%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP123^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:44 AM UTC 2026
% Result : Theorem 2.50s 1.56s
% Output : Refutation 2.50s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 6
% Syntax : Number of formulae : 35 ( 21 unt; 0 typ; 0 def)
% Number of atoms : 146 ( 46 equ; 0 cnn)
% Maximal formula atoms : 6 ( 4 avg)
% Number of connectives : 262 ( 25 ~; 14 |; 7 &; 212 @)
% ( 1 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 11 con; 0-6 aty)
% Number of variables : 47 ( 0 ^; 47 !; 0 ?; 47 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
a: $tType ).
thf(type_def_7,type,
set_a: $tType ).
thf(func_def_0,type,
set_a: $tType ).
thf(func_def_1,type,
a: $tType ).
thf(func_def_2,type,
condit1627435690bove_a: ( a > a > $o ) > set_a > $o ).
thf(func_def_3,type,
condit1001553558elow_a: ( a > a > $o ) > set_a > $o ).
thf(func_def_4,type,
finite40241356em_a_a: ( a > a > a ) > $o ).
thf(func_def_5,type,
abel_semigroup_a: ( a > a > a ) > $o ).
thf(func_def_6,type,
semigroup_a: ( a > a > a ) > $o ).
thf(func_def_7,type,
lattice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > ( a > a > a ) > $o ).
thf(func_def_8,type,
semilattice_a: ( a > a > a ) > $o ).
thf(func_def_9,type,
semilattice_order_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(func_def_10,type,
lattic655834328_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(func_def_11,type,
lattic1885654924_set_a: ( a > a > a ) > $o ).
thf(func_def_12,type,
modula1376131916tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
thf(func_def_13,type,
modula397570059tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
thf(func_def_14,type,
modula17988509_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_15,type,
modula1373251614_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_16,type,
modula581031071_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_17,type,
modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_18,type,
modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_19,type,
modula1727524044comp_a: ( a > a > $o ) > a > a > $o ).
thf(func_def_20,type,
modula1962211574trib_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
thf(func_def_21,type,
least_a: ( a > a > $o ) > ( a > $o ) > a ).
thf(func_def_22,type,
max_a: ( a > a > $o ) > a > a > a ).
thf(func_def_23,type,
min_a: ( a > a > $o ) > a > a > a ).
thf(func_def_24,type,
greatest_a: ( a > a > $o ) > ( a > $o ) > a ).
thf(func_def_25,type,
ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(func_def_26,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_27,type,
set_gr1491433118Than_a: ( a > a > $o ) > a > a > set_a ).
thf(func_def_28,type,
member_a: a > set_a > $o ).
thf(func_def_29,type,
inf: a > a > a ).
thf(func_def_30,type,
less: a > a > $o ).
thf(func_def_31,type,
less_eq: a > a > $o ).
thf(func_def_32,type,
sup: a > a > a ).
thf(func_def_33,type,
x: a ).
thf(func_def_34,type,
y: a ).
thf(func_def_35,type,
z: a ).
thf(func_def_47,type,
sK0: a ).
thf(func_def_48,type,
sK1: a ).
thf(func_def_49,type,
sK2: a ).
thf(func_def_50,type,
sK3: ( a > a > a ) > a ).
thf(func_def_51,type,
sK4: ( a > a > a ) > a ).
thf(func_def_52,type,
sK5: ( a > a > a ) > a ).
thf(func_def_53,type,
sK6: ( a > a > a ) > a ).
thf(func_def_54,type,
sK7: ( a > a > a ) > a ).
thf(func_def_55,type,
sK8: ( a > a > a ) > a ).
thf(func_def_56,type,
sK9: ( a > a > a ) > a ).
thf(func_def_57,type,
sK10: ( a > a > a ) > a ).
thf(func_def_58,type,
sK11: ( a > a > a ) > a ).
thf(func_def_59,type,
sK12: ( a > a > a ) > a ).
thf(func_def_60,type,
sK13: ( a > a > a ) > a ).
thf(func_def_61,type,
sK14: ( a > a > a ) > a ).
thf(func_def_62,type,
sK15: ( a > a > a ) > a ).
thf(func_def_63,type,
sK16: ( a > a > a ) > a ).
thf(func_def_64,type,
sK17: a ).
thf(func_def_65,type,
sK18: a ).
thf(func_def_66,type,
sK19: a ).
thf(func_def_68,type,
ph21:
!>[X0: $tType] : X0 ).
thf(f2381,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2380,f1005]) ).
thf(f1005,plain,
( ( less_eq @ x @ y )
= $true ),
inference(cnf_transformation,[status(thm)],[f527]) ).
thf(f527,plain,
( ( less_eq @ x @ y )
= $true ),
inference(fool_elimination,[status(thm)],[f526]) ).
thf(f526,plain,
less_eq @ x @ y,
inference(rectify,[status(thm)],[f251]) ).
thf(f251,axiom,
less_eq @ x @ y,
file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',conj_1) ).
thf(f2380,plain,
( ( less_eq @ x @ y )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f2366,f1027]) ).
thf(f1027,plain,
! [X0: a,X1: a] :
( ( less_eq @ X1 @ ( sup @ X1 @ X0 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f308]) ).
thf(f308,plain,
! [X0: a,X1: a] :
( ( less_eq @ X1 @ ( sup @ X1 @ X0 ) )
= $true ),
inference(fool_elimination,[status(thm)],[f307]) ).
thf(f307,plain,
! [X0: a,X1: a] : ( less_eq @ X1 @ ( sup @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f107]) ).
thf(f107,axiom,
! [X2: a,X3: a] : ( less_eq @ X3 @ ( sup @ X3 @ X2 ) ),
file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_106_local_Osup_Ocobounded1) ).
thf(f2366,plain,
( ( ( less_eq @ x @ ( sup @ x @ z ) )
!= $true )
| ( ( less_eq @ x @ y )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2317]) ).
thf(f2317,plain,
( ( $true != $true )
| ( ( less_eq @ x @ ( sup @ x @ z ) )
!= $true )
| ( ( less_eq @ x @ y )
!= $true ) ),
inference(superposition,[status(thm)],[f1191,f980]) ).
thf(f980,plain,
! [X2: a,X0: a,X1: a] :
( ( $true
= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) )
| ( ( less_eq @ X2 @ X0 )
!= $true )
| ( ( less_eq @ X2 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f841]) ).
thf(f841,plain,
! [X0: a,X1: a,X2: a] :
( ( ( $true
= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) )
| ( ( less_eq @ X2 @ X1 )
!= $true )
| ( ( less_eq @ X2 @ X0 )
!= $true ) )
& ( ( ( ( less_eq @ X2 @ X1 )
= $true )
& ( ( less_eq @ X2 @ X0 )
= $true ) )
| ( $true
!= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) ) ) ),
inference(flattening,[status(thm)],[f840]) ).
thf(f840,plain,
! [X0: a,X1: a,X2: a] :
( ( ( $true
= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) )
| ( ( less_eq @ X2 @ X1 )
!= $true )
| ( ( less_eq @ X2 @ X0 )
!= $true ) )
& ( ( ( ( less_eq @ X2 @ X1 )
= $true )
& ( ( less_eq @ X2 @ X0 )
= $true ) )
| ( $true
!= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f533]) ).
thf(f533,plain,
! [X0: a,X1: a,X2: a] :
( ( $true
= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) )
<=> ( ( ( less_eq @ X2 @ X1 )
= $true )
& ( ( less_eq @ X2 @ X0 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f532]) ).
thf(f532,plain,
! [X0: a,X1: a,X2: a] :
( ( less_eq @ X2 @ X1 )
& ( ( less_eq @ X2 @ X0 )
= ( less_eq @ X2 @ ( inf @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f158]) ).
thf(f158,axiom,
! [X8: a,X2: a,X3: a] :
( ( less_eq @ X3 @ X2 )
& ( ( less_eq @ X3 @ X8 )
= ( less_eq @ X3 @ ( inf @ X2 @ X8 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_157_local_Oinf_Obounded__iff) ).
thf(f1191,plain,
( ( less_eq @ x @ ( inf @ y @ ( sup @ x @ z ) ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1145]) ).
thf(f1145,plain,
( ( ( less_eq @ x @ ( inf @ y @ ( sup @ x @ z ) ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1089,f933]) ).
thf(f933,plain,
! [X2: a,X0: a,X1: a] :
( ( $true
= ( less_eq @ X1 @ ( sup @ X0 @ X2 ) ) )
| ( ( less_eq @ X1 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f703]) ).
thf(f703,plain,
! [X0: a,X1: a,X2: a] :
( ( $true
= ( less_eq @ X1 @ ( sup @ X0 @ X2 ) ) )
| ( ( less_eq @ X1 @ X2 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f483]) ).
thf(f483,plain,
! [X0: a,X1: a,X2: a] :
( ( ( less_eq @ X1 @ X2 )
= $true )
=> ( $true
= ( less_eq @ X1 @ ( sup @ X0 @ X2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f482]) ).
thf(f482,plain,
! [X0: a,X1: a,X2: a] :
( ( less_eq @ X1 @ X2 )
=> ( less_eq @ X1 @ ( sup @ X0 @ X2 ) ) ),
inference(rectify,[status(thm)],[f100]) ).
thf(f100,axiom,
! [X3: a,X0: a,X2: a] :
( ( less_eq @ X0 @ X2 )
=> ( less_eq @ X0 @ ( sup @ X3 @ X2 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_99_local_Ole__supI2) ).
thf(f1089,plain,
( ( less_eq @ x @ ( sup @ z @ ( inf @ y @ ( sup @ x @ z ) ) ) )
!= $true ),
inference(forward_demodulation,[status(thm)],[f941,f1006]) ).
thf(f1006,plain,
! [X0: a,X1: a] :
( ( sup @ X0 @ X1 )
= ( sup @ X1 @ X0 ) ),
inference(cnf_transformation,[status(thm)],[f690]) ).
thf(f690,plain,
! [X0: a,X1: a] :
( ( sup @ X0 @ X1 )
= ( sup @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f40]) ).
thf(f40,axiom,
! [X2: a,X3: a] :
( ( sup @ X3 @ X2 )
= ( sup @ X2 @ X3 ) ),
file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_39_local_Osup_Ocommute) ).
thf(f941,plain,
( ( less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f650]) ).
thf(f650,plain,
( ( less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) )
!= $true ),
inference(flattening,[status(thm)],[f575]) ).
thf(f575,plain,
( ( less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f574]) ).
thf(f574,plain,
~ ( less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ),
inference(rectify,[status(thm)],[f255]) ).
thf(f255,negated_conjecture,
~ ( less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ),
inference(negated_conjecture,[status(cth)],[f254]) ).
thf(f254,conjecture,
less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ),
file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',conj_4) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : ITP123^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.10 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.30 % Computer : n011.cluster.edu
% 0.13/0.30 % Model : x86_64 x86_64
% 0.13/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.30 % Memory : 8042.1875MB
% 0.13/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.30 % CPULimit : 300
% 0.13/0.30 % WCLimit : 300
% 0.13/0.30 % DateTime : Tue Feb 24 08:26:47 EST 2026
% 0.13/0.30 % CPUTime :
% 0.13/0.30 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.41 ---- Original DTF file ---
% 0.18/0.41 thf(spec,logic,$$dhol).
% 0.18/0.41 %------------------------------------------------------------------------------
% 0.18/0.41 % File : ITP123^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.41 % Domain : Interactive Theorem Proving
% 0.18/0.41 % Problem : Sledgehammer Modular_Distrib_Lattice problem prob_319__3264956_1
% 0.18/0.41 % Version : Especial.
% 0.18/0.41 % English :
% 0.18/0.41
% 0.18/0.41 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.41 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.41 % Source : [Des21]
% 0.18/0.41 % Names : Modular_Distrib_Lattice/prob_319__3264956_1 [Des21]
% 0.18/0.41
% 0.18/0.41 % Status : Theorem
% 0.18/0.41 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.18/0.41 % Syntax : Number of formulae : 290 ( 123 unt; 36 typ; 0 def)
% 0.18/0.41 % Number of atoms : 579 ( 186 equ; 0 cnn)
% 0.18/0.41 % Maximal formula atoms : 6 ( 2 avg)
% 0.18/0.41 % Number of connectives : 1863 ( 47 ~; 4 |; 31 &;1553 @)
% 0.18/0.41 % ( 0 <=>; 228 =>; 0 <=; 0 <~>)
% 0.18/0.41 % Maximal formula depth : 16 ( 6 avg)
% 0.18/0.41 % Number of types : 3 ( 2 usr)
% 0.18/0.41 % Number of type conns : 365 ( 365 >; 0 *; 0 +; 0 <<)
% 0.18/0.41 % Number of symbols : 35 ( 34 usr; 3 con; 0-6 aty)
% 0.18/0.41 % Number of variables : 660 ( 17 ^; 639 !; 4 ?; 660 :)
% 0.18/0.41 % SPC : TH0_THM_EQU_NAR
% 0.18/0.41
% 0.18/0.41 % Comments : This file was generated by Sledgehammer 2021-02-23 15:45:08.056
% 0.18/0.41 %------------------------------------------------------------------------------
% 0.18/0.41 % Could-be-implicit typings (2)
% 0.18/0.41 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.18/0.41 set_a: $tType ).
% 0.18/0.41
% 0.18/0.41 thf(ty_n_tf__a,type,
% 0.18/0.41 a: $tType ).
% 0.18/0.41
% 0.18/0.41 % Explicit typings (34)
% 0.18/0.41 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above_001tf__a,type,
% 0.18/0.41 condit1627435690bove_a: ( a > a > $o ) > set_a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below_001tf__a,type,
% 0.18/0.41 condit1001553558elow_a: ( a > a > $o ) > set_a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Finite__Set_Ocomp__fun__idem_001tf__a_001tf__a,type,
% 0.18/0.41 finite40241356em_a_a: ( a > a > a ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Groups_Oabel__semigroup_001tf__a,type,
% 0.18/0.41 abel_semigroup_a: ( a > a > a ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Groups_Osemigroup_001tf__a,type,
% 0.18/0.41 semigroup_a: ( a > a > a ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Lattices_Oclass_Olattice_001tf__a,type,
% 0.18/0.41 lattice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > ( a > a > a ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Lattices_Osemilattice_001tf__a,type,
% 0.18/0.41 semilattice_a: ( a > a > a ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Lattices_Osemilattice__order_001tf__a,type,
% 0.18/0.41 semilattice_order_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
% 0.18/0.41 lattic655834328_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
% 0.18/0.41 lattic1885654924_set_a: ( a > a > a ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_OM5__lattice_001tf__a,type,
% 0.18/0.41 modula1376131916tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_ON5__lattice_001tf__a,type,
% 0.18/0.41 modula397570059tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oa__aux_001tf__a,type,
% 0.18/0.41 modula17988509_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Ob__aux_001tf__a,type,
% 0.18/0.41 modula1373251614_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oc__aux_001tf__a,type,
% 0.18/0.41 modula581031071_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Od__aux_001tf__a,type,
% 0.18/0.41 modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oe__aux_001tf__a,type,
% 0.18/0.41 modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oincomp_001tf__a,type,
% 0.18/0.41 modula1727524044comp_a: ( a > a > $o ) > a > a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Ono__distrib_001tf__a,type,
% 0.18/0.41 modula1962211574trib_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Orderings_Oord_OLeast_001tf__a,type,
% 0.18/0.41 least_a: ( a > a > $o ) > ( a > $o ) > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Orderings_Oord_Omax_001tf__a,type,
% 0.18/0.41 max_a: ( a > a > $o ) > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Orderings_Oord_Omin_001tf__a,type,
% 0.18/0.41 min_a: ( a > a > $o ) > a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Orderings_Oorder_OGreatest_001tf__a,type,
% 0.18/0.41 greatest_a: ( a > a > $o ) > ( a > $o ) > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Orderings_Oordering_001tf__a,type,
% 0.18/0.41 ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.18/0.41 collect_a: ( a > $o ) > set_a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001tf__a,type,
% 0.18/0.41 set_gr1491433118Than_a: ( a > a > $o ) > a > a > set_a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_c_member_001tf__a,type,
% 0.18/0.41 member_a: a > set_a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_inf,type,
% 0.18/0.41 inf: a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_less,type,
% 0.18/0.41 less: a > a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_less__eq,type,
% 0.18/0.41 less_eq: a > a > $o ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_sup,type,
% 0.18/0.41 sup: a > a > a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_x,type,
% 0.18/0.41 x: a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_y,type,
% 0.18/0.41 y: a ).
% 0.18/0.41
% 0.18/0.41 thf(sy_v_z,type,
% 0.18/0.41 z: a ).
% 0.18/0.41
% 0.18/0.41 % Relevant facts (249)
% 0.18/0.41 thf(fact_0_local_Oantisym,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( less_eq @ Y @ X )
% 0.18/0.41 => ( X = Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.antisym
% 0.18/0.41 thf(fact_1_local_Oantisym__conv,axiom,
% 0.18/0.41 ! [Y: a,X: a] :
% 0.18/0.41 ( ( less_eq @ Y @ X )
% 0.18/0.41 => ( ( less_eq @ X @ Y )
% 0.18/0.41 = ( X = Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.antisym_conv
% 0.18/0.41 thf(fact_2_local_Odual__order_Oantisym,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( ( less_eq @ A @ B )
% 0.18/0.41 => ( A = B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.antisym
% 0.18/0.41 thf(fact_3_local_Odual__order_Oeq__iff,axiom,
% 0.18/0.41 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 0.18/0.41 = ( ^ [A2: a,B2: a] :
% 0.18/0.41 ( ( less_eq @ B2 @ A2 )
% 0.18/0.41 & ( less_eq @ A2 @ B2 ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.eq_iff
% 0.18/0.41 thf(fact_4_local_Odual__order_Otrans,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( ( less_eq @ C @ B )
% 0.18/0.41 => ( less_eq @ C @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.trans
% 0.18/0.41 thf(fact_5_local_Oeq__iff,axiom,
% 0.18/0.41 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 0.18/0.41 = ( ^ [X2: a,Y3: a] :
% 0.18/0.41 ( ( less_eq @ X2 @ Y3 )
% 0.18/0.41 & ( less_eq @ Y3 @ X2 ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.eq_iff
% 0.18/0.41 thf(fact_6_local_Oeq__refl,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( X = Y )
% 0.18/0.41 => ( less_eq @ X @ Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.eq_refl
% 0.18/0.41 thf(fact_7_local_Oord__eq__le__trans,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( A = B )
% 0.18/0.41 => ( ( less_eq @ B @ C )
% 0.18/0.41 => ( less_eq @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.ord_eq_le_trans
% 0.18/0.41 thf(fact_8_local_Oord__le__eq__trans,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( B = C )
% 0.18/0.41 => ( less_eq @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.ord_le_eq_trans
% 0.18/0.41 thf(fact_9_local_Oorder_Oantisym,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( less_eq @ B @ A )
% 0.18/0.41 => ( A = B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.antisym
% 0.18/0.41 thf(fact_10_local_Oorder_Oeq__iff,axiom,
% 0.18/0.41 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 0.18/0.41 = ( ^ [A2: a,B2: a] :
% 0.18/0.41 ( ( less_eq @ A2 @ B2 )
% 0.18/0.41 & ( less_eq @ B2 @ A2 ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.eq_iff
% 0.18/0.41 thf(fact_11_local_Oorder_Otrans,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( less_eq @ B @ C )
% 0.18/0.41 => ( less_eq @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.trans
% 0.18/0.41 thf(fact_12_local_Oorder__trans,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( less_eq @ Y @ Z2 )
% 0.18/0.41 => ( less_eq @ X @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order_trans
% 0.18/0.41 thf(fact_13_local_Odual__order_Oasym,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less @ B @ A )
% 0.18/0.41 => ~ ( less @ A @ B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.asym
% 0.18/0.41 thf(fact_14_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less @ B @ A )
% 0.18/0.41 => ( A != B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.strict_implies_not_eq
% 0.18/0.41 thf(fact_15_local_Odual__order_Ostrict__trans,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( less @ B @ A )
% 0.18/0.41 => ( ( less @ C @ B )
% 0.18/0.41 => ( less @ C @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.strict_trans
% 0.18/0.41 thf(fact_16_local_Oless__asym,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ~ ( less @ Y @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_asym
% 0.18/0.41 thf(fact_17_local_Oless__asym_H,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ~ ( less @ B @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_asym'
% 0.18/0.41 thf(fact_18_local_Oless__imp__neq,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( X != Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_imp_neq
% 0.18/0.41 thf(fact_19_local_Oless__imp__not__eq,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( X != Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_imp_not_eq
% 0.18/0.41 thf(fact_20_local_Oless__imp__not__eq2,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( Y != X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_imp_not_eq2
% 0.18/0.41 thf(fact_21_local_Oless__imp__not__less,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ~ ( less @ Y @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_imp_not_less
% 0.18/0.41 thf(fact_22_local_Oless__imp__triv,axiom,
% 0.18/0.41 ! [X: a,Y: a,P: $o] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( ( less @ Y @ X )
% 0.18/0.41 => P ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_imp_triv
% 0.18/0.41 thf(fact_23_local_Oless__irrefl,axiom,
% 0.18/0.41 ! [X: a] :
% 0.18/0.41 ~ ( less @ X @ X ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_irrefl
% 0.18/0.41 thf(fact_24_local_Oless__not__sym,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ~ ( less @ Y @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_not_sym
% 0.18/0.41 thf(fact_25_local_Oless__trans,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( ( less @ Y @ Z2 )
% 0.18/0.41 => ( less @ X @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_trans
% 0.18/0.41 thf(fact_26_local_Oord__eq__less__trans,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( A = B )
% 0.18/0.41 => ( ( less @ B @ C )
% 0.18/0.41 => ( less @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.ord_eq_less_trans
% 0.18/0.41 thf(fact_27_local_Oord__less__eq__trans,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ( ( B = C )
% 0.18/0.41 => ( less @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.ord_less_eq_trans
% 0.18/0.41 thf(fact_28_local_Oorder_Oasym,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ~ ( less @ B @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.asym
% 0.18/0.41 thf(fact_29_local_Oorder_Oirrefl,axiom,
% 0.18/0.41 ! [A: a] :
% 0.18/0.41 ~ ( less @ A @ A ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.irrefl
% 0.18/0.41 thf(fact_30_local_Oorder_Ostrict__implies__not__eq,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ( A != B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.strict_implies_not_eq
% 0.18/0.41 thf(fact_31_local_Oorder_Ostrict__trans,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ( ( less @ B @ C )
% 0.18/0.41 => ( less @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.strict_trans
% 0.18/0.41 thf(fact_32_local_Oinf_Oassoc,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( inf @ ( inf @ A @ B ) @ C )
% 0.18/0.41 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.assoc
% 0.18/0.41 thf(fact_33_local_Oinf_Ocommute,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( inf @ A @ B )
% 0.18/0.41 = ( inf @ B @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.commute
% 0.18/0.41 thf(fact_34_local_Oinf_Oleft__commute,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( inf @ B @ ( inf @ A @ C ) )
% 0.18/0.41 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.left_commute
% 0.18/0.41 thf(fact_35_local_Oinf__assoc,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( inf @ ( inf @ X @ Y ) @ Z2 )
% 0.18/0.41 = ( inf @ X @ ( inf @ Y @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_assoc
% 0.18/0.41 thf(fact_36_local_Oinf__commute,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( inf @ X @ Y )
% 0.18/0.41 = ( inf @ Y @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_commute
% 0.18/0.41 thf(fact_37_local_Oinf__left__commute,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( inf @ X @ ( inf @ Y @ Z2 ) )
% 0.18/0.41 = ( inf @ Y @ ( inf @ X @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_left_commute
% 0.18/0.41 thf(fact_38_local_Osup_Oassoc,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( sup @ ( sup @ A @ B ) @ C )
% 0.18/0.41 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.assoc
% 0.18/0.41 thf(fact_39_local_Osup_Ocommute,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( sup @ A @ B )
% 0.18/0.41 = ( sup @ B @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.commute
% 0.18/0.41 thf(fact_40_local_Osup_Oleft__commute,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( sup @ B @ ( sup @ A @ C ) )
% 0.18/0.41 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.left_commute
% 0.18/0.41 thf(fact_41_local_Osup__assoc,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( sup @ ( sup @ X @ Y ) @ Z2 )
% 0.18/0.41 = ( sup @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_assoc
% 0.18/0.41 thf(fact_42_local_Osup__commute,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( sup @ X @ Y )
% 0.18/0.41 = ( sup @ Y @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_commute
% 0.18/0.41 thf(fact_43_local_Osup__left__commute,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( sup @ X @ ( sup @ Y @ Z2 ) )
% 0.18/0.41 = ( sup @ Y @ ( sup @ X @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_left_commute
% 0.18/0.41 thf(fact_44_local_Oantisym__conv1,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ~ ( less @ X @ Y )
% 0.18/0.41 => ( ( less_eq @ X @ Y )
% 0.18/0.41 = ( X = Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.antisym_conv1
% 0.18/0.41 thf(fact_45_mem__Collect__eq,axiom,
% 0.18/0.41 ! [A: a,P: a > $o] :
% 0.18/0.41 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.18/0.41 = ( P @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % mem_Collect_eq
% 0.18/0.41 thf(fact_46_Collect__mem__eq,axiom,
% 0.18/0.41 ! [A3: set_a] :
% 0.18/0.41 ( ( collect_a
% 0.18/0.41 @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
% 0.18/0.41 = A3 ) ).
% 0.18/0.41
% 0.18/0.41 % Collect_mem_eq
% 0.18/0.41 thf(fact_47_Collect__cong,axiom,
% 0.18/0.41 ! [P: a > $o,Q: a > $o] :
% 0.18/0.41 ( ! [X3: a] :
% 0.18/0.41 ( ( P @ X3 )
% 0.18/0.41 = ( Q @ X3 ) )
% 0.18/0.41 => ( ( collect_a @ P )
% 0.18/0.41 = ( collect_a @ Q ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % Collect_cong
% 0.18/0.41 thf(fact_48_local_Oantisym__conv2,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( ~ ( less @ X @ Y ) )
% 0.18/0.41 = ( X = Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.antisym_conv2
% 0.18/0.41 thf(fact_49_local_Odual__order_Onot__eq__order__implies__strict,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( A != B )
% 0.18/0.41 => ( ( less_eq @ B @ A )
% 0.18/0.41 => ( less @ B @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.not_eq_order_implies_strict
% 0.18/0.41 thf(fact_50_local_Odual__order_Oorder__iff__strict,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 = ( ( less @ B @ A )
% 0.18/0.41 | ( A = B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.order_iff_strict
% 0.18/0.41 thf(fact_51_local_Odual__order_Ostrict__iff__order,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less @ B @ A )
% 0.18/0.41 = ( ( less_eq @ B @ A )
% 0.18/0.41 & ( A != B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.strict_iff_order
% 0.18/0.41 thf(fact_52_local_Odual__order_Ostrict__implies__order,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less @ B @ A )
% 0.18/0.41 => ( less_eq @ B @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.strict_implies_order
% 0.18/0.41 thf(fact_53_local_Odual__order_Ostrict__trans1,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( ( less @ C @ B )
% 0.18/0.41 => ( less @ C @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.strict_trans1
% 0.18/0.41 thf(fact_54_local_Odual__order_Ostrict__trans2,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( less @ B @ A )
% 0.18/0.41 => ( ( less_eq @ C @ B )
% 0.18/0.41 => ( less @ C @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.dual_order.strict_trans2
% 0.18/0.41 thf(fact_55_local_OleD,axiom,
% 0.18/0.41 ! [Y: a,X: a] :
% 0.18/0.41 ( ( less_eq @ Y @ X )
% 0.18/0.41 => ~ ( less @ X @ Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.leD
% 0.18/0.41 thf(fact_56_local_Ole__imp__less__or__eq,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( less @ X @ Y )
% 0.18/0.41 | ( X = Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_imp_less_or_eq
% 0.18/0.41 thf(fact_57_local_Ole__less,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 = ( ( less @ X @ Y )
% 0.18/0.41 | ( X = Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_less
% 0.18/0.41 thf(fact_58_local_Ole__less__trans,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( less @ Y @ Z2 )
% 0.18/0.41 => ( less @ X @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_less_trans
% 0.18/0.41 thf(fact_59_local_Ole__neq__trans,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( A != B )
% 0.18/0.41 => ( less @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_neq_trans
% 0.18/0.41 thf(fact_60_local_Oless__imp__le,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( less_eq @ X @ Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_imp_le
% 0.18/0.41 thf(fact_61_local_Oless__le,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 = ( ( less_eq @ X @ Y )
% 0.18/0.41 & ( X != Y ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_le
% 0.18/0.41 thf(fact_62_local_Oless__le__not__le,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 = ( ( less_eq @ X @ Y )
% 0.18/0.41 & ~ ( less_eq @ Y @ X ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_le_not_le
% 0.18/0.41 thf(fact_63_local_Oless__le__trans,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( less @ X @ Y )
% 0.18/0.41 => ( ( less_eq @ Y @ Z2 )
% 0.18/0.41 => ( less @ X @ Z2 ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.less_le_trans
% 0.18/0.41 thf(fact_64_local_Oorder_Onot__eq__order__implies__strict,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( A != B )
% 0.18/0.41 => ( ( less_eq @ A @ B )
% 0.18/0.41 => ( less @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.not_eq_order_implies_strict
% 0.18/0.41 thf(fact_65_local_Oorder_Oorder__iff__strict,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 = ( ( less @ A @ B )
% 0.18/0.41 | ( A = B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.order_iff_strict
% 0.18/0.41 thf(fact_66_local_Oorder_Ostrict__iff__order,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 = ( ( less_eq @ A @ B )
% 0.18/0.41 & ( A != B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.strict_iff_order
% 0.18/0.41 thf(fact_67_local_Oorder_Ostrict__implies__order,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ( less_eq @ A @ B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.strict_implies_order
% 0.18/0.41 thf(fact_68_local_Oorder_Ostrict__trans1,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( less @ B @ C )
% 0.18/0.41 => ( less @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.strict_trans1
% 0.18/0.41 thf(fact_69_local_Oorder_Ostrict__trans2,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less @ A @ B )
% 0.18/0.41 => ( ( less_eq @ B @ C )
% 0.18/0.41 => ( less @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.order.strict_trans2
% 0.18/0.41 thf(fact_70_local_Oinf_Oabsorb1,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( inf @ A @ B )
% 0.18/0.41 = A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.absorb1
% 0.18/0.41 thf(fact_71_local_Oinf_Oabsorb2,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( ( inf @ A @ B )
% 0.18/0.41 = B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.absorb2
% 0.18/0.41 thf(fact_72_local_Oinf_Oabsorb__iff1,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 = ( ( inf @ A @ B )
% 0.18/0.41 = A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.absorb_iff1
% 0.18/0.41 thf(fact_73_local_Oinf_Oabsorb__iff2,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 = ( ( inf @ A @ B )
% 0.18/0.41 = B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.absorb_iff2
% 0.18/0.41 thf(fact_74_local_Oinf_OboundedE,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 0.18/0.41 => ~ ( ( less_eq @ A @ B )
% 0.18/0.41 => ~ ( less_eq @ A @ C ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.boundedE
% 0.18/0.41 thf(fact_75_local_Oinf_OboundedI,axiom,
% 0.18/0.41 ! [A: a,B: a,C: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( less_eq @ A @ C )
% 0.18/0.41 => ( less_eq @ A @ ( inf @ B @ C ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.boundedI
% 0.18/0.41 thf(fact_76_local_Oinf_Ocobounded1,axiom,
% 0.18/0.41 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ A ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.cobounded1
% 0.18/0.41 thf(fact_77_local_Oinf_Ocobounded2,axiom,
% 0.18/0.41 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ B ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.cobounded2
% 0.18/0.41 thf(fact_78_local_Oinf_OcoboundedI1,axiom,
% 0.18/0.41 ! [A: a,C: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ C )
% 0.18/0.41 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.coboundedI1
% 0.18/0.41 thf(fact_79_local_Oinf_OcoboundedI2,axiom,
% 0.18/0.41 ! [B: a,C: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ C )
% 0.18/0.41 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.coboundedI2
% 0.18/0.41 thf(fact_80_local_Oinf_OorderE,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( A
% 0.18/0.41 = ( inf @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.orderE
% 0.18/0.41 thf(fact_81_local_Oinf_OorderI,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( A
% 0.18/0.41 = ( inf @ A @ B ) )
% 0.18/0.41 => ( less_eq @ A @ B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.orderI
% 0.18/0.41 thf(fact_82_local_Oinf_Oorder__iff,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 = ( A
% 0.18/0.41 = ( inf @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf.order_iff
% 0.18/0.41 thf(fact_83_local_Oinf__absorb1,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( inf @ X @ Y )
% 0.18/0.41 = X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_absorb1
% 0.18/0.41 thf(fact_84_local_Oinf__absorb2,axiom,
% 0.18/0.41 ! [Y: a,X: a] :
% 0.18/0.41 ( ( less_eq @ Y @ X )
% 0.18/0.41 => ( ( inf @ X @ Y )
% 0.18/0.41 = Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_absorb2
% 0.18/0.41 thf(fact_85_local_Oinf__greatest,axiom,
% 0.18/0.41 ! [X: a,Y: a,Z2: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( less_eq @ X @ Z2 )
% 0.18/0.41 => ( less_eq @ X @ ( inf @ Y @ Z2 ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_greatest
% 0.18/0.41 thf(fact_86_local_Oinf__le1,axiom,
% 0.18/0.41 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_le1
% 0.18/0.41 thf(fact_87_local_Oinf__le2,axiom,
% 0.18/0.41 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_le2
% 0.18/0.41 thf(fact_88_local_Oinf__mono,axiom,
% 0.18/0.41 ! [A: a,C: a,B: a,D: a] :
% 0.18/0.41 ( ( less_eq @ A @ C )
% 0.18/0.41 => ( ( less_eq @ B @ D )
% 0.18/0.41 => ( less_eq @ ( inf @ A @ B ) @ ( inf @ C @ D ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_mono
% 0.18/0.41 thf(fact_89_local_Oinf__unique,axiom,
% 0.18/0.41 ! [F: a > a > a,X: a,Y: a] :
% 0.18/0.41 ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ X3 )
% 0.18/0.41 => ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ Y4 )
% 0.18/0.41 => ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.41 ( ( less_eq @ X3 @ Y4 )
% 0.18/0.41 => ( ( less_eq @ X3 @ Z3 )
% 0.18/0.41 => ( less_eq @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
% 0.18/0.41 => ( ( inf @ X @ Y )
% 0.18/0.41 = ( F @ X @ Y ) ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.inf_unique
% 0.18/0.41 thf(fact_90_local_Ole__iff__inf,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 = ( ( inf @ X @ Y )
% 0.18/0.41 = X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_iff_inf
% 0.18/0.41 thf(fact_91_local_Ole__infE,axiom,
% 0.18/0.41 ! [X: a,A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ X @ ( inf @ A @ B ) )
% 0.18/0.41 => ~ ( ( less_eq @ X @ A )
% 0.18/0.41 => ~ ( less_eq @ X @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_infE
% 0.18/0.41 thf(fact_92_local_Ole__infI,axiom,
% 0.18/0.41 ! [X: a,A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ X @ A )
% 0.18/0.41 => ( ( less_eq @ X @ B )
% 0.18/0.41 => ( less_eq @ X @ ( inf @ A @ B ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_infI
% 0.18/0.41 thf(fact_93_local_Ole__infI1,axiom,
% 0.18/0.41 ! [A: a,X: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ X )
% 0.18/0.41 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_infI1
% 0.18/0.41 thf(fact_94_local_Ole__infI2,axiom,
% 0.18/0.41 ! [B: a,X: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ X )
% 0.18/0.41 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_infI2
% 0.18/0.41 thf(fact_95_local_Ole__iff__sup,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 = ( ( sup @ X @ Y )
% 0.18/0.41 = Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_iff_sup
% 0.18/0.41 thf(fact_96_local_Ole__supE,axiom,
% 0.18/0.41 ! [A: a,B: a,X: a] :
% 0.18/0.41 ( ( less_eq @ ( sup @ A @ B ) @ X )
% 0.18/0.41 => ~ ( ( less_eq @ A @ X )
% 0.18/0.41 => ~ ( less_eq @ B @ X ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_supE
% 0.18/0.41 thf(fact_97_local_Ole__supI,axiom,
% 0.18/0.41 ! [A: a,X: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ X )
% 0.18/0.41 => ( ( less_eq @ B @ X )
% 0.18/0.41 => ( less_eq @ ( sup @ A @ B ) @ X ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_supI
% 0.18/0.41 thf(fact_98_local_Ole__supI1,axiom,
% 0.18/0.41 ! [X: a,A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ X @ A )
% 0.18/0.41 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_supI1
% 0.18/0.41 thf(fact_99_local_Ole__supI2,axiom,
% 0.18/0.41 ! [X: a,B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ X @ B )
% 0.18/0.41 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.le_supI2
% 0.18/0.41 thf(fact_100_local_Osup_Oabsorb1,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( ( sup @ A @ B )
% 0.18/0.41 = A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.absorb1
% 0.18/0.41 thf(fact_101_local_Osup_Oabsorb2,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 => ( ( sup @ A @ B )
% 0.18/0.41 = B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.absorb2
% 0.18/0.41 thf(fact_102_local_Osup_Oabsorb__iff1,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 = ( ( sup @ A @ B )
% 0.18/0.41 = A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.absorb_iff1
% 0.18/0.41 thf(fact_103_local_Osup_Oabsorb__iff2,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ A @ B )
% 0.18/0.41 = ( ( sup @ A @ B )
% 0.18/0.41 = B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.absorb_iff2
% 0.18/0.41 thf(fact_104_local_Osup_OboundedE,axiom,
% 0.18/0.41 ! [B: a,C: a,A: a] :
% 0.18/0.41 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 0.18/0.41 => ~ ( ( less_eq @ B @ A )
% 0.18/0.41 => ~ ( less_eq @ C @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.boundedE
% 0.18/0.41 thf(fact_105_local_Osup_OboundedI,axiom,
% 0.18/0.41 ! [B: a,A: a,C: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( ( less_eq @ C @ A )
% 0.18/0.41 => ( less_eq @ ( sup @ B @ C ) @ A ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.boundedI
% 0.18/0.41 thf(fact_106_local_Osup_Ocobounded1,axiom,
% 0.18/0.41 ! [A: a,B: a] : ( less_eq @ A @ ( sup @ A @ B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.cobounded1
% 0.18/0.41 thf(fact_107_local_Osup_Ocobounded2,axiom,
% 0.18/0.41 ! [B: a,A: a] : ( less_eq @ B @ ( sup @ A @ B ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.cobounded2
% 0.18/0.41 thf(fact_108_local_Osup_OcoboundedI1,axiom,
% 0.18/0.41 ! [C: a,A: a,B: a] :
% 0.18/0.41 ( ( less_eq @ C @ A )
% 0.18/0.41 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.coboundedI1
% 0.18/0.41 thf(fact_109_local_Osup_OcoboundedI2,axiom,
% 0.18/0.41 ! [C: a,B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ C @ B )
% 0.18/0.41 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.coboundedI2
% 0.18/0.41 thf(fact_110_local_Osup_Omono,axiom,
% 0.18/0.41 ! [C: a,A: a,D: a,B: a] :
% 0.18/0.41 ( ( less_eq @ C @ A )
% 0.18/0.41 => ( ( less_eq @ D @ B )
% 0.18/0.41 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A @ B ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.mono
% 0.18/0.41 thf(fact_111_local_Osup_OorderE,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 => ( A
% 0.18/0.41 = ( sup @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.orderE
% 0.18/0.41 thf(fact_112_local_Osup_OorderI,axiom,
% 0.18/0.41 ! [A: a,B: a] :
% 0.18/0.41 ( ( A
% 0.18/0.41 = ( sup @ A @ B ) )
% 0.18/0.41 => ( less_eq @ B @ A ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.orderI
% 0.18/0.41 thf(fact_113_local_Osup_Oorder__iff,axiom,
% 0.18/0.41 ! [B: a,A: a] :
% 0.18/0.41 ( ( less_eq @ B @ A )
% 0.18/0.41 = ( A
% 0.18/0.41 = ( sup @ A @ B ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup.order_iff
% 0.18/0.41 thf(fact_114_local_Osup__absorb1,axiom,
% 0.18/0.41 ! [Y: a,X: a] :
% 0.18/0.41 ( ( less_eq @ Y @ X )
% 0.18/0.41 => ( ( sup @ X @ Y )
% 0.18/0.41 = X ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_absorb1
% 0.18/0.41 thf(fact_115_local_Osup__absorb2,axiom,
% 0.18/0.41 ! [X: a,Y: a] :
% 0.18/0.41 ( ( less_eq @ X @ Y )
% 0.18/0.41 => ( ( sup @ X @ Y )
% 0.18/0.41 = Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_absorb2
% 0.18/0.41 thf(fact_116_local_Osup__ge1,axiom,
% 0.18/0.41 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_ge1
% 0.18/0.41 thf(fact_117_local_Osup__ge2,axiom,
% 0.18/0.41 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_ge2
% 0.18/0.41 thf(fact_118_local_Osup__least,axiom,
% 0.18/0.41 ! [Y: a,X: a,Z2: a] :
% 0.18/0.41 ( ( less_eq @ Y @ X )
% 0.18/0.41 => ( ( less_eq @ Z2 @ X )
% 0.18/0.41 => ( less_eq @ ( sup @ Y @ Z2 ) @ X ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_least
% 0.18/0.41 thf(fact_119_local_Osup__mono,axiom,
% 0.18/0.41 ! [A: a,C: a,B: a,D: a] :
% 0.18/0.41 ( ( less_eq @ A @ C )
% 0.18/0.41 => ( ( less_eq @ B @ D )
% 0.18/0.41 => ( less_eq @ ( sup @ A @ B ) @ ( sup @ C @ D ) ) ) ) ).
% 0.18/0.41
% 0.18/0.41 % local.sup_mono
% 0.18/0.41 thf(fact_120_local_Osup__unique,axiom,
% 0.18/0.41 ! [F: a > a > a,X: a,Y: a] :
% 0.18/0.41 ( ! [X3: a,Y4: a] : ( less_eq @ X3 @ ( F @ X3 @ Y4 ) )
% 0.18/0.42 => ( ! [X3: a,Y4: a] : ( less_eq @ Y4 @ ( F @ X3 @ Y4 ) )
% 0.18/0.42 => ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.42 ( ( less_eq @ Y4 @ X3 )
% 0.18/0.42 => ( ( less_eq @ Z3 @ X3 )
% 0.18/0.42 => ( less_eq @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
% 0.18/0.42 => ( ( sup @ X @ Y )
% 0.18/0.42 = ( F @ X @ Y ) ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup_unique
% 0.18/0.42 thf(fact_121_local_Oinf_Ostrict__boundedE,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( less @ A @ ( inf @ B @ C ) )
% 0.18/0.42 => ~ ( ( less @ A @ B )
% 0.18/0.42 => ~ ( less @ A @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.strict_boundedE
% 0.18/0.42 thf(fact_122_local_Oinf_Ostrict__coboundedI1,axiom,
% 0.18/0.42 ! [A: a,C: a,B: a] :
% 0.18/0.42 ( ( less @ A @ C )
% 0.18/0.42 => ( less @ ( inf @ A @ B ) @ C ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.strict_coboundedI1
% 0.18/0.42 thf(fact_123_local_Oinf_Ostrict__coboundedI2,axiom,
% 0.18/0.42 ! [B: a,C: a,A: a] :
% 0.18/0.42 ( ( less @ B @ C )
% 0.18/0.42 => ( less @ ( inf @ A @ B ) @ C ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.strict_coboundedI2
% 0.18/0.42 thf(fact_124_local_Oinf_Ostrict__order__iff,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( less @ A @ B )
% 0.18/0.42 = ( ( A
% 0.18/0.42 = ( inf @ A @ B ) )
% 0.18/0.42 & ( A != B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.strict_order_iff
% 0.18/0.42 thf(fact_125_local_Oless__infI1,axiom,
% 0.18/0.42 ! [A: a,X: a,B: a] :
% 0.18/0.42 ( ( less @ A @ X )
% 0.18/0.42 => ( less @ ( inf @ A @ B ) @ X ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.less_infI1
% 0.18/0.42 thf(fact_126_local_Oless__infI2,axiom,
% 0.18/0.42 ! [B: a,X: a,A: a] :
% 0.18/0.42 ( ( less @ B @ X )
% 0.18/0.42 => ( less @ ( inf @ A @ B ) @ X ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.less_infI2
% 0.18/0.42 thf(fact_127_local_Oless__supI1,axiom,
% 0.18/0.42 ! [X: a,A: a,B: a] :
% 0.18/0.42 ( ( less @ X @ A )
% 0.18/0.42 => ( less @ X @ ( sup @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.less_supI1
% 0.18/0.42 thf(fact_128_local_Oless__supI2,axiom,
% 0.18/0.42 ! [X: a,B: a,A: a] :
% 0.18/0.42 ( ( less @ X @ B )
% 0.18/0.42 => ( less @ X @ ( sup @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.less_supI2
% 0.18/0.42 thf(fact_129_local_Osup_Ostrict__boundedE,axiom,
% 0.18/0.42 ! [B: a,C: a,A: a] :
% 0.18/0.42 ( ( less @ ( sup @ B @ C ) @ A )
% 0.18/0.42 => ~ ( ( less @ B @ A )
% 0.18/0.42 => ~ ( less @ C @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.strict_boundedE
% 0.18/0.42 thf(fact_130_local_Osup_Ostrict__coboundedI1,axiom,
% 0.18/0.42 ! [C: a,A: a,B: a] :
% 0.18/0.42 ( ( less @ C @ A )
% 0.18/0.42 => ( less @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.strict_coboundedI1
% 0.18/0.42 thf(fact_131_local_Osup_Ostrict__coboundedI2,axiom,
% 0.18/0.42 ! [C: a,B: a,A: a] :
% 0.18/0.42 ( ( less @ C @ B )
% 0.18/0.42 => ( less @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.strict_coboundedI2
% 0.18/0.42 thf(fact_132_local_Osup_Ostrict__order__iff,axiom,
% 0.18/0.42 ! [B: a,A: a] :
% 0.18/0.42 ( ( less @ B @ A )
% 0.18/0.42 = ( ( A
% 0.18/0.42 = ( sup @ A @ B ) )
% 0.18/0.42 & ( A != B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.strict_order_iff
% 0.18/0.42 thf(fact_133_local_Odistrib__imp1,axiom,
% 0.18/0.42 ! [X: a,Y: a,Z2: a] :
% 0.18/0.42 ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.42 ( ( inf @ X3 @ ( sup @ Y4 @ Z3 ) )
% 0.18/0.42 = ( sup @ ( inf @ X3 @ Y4 ) @ ( inf @ X3 @ Z3 ) ) )
% 0.18/0.42 => ( ( sup @ X @ ( inf @ Y @ Z2 ) )
% 0.18/0.42 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.distrib_imp1
% 0.18/0.42 thf(fact_134_local_Odistrib__imp2,axiom,
% 0.18/0.42 ! [X: a,Y: a,Z2: a] :
% 0.18/0.42 ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.42 ( ( sup @ X3 @ ( inf @ Y4 @ Z3 ) )
% 0.18/0.42 = ( inf @ ( sup @ X3 @ Y4 ) @ ( sup @ X3 @ Z3 ) ) )
% 0.18/0.42 => ( ( inf @ X @ ( sup @ Y @ Z2 ) )
% 0.18/0.42 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.distrib_imp2
% 0.18/0.42 thf(fact_135_local_Odistrib__inf__le,axiom,
% 0.18/0.42 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) @ ( inf @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.distrib_inf_le
% 0.18/0.42 thf(fact_136_local_Odistrib__sup__le,axiom,
% 0.18/0.42 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z2 ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.distrib_sup_le
% 0.18/0.42 thf(fact_137_local_Oincomp__def,axiom,
% 0.18/0.42 ! [X: a,Y: a] :
% 0.18/0.42 ( ( modula1727524044comp_a @ less_eq @ X @ Y )
% 0.18/0.42 = ( ~ ( less_eq @ X @ Y )
% 0.18/0.42 & ~ ( less_eq @ Y @ X ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.incomp_def
% 0.18/0.42 thf(fact_138_local_Oorder_Orefl,axiom,
% 0.18/0.42 ! [A: a] : ( less_eq @ A @ A ) ).
% 0.18/0.42
% 0.18/0.42 % local.order.refl
% 0.18/0.42 thf(fact_139_local_Oorder__refl,axiom,
% 0.18/0.42 ! [X: a] : ( less_eq @ X @ X ) ).
% 0.18/0.42
% 0.18/0.42 % local.order_refl
% 0.18/0.42 thf(fact_140_local_Oinf_Oidem,axiom,
% 0.18/0.42 ! [A: a] :
% 0.18/0.42 ( ( inf @ A @ A )
% 0.18/0.42 = A ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.idem
% 0.18/0.42 thf(fact_141_local_Oinf_Oleft__idem,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( inf @ A @ ( inf @ A @ B ) )
% 0.18/0.42 = ( inf @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.left_idem
% 0.18/0.42 thf(fact_142_local_Oinf_Oright__idem,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( inf @ ( inf @ A @ B ) @ B )
% 0.18/0.42 = ( inf @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.right_idem
% 0.18/0.42 thf(fact_143_local_Oinf__idem,axiom,
% 0.18/0.42 ! [X: a] :
% 0.18/0.42 ( ( inf @ X @ X )
% 0.18/0.42 = X ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf_idem
% 0.18/0.42 thf(fact_144_local_Oinf__left__idem,axiom,
% 0.18/0.42 ! [X: a,Y: a] :
% 0.18/0.42 ( ( inf @ X @ ( inf @ X @ Y ) )
% 0.18/0.42 = ( inf @ X @ Y ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf_left_idem
% 0.18/0.42 thf(fact_145_local_Oinf__right__idem,axiom,
% 0.18/0.42 ! [X: a,Y: a] :
% 0.18/0.42 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 0.18/0.42 = ( inf @ X @ Y ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf_right_idem
% 0.18/0.42 thf(fact_146_local_Osup_Oidem,axiom,
% 0.18/0.42 ! [A: a] :
% 0.18/0.42 ( ( sup @ A @ A )
% 0.18/0.42 = A ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.idem
% 0.18/0.42 thf(fact_147_local_Osup_Oleft__idem,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( sup @ A @ ( sup @ A @ B ) )
% 0.18/0.42 = ( sup @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.left_idem
% 0.18/0.42 thf(fact_148_local_Osup_Oright__idem,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( sup @ ( sup @ A @ B ) @ B )
% 0.18/0.42 = ( sup @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.right_idem
% 0.18/0.42 thf(fact_149_local_Osup__idem,axiom,
% 0.18/0.42 ! [X: a] :
% 0.18/0.42 ( ( sup @ X @ X )
% 0.18/0.42 = X ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup_idem
% 0.18/0.42 thf(fact_150_local_Osup__left__idem,axiom,
% 0.18/0.42 ! [X: a,Y: a] :
% 0.18/0.42 ( ( sup @ X @ ( sup @ X @ Y ) )
% 0.18/0.42 = ( sup @ X @ Y ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup_left_idem
% 0.18/0.42 thf(fact_151_local_OGreatestI2__order,axiom,
% 0.18/0.42 ! [P: a > $o,X: a,Q: a > $o] :
% 0.18/0.42 ( ( P @ X )
% 0.18/0.42 => ( ! [Y4: a] :
% 0.18/0.42 ( ( P @ Y4 )
% 0.18/0.42 => ( less_eq @ Y4 @ X ) )
% 0.18/0.42 => ( ! [X3: a] :
% 0.18/0.42 ( ( P @ X3 )
% 0.18/0.42 => ( ! [Y5: a] :
% 0.18/0.42 ( ( P @ Y5 )
% 0.18/0.42 => ( less_eq @ Y5 @ X3 ) )
% 0.18/0.42 => ( Q @ X3 ) ) )
% 0.18/0.42 => ( Q @ ( greatest_a @ less_eq @ P ) ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.GreatestI2_order
% 0.18/0.42 thf(fact_152_local_OGreatest__equality,axiom,
% 0.18/0.42 ! [P: a > $o,X: a] :
% 0.18/0.42 ( ( P @ X )
% 0.18/0.42 => ( ! [Y4: a] :
% 0.18/0.42 ( ( P @ Y4 )
% 0.18/0.42 => ( less_eq @ Y4 @ X ) )
% 0.18/0.42 => ( ( greatest_a @ less_eq @ P )
% 0.18/0.42 = X ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.Greatest_equality
% 0.18/0.42 thf(fact_153_local_Omax__def,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( ( less_eq @ A @ B )
% 0.18/0.42 => ( ( max_a @ less_eq @ A @ B )
% 0.18/0.42 = B ) )
% 0.18/0.42 & ( ~ ( less_eq @ A @ B )
% 0.18/0.42 => ( ( max_a @ less_eq @ A @ B )
% 0.18/0.42 = A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.max_def
% 0.18/0.42 thf(fact_154_local_Omin__def,axiom,
% 0.18/0.42 ! [A: a,B: a] :
% 0.18/0.42 ( ( ( less_eq @ A @ B )
% 0.18/0.42 => ( ( min_a @ less_eq @ A @ B )
% 0.18/0.42 = A ) )
% 0.18/0.42 & ( ~ ( less_eq @ A @ B )
% 0.18/0.42 => ( ( min_a @ less_eq @ A @ B )
% 0.18/0.42 = B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.min_def
% 0.18/0.42 thf(fact_155_local_ON5__lattice__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula397570059tice_a @ inf @ less @ sup @ A @ B @ C )
% 0.18/0.42 = ( ( ( inf @ A @ C )
% 0.18/0.42 = ( inf @ B @ C ) )
% 0.18/0.42 & ( less @ A @ B )
% 0.18/0.42 & ( ( sup @ A @ C )
% 0.18/0.42 = ( sup @ B @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.N5_lattice_def
% 0.18/0.42 thf(fact_156_local_Ono__distrib__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1962211574trib_a @ inf @ less @ sup @ A @ B @ C )
% 0.18/0.42 = ( less @ ( sup @ ( inf @ A @ B ) @ ( inf @ C @ A ) ) @ ( inf @ A @ ( sup @ B @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.no_distrib_def
% 0.18/0.42 thf(fact_157_local_Oinf_Obounded__iff,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 0.18/0.42 = ( ( less_eq @ A @ B )
% 0.18/0.42 & ( less_eq @ A @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf.bounded_iff
% 0.18/0.42 thf(fact_158_local_Ole__inf__iff,axiom,
% 0.18/0.42 ! [X: a,Y: a,Z2: a] :
% 0.18/0.42 ( ( less_eq @ X @ ( inf @ Y @ Z2 ) )
% 0.18/0.42 = ( ( less_eq @ X @ Y )
% 0.18/0.42 & ( less_eq @ X @ Z2 ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.le_inf_iff
% 0.18/0.42 thf(fact_159_local_Ole__sup__iff,axiom,
% 0.18/0.42 ! [X: a,Y: a,Z2: a] :
% 0.18/0.42 ( ( less_eq @ ( sup @ X @ Y ) @ Z2 )
% 0.18/0.42 = ( ( less_eq @ X @ Z2 )
% 0.18/0.42 & ( less_eq @ Y @ Z2 ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.le_sup_iff
% 0.18/0.42 thf(fact_160_local_Osup_Obounded__iff,axiom,
% 0.18/0.42 ! [B: a,C: a,A: a] :
% 0.18/0.42 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 0.18/0.42 = ( ( less_eq @ B @ A )
% 0.18/0.42 & ( less_eq @ C @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup.bounded_iff
% 0.18/0.42 thf(fact_161_local_Oinf__sup__absorb,axiom,
% 0.18/0.42 ! [X: a,Y: a] :
% 0.18/0.42 ( ( inf @ X @ ( sup @ X @ Y ) )
% 0.18/0.42 = X ) ).
% 0.18/0.42
% 0.18/0.42 % local.inf_sup_absorb
% 0.18/0.42 thf(fact_162_local_Osup__inf__absorb,axiom,
% 0.18/0.42 ! [X: a,Y: a] :
% 0.18/0.42 ( ( sup @ X @ ( inf @ X @ Y ) )
% 0.18/0.42 = X ) ).
% 0.18/0.42
% 0.18/0.42 % local.sup_inf_absorb
% 0.18/0.42 thf(fact_163_local_OM5__lattice__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1376131916tice_a @ inf @ less @ sup @ A @ B @ C )
% 0.18/0.42 = ( ( ( inf @ A @ B )
% 0.18/0.42 = ( inf @ B @ C ) )
% 0.18/0.42 & ( ( inf @ C @ A )
% 0.18/0.42 = ( inf @ B @ C ) )
% 0.18/0.42 & ( ( sup @ A @ B )
% 0.18/0.42 = ( sup @ B @ C ) )
% 0.18/0.42 & ( ( sup @ C @ A )
% 0.18/0.42 = ( sup @ B @ C ) )
% 0.18/0.42 & ( less @ ( inf @ A @ B ) @ ( sup @ A @ B ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.M5_lattice_def
% 0.18/0.42 thf(fact_164_local_OLeast1I,axiom,
% 0.18/0.42 ! [P: a > $o] :
% 0.18/0.42 ( ? [X4: a] :
% 0.18/0.42 ( ( P @ X4 )
% 0.18/0.42 & ! [Y4: a] :
% 0.18/0.42 ( ( P @ Y4 )
% 0.18/0.42 => ( less_eq @ X4 @ Y4 ) )
% 0.18/0.42 & ! [Y4: a] :
% 0.18/0.42 ( ( ( P @ Y4 )
% 0.18/0.42 & ! [Ya: a] :
% 0.18/0.42 ( ( P @ Ya )
% 0.18/0.42 => ( less_eq @ Y4 @ Ya ) ) )
% 0.18/0.42 => ( Y4 = X4 ) ) )
% 0.18/0.42 => ( P @ ( least_a @ less_eq @ P ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.Least1I
% 0.18/0.42 thf(fact_165_local_OLeast1__le,axiom,
% 0.18/0.42 ! [P: a > $o,Z2: a] :
% 0.18/0.42 ( ? [X4: a] :
% 0.18/0.42 ( ( P @ X4 )
% 0.18/0.42 & ! [Y4: a] :
% 0.18/0.42 ( ( P @ Y4 )
% 0.18/0.42 => ( less_eq @ X4 @ Y4 ) )
% 0.18/0.42 & ! [Y4: a] :
% 0.18/0.42 ( ( ( P @ Y4 )
% 0.18/0.42 & ! [Ya: a] :
% 0.18/0.42 ( ( P @ Ya )
% 0.18/0.42 => ( less_eq @ Y4 @ Ya ) ) )
% 0.18/0.42 => ( Y4 = X4 ) ) )
% 0.18/0.42 => ( ( P @ Z2 )
% 0.18/0.42 => ( less_eq @ ( least_a @ less_eq @ P ) @ Z2 ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.Least1_le
% 0.18/0.42 thf(fact_166_local_OLeastI2__order,axiom,
% 0.18/0.42 ! [P: a > $o,X: a,Q: a > $o] :
% 0.18/0.42 ( ( P @ X )
% 0.18/0.42 => ( ! [Y4: a] :
% 0.18/0.42 ( ( P @ Y4 )
% 0.18/0.42 => ( less_eq @ X @ Y4 ) )
% 0.18/0.42 => ( ! [X3: a] :
% 0.18/0.42 ( ( P @ X3 )
% 0.18/0.42 => ( ! [Y5: a] :
% 0.18/0.42 ( ( P @ Y5 )
% 0.18/0.42 => ( less_eq @ X3 @ Y5 ) )
% 0.18/0.42 => ( Q @ X3 ) ) )
% 0.18/0.42 => ( Q @ ( least_a @ less_eq @ P ) ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.LeastI2_order
% 0.18/0.42 thf(fact_167_local_OLeast__equality,axiom,
% 0.18/0.42 ! [P: a > $o,X: a] :
% 0.18/0.42 ( ( P @ X )
% 0.18/0.42 => ( ! [Y4: a] :
% 0.18/0.42 ( ( P @ Y4 )
% 0.18/0.42 => ( less_eq @ X @ Y4 ) )
% 0.18/0.42 => ( ( least_a @ less_eq @ P )
% 0.18/0.42 = X ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.Least_equality
% 0.18/0.42 thf(fact_168_local_Oa__join__d,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( sup @ A @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) )
% 0.18/0.42 = ( sup @ A @ ( inf @ B @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.a_join_d
% 0.18/0.42 thf(fact_169_local_Ob__join__d,axiom,
% 0.18/0.42 ! [B: a,A: a,C: a] :
% 0.18/0.42 ( ( sup @ B @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) )
% 0.18/0.42 = ( sup @ B @ ( inf @ C @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.b_join_d
% 0.18/0.42 thf(fact_170_local_Od__aux__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( sup @ ( sup @ ( inf @ A @ B ) @ ( inf @ B @ C ) ) @ ( inf @ C @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.d_aux_def
% 0.18/0.42 thf(fact_171_local_Od__b__c__a,axiom,
% 0.18/0.42 ! [B: a,C: a,A: a] :
% 0.18/0.42 ( ( modula1936294176_aux_a @ inf @ sup @ B @ C @ A )
% 0.18/0.42 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.d_b_c_a
% 0.18/0.42 thf(fact_172_local_Od__c__a__b,axiom,
% 0.18/0.42 ! [C: a,A: a,B: a] :
% 0.18/0.42 ( ( modula1936294176_aux_a @ inf @ sup @ C @ A @ B )
% 0.18/0.42 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.d_c_a_b
% 0.18/0.42 thf(fact_173_local_Oa__meet__e,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( inf @ A @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
% 0.18/0.42 = ( inf @ A @ ( sup @ B @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.a_meet_e
% 0.18/0.42 thf(fact_174_local_Ob__meet__e,axiom,
% 0.18/0.42 ! [B: a,A: a,C: a] :
% 0.18/0.42 ( ( inf @ B @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
% 0.18/0.42 = ( inf @ B @ ( sup @ C @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.b_meet_e
% 0.18/0.42 thf(fact_175_local_Oc__meet__e,axiom,
% 0.18/0.42 ! [C: a,A: a,B: a] :
% 0.18/0.42 ( ( inf @ C @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
% 0.18/0.42 = ( inf @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.c_meet_e
% 0.18/0.42 thf(fact_176_local_Oe__aux__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( inf @ ( inf @ ( sup @ A @ B ) @ ( sup @ B @ C ) ) @ ( sup @ C @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.e_aux_def
% 0.18/0.42 thf(fact_177_local_Oe__b__c__a,axiom,
% 0.18/0.42 ! [B: a,C: a,A: a] :
% 0.18/0.42 ( ( modula1144073633_aux_a @ inf @ sup @ B @ C @ A )
% 0.18/0.42 = ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.e_b_c_a
% 0.18/0.42 thf(fact_178_local_Oe__c__a__b,axiom,
% 0.18/0.42 ! [C: a,A: a,B: a] :
% 0.18/0.42 ( ( modula1144073633_aux_a @ inf @ sup @ C @ A @ B )
% 0.18/0.42 = ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.e_c_a_b
% 0.18/0.42 thf(fact_179_local_OInf__fin_Osemilattice__order__set__axioms,axiom,
% 0.18/0.42 lattic655834328_set_a @ inf @ less_eq @ less ).
% 0.18/0.42
% 0.18/0.42 % local.Inf_fin.semilattice_order_set_axioms
% 0.18/0.42 thf(fact_180_local_OM5__lattice__incomp,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1376131916tice_a @ inf @ less @ sup @ A @ B @ C )
% 0.18/0.42 => ( modula1727524044comp_a @ less_eq @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.M5_lattice_incomp
% 0.18/0.42 thf(fact_181_local_Oc__aux__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula581031071_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( sup @ ( inf @ C @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.c_aux_def
% 0.18/0.42 thf(fact_182_local_Ob__aux__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1373251614_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( sup @ ( inf @ B @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.b_aux_def
% 0.18/0.42 thf(fact_183_local_Oa__aux__def,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula17988509_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( sup @ ( inf @ A @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.a_aux_def
% 0.18/0.42 thf(fact_184_lattice_Ono__distrib_Ocong,axiom,
% 0.18/0.42 modula1962211574trib_a = modula1962211574trib_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.no_distrib.cong
% 0.18/0.42 thf(fact_185_lattice_ON5__lattice_Ocong,axiom,
% 0.18/0.42 modula397570059tice_a = modula397570059tice_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.N5_lattice.cong
% 0.18/0.42 thf(fact_186_lattice_OM5__lattice_Ocong,axiom,
% 0.18/0.42 modula1376131916tice_a = modula1376131916tice_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.M5_lattice.cong
% 0.18/0.42 thf(fact_187_lattice_Oincomp_Ocong,axiom,
% 0.18/0.42 modula1727524044comp_a = modula1727524044comp_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.incomp.cong
% 0.18/0.42 thf(fact_188_lattice_Oe__aux_Ocong,axiom,
% 0.18/0.42 modula1144073633_aux_a = modula1144073633_aux_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.e_aux.cong
% 0.18/0.42 thf(fact_189_lattice_Od__aux_Ocong,axiom,
% 0.18/0.42 modula1936294176_aux_a = modula1936294176_aux_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.d_aux.cong
% 0.18/0.42 thf(fact_190_local_Oinf_Osemilattice__order__axioms,axiom,
% 0.18/0.42 semilattice_order_a @ inf @ less_eq @ less ).
% 0.18/0.42
% 0.18/0.42 % local.inf.semilattice_order_axioms
% 0.18/0.42 thf(fact_191_local_Oorder_Oordering__axioms,axiom,
% 0.18/0.42 ordering_a @ less_eq @ less ).
% 0.18/0.42
% 0.18/0.42 % local.order.ordering_axioms
% 0.18/0.42 thf(fact_192_local_Ocomp__fun__idem__sup,axiom,
% 0.18/0.42 finite40241356em_a_a @ sup ).
% 0.18/0.42
% 0.18/0.42 % local.comp_fun_idem_sup
% 0.18/0.42 thf(fact_193_local_Ocomp__fun__idem__inf,axiom,
% 0.18/0.42 finite40241356em_a_a @ inf ).
% 0.18/0.42
% 0.18/0.42 % local.comp_fun_idem_inf
% 0.18/0.42 thf(fact_194_local_Ob__a,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula1373251614_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( modula17988509_aux_a @ inf @ sup @ B @ C @ A ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.b_a
% 0.18/0.42 thf(fact_195_local_Oc__a,axiom,
% 0.18/0.42 ! [A: a,B: a,C: a] :
% 0.18/0.42 ( ( modula581031071_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.42 = ( modula17988509_aux_a @ inf @ sup @ C @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.c_a
% 0.18/0.42 thf(fact_196_lattice_Oc__aux_Ocong,axiom,
% 0.18/0.42 modula581031071_aux_a = modula581031071_aux_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.c_aux.cong
% 0.18/0.42 thf(fact_197_lattice_Ob__aux_Ocong,axiom,
% 0.18/0.42 modula1373251614_aux_a = modula1373251614_aux_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.b_aux.cong
% 0.18/0.42 thf(fact_198_lattice_Oa__aux_Ocong,axiom,
% 0.18/0.42 modula17988509_aux_a = modula17988509_aux_a ).
% 0.18/0.42
% 0.18/0.42 % lattice.a_aux.cong
% 0.18/0.42 thf(fact_199_local_Osup_Osemigroup__axioms,axiom,
% 0.18/0.42 semigroup_a @ sup ).
% 0.18/0.42
% 0.18/0.42 % local.sup.semigroup_axioms
% 0.18/0.42 thf(fact_200_local_Oinf_Osemigroup__axioms,axiom,
% 0.18/0.42 semigroup_a @ inf ).
% 0.18/0.42
% 0.18/0.42 % local.inf.semigroup_axioms
% 0.18/0.42 thf(fact_201_local_Osup_Osemilattice__axioms,axiom,
% 0.18/0.42 semilattice_a @ sup ).
% 0.18/0.42
% 0.18/0.42 % local.sup.semilattice_axioms
% 0.18/0.42 thf(fact_202_local_Oinf_Osemilattice__axioms,axiom,
% 0.18/0.42 semilattice_a @ inf ).
% 0.18/0.42
% 0.18/0.42 % local.inf.semilattice_axioms
% 0.18/0.42 thf(fact_203_local_Osup_Oabel__semigroup__axioms,axiom,
% 0.18/0.42 abel_semigroup_a @ sup ).
% 0.18/0.42
% 0.18/0.42 % local.sup.abel_semigroup_axioms
% 0.18/0.42 thf(fact_204_local_Oinf_Oabel__semigroup__axioms,axiom,
% 0.18/0.42 abel_semigroup_a @ inf ).
% 0.18/0.42
% 0.18/0.42 % local.inf.abel_semigroup_axioms
% 0.18/0.42 thf(fact_205_semilattice__order__set_Oaxioms_I1_J,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 0.18/0.42 ( ( lattic655834328_set_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( semilattice_order_a @ F @ Less_eq @ Less ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order_set.axioms(1)
% 0.18/0.42 thf(fact_206_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.18/0.42 ! [F: a > a > a] :
% 0.18/0.42 ( ( abel_semigroup_a @ F )
% 0.18/0.42 => ( semigroup_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % abel_semigroup.axioms(1)
% 0.18/0.42 thf(fact_207_abel__semigroup_Oleft__commute,axiom,
% 0.18/0.42 ! [F: a > a > a,B: a,A: a,C: a] :
% 0.18/0.42 ( ( abel_semigroup_a @ F )
% 0.18/0.42 => ( ( F @ B @ ( F @ A @ C ) )
% 0.18/0.42 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % abel_semigroup.left_commute
% 0.18/0.42 thf(fact_208_abel__semigroup_Ocommute,axiom,
% 0.18/0.42 ! [F: a > a > a,A: a,B: a] :
% 0.18/0.42 ( ( abel_semigroup_a @ F )
% 0.18/0.42 => ( ( F @ A @ B )
% 0.18/0.42 = ( F @ B @ A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % abel_semigroup.commute
% 0.18/0.42 thf(fact_209_semigroup_Ointro,axiom,
% 0.18/0.42 ! [F: a > a > a] :
% 0.18/0.42 ( ! [A4: a,B3: a,C2: a] :
% 0.18/0.42 ( ( F @ ( F @ A4 @ B3 ) @ C2 )
% 0.18/0.42 = ( F @ A4 @ ( F @ B3 @ C2 ) ) )
% 0.18/0.42 => ( semigroup_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % semigroup.intro
% 0.18/0.42 thf(fact_210_semigroup_Oassoc,axiom,
% 0.18/0.42 ! [F: a > a > a,A: a,B: a,C: a] :
% 0.18/0.42 ( ( semigroup_a @ F )
% 0.18/0.42 => ( ( F @ ( F @ A @ B ) @ C )
% 0.18/0.42 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semigroup.assoc
% 0.18/0.42 thf(fact_211_semigroup__def,axiom,
% 0.18/0.42 ( semigroup_a
% 0.18/0.42 = ( ^ [F2: a > a > a] :
% 0.18/0.42 ! [A2: a,B2: a,C3: a] :
% 0.18/0.42 ( ( F2 @ ( F2 @ A2 @ B2 ) @ C3 )
% 0.18/0.42 = ( F2 @ A2 @ ( F2 @ B2 @ C3 ) ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semigroup_def
% 0.18/0.42 thf(fact_212_semilattice_Oaxioms_I1_J,axiom,
% 0.18/0.42 ! [F: a > a > a] :
% 0.18/0.42 ( ( semilattice_a @ F )
% 0.18/0.42 => ( abel_semigroup_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice.axioms(1)
% 0.18/0.42 thf(fact_213_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 0.18/0.42 lattic1885654924_set_a @ sup ).
% 0.18/0.42
% 0.18/0.42 % local.Sup_fin.semilattice_set_axioms
% 0.18/0.42 thf(fact_214_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 0.18/0.42 lattic1885654924_set_a @ inf ).
% 0.18/0.42
% 0.18/0.42 % local.Inf_fin.semilattice_set_axioms
% 0.18/0.42 thf(fact_215_semilattice__order__set_Oaxioms_I2_J,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 0.18/0.42 ( ( lattic655834328_set_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( lattic1885654924_set_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order_set.axioms(2)
% 0.18/0.42 thf(fact_216_semilattice__set__def,axiom,
% 0.18/0.42 lattic1885654924_set_a = semilattice_a ).
% 0.18/0.42
% 0.18/0.42 % semilattice_set_def
% 0.18/0.42 thf(fact_217_semilattice__set_Ointro,axiom,
% 0.18/0.42 ! [F: a > a > a] :
% 0.18/0.42 ( ( semilattice_a @ F )
% 0.18/0.42 => ( lattic1885654924_set_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_set.intro
% 0.18/0.42 thf(fact_218_semilattice__set_Oaxioms,axiom,
% 0.18/0.42 ! [F: a > a > a] :
% 0.18/0.42 ( ( lattic1885654924_set_a @ F )
% 0.18/0.42 => ( semilattice_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_set.axioms
% 0.18/0.42 thf(fact_219_semilattice__order__set__def,axiom,
% 0.18/0.42 ( lattic655834328_set_a
% 0.18/0.42 = ( ^ [F2: a > a > a,Less_eq2: a > a > $o,Less2: a > a > $o] :
% 0.18/0.42 ( ( semilattice_order_a @ F2 @ Less_eq2 @ Less2 )
% 0.18/0.42 & ( lattic1885654924_set_a @ F2 ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order_set_def
% 0.18/0.42 thf(fact_220_semilattice__order__set_Ointro,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( lattic1885654924_set_a @ F )
% 0.18/0.42 => ( lattic655834328_set_a @ F @ Less_eq @ Less ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order_set.intro
% 0.18/0.42 thf(fact_221_semilattice__order_Omono,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,C: a,B: a,D: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ C )
% 0.18/0.42 => ( ( Less_eq @ B @ D )
% 0.18/0.42 => ( Less_eq @ ( F @ A @ B ) @ ( F @ C @ D ) ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.mono
% 0.18/0.42 thf(fact_222_semilattice__order_OorderE,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ B )
% 0.18/0.42 => ( A
% 0.18/0.42 = ( F @ A @ B ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.orderE
% 0.18/0.42 thf(fact_223_semilattice__order_OorderI,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( A
% 0.18/0.42 = ( F @ A @ B ) )
% 0.18/0.42 => ( Less_eq @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.orderI
% 0.18/0.42 thf(fact_224_semilattice__order_Oabsorb1,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ B )
% 0.18/0.42 => ( ( F @ A @ B )
% 0.18/0.42 = A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.absorb1
% 0.18/0.42 thf(fact_225_semilattice__order_Oabsorb2,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,A: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ B @ A )
% 0.18/0.42 => ( ( F @ A @ B )
% 0.18/0.42 = B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.absorb2
% 0.18/0.42 thf(fact_226_semilattice__order_OboundedE,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ ( F @ B @ C ) )
% 0.18/0.42 => ~ ( ( Less_eq @ A @ B )
% 0.18/0.42 => ~ ( Less_eq @ A @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.boundedE
% 0.18/0.42 thf(fact_227_semilattice__order_OboundedI,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ B )
% 0.18/0.42 => ( ( Less_eq @ A @ C )
% 0.18/0.42 => ( Less_eq @ A @ ( F @ B @ C ) ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.boundedI
% 0.18/0.42 thf(fact_228_semilattice__order_Oorder__iff,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ B )
% 0.18/0.42 = ( A
% 0.18/0.42 = ( F @ A @ B ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.order_iff
% 0.18/0.42 thf(fact_229_semilattice__order_Ocobounded1,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( Less_eq @ ( F @ A @ B ) @ A ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.cobounded1
% 0.18/0.42 thf(fact_230_semilattice__order_Ocobounded2,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( Less_eq @ ( F @ A @ B ) @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.cobounded2
% 0.18/0.42 thf(fact_231_semilattice__order_Oabsorb__iff1,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ B )
% 0.18/0.42 = ( ( F @ A @ B )
% 0.18/0.42 = A ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.absorb_iff1
% 0.18/0.42 thf(fact_232_semilattice__order_Oabsorb__iff2,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,A: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ B @ A )
% 0.18/0.42 = ( ( F @ A @ B )
% 0.18/0.42 = B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.absorb_iff2
% 0.18/0.42 thf(fact_233_semilattice__order_Obounded__iff,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ ( F @ B @ C ) )
% 0.18/0.42 = ( ( Less_eq @ A @ B )
% 0.18/0.42 & ( Less_eq @ A @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.bounded_iff
% 0.18/0.42 thf(fact_234_semilattice__order_OcoboundedI1,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,C: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ A @ C )
% 0.18/0.42 => ( Less_eq @ ( F @ A @ B ) @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.coboundedI1
% 0.18/0.42 thf(fact_235_semilattice__order_OcoboundedI2,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,C: a,A: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less_eq @ B @ C )
% 0.18/0.42 => ( Less_eq @ ( F @ A @ B ) @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.coboundedI2
% 0.18/0.42 thf(fact_236_semilattice__order_Ostrict__boundedE,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less @ A @ ( F @ B @ C ) )
% 0.18/0.42 => ~ ( ( Less @ A @ B )
% 0.18/0.42 => ~ ( Less @ A @ C ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.strict_boundedE
% 0.18/0.42 thf(fact_237_semilattice__order_Ostrict__order__iff,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less @ A @ B )
% 0.18/0.42 = ( ( A
% 0.18/0.42 = ( F @ A @ B ) )
% 0.18/0.42 & ( A != B ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.strict_order_iff
% 0.18/0.42 thf(fact_238_semilattice__order_Ostrict__coboundedI1,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,C: a,B: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less @ A @ C )
% 0.18/0.42 => ( Less @ ( F @ A @ B ) @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.strict_coboundedI1
% 0.18/0.42 thf(fact_239_semilattice__order_Ostrict__coboundedI2,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,C: a,A: a] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( ( Less @ B @ C )
% 0.18/0.42 => ( Less @ ( F @ A @ B ) @ C ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.strict_coboundedI2
% 0.18/0.42 thf(fact_240_semilattice_Oidem,axiom,
% 0.18/0.42 ! [F: a > a > a,A: a] :
% 0.18/0.42 ( ( semilattice_a @ F )
% 0.18/0.42 => ( ( F @ A @ A )
% 0.18/0.42 = A ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice.idem
% 0.18/0.42 thf(fact_241_semilattice_Oleft__idem,axiom,
% 0.18/0.42 ! [F: a > a > a,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_a @ F )
% 0.18/0.42 => ( ( F @ A @ ( F @ A @ B ) )
% 0.18/0.42 = ( F @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice.left_idem
% 0.18/0.42 thf(fact_242_semilattice_Oright__idem,axiom,
% 0.18/0.42 ! [F: a > a > a,A: a,B: a] :
% 0.18/0.42 ( ( semilattice_a @ F )
% 0.18/0.42 => ( ( F @ ( F @ A @ B ) @ B )
% 0.18/0.42 = ( F @ A @ B ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice.right_idem
% 0.18/0.42 thf(fact_243_semilattice__order_Oaxioms_I1_J,axiom,
% 0.18/0.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 0.18/0.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 0.18/0.42 => ( semilattice_a @ F ) ) ).
% 0.18/0.42
% 0.18/0.42 % semilattice_order.axioms(1)
% 0.18/0.42 thf(fact_244_local_Obdd__below__def,axiom,
% 0.18/0.42 ! [A3: set_a] :
% 0.18/0.42 ( ( condit1001553558elow_a @ less_eq @ A3 )
% 0.18/0.42 = ( ? [M: a] :
% 0.18/0.42 ! [X2: a] :
% 0.18/0.42 ( ( member_a @ X2 @ A3 )
% 0.18/0.42 => ( less_eq @ M @ X2 ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.bdd_below_def
% 0.18/0.42 thf(fact_245_local_Obdd__above__def,axiom,
% 0.18/0.42 ! [A3: set_a] :
% 0.18/0.42 ( ( condit1627435690bove_a @ less_eq @ A3 )
% 0.18/0.42 = ( ? [M2: a] :
% 0.18/0.42 ! [X2: a] :
% 0.18/0.42 ( ( member_a @ X2 @ A3 )
% 0.18/0.42 => ( less_eq @ X2 @ M2 ) ) ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.bdd_above_def
% 0.18/0.42 thf(fact_246_local_Obdd__belowI,axiom,
% 0.18/0.42 ! [A3: set_a,M3: a] :
% 0.18/0.42 ( ! [X3: a] :
% 0.18/0.42 ( ( member_a @ X3 @ A3 )
% 0.18/0.42 => ( less_eq @ M3 @ X3 ) )
% 0.18/0.42 => ( condit1001553558elow_a @ less_eq @ A3 ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.bdd_belowI
% 0.18/0.42 thf(fact_247_local_Obdd__aboveI,axiom,
% 0.18/0.42 ! [A3: set_a,M4: a] :
% 0.18/0.42 ( ! [X3: a] :
% 0.18/0.42 ( ( member_a @ X3 @ A3 )
% 0.18/0.42 => ( less_eq @ X3 @ M4 ) )
% 0.18/0.42 => ( condit1627435690bove_a @ less_eq @ A3 ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.bdd_aboveI
% 0.18/0.42 thf(fact_248_local_Obdd__below__Ioo,axiom,
% 0.18/0.42 ! [A: a,B: a] : ( condit1001553558elow_a @ less_eq @ ( set_gr1491433118Than_a @ less @ A @ B ) ) ).
% 0.18/0.42
% 0.18/0.42 % local.bdd_below_Ioo
% 0.18/0.42
% 0.18/0.42 % Conjectures (5)
% 0.18/0.42 thf(conj_0,hypothesis,
% 0.18/0.42 lattice_a @ inf @ less_eq @ less @ sup ).
% 0.18/0.42
% 0.18/0.42 thf(conj_1,hypothesis,
% 0.18/0.42 less_eq @ x @ y ).
% 0.18/0.42
% 0.18/0.42 thf(conj_2,hypothesis,
% 0.18/0.42 ( ( sup @ x @ ( inf @ y @ z ) )
% 0.18/0.42 != ( inf @ y @ ( sup @ x @ z ) ) ) ).
% 0.18/0.42
% 0.18/0.42 thf(conj_3,hypothesis,
% 0.18/0.42 less @ ( sup @ x @ ( inf @ y @ z ) ) @ ( inf @ y @ ( sup @ x @ z ) ) ).
% 0.18/0.42
% 0.18/0.42 thf(conj_4,conjecture,
% 0.18/0.42 less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ).
% 0.18/0.42
% 0.18/0.42 %------------------------------------------------------------------------------
% 0.18/0.42 ------------------------
% 2.19/1.38 ---- Embedded in THF ---
% 2.19/1.38 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.19/1.38 %%% Generated on Tue Feb 24 08:26:48 EST 2026
% 2.19/1.38 %%% using '$$dhol' embedding, version 1.3.0.
% 2.19/1.38 %%% Logic specification used:
% 2.19/1.38 %%% thf(spec, logic, $$dhol).
% 2.19/1.38
% 2.19/1.38 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2DTF31975.p
% See solution above
% 2.19/1.39 ------------------------
% 2.47/1.42 ---- Cleaned THF ---
% 2.47/1.42 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.47/1.42 set_a: $tType ).
% 2.47/1.42
% 2.47/1.42 thf(ty_n_tf__a,type,
% 2.47/1.42 a: $tType ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above_001tf__a,type,
% 2.47/1.42 condit1627435690bove_a: ( a > a > $o ) > set_a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below_001tf__a,type,
% 2.47/1.42 condit1001553558elow_a: ( a > a > $o ) > set_a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Finite__Set_Ocomp__fun__idem_001tf__a_001tf__a,type,
% 2.47/1.42 finite40241356em_a_a: ( a > a > a ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Groups_Oabel__semigroup_001tf__a,type,
% 2.47/1.42 abel_semigroup_a: ( a > a > a ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Groups_Osemigroup_001tf__a,type,
% 2.47/1.42 semigroup_a: ( a > a > a ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Lattices_Oclass_Olattice_001tf__a,type,
% 2.47/1.42 lattice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > ( a > a > a ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Lattices_Osemilattice_001tf__a,type,
% 2.47/1.42 semilattice_a: ( a > a > a ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Lattices_Osemilattice__order_001tf__a,type,
% 2.47/1.42 semilattice_order_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
% 2.47/1.42 lattic655834328_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
% 2.47/1.42 lattic1885654924_set_a: ( a > a > a ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_OM5__lattice_001tf__a,type,
% 2.47/1.42 modula1376131916tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_ON5__lattice_001tf__a,type,
% 2.47/1.42 modula397570059tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oa__aux_001tf__a,type,
% 2.47/1.42 modula17988509_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Ob__aux_001tf__a,type,
% 2.47/1.42 modula1373251614_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oc__aux_001tf__a,type,
% 2.47/1.42 modula581031071_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Od__aux_001tf__a,type,
% 2.47/1.42 modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oe__aux_001tf__a,type,
% 2.47/1.42 modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oincomp_001tf__a,type,
% 2.47/1.42 modula1727524044comp_a: ( a > a > $o ) > a > a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Ono__distrib_001tf__a,type,
% 2.47/1.42 modula1962211574trib_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Orderings_Oord_OLeast_001tf__a,type,
% 2.47/1.42 least_a: ( a > a > $o ) > ( a > $o ) > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Orderings_Oord_Omax_001tf__a,type,
% 2.47/1.42 max_a: ( a > a > $o ) > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Orderings_Oord_Omin_001tf__a,type,
% 2.47/1.42 min_a: ( a > a > $o ) > a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Orderings_Oorder_OGreatest_001tf__a,type,
% 2.47/1.42 greatest_a: ( a > a > $o ) > ( a > $o ) > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Orderings_Oordering_001tf__a,type,
% 2.47/1.42 ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.47/1.42 collect_a: ( a > $o ) > set_a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001tf__a,type,
% 2.47/1.42 set_gr1491433118Than_a: ( a > a > $o ) > a > a > set_a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_c_member_001tf__a,type,
% 2.47/1.42 member_a: a > set_a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_inf,type,
% 2.47/1.42 inf: a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_less,type,
% 2.47/1.42 less: a > a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_less__eq,type,
% 2.47/1.42 less_eq: a > a > $o ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_sup,type,
% 2.47/1.42 sup: a > a > a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_x,type,
% 2.47/1.42 x: a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_y,type,
% 2.47/1.42 y: a ).
% 2.47/1.42
% 2.47/1.42 thf(sy_v_z,type,
% 2.47/1.42 z: a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_0_local_Oantisym,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( less_eq @ Y @ X )
% 2.47/1.42 => ( X = Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_1_local_Oantisym__conv,axiom,
% 2.47/1.42 ! [Y: a,X: a] :
% 2.47/1.42 ( ( less_eq @ Y @ X )
% 2.47/1.42 => ( ( less_eq @ X @ Y )
% 2.47/1.42 = ( X = Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_2_local_Odual__order_Oantisym,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( ( less_eq @ A @ B )
% 2.47/1.42 => ( A = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_3_local_Odual__order_Oeq__iff,axiom,
% 2.47/1.42 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 2.47/1.42 = ( ^ [A2: a,B2: a] :
% 2.47/1.42 ( ( less_eq @ B2 @ A2 )
% 2.47/1.42 & ( less_eq @ A2 @ B2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_4_local_Odual__order_Otrans,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( ( less_eq @ C @ B )
% 2.47/1.42 => ( less_eq @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_5_local_Oeq__iff,axiom,
% 2.47/1.42 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 2.47/1.42 = ( ^ [X2: a,Y3: a] :
% 2.47/1.42 ( ( less_eq @ X2 @ Y3 )
% 2.47/1.42 & ( less_eq @ Y3 @ X2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_6_local_Oeq__refl,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( X = Y )
% 2.47/1.42 => ( less_eq @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_7_local_Oord__eq__le__trans,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( A = B )
% 2.47/1.42 => ( ( less_eq @ B @ C )
% 2.47/1.42 => ( less_eq @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_8_local_Oord__le__eq__trans,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( B = C )
% 2.47/1.42 => ( less_eq @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_9_local_Oorder_Oantisym,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( less_eq @ B @ A )
% 2.47/1.42 => ( A = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_10_local_Oorder_Oeq__iff,axiom,
% 2.47/1.42 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 2.47/1.42 = ( ^ [A2: a,B2: a] :
% 2.47/1.42 ( ( less_eq @ A2 @ B2 )
% 2.47/1.42 & ( less_eq @ B2 @ A2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_11_local_Oorder_Otrans,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( less_eq @ B @ C )
% 2.47/1.42 => ( less_eq @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_12_local_Oorder__trans,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( less_eq @ Y @ Z2 )
% 2.47/1.42 => ( less_eq @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_13_local_Odual__order_Oasym,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 => ~ ( less @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_14_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 => ( A != B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_15_local_Odual__order_Ostrict__trans,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 => ( ( less @ C @ B )
% 2.47/1.42 => ( less @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_16_local_Oless__asym,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ~ ( less @ Y @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_17_local_Oless__asym_H,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ~ ( less @ B @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_18_local_Oless__imp__neq,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( X != Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_19_local_Oless__imp__not__eq,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( X != Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_20_local_Oless__imp__not__eq2,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( Y != X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_21_local_Oless__imp__not__less,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ~ ( less @ Y @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_22_local_Oless__imp__triv,axiom,
% 2.47/1.42 ! [X: a,Y: a,P: $o] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( ( less @ Y @ X )
% 2.47/1.42 => P ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_23_local_Oless__irrefl,axiom,
% 2.47/1.42 ! [X: a] :
% 2.47/1.42 ~ ( less @ X @ X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_24_local_Oless__not__sym,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ~ ( less @ Y @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_25_local_Oless__trans,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( ( less @ Y @ Z2 )
% 2.47/1.42 => ( less @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_26_local_Oord__eq__less__trans,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( A = B )
% 2.47/1.42 => ( ( less @ B @ C )
% 2.47/1.42 => ( less @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_27_local_Oord__less__eq__trans,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ( ( B = C )
% 2.47/1.42 => ( less @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_28_local_Oorder_Oasym,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ~ ( less @ B @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_29_local_Oorder_Oirrefl,axiom,
% 2.47/1.42 ! [A: a] :
% 2.47/1.42 ~ ( less @ A @ A ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_30_local_Oorder_Ostrict__implies__not__eq,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ( A != B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_31_local_Oorder_Ostrict__trans,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ( ( less @ B @ C )
% 2.47/1.42 => ( less @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_32_local_Oinf_Oassoc,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( inf @ ( inf @ A @ B ) @ C )
% 2.47/1.42 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_33_local_Oinf_Ocommute,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( inf @ A @ B )
% 2.47/1.42 = ( inf @ B @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_34_local_Oinf_Oleft__commute,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( inf @ B @ ( inf @ A @ C ) )
% 2.47/1.42 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_35_local_Oinf__assoc,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( inf @ ( inf @ X @ Y ) @ Z2 )
% 2.47/1.42 = ( inf @ X @ ( inf @ Y @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_36_local_Oinf__commute,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( inf @ X @ Y )
% 2.47/1.42 = ( inf @ Y @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_37_local_Oinf__left__commute,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( inf @ X @ ( inf @ Y @ Z2 ) )
% 2.47/1.42 = ( inf @ Y @ ( inf @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_38_local_Osup_Oassoc,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( sup @ ( sup @ A @ B ) @ C )
% 2.47/1.42 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_39_local_Osup_Ocommute,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( sup @ A @ B )
% 2.47/1.42 = ( sup @ B @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_40_local_Osup_Oleft__commute,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( sup @ B @ ( sup @ A @ C ) )
% 2.47/1.42 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_41_local_Osup__assoc,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( sup @ ( sup @ X @ Y ) @ Z2 )
% 2.47/1.42 = ( sup @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_42_local_Osup__commute,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( sup @ X @ Y )
% 2.47/1.42 = ( sup @ Y @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_43_local_Osup__left__commute,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( sup @ X @ ( sup @ Y @ Z2 ) )
% 2.47/1.42 = ( sup @ Y @ ( sup @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_44_local_Oantisym__conv1,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ~ ( less @ X @ Y )
% 2.47/1.42 => ( ( less_eq @ X @ Y )
% 2.47/1.42 = ( X = Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_45_mem__Collect__eq,axiom,
% 2.47/1.42 ! [A: a,P: a > $o] :
% 2.47/1.42 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.47/1.42 = ( P @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_46_Collect__mem__eq,axiom,
% 2.47/1.42 ! [A3: set_a] :
% 2.47/1.42 ( ( collect_a
% 2.47/1.42 @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
% 2.47/1.42 = A3 ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_47_Collect__cong,axiom,
% 2.47/1.42 ! [P: a > $o,Q: a > $o] :
% 2.47/1.42 ( ! [X3: a] :
% 2.47/1.42 ( ( P @ X3 )
% 2.47/1.42 = ( Q @ X3 ) )
% 2.47/1.42 => ( ( collect_a @ P )
% 2.47/1.42 = ( collect_a @ Q ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_48_local_Oantisym__conv2,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( ~ ( less @ X @ Y ) )
% 2.47/1.42 = ( X = Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_49_local_Odual__order_Onot__eq__order__implies__strict,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( A != B )
% 2.47/1.42 => ( ( less_eq @ B @ A )
% 2.47/1.42 => ( less @ B @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_50_local_Odual__order_Oorder__iff__strict,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 = ( ( less @ B @ A )
% 2.47/1.42 | ( A = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_51_local_Odual__order_Ostrict__iff__order,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 = ( ( less_eq @ B @ A )
% 2.47/1.42 & ( A != B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_52_local_Odual__order_Ostrict__implies__order,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 => ( less_eq @ B @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_53_local_Odual__order_Ostrict__trans1,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( ( less @ C @ B )
% 2.47/1.42 => ( less @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_54_local_Odual__order_Ostrict__trans2,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 => ( ( less_eq @ C @ B )
% 2.47/1.42 => ( less @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_55_local_OleD,axiom,
% 2.47/1.42 ! [Y: a,X: a] :
% 2.47/1.42 ( ( less_eq @ Y @ X )
% 2.47/1.42 => ~ ( less @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_56_local_Ole__imp__less__or__eq,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( less @ X @ Y )
% 2.47/1.42 | ( X = Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_57_local_Ole__less,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 = ( ( less @ X @ Y )
% 2.47/1.42 | ( X = Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_58_local_Ole__less__trans,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( less @ Y @ Z2 )
% 2.47/1.42 => ( less @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_59_local_Ole__neq__trans,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( A != B )
% 2.47/1.42 => ( less @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_60_local_Oless__imp__le,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( less_eq @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_61_local_Oless__le,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 = ( ( less_eq @ X @ Y )
% 2.47/1.42 & ( X != Y ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_62_local_Oless__le__not__le,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 = ( ( less_eq @ X @ Y )
% 2.47/1.42 & ~ ( less_eq @ Y @ X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_63_local_Oless__le__trans,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less @ X @ Y )
% 2.47/1.42 => ( ( less_eq @ Y @ Z2 )
% 2.47/1.42 => ( less @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_64_local_Oorder_Onot__eq__order__implies__strict,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( A != B )
% 2.47/1.42 => ( ( less_eq @ A @ B )
% 2.47/1.42 => ( less @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_65_local_Oorder_Oorder__iff__strict,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 = ( ( less @ A @ B )
% 2.47/1.42 | ( A = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_66_local_Oorder_Ostrict__iff__order,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 = ( ( less_eq @ A @ B )
% 2.47/1.42 & ( A != B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_67_local_Oorder_Ostrict__implies__order,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ( less_eq @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_68_local_Oorder_Ostrict__trans1,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( less @ B @ C )
% 2.47/1.42 => ( less @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_69_local_Oorder_Ostrict__trans2,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 => ( ( less_eq @ B @ C )
% 2.47/1.42 => ( less @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_70_local_Oinf_Oabsorb1,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( inf @ A @ B )
% 2.47/1.42 = A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_71_local_Oinf_Oabsorb2,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( ( inf @ A @ B )
% 2.47/1.42 = B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_72_local_Oinf_Oabsorb__iff1,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 = ( ( inf @ A @ B )
% 2.47/1.42 = A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_73_local_Oinf_Oabsorb__iff2,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 = ( ( inf @ A @ B )
% 2.47/1.42 = B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_74_local_Oinf_OboundedE,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 2.47/1.42 => ~ ( ( less_eq @ A @ B )
% 2.47/1.42 => ~ ( less_eq @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_75_local_Oinf_OboundedI,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( less_eq @ A @ C )
% 2.47/1.42 => ( less_eq @ A @ ( inf @ B @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_76_local_Oinf_Ocobounded1,axiom,
% 2.47/1.42 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ A ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_77_local_Oinf_Ocobounded2,axiom,
% 2.47/1.42 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ B ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_78_local_Oinf_OcoboundedI1,axiom,
% 2.47/1.42 ! [A: a,C: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ C )
% 2.47/1.42 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_79_local_Oinf_OcoboundedI2,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ C )
% 2.47/1.42 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_80_local_Oinf_OorderE,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( A
% 2.47/1.42 = ( inf @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_81_local_Oinf_OorderI,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( A
% 2.47/1.42 = ( inf @ A @ B ) )
% 2.47/1.42 => ( less_eq @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_82_local_Oinf_Oorder__iff,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 = ( A
% 2.47/1.42 = ( inf @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_83_local_Oinf__absorb1,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( inf @ X @ Y )
% 2.47/1.42 = X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_84_local_Oinf__absorb2,axiom,
% 2.47/1.42 ! [Y: a,X: a] :
% 2.47/1.42 ( ( less_eq @ Y @ X )
% 2.47/1.42 => ( ( inf @ X @ Y )
% 2.47/1.42 = Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_85_local_Oinf__greatest,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( less_eq @ X @ Z2 )
% 2.47/1.42 => ( less_eq @ X @ ( inf @ Y @ Z2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_86_local_Oinf__le1,axiom,
% 2.47/1.42 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_87_local_Oinf__le2,axiom,
% 2.47/1.42 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_88_local_Oinf__mono,axiom,
% 2.47/1.42 ! [A: a,C: a,B: a,D: a] :
% 2.47/1.42 ( ( less_eq @ A @ C )
% 2.47/1.42 => ( ( less_eq @ B @ D )
% 2.47/1.42 => ( less_eq @ ( inf @ A @ B ) @ ( inf @ C @ D ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_89_local_Oinf__unique,axiom,
% 2.47/1.42 ! [F: a > a > a,X: a,Y: a] :
% 2.47/1.42 ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ X3 )
% 2.47/1.42 => ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ Y4 )
% 2.47/1.42 => ( ! [X3: a,Y4: a,Z3: a] :
% 2.47/1.42 ( ( less_eq @ X3 @ Y4 )
% 2.47/1.42 => ( ( less_eq @ X3 @ Z3 )
% 2.47/1.42 => ( less_eq @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
% 2.47/1.42 => ( ( inf @ X @ Y )
% 2.47/1.42 = ( F @ X @ Y ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_90_local_Ole__iff__inf,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 = ( ( inf @ X @ Y )
% 2.47/1.42 = X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_91_local_Ole__infE,axiom,
% 2.47/1.42 ! [X: a,A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ X @ ( inf @ A @ B ) )
% 2.47/1.42 => ~ ( ( less_eq @ X @ A )
% 2.47/1.42 => ~ ( less_eq @ X @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_92_local_Ole__infI,axiom,
% 2.47/1.42 ! [X: a,A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ X @ A )
% 2.47/1.42 => ( ( less_eq @ X @ B )
% 2.47/1.42 => ( less_eq @ X @ ( inf @ A @ B ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_93_local_Ole__infI1,axiom,
% 2.47/1.42 ! [A: a,X: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ X )
% 2.47/1.42 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_94_local_Ole__infI2,axiom,
% 2.47/1.42 ! [B: a,X: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ X )
% 2.47/1.42 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_95_local_Ole__iff__sup,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 = ( ( sup @ X @ Y )
% 2.47/1.42 = Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_96_local_Ole__supE,axiom,
% 2.47/1.42 ! [A: a,B: a,X: a] :
% 2.47/1.42 ( ( less_eq @ ( sup @ A @ B ) @ X )
% 2.47/1.42 => ~ ( ( less_eq @ A @ X )
% 2.47/1.42 => ~ ( less_eq @ B @ X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_97_local_Ole__supI,axiom,
% 2.47/1.42 ! [A: a,X: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ X )
% 2.47/1.42 => ( ( less_eq @ B @ X )
% 2.47/1.42 => ( less_eq @ ( sup @ A @ B ) @ X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_98_local_Ole__supI1,axiom,
% 2.47/1.42 ! [X: a,A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ X @ A )
% 2.47/1.42 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_99_local_Ole__supI2,axiom,
% 2.47/1.42 ! [X: a,B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ X @ B )
% 2.47/1.42 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_100_local_Osup_Oabsorb1,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( ( sup @ A @ B )
% 2.47/1.42 = A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_101_local_Osup_Oabsorb2,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( sup @ A @ B )
% 2.47/1.42 = B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_102_local_Osup_Oabsorb__iff1,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 = ( ( sup @ A @ B )
% 2.47/1.42 = A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_103_local_Osup_Oabsorb__iff2,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ A @ B )
% 2.47/1.42 = ( ( sup @ A @ B )
% 2.47/1.42 = B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_104_local_Osup_OboundedE,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 2.47/1.42 => ~ ( ( less_eq @ B @ A )
% 2.47/1.42 => ~ ( less_eq @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_105_local_Osup_OboundedI,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( ( less_eq @ C @ A )
% 2.47/1.42 => ( less_eq @ ( sup @ B @ C ) @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_106_local_Osup_Ocobounded1,axiom,
% 2.47/1.42 ! [A: a,B: a] : ( less_eq @ A @ ( sup @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_107_local_Osup_Ocobounded2,axiom,
% 2.47/1.42 ! [B: a,A: a] : ( less_eq @ B @ ( sup @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_108_local_Osup_OcoboundedI1,axiom,
% 2.47/1.42 ! [C: a,A: a,B: a] :
% 2.47/1.42 ( ( less_eq @ C @ A )
% 2.47/1.42 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_109_local_Osup_OcoboundedI2,axiom,
% 2.47/1.42 ! [C: a,B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ C @ B )
% 2.47/1.42 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_110_local_Osup_Omono,axiom,
% 2.47/1.42 ! [C: a,A: a,D: a,B: a] :
% 2.47/1.42 ( ( less_eq @ C @ A )
% 2.47/1.42 => ( ( less_eq @ D @ B )
% 2.47/1.42 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A @ B ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_111_local_Osup_OorderE,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 => ( A
% 2.47/1.42 = ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_112_local_Osup_OorderI,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( A
% 2.47/1.42 = ( sup @ A @ B ) )
% 2.47/1.42 => ( less_eq @ B @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_113_local_Osup_Oorder__iff,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less_eq @ B @ A )
% 2.47/1.42 = ( A
% 2.47/1.42 = ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_114_local_Osup__absorb1,axiom,
% 2.47/1.42 ! [Y: a,X: a] :
% 2.47/1.42 ( ( less_eq @ Y @ X )
% 2.47/1.42 => ( ( sup @ X @ Y )
% 2.47/1.42 = X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_115_local_Osup__absorb2,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( less_eq @ X @ Y )
% 2.47/1.42 => ( ( sup @ X @ Y )
% 2.47/1.42 = Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_116_local_Osup__ge1,axiom,
% 2.47/1.42 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_117_local_Osup__ge2,axiom,
% 2.47/1.42 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_118_local_Osup__least,axiom,
% 2.47/1.42 ! [Y: a,X: a,Z2: a] :
% 2.47/1.42 ( ( less_eq @ Y @ X )
% 2.47/1.42 => ( ( less_eq @ Z2 @ X )
% 2.47/1.42 => ( less_eq @ ( sup @ Y @ Z2 ) @ X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_119_local_Osup__mono,axiom,
% 2.47/1.42 ! [A: a,C: a,B: a,D: a] :
% 2.47/1.42 ( ( less_eq @ A @ C )
% 2.47/1.42 => ( ( less_eq @ B @ D )
% 2.47/1.42 => ( less_eq @ ( sup @ A @ B ) @ ( sup @ C @ D ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_120_local_Osup__unique,axiom,
% 2.47/1.42 ! [F: a > a > a,X: a,Y: a] :
% 2.47/1.42 ( ! [X3: a,Y4: a] : ( less_eq @ X3 @ ( F @ X3 @ Y4 ) )
% 2.47/1.42 => ( ! [X3: a,Y4: a] : ( less_eq @ Y4 @ ( F @ X3 @ Y4 ) )
% 2.47/1.42 => ( ! [X3: a,Y4: a,Z3: a] :
% 2.47/1.42 ( ( less_eq @ Y4 @ X3 )
% 2.47/1.42 => ( ( less_eq @ Z3 @ X3 )
% 2.47/1.42 => ( less_eq @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
% 2.47/1.42 => ( ( sup @ X @ Y )
% 2.47/1.42 = ( F @ X @ Y ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_121_local_Oinf_Ostrict__boundedE,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less @ A @ ( inf @ B @ C ) )
% 2.47/1.42 => ~ ( ( less @ A @ B )
% 2.47/1.42 => ~ ( less @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_122_local_Oinf_Ostrict__coboundedI1,axiom,
% 2.47/1.42 ! [A: a,C: a,B: a] :
% 2.47/1.42 ( ( less @ A @ C )
% 2.47/1.42 => ( less @ ( inf @ A @ B ) @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_123_local_Oinf_Ostrict__coboundedI2,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( less @ B @ C )
% 2.47/1.42 => ( less @ ( inf @ A @ B ) @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_124_local_Oinf_Ostrict__order__iff,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( less @ A @ B )
% 2.47/1.42 = ( ( A
% 2.47/1.42 = ( inf @ A @ B ) )
% 2.47/1.42 & ( A != B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_125_local_Oless__infI1,axiom,
% 2.47/1.42 ! [A: a,X: a,B: a] :
% 2.47/1.42 ( ( less @ A @ X )
% 2.47/1.42 => ( less @ ( inf @ A @ B ) @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_126_local_Oless__infI2,axiom,
% 2.47/1.42 ! [B: a,X: a,A: a] :
% 2.47/1.42 ( ( less @ B @ X )
% 2.47/1.42 => ( less @ ( inf @ A @ B ) @ X ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_127_local_Oless__supI1,axiom,
% 2.47/1.42 ! [X: a,A: a,B: a] :
% 2.47/1.42 ( ( less @ X @ A )
% 2.47/1.42 => ( less @ X @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_128_local_Oless__supI2,axiom,
% 2.47/1.42 ! [X: a,B: a,A: a] :
% 2.47/1.42 ( ( less @ X @ B )
% 2.47/1.42 => ( less @ X @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_129_local_Osup_Ostrict__boundedE,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( less @ ( sup @ B @ C ) @ A )
% 2.47/1.42 => ~ ( ( less @ B @ A )
% 2.47/1.42 => ~ ( less @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_130_local_Osup_Ostrict__coboundedI1,axiom,
% 2.47/1.42 ! [C: a,A: a,B: a] :
% 2.47/1.42 ( ( less @ C @ A )
% 2.47/1.42 => ( less @ C @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_131_local_Osup_Ostrict__coboundedI2,axiom,
% 2.47/1.42 ! [C: a,B: a,A: a] :
% 2.47/1.42 ( ( less @ C @ B )
% 2.47/1.42 => ( less @ C @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_132_local_Osup_Ostrict__order__iff,axiom,
% 2.47/1.42 ! [B: a,A: a] :
% 2.47/1.42 ( ( less @ B @ A )
% 2.47/1.42 = ( ( A
% 2.47/1.42 = ( sup @ A @ B ) )
% 2.47/1.42 & ( A != B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_133_local_Odistrib__imp1,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ! [X3: a,Y4: a,Z3: a] :
% 2.47/1.42 ( ( inf @ X3 @ ( sup @ Y4 @ Z3 ) )
% 2.47/1.42 = ( sup @ ( inf @ X3 @ Y4 ) @ ( inf @ X3 @ Z3 ) ) )
% 2.47/1.42 => ( ( sup @ X @ ( inf @ Y @ Z2 ) )
% 2.47/1.42 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_134_local_Odistrib__imp2,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ! [X3: a,Y4: a,Z3: a] :
% 2.47/1.42 ( ( sup @ X3 @ ( inf @ Y4 @ Z3 ) )
% 2.47/1.42 = ( inf @ ( sup @ X3 @ Y4 ) @ ( sup @ X3 @ Z3 ) ) )
% 2.47/1.42 => ( ( inf @ X @ ( sup @ Y @ Z2 ) )
% 2.47/1.42 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_135_local_Odistrib__inf__le,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) @ ( inf @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_136_local_Odistrib__sup__le,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z2 ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_137_local_Oincomp__def,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( modula1727524044comp_a @ less_eq @ X @ Y )
% 2.47/1.42 = ( ~ ( less_eq @ X @ Y )
% 2.47/1.42 & ~ ( less_eq @ Y @ X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_138_local_Oorder_Orefl,axiom,
% 2.47/1.42 ! [A: a] : ( less_eq @ A @ A ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_139_local_Oorder__refl,axiom,
% 2.47/1.42 ! [X: a] : ( less_eq @ X @ X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_140_local_Oinf_Oidem,axiom,
% 2.47/1.42 ! [A: a] :
% 2.47/1.42 ( ( inf @ A @ A )
% 2.47/1.42 = A ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_141_local_Oinf_Oleft__idem,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( inf @ A @ ( inf @ A @ B ) )
% 2.47/1.42 = ( inf @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_142_local_Oinf_Oright__idem,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( inf @ ( inf @ A @ B ) @ B )
% 2.47/1.42 = ( inf @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_143_local_Oinf__idem,axiom,
% 2.47/1.42 ! [X: a] :
% 2.47/1.42 ( ( inf @ X @ X )
% 2.47/1.42 = X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_144_local_Oinf__left__idem,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( inf @ X @ ( inf @ X @ Y ) )
% 2.47/1.42 = ( inf @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_145_local_Oinf__right__idem,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 2.47/1.42 = ( inf @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_146_local_Osup_Oidem,axiom,
% 2.47/1.42 ! [A: a] :
% 2.47/1.42 ( ( sup @ A @ A )
% 2.47/1.42 = A ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_147_local_Osup_Oleft__idem,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( sup @ A @ ( sup @ A @ B ) )
% 2.47/1.42 = ( sup @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_148_local_Osup_Oright__idem,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( sup @ ( sup @ A @ B ) @ B )
% 2.47/1.42 = ( sup @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_149_local_Osup__idem,axiom,
% 2.47/1.42 ! [X: a] :
% 2.47/1.42 ( ( sup @ X @ X )
% 2.47/1.42 = X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_150_local_Osup__left__idem,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( sup @ X @ ( sup @ X @ Y ) )
% 2.47/1.42 = ( sup @ X @ Y ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_151_local_OGreatestI2__order,axiom,
% 2.47/1.42 ! [P: a > $o,X: a,Q: a > $o] :
% 2.47/1.42 ( ( P @ X )
% 2.47/1.42 => ( ! [Y4: a] :
% 2.47/1.42 ( ( P @ Y4 )
% 2.47/1.42 => ( less_eq @ Y4 @ X ) )
% 2.47/1.42 => ( ! [X3: a] :
% 2.47/1.42 ( ( P @ X3 )
% 2.47/1.42 => ( ! [Y5: a] :
% 2.47/1.42 ( ( P @ Y5 )
% 2.47/1.42 => ( less_eq @ Y5 @ X3 ) )
% 2.47/1.42 => ( Q @ X3 ) ) )
% 2.47/1.42 => ( Q @ ( greatest_a @ less_eq @ P ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_152_local_OGreatest__equality,axiom,
% 2.47/1.42 ! [P: a > $o,X: a] :
% 2.47/1.42 ( ( P @ X )
% 2.47/1.42 => ( ! [Y4: a] :
% 2.47/1.42 ( ( P @ Y4 )
% 2.47/1.42 => ( less_eq @ Y4 @ X ) )
% 2.47/1.42 => ( ( greatest_a @ less_eq @ P )
% 2.47/1.42 = X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_153_local_Omax__def,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( max_a @ less_eq @ A @ B )
% 2.47/1.42 = B ) )
% 2.47/1.42 & ( ~ ( less_eq @ A @ B )
% 2.47/1.42 => ( ( max_a @ less_eq @ A @ B )
% 2.47/1.42 = A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_154_local_Omin__def,axiom,
% 2.47/1.42 ! [A: a,B: a] :
% 2.47/1.42 ( ( ( less_eq @ A @ B )
% 2.47/1.42 => ( ( min_a @ less_eq @ A @ B )
% 2.47/1.42 = A ) )
% 2.47/1.42 & ( ~ ( less_eq @ A @ B )
% 2.47/1.42 => ( ( min_a @ less_eq @ A @ B )
% 2.47/1.42 = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_155_local_ON5__lattice__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula397570059tice_a @ inf @ less @ sup @ A @ B @ C )
% 2.47/1.42 = ( ( ( inf @ A @ C )
% 2.47/1.42 = ( inf @ B @ C ) )
% 2.47/1.42 & ( less @ A @ B )
% 2.47/1.42 & ( ( sup @ A @ C )
% 2.47/1.42 = ( sup @ B @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_156_local_Ono__distrib__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1962211574trib_a @ inf @ less @ sup @ A @ B @ C )
% 2.47/1.42 = ( less @ ( sup @ ( inf @ A @ B ) @ ( inf @ C @ A ) ) @ ( inf @ A @ ( sup @ B @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_157_local_Oinf_Obounded__iff,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 2.47/1.42 = ( ( less_eq @ A @ B )
% 2.47/1.42 & ( less_eq @ A @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_158_local_Ole__inf__iff,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less_eq @ X @ ( inf @ Y @ Z2 ) )
% 2.47/1.42 = ( ( less_eq @ X @ Y )
% 2.47/1.42 & ( less_eq @ X @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_159_local_Ole__sup__iff,axiom,
% 2.47/1.42 ! [X: a,Y: a,Z2: a] :
% 2.47/1.42 ( ( less_eq @ ( sup @ X @ Y ) @ Z2 )
% 2.47/1.42 = ( ( less_eq @ X @ Z2 )
% 2.47/1.42 & ( less_eq @ Y @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_160_local_Osup_Obounded__iff,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 2.47/1.42 = ( ( less_eq @ B @ A )
% 2.47/1.42 & ( less_eq @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_161_local_Oinf__sup__absorb,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( inf @ X @ ( sup @ X @ Y ) )
% 2.47/1.42 = X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_162_local_Osup__inf__absorb,axiom,
% 2.47/1.42 ! [X: a,Y: a] :
% 2.47/1.42 ( ( sup @ X @ ( inf @ X @ Y ) )
% 2.47/1.42 = X ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_163_local_OM5__lattice__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1376131916tice_a @ inf @ less @ sup @ A @ B @ C )
% 2.47/1.42 = ( ( ( inf @ A @ B )
% 2.47/1.42 = ( inf @ B @ C ) )
% 2.47/1.42 & ( ( inf @ C @ A )
% 2.47/1.42 = ( inf @ B @ C ) )
% 2.47/1.42 & ( ( sup @ A @ B )
% 2.47/1.42 = ( sup @ B @ C ) )
% 2.47/1.42 & ( ( sup @ C @ A )
% 2.47/1.42 = ( sup @ B @ C ) )
% 2.47/1.42 & ( less @ ( inf @ A @ B ) @ ( sup @ A @ B ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_164_local_OLeast1I,axiom,
% 2.47/1.42 ! [P: a > $o] :
% 2.47/1.42 ( ? [X4: a] :
% 2.47/1.42 ( ( P @ X4 )
% 2.47/1.42 & ! [Y4: a] :
% 2.47/1.42 ( ( P @ Y4 )
% 2.47/1.42 => ( less_eq @ X4 @ Y4 ) )
% 2.47/1.42 & ! [Y4: a] :
% 2.47/1.42 ( ( ( P @ Y4 )
% 2.47/1.42 & ! [Ya: a] :
% 2.47/1.42 ( ( P @ Ya )
% 2.47/1.42 => ( less_eq @ Y4 @ Ya ) ) )
% 2.47/1.42 => ( Y4 = X4 ) ) )
% 2.47/1.42 => ( P @ ( least_a @ less_eq @ P ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_165_local_OLeast1__le,axiom,
% 2.47/1.42 ! [P: a > $o,Z2: a] :
% 2.47/1.42 ( ? [X4: a] :
% 2.47/1.42 ( ( P @ X4 )
% 2.47/1.42 & ! [Y4: a] :
% 2.47/1.42 ( ( P @ Y4 )
% 2.47/1.42 => ( less_eq @ X4 @ Y4 ) )
% 2.47/1.42 & ! [Y4: a] :
% 2.47/1.42 ( ( ( P @ Y4 )
% 2.47/1.42 & ! [Ya: a] :
% 2.47/1.42 ( ( P @ Ya )
% 2.47/1.42 => ( less_eq @ Y4 @ Ya ) ) )
% 2.47/1.42 => ( Y4 = X4 ) ) )
% 2.47/1.42 => ( ( P @ Z2 )
% 2.47/1.42 => ( less_eq @ ( least_a @ less_eq @ P ) @ Z2 ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_166_local_OLeastI2__order,axiom,
% 2.47/1.42 ! [P: a > $o,X: a,Q: a > $o] :
% 2.47/1.42 ( ( P @ X )
% 2.47/1.42 => ( ! [Y4: a] :
% 2.47/1.42 ( ( P @ Y4 )
% 2.47/1.42 => ( less_eq @ X @ Y4 ) )
% 2.47/1.42 => ( ! [X3: a] :
% 2.47/1.42 ( ( P @ X3 )
% 2.47/1.42 => ( ! [Y5: a] :
% 2.47/1.42 ( ( P @ Y5 )
% 2.47/1.42 => ( less_eq @ X3 @ Y5 ) )
% 2.47/1.42 => ( Q @ X3 ) ) )
% 2.47/1.42 => ( Q @ ( least_a @ less_eq @ P ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_167_local_OLeast__equality,axiom,
% 2.47/1.42 ! [P: a > $o,X: a] :
% 2.47/1.42 ( ( P @ X )
% 2.47/1.42 => ( ! [Y4: a] :
% 2.47/1.42 ( ( P @ Y4 )
% 2.47/1.42 => ( less_eq @ X @ Y4 ) )
% 2.47/1.42 => ( ( least_a @ less_eq @ P )
% 2.47/1.42 = X ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_168_local_Oa__join__d,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( sup @ A @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) )
% 2.47/1.42 = ( sup @ A @ ( inf @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_169_local_Ob__join__d,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( sup @ B @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) )
% 2.47/1.42 = ( sup @ B @ ( inf @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_170_local_Od__aux__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( sup @ ( sup @ ( inf @ A @ B ) @ ( inf @ B @ C ) ) @ ( inf @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_171_local_Od__b__c__a,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( modula1936294176_aux_a @ inf @ sup @ B @ C @ A )
% 2.47/1.42 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_172_local_Od__c__a__b,axiom,
% 2.47/1.42 ! [C: a,A: a,B: a] :
% 2.47/1.42 ( ( modula1936294176_aux_a @ inf @ sup @ C @ A @ B )
% 2.47/1.42 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_173_local_Oa__meet__e,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( inf @ A @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
% 2.47/1.42 = ( inf @ A @ ( sup @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_174_local_Ob__meet__e,axiom,
% 2.47/1.42 ! [B: a,A: a,C: a] :
% 2.47/1.42 ( ( inf @ B @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
% 2.47/1.42 = ( inf @ B @ ( sup @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_175_local_Oc__meet__e,axiom,
% 2.47/1.42 ! [C: a,A: a,B: a] :
% 2.47/1.42 ( ( inf @ C @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
% 2.47/1.42 = ( inf @ C @ ( sup @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_176_local_Oe__aux__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( inf @ ( inf @ ( sup @ A @ B ) @ ( sup @ B @ C ) ) @ ( sup @ C @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_177_local_Oe__b__c__a,axiom,
% 2.47/1.42 ! [B: a,C: a,A: a] :
% 2.47/1.42 ( ( modula1144073633_aux_a @ inf @ sup @ B @ C @ A )
% 2.47/1.42 = ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_178_local_Oe__c__a__b,axiom,
% 2.47/1.42 ! [C: a,A: a,B: a] :
% 2.47/1.42 ( ( modula1144073633_aux_a @ inf @ sup @ C @ A @ B )
% 2.47/1.42 = ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_179_local_OInf__fin_Osemilattice__order__set__axioms,axiom,
% 2.47/1.42 lattic655834328_set_a @ inf @ less_eq @ less ).
% 2.47/1.42
% 2.47/1.42 thf(fact_180_local_OM5__lattice__incomp,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1376131916tice_a @ inf @ less @ sup @ A @ B @ C )
% 2.47/1.42 => ( modula1727524044comp_a @ less_eq @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_181_local_Oc__aux__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula581031071_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( sup @ ( inf @ C @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_182_local_Ob__aux__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1373251614_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( sup @ ( inf @ B @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_183_local_Oa__aux__def,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula17988509_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( sup @ ( inf @ A @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_184_lattice_Ono__distrib_Ocong,axiom,
% 2.47/1.42 modula1962211574trib_a = modula1962211574trib_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_185_lattice_ON5__lattice_Ocong,axiom,
% 2.47/1.42 modula397570059tice_a = modula397570059tice_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_186_lattice_OM5__lattice_Ocong,axiom,
% 2.47/1.42 modula1376131916tice_a = modula1376131916tice_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_187_lattice_Oincomp_Ocong,axiom,
% 2.47/1.42 modula1727524044comp_a = modula1727524044comp_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_188_lattice_Oe__aux_Ocong,axiom,
% 2.47/1.42 modula1144073633_aux_a = modula1144073633_aux_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_189_lattice_Od__aux_Ocong,axiom,
% 2.47/1.42 modula1936294176_aux_a = modula1936294176_aux_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_190_local_Oinf_Osemilattice__order__axioms,axiom,
% 2.47/1.42 semilattice_order_a @ inf @ less_eq @ less ).
% 2.47/1.42
% 2.47/1.42 thf(fact_191_local_Oorder_Oordering__axioms,axiom,
% 2.47/1.42 ordering_a @ less_eq @ less ).
% 2.47/1.42
% 2.47/1.42 thf(fact_192_local_Ocomp__fun__idem__sup,axiom,
% 2.47/1.42 finite40241356em_a_a @ sup ).
% 2.47/1.42
% 2.47/1.42 thf(fact_193_local_Ocomp__fun__idem__inf,axiom,
% 2.47/1.42 finite40241356em_a_a @ inf ).
% 2.47/1.42
% 2.47/1.42 thf(fact_194_local_Ob__a,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula1373251614_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( modula17988509_aux_a @ inf @ sup @ B @ C @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_195_local_Oc__a,axiom,
% 2.47/1.42 ! [A: a,B: a,C: a] :
% 2.47/1.42 ( ( modula581031071_aux_a @ inf @ sup @ A @ B @ C )
% 2.47/1.42 = ( modula17988509_aux_a @ inf @ sup @ C @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_196_lattice_Oc__aux_Ocong,axiom,
% 2.47/1.42 modula581031071_aux_a = modula581031071_aux_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_197_lattice_Ob__aux_Ocong,axiom,
% 2.47/1.42 modula1373251614_aux_a = modula1373251614_aux_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_198_lattice_Oa__aux_Ocong,axiom,
% 2.47/1.42 modula17988509_aux_a = modula17988509_aux_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_199_local_Osup_Osemigroup__axioms,axiom,
% 2.47/1.42 semigroup_a @ sup ).
% 2.47/1.42
% 2.47/1.42 thf(fact_200_local_Oinf_Osemigroup__axioms,axiom,
% 2.47/1.42 semigroup_a @ inf ).
% 2.47/1.42
% 2.47/1.42 thf(fact_201_local_Osup_Osemilattice__axioms,axiom,
% 2.47/1.42 semilattice_a @ sup ).
% 2.47/1.42
% 2.47/1.42 thf(fact_202_local_Oinf_Osemilattice__axioms,axiom,
% 2.47/1.42 semilattice_a @ inf ).
% 2.47/1.42
% 2.47/1.42 thf(fact_203_local_Osup_Oabel__semigroup__axioms,axiom,
% 2.47/1.42 abel_semigroup_a @ sup ).
% 2.47/1.42
% 2.47/1.42 thf(fact_204_local_Oinf_Oabel__semigroup__axioms,axiom,
% 2.47/1.42 abel_semigroup_a @ inf ).
% 2.47/1.42
% 2.47/1.42 thf(fact_205_semilattice__order__set_Oaxioms_I1_J,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 2.47/1.42 ( ( lattic655834328_set_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( semilattice_order_a @ F @ Less_eq @ Less ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_206_abel__semigroup_Oaxioms_I1_J,axiom,
% 2.47/1.42 ! [F: a > a > a] :
% 2.47/1.42 ( ( abel_semigroup_a @ F )
% 2.47/1.42 => ( semigroup_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_207_abel__semigroup_Oleft__commute,axiom,
% 2.47/1.42 ! [F: a > a > a,B: a,A: a,C: a] :
% 2.47/1.42 ( ( abel_semigroup_a @ F )
% 2.47/1.42 => ( ( F @ B @ ( F @ A @ C ) )
% 2.47/1.42 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_208_abel__semigroup_Ocommute,axiom,
% 2.47/1.42 ! [F: a > a > a,A: a,B: a] :
% 2.47/1.42 ( ( abel_semigroup_a @ F )
% 2.47/1.42 => ( ( F @ A @ B )
% 2.47/1.42 = ( F @ B @ A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_209_semigroup_Ointro,axiom,
% 2.47/1.42 ! [F: a > a > a] :
% 2.47/1.42 ( ! [A4: a,B3: a,C2: a] :
% 2.47/1.42 ( ( F @ ( F @ A4 @ B3 ) @ C2 )
% 2.47/1.42 = ( F @ A4 @ ( F @ B3 @ C2 ) ) )
% 2.47/1.42 => ( semigroup_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_210_semigroup_Oassoc,axiom,
% 2.47/1.42 ! [F: a > a > a,A: a,B: a,C: a] :
% 2.47/1.42 ( ( semigroup_a @ F )
% 2.47/1.42 => ( ( F @ ( F @ A @ B ) @ C )
% 2.47/1.42 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_211_semigroup__def,axiom,
% 2.47/1.42 ( semigroup_a
% 2.47/1.42 = ( ^ [F2: a > a > a] :
% 2.47/1.42 ! [A2: a,B2: a,C3: a] :
% 2.47/1.42 ( ( F2 @ ( F2 @ A2 @ B2 ) @ C3 )
% 2.47/1.42 = ( F2 @ A2 @ ( F2 @ B2 @ C3 ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_212_semilattice_Oaxioms_I1_J,axiom,
% 2.47/1.42 ! [F: a > a > a] :
% 2.47/1.42 ( ( semilattice_a @ F )
% 2.47/1.42 => ( abel_semigroup_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_213_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 2.47/1.42 lattic1885654924_set_a @ sup ).
% 2.47/1.42
% 2.47/1.42 thf(fact_214_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 2.47/1.42 lattic1885654924_set_a @ inf ).
% 2.47/1.42
% 2.47/1.42 thf(fact_215_semilattice__order__set_Oaxioms_I2_J,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 2.47/1.42 ( ( lattic655834328_set_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( lattic1885654924_set_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_216_semilattice__set__def,axiom,
% 2.47/1.42 lattic1885654924_set_a = semilattice_a ).
% 2.47/1.42
% 2.47/1.42 thf(fact_217_semilattice__set_Ointro,axiom,
% 2.47/1.42 ! [F: a > a > a] :
% 2.47/1.42 ( ( semilattice_a @ F )
% 2.47/1.42 => ( lattic1885654924_set_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_218_semilattice__set_Oaxioms,axiom,
% 2.47/1.42 ! [F: a > a > a] :
% 2.47/1.42 ( ( lattic1885654924_set_a @ F )
% 2.47/1.42 => ( semilattice_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_219_semilattice__order__set__def,axiom,
% 2.47/1.42 ( lattic655834328_set_a
% 2.47/1.42 = ( ^ [F2: a > a > a,Less_eq2: a > a > $o,Less2: a > a > $o] :
% 2.47/1.42 ( ( semilattice_order_a @ F2 @ Less_eq2 @ Less2 )
% 2.47/1.42 & ( lattic1885654924_set_a @ F2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_220_semilattice__order__set_Ointro,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( lattic1885654924_set_a @ F )
% 2.47/1.42 => ( lattic655834328_set_a @ F @ Less_eq @ Less ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_221_semilattice__order_Omono,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,C: a,B: a,D: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ C )
% 2.47/1.42 => ( ( Less_eq @ B @ D )
% 2.47/1.42 => ( Less_eq @ ( F @ A @ B ) @ ( F @ C @ D ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_222_semilattice__order_OorderE,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ B )
% 2.47/1.42 => ( A
% 2.47/1.42 = ( F @ A @ B ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_223_semilattice__order_OorderI,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( A
% 2.47/1.42 = ( F @ A @ B ) )
% 2.47/1.42 => ( Less_eq @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_224_semilattice__order_Oabsorb1,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ B )
% 2.47/1.42 => ( ( F @ A @ B )
% 2.47/1.42 = A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_225_semilattice__order_Oabsorb2,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,A: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ B @ A )
% 2.47/1.42 => ( ( F @ A @ B )
% 2.47/1.42 = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_226_semilattice__order_OboundedE,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ ( F @ B @ C ) )
% 2.47/1.42 => ~ ( ( Less_eq @ A @ B )
% 2.47/1.42 => ~ ( Less_eq @ A @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_227_semilattice__order_OboundedI,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ B )
% 2.47/1.42 => ( ( Less_eq @ A @ C )
% 2.47/1.42 => ( Less_eq @ A @ ( F @ B @ C ) ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_228_semilattice__order_Oorder__iff,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ B )
% 2.47/1.42 = ( A
% 2.47/1.42 = ( F @ A @ B ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_229_semilattice__order_Ocobounded1,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( Less_eq @ ( F @ A @ B ) @ A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_230_semilattice__order_Ocobounded2,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( Less_eq @ ( F @ A @ B ) @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_231_semilattice__order_Oabsorb__iff1,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ B )
% 2.47/1.42 = ( ( F @ A @ B )
% 2.47/1.42 = A ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_232_semilattice__order_Oabsorb__iff2,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,A: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ B @ A )
% 2.47/1.42 = ( ( F @ A @ B )
% 2.47/1.42 = B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_233_semilattice__order_Obounded__iff,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ ( F @ B @ C ) )
% 2.47/1.42 = ( ( Less_eq @ A @ B )
% 2.47/1.42 & ( Less_eq @ A @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_234_semilattice__order_OcoboundedI1,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,C: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ A @ C )
% 2.47/1.42 => ( Less_eq @ ( F @ A @ B ) @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_235_semilattice__order_OcoboundedI2,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,C: a,A: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less_eq @ B @ C )
% 2.47/1.42 => ( Less_eq @ ( F @ A @ B ) @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_236_semilattice__order_Ostrict__boundedE,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less @ A @ ( F @ B @ C ) )
% 2.47/1.42 => ~ ( ( Less @ A @ B )
% 2.47/1.42 => ~ ( Less @ A @ C ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_237_semilattice__order_Ostrict__order__iff,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less @ A @ B )
% 2.47/1.42 = ( ( A
% 2.47/1.42 = ( F @ A @ B ) )
% 2.47/1.42 & ( A != B ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_238_semilattice__order_Ostrict__coboundedI1,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: a,C: a,B: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less @ A @ C )
% 2.47/1.42 => ( Less @ ( F @ A @ B ) @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_239_semilattice__order_Ostrict__coboundedI2,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,B: a,C: a,A: a] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( ( Less @ B @ C )
% 2.47/1.42 => ( Less @ ( F @ A @ B ) @ C ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_240_semilattice_Oidem,axiom,
% 2.47/1.42 ! [F: a > a > a,A: a] :
% 2.47/1.42 ( ( semilattice_a @ F )
% 2.47/1.42 => ( ( F @ A @ A )
% 2.47/1.42 = A ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_241_semilattice_Oleft__idem,axiom,
% 2.47/1.42 ! [F: a > a > a,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_a @ F )
% 2.47/1.42 => ( ( F @ A @ ( F @ A @ B ) )
% 2.47/1.42 = ( F @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_242_semilattice_Oright__idem,axiom,
% 2.47/1.42 ! [F: a > a > a,A: a,B: a] :
% 2.47/1.42 ( ( semilattice_a @ F )
% 2.47/1.42 => ( ( F @ ( F @ A @ B ) @ B )
% 2.47/1.42 = ( F @ A @ B ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_243_semilattice__order_Oaxioms_I1_J,axiom,
% 2.47/1.42 ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o] :
% 2.47/1.42 ( ( semilattice_order_a @ F @ Less_eq @ Less )
% 2.47/1.42 => ( semilattice_a @ F ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_244_local_Obdd__below__def,axiom,
% 2.47/1.42 ! [A3: set_a] :
% 2.47/1.42 ( ( condit1001553558elow_a @ less_eq @ A3 )
% 2.47/1.42 = ( ? [M: a] :
% 2.47/1.42 ! [X2: a] :
% 2.47/1.42 ( ( member_a @ X2 @ A3 )
% 2.47/1.42 => ( less_eq @ M @ X2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_245_local_Obdd__above__def,axiom,
% 2.47/1.42 ! [A3: set_a] :
% 2.47/1.42 ( ( condit1627435690bove_a @ less_eq @ A3 )
% 2.47/1.42 = ( ? [M2: a] :
% 2.47/1.42 ! [X2: a] :
% 2.47/1.42 ( ( member_a @ X2 @ A3 )
% 2.47/1.42 => ( less_eq @ X2 @ M2 ) ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_246_local_Obdd__belowI,axiom,
% 2.47/1.42 ! [A3: set_a,M3: a] :
% 2.47/1.42 ( ! [X3: a] :
% 2.47/1.42 ( ( member_a @ X3 @ A3 )
% 2.47/1.42 => ( less_eq @ M3 @ X3 ) )
% 2.47/1.42 => ( condit1001553558elow_a @ less_eq @ A3 ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_247_local_Obdd__aboveI,axiom,
% 2.47/1.42 ! [A3: set_a,M4: a] :
% 2.47/1.42 ( ! [X3: a] :
% 2.47/1.42 ( ( member_a @ X3 @ A3 )
% 2.47/1.42 => ( less_eq @ X3 @ M4 ) )
% 2.47/1.42 => ( condit1627435690bove_a @ less_eq @ A3 ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(fact_248_local_Obdd__below__Ioo,axiom,
% 2.47/1.42 ! [A: a,B: a] : ( condit1001553558elow_a @ less_eq @ ( set_gr1491433118Than_a @ less @ A @ B ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(conj_0,hypothesis,
% 2.47/1.42 lattice_a @ inf @ less_eq @ less @ sup ).
% 2.47/1.42
% 2.47/1.42 thf(conj_1,hypothesis,
% 2.47/1.42 less_eq @ x @ y ).
% 2.47/1.42
% 2.47/1.42 thf(conj_2,hypothesis,
% 2.47/1.42 ( ( sup @ x @ ( inf @ y @ z ) )
% 2.47/1.42 != ( inf @ y @ ( sup @ x @ z ) ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(conj_3,hypothesis,
% 2.47/1.42 less @ ( sup @ x @ ( inf @ y @ z ) ) @ ( inf @ y @ ( sup @ x @ z ) ) ).
% 2.47/1.42
% 2.47/1.42 thf(conj_4,conjecture,
% 2.47/1.42 less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ).
% 2.47/1.42 ------------------------
% 2.47/1.44 % (32199)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/183Mi)
% 2.47/1.44 % (32202)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/2Mi)
% 2.47/1.44 % (32202)Instruction limit reached!
% 2.47/1.44 % (32202)------------------------------
% 2.47/1.44 % (32202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (32202)Termination reason: Unknown
% 2.47/1.44 % (32202)Termination phase: shuffling
% 2.47/1.44
% 2.47/1.44 % (32202)Memory used [KB]: 1279
% 2.47/1.44 % (32202)Time elapsed: 0.002 s
% 2.47/1.44 % (32202)Instructions burned: 2 (million)
% 2.47/1.44 % (32202)------------------------------
% 2.47/1.44 % (32202)------------------------------
% 2.47/1.44 % (32201)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/27Mi)
% 2.47/1.45 % (32200)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/4Mi)
% 2.47/1.45 % (32203)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/2Mi)
% 2.47/1.45 % (32204)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/275Mi)
% 2.47/1.45 % (32205)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/18Mi)
% 2.47/1.45 % (32203)Instruction limit reached!
% 2.47/1.45 % (32203)------------------------------
% 2.47/1.45 % (32203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (32203)Termination reason: Unknown
% 2.47/1.45 % (32203)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (32203)Memory used [KB]: 1279
% 2.47/1.45 % (32203)Time elapsed: 0.004 s
% 2.47/1.45 % (32203)Instructions burned: 3 (million)
% 2.47/1.45 % (32203)------------------------------
% 2.47/1.45 % (32203)------------------------------
% 2.47/1.45 % (32200)Instruction limit reached!
% 2.47/1.45 % (32200)------------------------------
% 2.47/1.45 % (32200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (32200)Termination reason: Unknown
% 2.47/1.45 % (32200)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (32200)Memory used [KB]: 1279
% 2.47/1.45 % (32200)Time elapsed: 0.005 s
% 2.47/1.45 % (32200)Instructions burned: 5 (million)
% 2.47/1.45 % (32200)------------------------------
% 2.47/1.45 % (32200)------------------------------
% 2.47/1.45 % (32206)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/3Mi)
% 2.47/1.45 % (32206)Instruction limit reached!
% 2.47/1.45 % (32206)------------------------------
% 2.47/1.45 % (32206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (32206)Termination reason: Unknown
% 2.47/1.45 % (32206)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (32206)Memory used [KB]: 1279
% 2.47/1.45 % (32206)Time elapsed: 0.003 s
% 2.47/1.45 % (32206)Instructions burned: 4 (million)
% 2.47/1.45 % (32206)------------------------------
% 2.47/1.45 % (32206)------------------------------
% 2.47/1.46 % (32201)Instruction limit reached!
% 2.47/1.46 % (32201)------------------------------
% 2.47/1.46 % (32201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (32201)Termination reason: Unknown
% 2.47/1.46 % (32201)Termination phase: Unused predicate definition removal
% 2.47/1.46
% 2.47/1.46 % (32201)Memory used [KB]: 1535
% 2.47/1.46 % (32201)Time elapsed: 0.016 s
% 2.47/1.46 % (32201)Instructions burned: 28 (million)
% 2.47/1.46 % (32201)------------------------------
% 2.47/1.46 % (32201)------------------------------
% 2.47/1.46 % (32205)Instruction limit reached!
% 2.47/1.46 % (32205)------------------------------
% 2.47/1.46 % (32205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (32205)Termination reason: Unknown
% 2.47/1.46 % (32205)Termination phase: Property scanning
% 2.47/1.46
% 2.47/1.46 % (32205)Memory used [KB]: 1407
% 2.47/1.46 % (32205)Time elapsed: 0.013 s
% 2.47/1.46 % (32205)Instructions burned: 18 (million)
% 2.47/1.46 % (32205)------------------------------
% 2.47/1.46 % (32205)------------------------------
% 2.47/1.46 % (32209)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/3Mi)
% 2.47/1.46 % (32209)Instruction limit reached!
% 2.47/1.46 % (32209)------------------------------
% 2.47/1.46 % (32209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (32209)Termination reason: Unknown
% 2.47/1.46 % (32209)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (32209)Memory used [KB]: 1279
% 2.47/1.46 % (32209)Time elapsed: 0.003 s
% 2.47/1.46 % (32209)Instructions burned: 4 (million)
% 2.47/1.46 % (32209)------------------------------
% 2.47/1.46 % (32209)------------------------------
% 2.47/1.47 % (32207)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/37Mi)
% 2.47/1.47 % (32208)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/15Mi)
% 2.47/1.47 % (32210)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/1041Mi)
% 2.47/1.47 % (32212)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/16Mi)
% 2.47/1.47 % (32211)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/7Mi)
% 2.47/1.48 % (32208)Instruction limit reached!
% 2.47/1.48 % (32208)------------------------------
% 2.47/1.48 % (32208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (32208)Termination reason: Unknown
% 2.47/1.48 % (32208)Termination phase: Property scanning
% 2.47/1.48
% 2.47/1.48 % (32208)Memory used [KB]: 1407
% 2.47/1.48 % (32208)Time elapsed: 0.011 s
% 2.47/1.48 % (32208)Instructions burned: 16 (million)
% 2.47/1.48 % (32208)------------------------------
% 2.47/1.48 % (32208)------------------------------
% 2.47/1.48 % (32211)Instruction limit reached!
% 2.47/1.48 % (32211)------------------------------
% 2.47/1.48 % (32211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (32211)Termination reason: Unknown
% 2.47/1.48 % (32211)Termination phase: shuffling
% 2.47/1.48
% 2.47/1.48 % (32211)Memory used [KB]: 1279
% 2.47/1.48 % (32211)Time elapsed: 0.006 s
% 2.47/1.48 % (32211)Instructions burned: 7 (million)
% 2.47/1.48 % (32211)------------------------------
% 2.47/1.48 % (32211)------------------------------
% 2.47/1.48 % (32212)Instruction limit reached!
% 2.47/1.48 % (32212)------------------------------
% 2.47/1.48 % (32212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (32212)Termination reason: Unknown
% 2.47/1.48 % (32212)Termination phase: Property scanning
% 2.47/1.48
% 2.47/1.48 % (32212)Memory used [KB]: 1407
% 2.47/1.48 % (32212)Time elapsed: 0.008 s
% 2.47/1.48 % (32212)Instructions burned: 18 (million)
% 2.47/1.48 % (32212)------------------------------
% 2.47/1.48 % (32212)------------------------------
% 2.47/1.48 % (32207)Instruction limit reached!
% 2.47/1.48 % (32207)------------------------------
% 2.47/1.48 % (32207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (32207)Termination reason: Unknown
% 2.47/1.48 % (32207)Termination phase: Property scanning
% 2.47/1.48
% 2.47/1.48 % (32207)Memory used [KB]: 1791
% 2.47/1.48 % (32207)Time elapsed: 0.021 s
% 2.47/1.48 % (32207)Instructions burned: 38 (million)
% 2.47/1.48 % (32207)------------------------------
% 2.47/1.48 % (32207)------------------------------
% 2.47/1.49 % (32214)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/3Mi)
% 2.47/1.49 % (32213)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/3Mi)
% 2.47/1.49 % (32214)Instruction limit reached!
% 2.47/1.49 % (32214)------------------------------
% 2.47/1.49 % (32214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (32214)Termination reason: Unknown
% 2.47/1.49 % (32214)Termination phase: shuffling
% 2.47/1.49
% 2.47/1.49 % (32214)Memory used [KB]: 1279
% 2.47/1.49 % (32214)Time elapsed: 0.002 s
% 2.47/1.49 % (32214)Instructions burned: 3 (million)
% 2.47/1.49 % (32214)------------------------------
% 2.47/1.49 % (32214)------------------------------
% 2.47/1.49 % (32215)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/7Mi)
% 2.47/1.49 % (32213)Instruction limit reached!
% 2.47/1.49 % (32213)------------------------------
% 2.47/1.49 % (32213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (32213)Termination reason: Unknown
% 2.47/1.49 % (32213)Termination phase: shuffling
% 2.47/1.49
% 2.47/1.49 % (32213)Memory used [KB]: 1279
% 2.47/1.49 % (32213)Time elapsed: 0.003 s
% 2.47/1.49 % (32213)Instructions burned: 4 (million)
% 2.47/1.49 % (32213)------------------------------
% 2.47/1.49 % (32213)------------------------------
% 2.47/1.49 % (32215)Instruction limit reached!
% 2.47/1.49 % (32215)------------------------------
% 2.47/1.49 % (32215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (32215)Termination reason: Unknown
% 2.47/1.49 % (32215)Termination phase: shuffling
% 2.47/1.49
% 2.47/1.49 % (32215)Memory used [KB]: 1279
% 2.47/1.49 % (32215)Time elapsed: 0.004 s
% 2.47/1.49 % (32215)Instructions burned: 7 (million)
% 2.47/1.49 % (32215)------------------------------
% 2.47/1.49 % (32215)------------------------------
% 2.47/1.50 % (32199)Instruction limit reached!
% 2.47/1.50 % (32199)------------------------------
% 2.47/1.50 % (32199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.50 % (32199)Termination reason: Unknown
% 2.47/1.50 % (32199)Termination phase: Saturation
% 2.47/1.50
% 2.47/1.50 % (32199)Memory used [KB]: 6652
% 2.47/1.50 % (32199)Time elapsed: 0.058 s
% 2.47/1.50 % (32199)Instructions burned: 185 (million)
% 2.47/1.50 % (32199)------------------------------
% 2.47/1.50 % (32199)------------------------------
% 2.50/1.50 % (32217)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/4Mi)
% 2.50/1.50 % (32218)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/18Mi)
% 2.50/1.50 % (32217)Instruction limit reached!
% 2.50/1.50 % (32217)------------------------------
% 2.50/1.50 % (32217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.50 % (32217)Termination reason: Unknown
% 2.50/1.50 % (32217)Termination phase: shuffling
% 2.50/1.50
% 2.50/1.50 % (32217)Memory used [KB]: 1279
% 2.50/1.50 % (32217)Time elapsed: 0.003 s
% 2.50/1.50 % (32217)Instructions burned: 6 (million)
% 2.50/1.50 % (32217)------------------------------
% 2.50/1.50 % (32217)------------------------------
% 2.50/1.50 % (32216)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/3Mi)
% 2.50/1.50 % (32219)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/710Mi)
% 2.50/1.50 % (32216)Instruction limit reached!
% 2.50/1.50 % (32216)------------------------------
% 2.50/1.50 % (32216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.50 % (32216)Termination reason: Unknown
% 2.50/1.50 % (32216)Termination phase: shuffling
% 2.50/1.50
% 2.50/1.50 % (32216)Memory used [KB]: 1279
% 2.50/1.50 % (32216)Time elapsed: 0.004 s
% 2.50/1.50 % (32216)Instructions burned: 4 (million)
% 2.50/1.50 % (32216)------------------------------
% 2.50/1.50 % (32216)------------------------------
% 2.50/1.51 % (32220)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/6Mi)
% 2.50/1.51 % (32218)Instruction limit reached!
% 2.50/1.51 % (32218)------------------------------
% 2.50/1.51 % (32218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.51 % (32218)Termination reason: Unknown
% 2.50/1.51 % (32218)Termination phase: Property scanning
% 2.50/1.51
% 2.50/1.51 % (32218)Memory used [KB]: 1535
% 2.50/1.51 % (32218)Time elapsed: 0.008 s
% 2.50/1.51 % (32218)Instructions burned: 18 (million)
% 2.50/1.51 % (32218)------------------------------
% 2.50/1.51 % (32218)------------------------------
% 2.50/1.51 % (32220)Instruction limit reached!
% 2.50/1.51 % (32220)------------------------------
% 2.50/1.51 % (32220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.51 % (32220)Termination reason: Unknown
% 2.50/1.51 % (32220)Termination phase: shuffling
% 2.50/1.51
% 2.50/1.51 % (32220)Memory used [KB]: 1279
% 2.50/1.51 % (32220)Time elapsed: 0.003 s
% 2.50/1.51 % (32220)Instructions burned: 6 (million)
% 2.50/1.51 % (32220)------------------------------
% 2.50/1.51 % (32220)------------------------------
% 2.50/1.51 % (32221)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/902Mi)
% 2.50/1.51 % (32222)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/21Mi)
% 2.50/1.52 % (32223)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/5Mi)
% 2.50/1.52 % (32224)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/6Mi)
% 2.50/1.52 % (32223)Instruction limit reached!
% 2.50/1.52 % (32223)------------------------------
% 2.50/1.52 % (32223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.52 % (32223)Termination reason: Unknown
% 2.50/1.52 % (32223)Termination phase: shuffling
% 2.50/1.52
% 2.50/1.52 % (32223)Memory used [KB]: 1407
% 2.50/1.52 % (32223)Time elapsed: 0.004 s
% 2.50/1.52 % (32223)Instructions burned: 7 (million)
% 2.50/1.52 % (32223)------------------------------
% 2.50/1.52 % (32223)------------------------------
% 2.50/1.52 % (32224)Instruction limit reached!
% 2.50/1.52 % (32224)------------------------------
% 2.50/1.52 % (32224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.52 % (32224)Termination reason: Unknown
% 2.50/1.52 % (32224)Termination phase: shuffling
% 2.50/1.52
% 2.50/1.52 % (32224)Memory used [KB]: 1279
% 2.50/1.52 % (32224)Time elapsed: 0.003 s
% 2.50/1.52 % (32224)Instructions burned: 7 (million)
% 2.50/1.52 % (32224)------------------------------
% 2.50/1.52 % (32224)------------------------------
% 2.50/1.52 % (32222)Instruction limit reached!
% 2.50/1.52 % (32222)------------------------------
% 2.50/1.52 % (32222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.52 % (32222)Termination reason: Unknown
% 2.50/1.52 % (32222)Termination phase: Property scanning
% 2.50/1.52
% 2.50/1.52 % (32222)Memory used [KB]: 1535
% 2.50/1.52 % (32222)Time elapsed: 0.009 s
% 2.50/1.52 % (32222)Instructions burned: 23 (million)
% 2.50/1.52 % (32222)------------------------------
% 2.50/1.52 % (32222)------------------------------
% 2.50/1.53 % (32226)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/779Mi)
% 2.50/1.53 % (32227)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/19Mi)
% 2.50/1.54 % (32225)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31975 for (2999ds/377Mi)
% 2.50/1.54 % (32227)Instruction limit reached!
% 2.50/1.54 % (32227)------------------------------
% 2.50/1.54 % (32227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.54 % (32227)Termination reason: Unknown
% 2.50/1.54 % (32227)Termination phase: shuffling
% 2.50/1.54
% 2.50/1.54 % (32227)Memory used [KB]: 1663
% 2.50/1.54 % (32227)Time elapsed: 0.008 s
% 2.50/1.54 % (32227)Instructions burned: 22 (million)
% 2.50/1.54 % (32227)------------------------------
% 2.50/1.54 % (32227)------------------------------
% 2.50/1.55 % (32228)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31975 for (2998ds/879Mi)
% 2.50/1.56 % (32204)First to succeed.
% 2.50/1.56 % (32204)Refutation found. Thanks to Tanya!
% 2.50/1.56 % SZS status Theorem for DTF2THF_31975
% 2.50/1.56 % SZS output start Proof for DTF2THF_31975
% 2.50/1.56 thf(type_def_5, type, a: $tType).
% 2.50/1.56 thf(type_def_7, type, set_a: $tType).
% 2.50/1.56 thf(func_def_0, type, set_a: $tType).
% 2.50/1.56 thf(func_def_1, type, a: $tType).
% 2.50/1.56 thf(func_def_2, type, condit1627435690bove_a: (a > a > $o) > set_a > $o).
% 2.50/1.56 thf(func_def_3, type, condit1001553558elow_a: (a > a > $o) > set_a > $o).
% 2.50/1.56 thf(func_def_4, type, finite40241356em_a_a: (a > a > a) > $o).
% 2.50/1.56 thf(func_def_5, type, abel_semigroup_a: (a > a > a) > $o).
% 2.50/1.56 thf(func_def_6, type, semigroup_a: (a > a > a) > $o).
% 2.50/1.56 thf(func_def_7, type, lattice_a: (a > a > a) > (a > a > $o) > (a > a > $o) > (a > a > a) > $o).
% 2.50/1.56 thf(func_def_8, type, semilattice_a: (a > a > a) > $o).
% 2.50/1.56 thf(func_def_9, type, semilattice_order_a: (a > a > a) > (a > a > $o) > (a > a > $o) > $o).
% 2.50/1.56 thf(func_def_10, type, lattic655834328_set_a: (a > a > a) > (a > a > $o) > (a > a > $o) > $o).
% 2.50/1.56 thf(func_def_11, type, lattic1885654924_set_a: (a > a > a) > $o).
% 2.50/1.56 thf(func_def_12, type, modula1376131916tice_a: (a > a > a) > (a > a > $o) > (a > a > a) > a > a > a > $o).
% 2.50/1.56 thf(func_def_13, type, modula397570059tice_a: (a > a > a) > (a > a > $o) > (a > a > a) > a > a > a > $o).
% 2.50/1.56 thf(func_def_14, type, modula17988509_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 2.50/1.56 thf(func_def_15, type, modula1373251614_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 2.50/1.56 thf(func_def_16, type, modula581031071_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 2.50/1.56 thf(func_def_17, type, modula1936294176_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 2.50/1.56 thf(func_def_18, type, modula1144073633_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 2.50/1.56 thf(func_def_19, type, modula1727524044comp_a: (a > a > $o) > a > a > $o).
% 2.50/1.56 thf(func_def_20, type, modula1962211574trib_a: (a > a > a) > (a > a > $o) > (a > a > a) > a > a > a > $o).
% 2.50/1.56 thf(func_def_21, type, least_a: (a > a > $o) > (a > $o) > a).
% 2.50/1.56 thf(func_def_22, type, max_a: (a > a > $o) > a > a > a).
% 2.50/1.56 thf(func_def_23, type, min_a: (a > a > $o) > a > a > a).
% 2.50/1.56 thf(func_def_24, type, greatest_a: (a > a > $o) > (a > $o) > a).
% 2.50/1.56 thf(func_def_25, type, ordering_a: (a > a > $o) > (a > a > $o) > $o).
% 2.50/1.56 thf(func_def_26, type, collect_a: (a > $o) > set_a).
% 2.50/1.56 thf(func_def_27, type, set_gr1491433118Than_a: (a > a > $o) > a > a > set_a).
% 2.50/1.56 thf(func_def_28, type, member_a: a > set_a > $o).
% 2.50/1.56 thf(func_def_29, type, inf: a > a > a).
% 2.50/1.56 thf(func_def_30, type, less: a > a > $o).
% 2.50/1.56 thf(func_def_31, type, less_eq: a > a > $o).
% 2.50/1.56 thf(func_def_32, type, sup: a > a > a).
% 2.50/1.56 thf(func_def_33, type, x: a).
% 2.50/1.56 thf(func_def_34, type, y: a).
% 2.50/1.56 thf(func_def_35, type, z: a).
% 2.50/1.56 thf(func_def_47, type, sK0: a).
% 2.50/1.56 thf(func_def_48, type, sK1: a).
% 2.50/1.56 thf(func_def_49, type, sK2: a).
% 2.50/1.56 thf(func_def_50, type, sK3: (a > a > a) > a).
% 2.50/1.56 thf(func_def_51, type, sK4: (a > a > a) > a).
% 2.50/1.56 thf(func_def_52, type, sK5: (a > a > a) > a).
% 2.50/1.56 thf(func_def_53, type, sK6: (a > a > a) > a).
% 2.50/1.56 thf(func_def_54, type, sK7: (a > a > a) > a).
% 2.50/1.56 thf(func_def_55, type, sK8: (a > a > a) > a).
% 2.50/1.56 thf(func_def_56, type, sK9: (a > a > a) > a).
% 2.50/1.56 thf(func_def_57, type, sK10: (a > a > a) > a).
% 2.50/1.56 thf(func_def_58, type, sK11: (a > a > a) > a).
% 2.50/1.56 thf(func_def_59, type, sK12: (a > a > a) > a).
% 2.50/1.56 thf(func_def_60, type, sK13: (a > a > a) > a).
% 2.50/1.56 thf(func_def_61, type, sK14: (a > a > a) > a).
% 2.50/1.56 thf(func_def_62, type, sK15: (a > a > a) > a).
% 2.50/1.56 thf(func_def_63, type, sK16: (a > a > a) > a).
% 2.50/1.56 thf(func_def_64, type, sK17: a).
% 2.50/1.56 thf(func_def_65, type, sK18: a).
% 2.50/1.56 thf(func_def_66, type, sK19: a).
% 2.50/1.56 thf(func_def_68, type, ph21: !>[X0: $tType]:(X0)).
% 2.50/1.56 thf(f2381,plain,(
% 2.50/1.56 $false),
% 2.50/1.56 inference(subsumption_resolution,[status(thm)],[f2380,f1005])).
% 2.50/1.56 thf(f1005,plain,(
% 2.50/1.56 ((less_eq @ x @ y) = $true)),
% 2.50/1.56 inference(cnf_transformation,[status(thm)],[f527])).
% 2.50/1.56 thf(f527,plain,(
% 2.50/1.56 ((less_eq @ x @ y) = $true)),
% 2.50/1.56 inference(fool_elimination,[status(thm)],[f526])).
% 2.50/1.56 thf(f526,plain,(
% 2.50/1.56 (less_eq @ x @ y)),
% 2.50/1.56 inference(rectify,[status(thm)],[f251])).
% 2.50/1.56 thf(f251,axiom,(
% 2.50/1.56 (less_eq @ x @ y)),
% 2.50/1.56 file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',conj_1)).
% 2.50/1.56 thf(f2380,plain,(
% 2.50/1.56 ((less_eq @ x @ y) != $true)),
% 2.50/1.56 inference(subsumption_resolution,[status(thm)],[f2366,f1027])).
% 2.50/1.56 thf(f1027,plain,(
% 2.50/1.56 ( ! [X0 : a,X1 : a] : (((less_eq @ X1 @ (sup @ X1 @ X0)) = $true)) )),
% 2.50/1.56 inference(cnf_transformation,[status(thm)],[f308])).
% 2.50/1.56 thf(f308,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a] : ((less_eq @ X1 @ (sup @ X1 @ X0)) = $true)),
% 2.50/1.56 inference(fool_elimination,[status(thm)],[f307])).
% 2.50/1.56 thf(f307,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a] : (less_eq @ X1 @ (sup @ X1 @ X0))),
% 2.50/1.56 inference(rectify,[status(thm)],[f107])).
% 2.50/1.56 thf(f107,axiom,(
% 2.50/1.56 ! [X2 : a,X3 : a] : (less_eq @ X3 @ (sup @ X3 @ X2))),
% 2.50/1.56 file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_106_local_Osup_Ocobounded1)).
% 2.50/1.56 thf(f2366,plain,(
% 2.50/1.56 ((less_eq @ x @ (sup @ x @ z)) != $true) | ((less_eq @ x @ y) != $true)),
% 2.50/1.56 inference(trivial_inequality_removal,[status(thm)],[f2317])).
% 2.50/1.56 thf(f2317,plain,(
% 2.50/1.56 ($true != $true) | ((less_eq @ x @ (sup @ x @ z)) != $true) | ((less_eq @ x @ y) != $true)),
% 2.50/1.56 inference(superposition,[status(thm)],[f1191,f980])).
% 2.50/1.56 thf(f980,plain,(
% 2.50/1.56 ( ! [X2 : a,X0 : a,X1 : a] : (($true = (less_eq @ X2 @ (inf @ X1 @ X0))) | ((less_eq @ X2 @ X0) != $true) | ((less_eq @ X2 @ X1) != $true)) )),
% 2.50/1.56 inference(cnf_transformation,[status(thm)],[f841])).
% 2.50/1.56 thf(f841,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : ((($true = (less_eq @ X2 @ (inf @ X1 @ X0))) | ((less_eq @ X2 @ X1) != $true) | ((less_eq @ X2 @ X0) != $true)) & ((((less_eq @ X2 @ X1) = $true) & ((less_eq @ X2 @ X0) = $true)) | ($true != (less_eq @ X2 @ (inf @ X1 @ X0)))))),
% 2.50/1.56 inference(flattening,[status(thm)],[f840])).
% 2.50/1.56 thf(f840,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : ((($true = (less_eq @ X2 @ (inf @ X1 @ X0))) | (((less_eq @ X2 @ X1) != $true) | ((less_eq @ X2 @ X0) != $true))) & ((((less_eq @ X2 @ X1) = $true) & ((less_eq @ X2 @ X0) = $true)) | ($true != (less_eq @ X2 @ (inf @ X1 @ X0)))))),
% 2.50/1.56 inference(nnf_transformation,[status(thm)],[f533])).
% 2.50/1.56 thf(f533,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : (($true = (less_eq @ X2 @ (inf @ X1 @ X0))) <=> (((less_eq @ X2 @ X1) = $true) & ((less_eq @ X2 @ X0) = $true)))),
% 2.50/1.56 inference(fool_elimination,[status(thm)],[f532])).
% 2.50/1.56 thf(f532,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : ((less_eq @ X2 @ X1) & (less_eq @ X2 @ X0) = (less_eq @ X2 @ (inf @ X1 @ X0)))),
% 2.50/1.56 inference(rectify,[status(thm)],[f158])).
% 2.50/1.56 thf(f158,axiom,(
% 2.50/1.56 ! [X8 : a,X2 : a,X3 : a] : ((less_eq @ X3 @ X2) & (less_eq @ X3 @ X8) = (less_eq @ X3 @ (inf @ X2 @ X8)))),
% 2.50/1.56 file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_157_local_Oinf_Obounded__iff)).
% 2.50/1.56 thf(f1191,plain,(
% 2.50/1.56 ((less_eq @ x @ (inf @ y @ (sup @ x @ z))) != $true)),
% 2.50/1.56 inference(trivial_inequality_removal,[status(thm)],[f1145])).
% 2.50/1.56 thf(f1145,plain,(
% 2.50/1.56 ((less_eq @ x @ (inf @ y @ (sup @ x @ z))) != $true) | ($true != $true)),
% 2.50/1.56 inference(superposition,[status(thm)],[f1089,f933])).
% 2.50/1.56 thf(f933,plain,(
% 2.50/1.56 ( ! [X2 : a,X0 : a,X1 : a] : (($true = (less_eq @ X1 @ (sup @ X0 @ X2))) | ((less_eq @ X1 @ X2) != $true)) )),
% 2.50/1.56 inference(cnf_transformation,[status(thm)],[f703])).
% 2.50/1.56 thf(f703,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : (($true = (less_eq @ X1 @ (sup @ X0 @ X2))) | ((less_eq @ X1 @ X2) != $true))),
% 2.50/1.56 inference(ennf_transformation,[status(thm)],[f483])).
% 2.50/1.56 thf(f483,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : (((less_eq @ X1 @ X2) = $true) => ($true = (less_eq @ X1 @ (sup @ X0 @ X2))))),
% 2.50/1.56 inference(fool_elimination,[status(thm)],[f482])).
% 2.50/1.56 thf(f482,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a,X2 : a] : ((less_eq @ X1 @ X2) => (less_eq @ X1 @ (sup @ X0 @ X2)))),
% 2.50/1.56 inference(rectify,[status(thm)],[f100])).
% 2.50/1.56 thf(f100,axiom,(
% 2.50/1.56 ! [X3 : a,X0 : a,X2 : a] : ((less_eq @ X0 @ X2) => (less_eq @ X0 @ (sup @ X3 @ X2)))),
% 2.50/1.56 file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_99_local_Ole__supI2)).
% 2.50/1.56 thf(f1089,plain,(
% 2.50/1.56 ((less_eq @ x @ (sup @ z @ (inf @ y @ (sup @ x @ z)))) != $true)),
% 2.50/1.56 inference(forward_demodulation,[status(thm)],[f941,f1006])).
% 2.50/1.56 thf(f1006,plain,(
% 2.50/1.56 ( ! [X0 : a,X1 : a] : (((sup @ X0 @ X1) = (sup @ X1 @ X0))) )),
% 2.50/1.56 inference(cnf_transformation,[status(thm)],[f690])).
% 2.50/1.56 thf(f690,plain,(
% 2.50/1.56 ! [X0 : a,X1 : a] : ((sup @ X0 @ X1) = (sup @ X1 @ X0))),
% 2.50/1.56 inference(rectify,[status(thm)],[f40])).
% 2.50/1.56 thf(f40,axiom,(
% 2.50/1.56 ! [X2 : a,X3 : a] : ((sup @ X3 @ X2) = (sup @ X2 @ X3))),
% 2.50/1.56 file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',fact_39_local_Osup_Ocommute)).
% 2.50/1.56 thf(f941,plain,(
% 2.50/1.56 ((less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z)) != $true)),
% 2.50/1.56 inference(cnf_transformation,[status(thm)],[f650])).
% 2.50/1.56 thf(f650,plain,(
% 2.50/1.56 ((less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z)) != $true)),
% 2.50/1.56 inference(flattening,[status(thm)],[f575])).
% 2.50/1.56 thf(f575,plain,(
% 2.50/1.56 ~((less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z)) = $true)),
% 2.50/1.56 inference(fool_elimination,[status(thm)],[f574])).
% 2.50/1.56 thf(f574,plain,(
% 2.50/1.56 ~(less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z))),
% 2.50/1.56 inference(rectify,[status(thm)],[f255])).
% 2.50/1.56 thf(f255,negated_conjecture,(
% 2.50/1.56 ~(less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z))),
% 2.50/1.56 inference(negated_conjecture,[status(cth)],[f254])).
% 2.50/1.56 thf(f254,conjecture,(
% 2.50/1.56 (less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z))),
% 2.50/1.56 file('/export/starexec/sandbox2/tmp/tmp.QmiEmqZUN5/DTF2THF_31975.p',conj_4)).
% 2.50/1.56 % SZS output end Proof for DTF2THF_31975
% 2.50/1.56 % (32204)------------------------------
% 2.50/1.56 % (32204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.56 % (32204)Termination reason: Refutation
% 2.50/1.56
% 2.50/1.56 % (32204)Memory used [KB]: 6524
% 2.50/1.56 % (32204)Time elapsed: 0.112 s
% 2.50/1.56 % (32204)Instructions burned: 240 (million)
% 2.50/1.56 % (32204)------------------------------
% 2.50/1.56 % (32204)------------------------------
% 2.50/1.56 % (32198)Success in time 0.124 s
%------------------------------------------------------------------------------