B2 @ C ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_235_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
= ( A2
= ( F @ A2 @ B2 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_236_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( Less_eq @ ( F @ A2 @ B2 ) @ A2 ) ) ).
% semilattice_order.cobounded1
thf(fact_237_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( Less_eq @ ( F @ A2 @ B2 ) @ B2 ) ) ).
% semilattice_order.cobounded2
thf(fact_238_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
= ( ( F @ A2 @ B2 )
= A2 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_239_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A2 )
= ( ( F @ A2 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_240_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ ( F @ B2 @ C ) )
= ( ( Less_eq @ A2 @ B2 )
& ( Less_eq @ A2 @ C ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_241_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ C )
=> ( Less_eq @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_242_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C: A,A2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ C )
=> ( Less_eq @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_243_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A2 @ ( F @ B2 @ C ) )
=> ~ ( ( Less @ A2 @ B2 )
=> ~ ( Less @ A2 @ C ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_244_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
= ( ( A2
= ( F @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_245_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A2 @ C )
=> ( Less @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_246_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C: A,A2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ B2 @ C )
=> ( Less @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_247_semilattice_Oidem,axiom,
! [A: $tType,F: A > A > A,A2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A2 @ A2 )
= A2 ) ) ).
% semilattice.idem
thf(fact_248_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_249_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_250_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semilattice @ A @ F ) ) ).
% semilattice_order.axioms(1)
thf(fact_251_local_Obdd__below__def,axiom,
! [A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ A4 )
= ( ? [M: a] :
! [X2: a] :
( ( member @ a @ X2 @ A4 )
=> ( less_eq @ M @ X2 ) ) ) ) ).
% local.bdd_below_def
thf(fact_252_local_Obdd__above__def,axiom,
! [A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ A4 )
= ( ? [M2: a] :
! [X2: a] :
( ( member @ a @ X2 @ A4 )
=> ( less_eq @ X2 @ M2 ) ) ) ) ).
% local.bdd_above_def
thf(fact_253_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_254_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_255_local_Obdd__below__Ioo,axiom,
! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% local.bdd_below_Ioo
% Conjectures (5)
thf(conj_0,hypothesis,
lattice @ a @ inf @ less_eq @ less @ sup ).
thf(conj_1,hypothesis,
less_eq @ x @ y ).
thf(conj_2,hypothesis,
( ( sup @ x @ ( inf @ y @ z ) )
!= ( inf @ y @ ( sup @ x @ z ) ) ) ).
thf(conj_3,hypothesis,
less @ ( sup @ x @ ( inf @ y @ z ) ) @ ( inf @ y @ ( sup @ x @ z ) ) ).
thf(conj_4,conjecture,
less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ).
%------------------------------------------------------------------------------