%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP031^1 : 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 : n001.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 08:54:59 AM UTC 2026
% Result : Theorem 2.70s 1.63s
% Output : Refutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 6
% Syntax : Number of formulae : 33 ( 26 unt; 0 typ; 0 def)
% Number of atoms : 83 ( 42 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 150 ( 12 ~; 0 |; 4 &; 129 @)
% ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 88 ( 85 usr; 9 con; 0-4 aty)
% Number of variables : 34 ( 17 ^; 15 !; 2 ?; 34 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
binary1439146945Tree_a: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_8,type,
set_a: $tType ).
thf(type_def_9,type,
int: $tType ).
thf(func_def_0,type,
binary1439146945Tree_a: $tType ).
thf(func_def_1,type,
set_a: $tType ).
thf(func_def_2,type,
int: $tType ).
thf(func_def_3,type,
a: $tType ).
thf(func_def_4,type,
binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_5,type,
binary476621312_Tip_a: binary1439146945Tree_a ).
thf(func_def_6,type,
binary256242811Tree_a: binary1439146945Tree_a > set_a ).
thf(func_def_7,type,
binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_8,type,
binary504661350_eqs_a: ( a > int ) > a > set_a ).
thf(func_def_9,type,
binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
thf(func_def_10,type,
binary945792244etOf_a: binary1439146945Tree_a > set_a ).
thf(func_def_11,type,
binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
thf(func_def_12,type,
binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
thf(func_def_13,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(func_def_14,type,
minus_minus_o: $o > $o > $o ).
thf(func_def_15,type,
minus_minus_int: int > int > int ).
thf(func_def_16,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_17,type,
the_a: ( a > $o ) > a ).
thf(func_def_18,type,
if_a: $o > a > a > a ).
thf(func_def_19,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(func_def_20,type,
sup_sup_o: $o > $o > $o ).
thf(func_def_21,type,
sup_sup_int: int > int > int ).
thf(func_def_22,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(func_def_23,type,
bot_bot_a_o: a > $o ).
thf(func_def_25,type,
bot_bot_set_a: set_a ).
thf(func_def_26,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(func_def_27,type,
ord_less_int: int > int > $o ).
thf(func_def_28,type,
ord_less_set_a: set_a > set_a > $o ).
thf(func_def_29,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(func_def_30,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_31,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_32,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_33,type,
insert_a: a > set_a > set_a ).
thf(func_def_34,type,
is_empty_a: set_a > $o ).
thf(func_def_35,type,
is_singleton_a: set_a > $o ).
thf(func_def_36,type,
pairwise_a: ( a > a > $o ) > set_a > $o ).
thf(func_def_37,type,
remove_a: a > set_a > set_a ).
thf(func_def_38,type,
the_elem_a: set_a > a ).
thf(func_def_39,type,
member_a: a > set_a > $o ).
thf(func_def_40,type,
e: a ).
thf(func_def_41,type,
h: a > int ).
thf(func_def_42,type,
t1: binary1439146945Tree_a ).
thf(func_def_43,type,
t2: binary1439146945Tree_a ).
thf(func_def_44,type,
x: a ).
thf(func_def_60,type,
sP0: set_a > a > set_a > $o ).
thf(func_def_61,type,
sP1: set_a > a > a > set_a > $o ).
thf(func_def_62,type,
sP2: a > set_a > a > set_a > $o ).
thf(func_def_63,type,
sP3: set_a > a > set_a > $o ).
thf(func_def_64,type,
sK4: ( a > $o ) > ( a > $o ) > a ).
thf(func_def_65,type,
sK5: set_a > a ).
thf(func_def_66,type,
sK6: ( a > $o ) > a ).
thf(func_def_67,type,
sK7: int > int ).
thf(func_def_68,type,
sK8: ( a > $o ) > set_a > a ).
thf(func_def_69,type,
sK9: ( a > $o ) > set_a > a ).
thf(func_def_70,type,
sK10: ( a > $o ) > set_a > set_a > a ).
thf(func_def_71,type,
sK11: int > int ).
thf(func_def_72,type,
sK12: set_a > set_a > a ).
thf(func_def_73,type,
sK13: set_a > a > set_a ).
thf(func_def_74,type,
sK14: ( int > int > $o ) > int ).
thf(func_def_75,type,
sK15: ( int > int > $o ) > int ).
thf(func_def_76,type,
sK16: ( int > int > $o ) > int ).
thf(func_def_77,type,
sK17: ( int > int > $o ) > int ).
thf(func_def_78,type,
sK18: ( int > int > $o ) > int ).
thf(func_def_79,type,
sK19: set_a > a ).
thf(func_def_80,type,
sK20: a > set_a > set_a ).
thf(func_def_81,type,
sK21: ( a > $o ) > a ).
thf(func_def_82,type,
sK22: ( binary1439146945Tree_a > $o ) > binary1439146945Tree_a ).
thf(func_def_83,type,
sK23: ( binary1439146945Tree_a > $o ) > a ).
thf(func_def_84,type,
sK24: ( binary1439146945Tree_a > $o ) > binary1439146945Tree_a ).
thf(func_def_85,type,
sK25: set_a > a > a > set_a > set_a ).
thf(func_def_86,type,
sK26: ( a > int ) > binary1439146945Tree_a > a > a ).
thf(func_def_87,type,
sK27: ( a > int ) > binary1439146945Tree_a > a > a ).
thf(func_def_88,type,
sK28: set_a > a ).
thf(func_def_89,type,
sK29: binary1439146945Tree_a > a ).
thf(func_def_90,type,
sK30: binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_91,type,
sK31: binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_92,type,
sK32: ( int > int ) > int ).
thf(func_def_93,type,
sK33: ( int > int ) > int ).
thf(func_def_94,type,
sK34: ( int > int ) > int ).
thf(func_def_95,type,
sK35: ( int > int ) > int ).
thf(func_def_96,type,
sK36: ( a > $o ) > set_a > set_a > a ).
thf(func_def_97,type,
sK37: ( a > $o ) > set_a > a ).
thf(func_def_98,type,
sK38: ( a > $o ) > set_a > a ).
thf(func_def_99,type,
sK39: ( int > int ) > int ).
thf(func_def_100,type,
sK40: ( int > int ) > int ).
thf(func_def_101,type,
sK41: a ).
thf(func_def_102,type,
sK42: ( int > int ) > int ).
thf(func_def_103,type,
sK43: ( int > int ) > int ).
thf(func_def_105,type,
ph45:
!>[X0: $tType] : X0 ).
thf(f1904,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1900,f1353]) ).
thf(f1353,plain,
! [X0: int] :
( ( ord_less_int @ X0 @ X0 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f972]) ).
thf(f972,plain,
! [X0: int] :
( ( ord_less_int @ X0 @ X0 )
!= $true ),
inference(flattening,[status(thm)],[f841]) ).
thf(f841,plain,
! [X0: int] :
( ( ord_less_int @ X0 @ X0 )
!= $true ),
inference(fool_elimination,[status(thm)],[f840]) ).
thf(f840,plain,
! [X0: int] :
~ ( ord_less_int @ X0 @ X0 ),
inference(rectify,[status(thm)],[f243]) ).
thf(f243,axiom,
! [X15: int] :
~ ( ord_less_int @ X15 @ X15 ),
file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_242_less__irrefl) ).
thf(f1900,plain,
( $true
= ( ord_less_int @ ( h @ e ) @ ( h @ e ) ) ),
inference(superposition,[status(thm)],[f1211,f1861]) ).
thf(f1861,plain,
( ( h @ x )
= ( h @ e ) ),
inference(superposition,[status(thm)],[f1855,f1429]) ).
thf(f1429,plain,
( ( h @ x )
= ( h @ sK41 ) ),
inference(cnf_transformation,[status(thm)],[f1177]) ).
thf(f1177,plain,
( ( ( h @ x )
= ( h @ sK41 ) )
& ( $true
= ( member_a @ sK41 @ ( binary504661350_eqs_a @ h @ e ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK41])],[f990,f1176]) ).
thf(f1176,plain,
( ? [X0: a] :
( ( ( h @ X0 )
= ( h @ x ) )
& ( ( member_a @ X0 @ ( binary504661350_eqs_a @ h @ e ) )
= $true ) )
=> ( ( ( h @ x )
= ( h @ sK41 ) )
& ( $true
= ( member_a @ sK41 @ ( binary504661350_eqs_a @ h @ e ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f990,plain,
? [X0: a] :
( ( ( h @ X0 )
= ( h @ x ) )
& ( ( member_a @ X0 @ ( binary504661350_eqs_a @ h @ e ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f715]) ).
thf(f715,plain,
~ ! [X0: a] :
( ( ( member_a @ X0 @ ( binary504661350_eqs_a @ h @ e ) )
= $true )
=> ( ( h @ X0 )
!= ( h @ x ) ) ),
inference(fool_elimination,[status(thm)],[f714]) ).
thf(f714,plain,
~ ! [X0: a] :
( ( member_a @ X0 @ ( binary504661350_eqs_a @ h @ e ) )
=> ( ( h @ X0 )
!= ( h @ x ) ) ),
inference(rectify,[status(thm)],[f360]) ).
thf(f360,negated_conjecture,
~ ! [X37: a] :
( ( member_a @ X37 @ ( binary504661350_eqs_a @ h @ e ) )
=> ( ( h @ x )
!= ( h @ X37 ) ) ),
inference(negated_conjecture,[status(cth)],[f359]) ).
thf(f359,conjecture,
! [X37: a] :
( ( member_a @ X37 @ ( binary504661350_eqs_a @ h @ e ) )
=> ( ( h @ x )
!= ( h @ X37 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',conj_0) ).
thf(f1855,plain,
( ( h @ e )
= ( h @ sK41 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1854]) ).
thf(f1854,plain,
( ( ( h @ e )
= ( h @ sK41 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1838]) ).
thf(f1838,plain,
( ( ^ [Y0: a] :
( ( h @ e )
= ( h @ Y0 ) )
@ sK41 )
= $true ),
inference(superposition,[status(thm)],[f1832,f1404]) ).
thf(f1404,plain,
! [X0: a > $o,X1: a] :
( ( X0 @ X1 )
= ( member_a @ X1 @ ( collect_a @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f916]) ).
thf(f916,plain,
! [X0: a > $o,X1: a] :
( ( X0 @ X1 )
= ( member_a @ X1 @ ( collect_a @ X0 ) ) ),
inference(rectify,[status(thm)],[f46]) ).
thf(f46,axiom,
! [X18: a > $o,X2: a] :
( ( member_a @ X2 @ ( collect_a @ X18 ) )
= ( X18 @ X2 ) ),
file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_45_mem__Collect__eq) ).
thf(f1832,plain,
( ( member_a @ sK41
@ ( collect_a
@ ^ [Y0: a] :
( ( h @ e )
= ( h @ Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1605]) ).
thf(f1605,plain,
( ( member_a @ sK41
@ ( ^ [Y0: a > int,Y1: a] :
( collect_a
@ ^ [Y2: a] :
( ( Y0 @ Y1 )
= ( Y0 @ Y2 ) ) )
@ h
@ e ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1428,f1326]) ).
thf(f1326,plain,
( binary504661350_eqs_a
= ( ^ [Y0: a > int,Y1: a] :
( collect_a
@ ^ [Y2: a] :
( ( Y0 @ Y1 )
= ( Y0 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f567]) ).
thf(f567,plain,
( binary504661350_eqs_a
= ( ^ [Y0: a > int,Y1: a] :
( collect_a
@ ^ [Y2: a] :
( ( Y0 @ Y1 )
= ( Y0 @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f566]) ).
thf(f566,plain,
( ( ^ [X0: a > int,X1: a] :
( collect_a
@ ^ [X2: a] :
( ( X0 @ X1 )
= ( X0 @ X2 ) ) ) )
= binary504661350_eqs_a ),
inference(rectify,[status(thm)],[f10]) ).
thf(f10,axiom,
( ( ^ [X5: a > int,X6: a] :
( collect_a
@ ^ [X7: a] :
( ( X5 @ X7 )
= ( X5 @ X6 ) ) ) )
= binary504661350_eqs_a ),
file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_9_eqs__def) ).
thf(f1428,plain,
( $true
= ( member_a @ sK41 @ ( binary504661350_eqs_a @ h @ e ) ) ),
inference(cnf_transformation,[status(thm)],[f1177]) ).
thf(f1211,plain,
( ( ord_less_int @ ( h @ e ) @ ( h @ x ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f471]) ).
thf(f471,plain,
( ( ord_less_int @ ( h @ e ) @ ( h @ x ) )
= $true ),
inference(fool_elimination,[status(thm)],[f470]) ).
thf(f470,plain,
ord_less_int @ ( h @ e ) @ ( h @ x ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
ord_less_int @ ( h @ e ) @ ( h @ x ),
file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_1_eLess) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP031^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n001.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:24:35 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : ITP031^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.46 % Domain : Interactive Theorem Proving
% 0.29/0.46 % Problem : Sledgehammer BinaryTree problem prob_301__3253080_1
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Des21]
% 0.29/0.46 % Names : BinaryTree/prob_301__3253080_1 [Des21]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.29/0.46 % Syntax : Number of formulae : 404 ( 207 unt; 45 typ; 0 def)
% 0.29/0.46 % Number of atoms : 982 ( 384 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 7 ( 2 avg)
% 0.29/0.46 % Number of connectives : 2678 ( 112 ~; 18 |; 67 &;2153 @)
% 0.29/0.46 % ( 0 <=>; 328 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 14 ( 6 avg)
% 0.29/0.46 % Number of types : 5 ( 4 usr)
% 0.29/0.46 % Number of type conns : 243 ( 243 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 44 ( 41 usr; 9 con; 0-4 aty)
% 0.29/0.46 % Number of variables : 972 ( 110 ^; 847 !; 15 ?; 972 :)
% 0.29/0.46 % SPC : TH0_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:30:37.987
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % Could-be-implicit typings (4)
% 0.29/0.46 thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
% 0.29/0.46 binary1439146945Tree_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 set_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Int__Oint,type,
% 0.29/0.46 int: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_tf__a,type,
% 0.29/0.46 a: $tType ).
% 0.29/0.46
% 0.29/0.46 % Explicit typings (41)
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001tf__a,type,
% 0.29/0.46 binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001tf__a,type,
% 0.29/0.46 binary476621312_Tip_a: binary1439146945Tree_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Oset__Tree_001tf__a,type,
% 0.29/0.46 binary256242811Tree_a: binary1439146945Tree_a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Obinsert_001tf__a,type,
% 0.29/0.46 binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oeqs_001tf__a,type,
% 0.29/0.46 binary504661350_eqs_a: ( a > int ) > a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Omemb_001tf__a,type,
% 0.29/0.46 binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001tf__a,type,
% 0.29/0.46 binary945792244etOf_a: binary1439146945Tree_a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001tf__a,type,
% 0.29/0.46 binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001tf__a,type,
% 0.29/0.46 binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.46 minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
% 0.29/0.46 minus_minus_o: $o > $o > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.29/0.46 minus_minus_int: int > int > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 minus_minus_set_a: set_a > set_a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_HOL_OThe_001tf__a,type,
% 0.29/0.46 the_a: ( a > $o ) > a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_If_001tf__a,type,
% 0.29/0.46 if_a: $o > a > a > a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.46 sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
% 0.29/0.46 sup_sup_o: $o > $o > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 0.29/0.46 sup_sup_int: int > int > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 sup_sup_set_a: set_a > set_a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.46 bot_bot_a_o: a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 0.29/0.46 bot_bot_o: $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 bot_bot_set_a: set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.46 ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.29/0.46 ord_less_int: int > int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 ord_less_set_a: set_a > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.46 ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.29/0.46 ord_less_eq_int: int > int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.29/0.46 collect_a: ( a > $o ) > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oinsert_001tf__a,type,
% 0.29/0.46 insert_a: a > set_a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 0.29/0.46 is_empty_a: set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Ois__singleton_001tf__a,type,
% 0.29/0.46 is_singleton_a: set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Opairwise_001tf__a,type,
% 0.29/0.46 pairwise_a: ( a > a > $o ) > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oremove_001tf__a,type,
% 0.29/0.46 remove_a: a > set_a > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 0.29/0.46 the_elem_a: set_a > a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001tf__a,type,
% 0.29/0.46 member_a: a > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_e,type,
% 0.29/0.46 e: a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_h,type,
% 0.29/0.46 h: a > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_t1____,type,
% 0.29/0.46 t1: binary1439146945Tree_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_t2____,type,
% 0.29/0.46 t2: binary1439146945Tree_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_x____,type,
% 0.29/0.46 x: a ).
% 0.29/0.46
% 0.29/0.46 % Relevant facts (355)
% 0.29/0.46 thf(fact_0_eqsLessX,axiom,
% 0.29/0.46 ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ ( binary504661350_eqs_a @ h @ e ) )
% 0.29/0.46 => ( ord_less_int @ ( h @ X ) @ ( h @ x ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eqsLessX
% 0.29/0.46 thf(fact_1_eLess,axiom,
% 0.29/0.46 ord_less_int @ ( h @ e ) @ ( h @ x ) ).
% 0.29/0.46
% 0.29/0.46 % eLess
% 0.29/0.46 thf(fact_2_res,axiom,
% 0.29/0.46 ( ( binary1226383794sert_a @ h @ e @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
% 0.29/0.46 = ( binary717961607le_T_a @ ( binary1226383794sert_a @ h @ e @ t1 ) @ x @ t2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % res
% 0.29/0.46 thf(fact_3_s1,axiom,
% 0.29/0.46 binary1721989714Tree_a @ h @ t1 ).
% 0.29/0.46
% 0.29/0.46 % s1
% 0.29/0.46 thf(fact_4_s2,axiom,
% 0.29/0.46 binary1721989714Tree_a @ h @ t2 ).
% 0.29/0.46
% 0.29/0.46 % s2
% 0.29/0.46 thf(fact_5_s,axiom,
% 0.29/0.46 binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% 0.29/0.46
% 0.29/0.46 % s
% 0.29/0.46 thf(fact_6_sorted__distinct,axiom,
% 0.29/0.46 ! [H: a > int,A: a,B: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A @ B @ T ) ).
% 0.29/0.46
% 0.29/0.46 % sorted_distinct
% 0.29/0.46 thf(fact_7_c1,axiom,
% 0.29/0.46 ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t1 ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t1 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % c1
% 0.29/0.46 thf(fact_8_c2,axiom,
% 0.29/0.46 ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t2 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % c2
% 0.29/0.46 thf(fact_9_eqs__def,axiom,
% 0.29/0.46 ( binary504661350_eqs_a
% 0.29/0.46 = ( ^ [H2: a > int,X2: a] :
% 0.29/0.46 ( collect_a
% 0.29/0.46 @ ^ [Y: a] :
% 0.29/0.46 ( ( H2 @ Y )
% 0.29/0.46 = ( H2 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eqs_def
% 0.29/0.46 thf(fact_10_h1,axiom,
% 0.29/0.46 ( ( binary1721989714Tree_a @ h @ t1 )
% 0.29/0.46 => ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t1 ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t1 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % h1
% 0.29/0.46 thf(fact_11_h2,axiom,
% 0.29/0.46 ( ( binary1721989714Tree_a @ h @ t2 )
% 0.29/0.46 => ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t2 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % h2
% 0.29/0.46 thf(fact_12__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062,axiom,
% 0.29/0.46 ( ( binary1721989714Tree_a @ h @ binary476621312_Tip_a )
% 0.29/0.46 => ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ binary476621312_Tip_a ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ binary476621312_Tip_a ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>sortedTree h Tip \<longrightarrow> setOf (binsert h e Tip) = setOf Tip - eqs h e \<union> {e}\<close>
% 0.29/0.46 thf(fact_13_Tree_Oinject,axiom,
% 0.29/0.46 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a,Y21: binary1439146945Tree_a,Y22: a,Y23: binary1439146945Tree_a] :
% 0.29/0.46 ( ( ( binary717961607le_T_a @ X21 @ X22 @ X23 )
% 0.29/0.46 = ( binary717961607le_T_a @ Y21 @ Y22 @ Y23 ) )
% 0.29/0.46 = ( ( X21 = Y21 )
% 0.29/0.46 & ( X22 = Y22 )
% 0.29/0.46 & ( X23 = Y23 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Tree.inject
% 0.29/0.46 thf(fact_14_setOf_Osimps_I2_J,axiom,
% 0.29/0.46 ! [T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 0.29/0.46 ( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % setOf.simps(2)
% 0.29/0.46 thf(fact_15_setOf_Osimps_I1_J,axiom,
% 0.29/0.46 ( ( binary945792244etOf_a @ binary476621312_Tip_a )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % setOf.simps(1)
% 0.29/0.46 thf(fact_16_Tree_Odistinct_I1_J,axiom,
% 0.29/0.46 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 0.29/0.46 ( binary476621312_Tip_a
% 0.29/0.46 != ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Tree.distinct(1)
% 0.29/0.46 thf(fact_17_binsert_Osimps_I2_J,axiom,
% 0.29/0.46 ! [H: a > int,E: a,X3: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
% 0.29/0.46 ( ( ( ord_less_int @ ( H @ E ) @ ( H @ X3 ) )
% 0.29/0.46 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 = ( binary717961607le_T_a @ ( binary1226383794sert_a @ H @ E @ T1 ) @ X3 @ T2 ) ) )
% 0.29/0.46 & ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X3 ) )
% 0.29/0.46 => ( ( ( ord_less_int @ ( H @ X3 ) @ ( H @ E ) )
% 0.29/0.46 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 = ( binary717961607le_T_a @ T1 @ X3 @ ( binary1226383794sert_a @ H @ E @ T2 ) ) ) )
% 0.29/0.46 & ( ~ ( ord_less_int @ ( H @ X3 ) @ ( H @ E ) )
% 0.29/0.46 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 = ( binary717961607le_T_a @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % binsert.simps(2)
% 0.29/0.46 thf(fact_18_binsert_Osimps_I1_J,axiom,
% 0.29/0.46 ! [H: a > int,E: a] :
% 0.29/0.46 ( ( binary1226383794sert_a @ H @ E @ binary476621312_Tip_a )
% 0.29/0.46 = ( binary717961607le_T_a @ binary476621312_Tip_a @ E @ binary476621312_Tip_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % binsert.simps(1)
% 0.29/0.46 thf(fact_19_sortedTree_Osimps_I2_J,axiom,
% 0.29/0.46 ! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 0.29/0.46 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 = ( ( binary1721989714Tree_a @ H @ T1 )
% 0.29/0.46 & ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ ( binary945792244etOf_a @ T1 ) )
% 0.29/0.46 => ( ord_less_int @ ( H @ X2 ) @ ( H @ X3 ) ) )
% 0.29/0.46 & ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ ( binary945792244etOf_a @ T2 ) )
% 0.29/0.46 => ( ord_less_int @ ( H @ X3 ) @ ( H @ X2 ) ) )
% 0.29/0.46 & ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sortedTree.simps(2)
% 0.29/0.46 thf(fact_20_sortedTree_Osimps_I1_J,axiom,
% 0.29/0.46 ! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% 0.29/0.46
% 0.29/0.46 % sortedTree.simps(1)
% 0.29/0.46 thf(fact_21_sortLemmaL,axiom,
% 0.29/0.46 ! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 0.29/0.46 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 => ( binary1721989714Tree_a @ H @ T1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sortLemmaL
% 0.29/0.46 thf(fact_22_sortLemmaR,axiom,
% 0.29/0.46 ! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 0.29/0.46 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 0.29/0.46 => ( binary1721989714Tree_a @ H @ T2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sortLemmaR
% 0.29/0.46 thf(fact_23_Tree_Oinduct,axiom,
% 0.29/0.46 ! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
% 0.29/0.46 ( ( P @ binary476621312_Tip_a )
% 0.29/0.46 => ( ! [X1: binary1439146945Tree_a,X24: a,X32: binary1439146945Tree_a] :
% 0.29/0.46 ( ( P @ X1 )
% 0.29/0.46 => ( ( P @ X32 )
% 0.29/0.46 => ( P @ ( binary717961607le_T_a @ X1 @ X24 @ X32 ) ) ) )
% 0.29/0.46 => ( P @ Tree ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Tree.induct
% 0.29/0.46 thf(fact_24_Tree_Oexhaust,axiom,
% 0.29/0.46 ! [Y2: binary1439146945Tree_a] :
% 0.29/0.46 ( ( Y2 != binary476621312_Tip_a )
% 0.29/0.46 => ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Tree.exhaust
% 0.29/0.46 thf(fact_25_sorted__distinct__pred__def,axiom,
% 0.29/0.46 ( binary670562003pred_a
% 0.29/0.46 = ( ^ [H2: a > int,A2: a,B2: a,T3: binary1439146945Tree_a] :
% 0.29/0.46 ( ( ( binary1721989714Tree_a @ H2 @ T3 )
% 0.29/0.46 & ( member_a @ A2 @ ( binary945792244etOf_a @ T3 ) )
% 0.29/0.46 & ( member_a @ B2 @ ( binary945792244etOf_a @ T3 ) )
% 0.29/0.46 & ( ( H2 @ A2 )
% 0.29/0.46 = ( H2 @ B2 ) ) )
% 0.29/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sorted_distinct_pred_def
% 0.29/0.46 thf(fact_26_insert__Diff__single,axiom,
% 0.29/0.46 ! [A: a,A3: set_a] :
% 0.29/0.46 ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
% 0.29/0.46 = ( insert_a @ A @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_single
% 0.29/0.46 thf(fact_27_singleton__conv,axiom,
% 0.29/0.46 ! [A: a] :
% 0.29/0.46 ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] : ( X2 = A ) )
% 0.29/0.46 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_conv
% 0.29/0.46 thf(fact_28_singleton__conv2,axiom,
% 0.29/0.46 ! [A: a] :
% 0.29/0.46 ( ( collect_a
% 0.29/0.46 @ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
% 0.29/0.46 @ A ) )
% 0.29/0.46 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_conv2
% 0.29/0.46 thf(fact_29_Un__Diff__cancel,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
% 0.29/0.46 = ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_Diff_cancel
% 0.29/0.46 thf(fact_30_Un__Diff__cancel2,axiom,
% 0.29/0.46 ! [B3: set_a,A3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( minus_minus_set_a @ B3 @ A3 ) @ A3 )
% 0.29/0.46 = ( sup_sup_set_a @ B3 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_Diff_cancel2
% 0.29/0.46 thf(fact_31_Diff__insert0,axiom,
% 0.29/0.46 ! [X3: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ~ ( member_a @ X3 @ A3 )
% 0.29/0.46 => ( ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ B3 ) )
% 0.29/0.46 = ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert0
% 0.29/0.46 thf(fact_32_insert__Diff1,axiom,
% 0.29/0.46 ! [X3: a,B3: set_a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ X3 @ B3 )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 0.29/0.46 = ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff1
% 0.29/0.46 thf(fact_33_Un__insert__left,axiom,
% 0.29/0.46 ! [A: a,B3: set_a,C: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( insert_a @ A @ B3 ) @ C )
% 0.29/0.46 = ( insert_a @ A @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_insert_left
% 0.29/0.46 thf(fact_34_Un__insert__right,axiom,
% 0.29/0.46 ! [A3: set_a,A: a,B3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B3 ) )
% 0.29/0.46 = ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_insert_right
% 0.29/0.46 thf(fact_35_Diff__empty,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
% 0.29/0.46 = A3 ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_empty
% 0.29/0.46 thf(fact_36_empty__Diff,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Diff
% 0.29/0.46 thf(fact_37_empty__Collect__eq,axiom,
% 0.29/0.46 ! [P: a > $o] :
% 0.29/0.46 ( ( bot_bot_set_a
% 0.29/0.46 = ( collect_a @ P ) )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Collect_eq
% 0.29/0.46 thf(fact_38_Collect__empty__eq,axiom,
% 0.29/0.46 ! [P: a > $o] :
% 0.29/0.46 ( ( ( collect_a @ P )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq
% 0.29/0.46 thf(fact_39_all__not__in__conv,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( ! [X2: a] :
% 0.29/0.46 ~ ( member_a @ X2 @ A3 ) )
% 0.29/0.46 = ( A3 = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % all_not_in_conv
% 0.29/0.46 thf(fact_40_empty__iff,axiom,
% 0.29/0.46 ! [C2: a] :
% 0.29/0.46 ~ ( member_a @ C2 @ bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % empty_iff
% 0.29/0.46 thf(fact_41_insert__absorb2,axiom,
% 0.29/0.46 ! [X3: a,A3: set_a] :
% 0.29/0.46 ( ( insert_a @ X3 @ ( insert_a @ X3 @ A3 ) )
% 0.29/0.46 = ( insert_a @ X3 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb2
% 0.29/0.46 thf(fact_42_insert__iff,axiom,
% 0.29/0.46 ! [A: a,B: a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 0.29/0.46 = ( ( A = B )
% 0.29/0.46 | ( member_a @ A @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_43_insertCI,axiom,
% 0.29/0.46 ! [A: a,B3: set_a,B: a] :
% 0.29/0.46 ( ( ~ ( member_a @ A @ B3 )
% 0.29/0.46 => ( A = B ) )
% 0.29/0.46 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_44_Un__iff,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.29/0.46 = ( ( member_a @ C2 @ A3 )
% 0.29/0.46 | ( member_a @ C2 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_iff
% 0.29/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A: a,P: a > $o] :
% 0.29/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.29/0.46 = ( P @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
% 0.29/0.46 = A3 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_47_Collect__cong,axiom,
% 0.29/0.46 ! [P: a > $o,Q: a > $o] :
% 0.29/0.46 ( ! [X4: a] :
% 0.29/0.46 ( ( P @ X4 )
% 0.29/0.46 = ( Q @ X4 ) )
% 0.29/0.46 => ( ( collect_a @ P )
% 0.29/0.46 = ( collect_a @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_48_UnCI,axiom,
% 0.29/0.46 ! [C2: a,B3: set_a,A3: set_a] :
% 0.29/0.46 ( ( ~ ( member_a @ C2 @ B3 )
% 0.29/0.46 => ( member_a @ C2 @ A3 ) )
% 0.29/0.46 => ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnCI
% 0.29/0.46 thf(fact_49_Diff__idemp,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ B3 )
% 0.29/0.46 = ( minus_minus_set_a @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_idemp
% 0.29/0.46 thf(fact_50_Diff__iff,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 0.29/0.46 = ( ( member_a @ C2 @ A3 )
% 0.29/0.46 & ~ ( member_a @ C2 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_iff
% 0.29/0.46 thf(fact_51_DiffI,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ A3 )
% 0.29/0.46 => ( ~ ( member_a @ C2 @ B3 )
% 0.29/0.46 => ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffI
% 0.29/0.46 thf(fact_52_singletonI,axiom,
% 0.29/0.46 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonI
% 0.29/0.46 thf(fact_53_Un__empty,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ( A3 = bot_bot_set_a )
% 0.29/0.46 & ( B3 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_empty
% 0.29/0.46 thf(fact_54_Diff__cancel,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A3 @ A3 )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_cancel
% 0.29/0.46 thf(fact_55_ex__in__conv,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
% 0.29/0.46 = ( A3 != bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_in_conv
% 0.29/0.46 thf(fact_56_equals0I,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ! [Y4: a] :
% 0.29/0.46 ~ ( member_a @ Y4 @ A3 )
% 0.29/0.46 => ( A3 = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0I
% 0.29/0.46 thf(fact_57_equals0D,axiom,
% 0.29/0.46 ! [A3: set_a,A: a] :
% 0.29/0.46 ( ( A3 = bot_bot_set_a )
% 0.29/0.46 => ~ ( member_a @ A @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0D
% 0.29/0.46 thf(fact_58_emptyE,axiom,
% 0.29/0.46 ! [A: a] :
% 0.29/0.46 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % emptyE
% 0.29/0.46 thf(fact_59_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A: a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ A @ A3 )
% 0.29/0.46 => ? [B4: set_a] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( insert_a @ A @ B4 ) )
% 0.29/0.46 & ~ ( member_a @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_60_insert__commute,axiom,
% 0.29/0.46 ! [X3: a,Y2: a,A3: set_a] :
% 0.29/0.46 ( ( insert_a @ X3 @ ( insert_a @ Y2 @ A3 ) )
% 0.29/0.46 = ( insert_a @ Y2 @ ( insert_a @ X3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_commute
% 0.29/0.46 thf(fact_61_insert__eq__iff,axiom,
% 0.29/0.46 ! [A: a,A3: set_a,B: a,B3: set_a] :
% 0.29/0.46 ( ~ ( member_a @ A @ A3 )
% 0.29/0.46 => ( ~ ( member_a @ B @ B3 )
% 0.29/0.46 => ( ( ( insert_a @ A @ A3 )
% 0.29/0.46 = ( insert_a @ B @ B3 ) )
% 0.29/0.46 = ( ( ( A = B )
% 0.29/0.46 => ( A3 = B3 ) )
% 0.29/0.46 & ( ( A != B )
% 0.29/0.46 => ? [C3: set_a] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( insert_a @ B @ C3 ) )
% 0.29/0.46 & ~ ( member_a @ B @ C3 )
% 0.29/0.46 & ( B3
% 0.29/0.46 = ( insert_a @ A @ C3 ) )
% 0.29/0.46 & ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_62_insert__absorb,axiom,
% 0.29/0.46 ! [A: a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ A @ A3 )
% 0.29/0.46 => ( ( insert_a @ A @ A3 )
% 0.29/0.46 = A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_63_insert__ident,axiom,
% 0.29/0.46 ! [X3: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ~ ( member_a @ X3 @ A3 )
% 0.29/0.46 => ( ~ ( member_a @ X3 @ B3 )
% 0.29/0.46 => ( ( ( insert_a @ X3 @ A3 )
% 0.29/0.46 = ( insert_a @ X3 @ B3 ) )
% 0.29/0.46 = ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_64_Set_Oset__insert,axiom,
% 0.29/0.46 ! [X3: a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ X3 @ A3 )
% 0.29/0.46 => ~ ! [B4: set_a] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( insert_a @ X3 @ B4 ) )
% 0.29/0.46 => ( member_a @ X3 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.set_insert
% 0.29/0.46 thf(fact_65_insertI2,axiom,
% 0.29/0.46 ! [A: a,B3: set_a,B: a] :
% 0.29/0.46 ( ( member_a @ A @ B3 )
% 0.29/0.46 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI2
% 0.29/0.46 thf(fact_66_insertI1,axiom,
% 0.29/0.46 ! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI1
% 0.29/0.46 thf(fact_67_insertE,axiom,
% 0.29/0.46 ! [A: a,B: a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 0.29/0.46 => ( ( A != B )
% 0.29/0.46 => ( member_a @ A @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertE
% 0.29/0.46 thf(fact_68_Un__left__commute,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a,C: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C ) )
% 0.29/0.46 = ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_left_commute
% 0.29/0.46 thf(fact_69_Un__left__absorb,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.29/0.46 = ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_left_absorb
% 0.29/0.46 thf(fact_70_Un__commute,axiom,
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 = ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_commute
% 0.29/0.46 thf(fact_71_Un__absorb,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A3 @ A3 )
% 0.29/0.46 = A3 ) ).
% 0.29/0.46
% 0.29/0.46 % Un_absorb
% 0.29/0.46 thf(fact_72_Un__assoc,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a,C: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
% 0.29/0.46 = ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_assoc
% 0.29/0.46 thf(fact_73_ball__Un,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a,P: a > $o] :
% 0.29/0.46 ( ( ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.29/0.46 => ( P @ X2 ) ) )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A3 )
% 0.29/0.46 => ( P @ X2 ) )
% 0.29/0.46 & ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ B3 )
% 0.29/0.46 => ( P @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ball_Un
% 0.29/0.46 thf(fact_74_bex__Un,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a,P: a > $o] :
% 0.29/0.46 ( ( ? [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 = ( ? [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A3 )
% 0.29/0.46 & ( P @ X2 ) )
% 0.29/0.46 | ? [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ B3 )
% 0.29/0.46 & ( P @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bex_Un
% 0.29/0.46 thf(fact_75_UnI2,axiom,
% 0.29/0.46 ! [C2: a,B3: set_a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ B3 )
% 0.29/0.46 => ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnI2
% 0.29/0.46 thf(fact_76_UnI1,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ A3 )
% 0.29/0.46 => ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnI1
% 0.29/0.46 thf(fact_77_UnE,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.29/0.46 => ( ~ ( member_a @ C2 @ A3 )
% 0.29/0.46 => ( member_a @ C2 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnE
% 0.29/0.46 thf(fact_78_DiffD2,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 0.29/0.46 => ~ ( member_a @ C2 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD2
% 0.29/0.46 thf(fact_79_DiffD1,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 0.29/0.46 => ( member_a @ C2 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD1
% 0.29/0.46 thf(fact_80_DiffE,axiom,
% 0.29/0.46 ! [C2: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 0.29/0.46 => ~ ( ( member_a @ C2 @ A3 )
% 0.29/0.46 => ( member_a @ C2 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffE
% 0.29/0.46 thf(fact_81_empty__def,axiom,
% 0.29/0.46 ( bot_bot_set_a
% 0.29/0.46 = ( collect_a
% 0.29/0.46 @ ^ [X2: a] : $false ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_def
% 0.29/0.46 thf(fact_82_insert__Collect,axiom,
% 0.29/0.46 ! [A: a,P: a > $o] :
% 0.29/0.46 ( ( insert_a @ A @ ( collect_a @ P ) )
% 0.29/0.46 = ( collect_a
% 0.29/0.46 @ ^ [U: a] :
% 0.29/0.46 ( ( U != A )
% 0.29/0.46 => ( P @ U ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Collect
% 0.29/0.46 thf(fact_83_insert__compr,axiom,
% 0.29/0.46 ( insert_a
% 0.29/0.46 = ( ^ [A2: a,B5: set_a] :
% 0.29/0.46 ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( X2 = A2 )
% 0.29/0.46 | ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_compr
% 0.29/0.46 thf(fact_84_Collect__disj__eq,axiom,
% 0.29/0.46 ! [P: a > $o,Q: a > $o] :
% 0.29/0.46 ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 | ( Q @ X2 ) ) )
% 0.29/0.46 = ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_disj_eq
% 0.29/0.46 thf(fact_85_Un__def,axiom,
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 = ( ^ [A4: set_a,B5: set_a] :
% 0.29/0.46 ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A4 )
% 0.29/0.46 | ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_def
% 0.29/0.46 thf(fact_86_set__diff__eq,axiom,
% 0.29/0.46 ( minus_minus_set_a
% 0.29/0.46 = ( ^ [A4: set_a,B5: set_a] :
% 0.29/0.46 ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A4 )
% 0.29/0.46 & ~ ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_diff_eq
% 0.29/0.46 thf(fact_87_singleton__inject,axiom,
% 0.29/0.46 ! [A: a,B: a] :
% 0.29/0.46 ( ( ( insert_a @ A @ bot_bot_set_a )
% 0.29/0.46 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.29/0.46 => ( A = B ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_inject
% 0.29/0.46 thf(fact_88_insert__not__empty,axiom,
% 0.29/0.46 ! [A: a,A3: set_a] :
% 0.29/0.46 ( ( insert_a @ A @ A3 )
% 0.29/0.46 != bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % insert_not_empty
% 0.29/0.46 thf(fact_89_doubleton__eq__iff,axiom,
% 0.29/0.46 ! [A: a,B: a,C2: a,D: a] :
% 0.29/0.46 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 0.29/0.46 = ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 0.29/0.46 = ( ( ( A = C2 )
% 0.29/0.46 & ( B = D ) )
% 0.29/0.46 | ( ( A = D )
% 0.29/0.46 & ( B = C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % doubleton_eq_iff
% 0.29/0.46 thf(fact_90_singleton__iff,axiom,
% 0.29/0.46 ! [B: a,A: a] :
% 0.29/0.46 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.29/0.46 = ( B = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_iff
% 0.29/0.46 thf(fact_91_singletonD,axiom,
% 0.29/0.46 ! [B: a,A: a] :
% 0.29/0.46 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.29/0.46 => ( B = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonD
% 0.29/0.46 thf(fact_92_Un__empty__right,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
% 0.29/0.46 = A3 ) ).
% 0.29/0.46
% 0.29/0.46 % Un_empty_right
% 0.29/0.46 thf(fact_93_Un__empty__left,axiom,
% 0.29/0.46 ! [B3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
% 0.29/0.46 = B3 ) ).
% 0.29/0.46
% 0.29/0.46 % Un_empty_left
% 0.29/0.46 thf(fact_94_insert__Diff__if,axiom,
% 0.29/0.46 ! [X3: a,B3: set_a,A3: set_a] :
% 0.29/0.46 ( ( ( member_a @ X3 @ B3 )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 0.29/0.46 = ( minus_minus_set_a @ A3 @ B3 ) ) )
% 0.29/0.46 & ( ~ ( member_a @ X3 @ B3 )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 0.29/0.46 = ( insert_a @ X3 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_if
% 0.29/0.46 thf(fact_95_Un__Diff,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a,C: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
% 0.29/0.46 = ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C ) @ ( minus_minus_set_a @ B3 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_Diff
% 0.29/0.46 thf(fact_96_Collect__conv__if2,axiom,
% 0.29/0.46 ! [P: a > $o,A: a] :
% 0.29/0.46 ( ( ( P @ A )
% 0.29/0.46 => ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( A = X2 )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 = ( insert_a @ A @ bot_bot_set_a ) ) )
% 0.29/0.46 & ( ~ ( P @ A )
% 0.29/0.46 => ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( A = X2 )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_conv_if2
% 0.29/0.46 thf(fact_97_Collect__conv__if,axiom,
% 0.29/0.46 ! [P: a > $o,A: a] :
% 0.29/0.46 ( ( ( P @ A )
% 0.29/0.46 => ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( X2 = A )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 = ( insert_a @ A @ bot_bot_set_a ) ) )
% 0.29/0.46 & ( ~ ( P @ A )
% 0.29/0.46 => ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( X2 = A )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_conv_if
% 0.29/0.46 thf(fact_98_insert__def,axiom,
% 0.29/0.46 ( insert_a
% 0.29/0.46 = ( ^ [A2: a] :
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [X2: a] : ( X2 = A2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_def
% 0.29/0.46 thf(fact_99_singleton__Un__iff,axiom,
% 0.29/0.46 ! [X3: a,A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( ( insert_a @ X3 @ bot_bot_set_a )
% 0.29/0.46 = ( sup_sup_set_a @ A3 @ B3 ) )
% 0.29/0.46 = ( ( ( A3 = bot_bot_set_a )
% 0.29/0.46 & ( B3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) ) )
% 0.29/0.46 | ( ( A3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 & ( B3 = bot_bot_set_a ) )
% 0.29/0.46 | ( ( A3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 & ( B3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_Un_iff
% 0.29/0.46 thf(fact_100_Un__singleton__iff,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a,X3: a] :
% 0.29/0.46 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 = ( ( ( A3 = bot_bot_set_a )
% 0.29/0.46 & ( B3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) ) )
% 0.29/0.46 | ( ( A3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 & ( B3 = bot_bot_set_a ) )
% 0.29/0.46 | ( ( A3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 & ( B3
% 0.29/0.46 = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_singleton_iff
% 0.29/0.46 thf(fact_101_insert__is__Un,axiom,
% 0.29/0.46 ( insert_a
% 0.29/0.46 = ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_is_Un
% 0.29/0.46 thf(fact_102_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X3: a,A3: set_a] :
% 0.29/0.46 ( ~ ( member_a @ X3 @ A3 )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 = A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert_absorb
% 0.29/0.46 thf(fact_103_Diff__insert2,axiom,
% 0.29/0.46 ! [A3: set_a,A: a,B3: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
% 0.29/0.46 = ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert2
% 0.29/0.46 thf(fact_104_insert__Diff,axiom,
% 0.29/0.46 ! [A: a,A3: set_a] :
% 0.29/0.46 ( ( member_a @ A @ A3 )
% 0.29/0.46 => ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
% 0.29/0.46 = A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff
% 0.29/0.46 thf(fact_105_Diff__insert,axiom,
% 0.29/0.46 ! [A3: set_a,A: a,B3: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
% 0.29/0.46 = ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert
% 0.29/0.46 thf(fact_106_sup__bot_Oright__neutral,axiom,
% 0.29/0.46 ! [A: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ A @ bot_bot_a_o )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.right_neutral
% 0.29/0.46 thf(fact_107_sup__bot_Oright__neutral,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A @ bot_bot_set_a )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.right_neutral
% 0.29/0.46 thf(fact_108_sup__bot_Oneutr__eq__iff,axiom,
% 0.29/0.46 ! [A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( bot_bot_a_o
% 0.29/0.46 = ( sup_sup_a_o @ A @ B ) )
% 0.29/0.46 = ( ( A = bot_bot_a_o )
% 0.29/0.46 & ( B = bot_bot_a_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.neutr_eq_iff
% 0.29/0.46 thf(fact_109_sup__bot_Oneutr__eq__iff,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( bot_bot_set_a
% 0.29/0.46 = ( sup_sup_set_a @ A @ B ) )
% 0.29/0.46 = ( ( A = bot_bot_set_a )
% 0.29/0.46 & ( B = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.neutr_eq_iff
% 0.29/0.46 thf(fact_110_sup__bot_Oleft__neutral,axiom,
% 0.29/0.46 ! [A: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ bot_bot_a_o @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.left_neutral
% 0.29/0.46 thf(fact_111_sup__bot_Oleft__neutral,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ bot_bot_set_a @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.left_neutral
% 0.29/0.46 thf(fact_112_sup__bot_Oeq__neutr__iff,axiom,
% 0.29/0.46 ! [A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( ( sup_sup_a_o @ A @ B )
% 0.29/0.46 = bot_bot_a_o )
% 0.29/0.46 = ( ( A = bot_bot_a_o )
% 0.29/0.46 & ( B = bot_bot_a_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.eq_neutr_iff
% 0.29/0.46 thf(fact_113_sup__bot_Oeq__neutr__iff,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ( sup_sup_set_a @ A @ B )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ( A = bot_bot_set_a )
% 0.29/0.46 & ( B = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot.eq_neutr_iff
% 0.29/0.46 thf(fact_114_sup__eq__bot__iff,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o] :
% 0.29/0.46 ( ( ( sup_sup_a_o @ X3 @ Y2 )
% 0.29/0.46 = bot_bot_a_o )
% 0.29/0.46 = ( ( X3 = bot_bot_a_o )
% 0.29/0.46 & ( Y2 = bot_bot_a_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_eq_bot_iff
% 0.29/0.46 thf(fact_115_sup__eq__bot__iff,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ( sup_sup_set_a @ X3 @ Y2 )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ( X3 = bot_bot_set_a )
% 0.29/0.46 & ( Y2 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_eq_bot_iff
% 0.29/0.46 thf(fact_116_bot__eq__sup__iff,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o] :
% 0.29/0.46 ( ( bot_bot_a_o
% 0.29/0.46 = ( sup_sup_a_o @ X3 @ Y2 ) )
% 0.29/0.46 = ( ( X3 = bot_bot_a_o )
% 0.29/0.46 & ( Y2 = bot_bot_a_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_eq_sup_iff
% 0.29/0.46 thf(fact_117_bot__eq__sup__iff,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( bot_bot_set_a
% 0.29/0.46 = ( sup_sup_set_a @ X3 @ Y2 ) )
% 0.29/0.46 = ( ( X3 = bot_bot_set_a )
% 0.29/0.46 & ( Y2 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_eq_sup_iff
% 0.29/0.46 thf(fact_118_sup__bot__right,axiom,
% 0.29/0.46 ! [X3: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ X3 @ bot_bot_a_o )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot_right
% 0.29/0.46 thf(fact_119_sup__bot__right,axiom,
% 0.29/0.46 ! [X3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot_right
% 0.29/0.46 thf(fact_120_sup__bot__left,axiom,
% 0.29/0.46 ! [X3: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ bot_bot_a_o @ X3 )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot_left
% 0.29/0.46 thf(fact_121_sup__bot__left,axiom,
% 0.29/0.46 ! [X3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_bot_left
% 0.29/0.46 thf(fact_122_memb__spec,axiom,
% 0.29/0.46 ! [H: a > int,T: binary1439146945Tree_a,X3: a] :
% 0.29/0.46 ( ( binary1721989714Tree_a @ H @ T )
% 0.29/0.46 => ( ( binary2053421120memb_a @ H @ X3 @ T )
% 0.29/0.46 = ( member_a @ X3 @ ( binary945792244etOf_a @ T ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % memb_spec
% 0.29/0.46 thf(fact_123_minus__apply,axiom,
% 0.29/0.46 ( minus_minus_a_o
% 0.29/0.46 = ( ^ [A4: a > $o,B5: a > $o,X2: a] : ( minus_minus_o @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_apply
% 0.29/0.46 thf(fact_124_sup__apply,axiom,
% 0.29/0.46 ( sup_sup_a_o
% 0.29/0.46 = ( ^ [F: a > $o,G: a > $o,X2: a] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_apply
% 0.29/0.46 thf(fact_125_sup_Oidem,axiom,
% 0.29/0.46 ! [A: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ A @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup.idem
% 0.29/0.46 thf(fact_126_sup_Oidem,axiom,
% 0.29/0.46 ! [A: int] :
% 0.29/0.46 ( ( sup_sup_int @ A @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup.idem
% 0.29/0.46 thf(fact_127_sup_Oidem,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % sup.idem
% 0.29/0.46 thf(fact_128_sup__idem,axiom,
% 0.29/0.46 ! [X3: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ X3 @ X3 )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_idem
% 0.29/0.46 thf(fact_129_sup__idem,axiom,
% 0.29/0.46 ! [X3: int] :
% 0.29/0.46 ( ( sup_sup_int @ X3 @ X3 )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_idem
% 0.29/0.46 thf(fact_130_sup__idem,axiom,
% 0.29/0.46 ! [X3: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ X3 @ X3 )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % sup_idem
% 0.29/0.46 thf(fact_131_sup_Oleft__idem,axiom,
% 0.29/0.46 ! [A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ A @ ( sup_sup_a_o @ A @ B ) )
% 0.29/0.46 = ( sup_sup_a_o @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_idem
% 0.29/0.46 thf(fact_132_sup_Oleft__idem,axiom,
% 0.29/0.46 ! [A: int,B: int] :
% 0.29/0.46 ( ( sup_sup_int @ A @ ( sup_sup_int @ A @ B ) )
% 0.29/0.46 = ( sup_sup_int @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_idem
% 0.29/0.46 thf(fact_133_sup_Oleft__idem,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
% 0.29/0.46 = ( sup_sup_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_idem
% 0.29/0.46 thf(fact_134_sup__left__idem,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y2 ) )
% 0.29/0.46 = ( sup_sup_a_o @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_idem
% 0.29/0.46 thf(fact_135_sup__left__idem,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y2 ) )
% 0.29/0.46 = ( sup_sup_int @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_idem
% 0.29/0.46 thf(fact_136_sup__left__idem,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_idem
% 0.29/0.46 thf(fact_137_sup_Oright__idem,axiom,
% 0.29/0.46 ! [A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B ) @ B )
% 0.29/0.46 = ( sup_sup_a_o @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.right_idem
% 0.29/0.46 thf(fact_138_sup_Oright__idem,axiom,
% 0.29/0.46 ! [A: int,B: int] :
% 0.29/0.46 ( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ B )
% 0.29/0.46 = ( sup_sup_int @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.right_idem
% 0.29/0.46 thf(fact_139_sup_Oright__idem,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
% 0.29/0.46 = ( sup_sup_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.right_idem
% 0.29/0.46 thf(fact_140_bot__set__def,axiom,
% 0.29/0.46 ( bot_bot_set_a
% 0.29/0.46 = ( collect_a @ bot_bot_a_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_set_def
% 0.29/0.46 thf(fact_141_sup__set__def,axiom,
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 = ( ^ [A4: set_a,B5: set_a] :
% 0.29/0.46 ( collect_a
% 0.29/0.46 @ ( sup_sup_a_o
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ A4 )
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_set_def
% 0.29/0.46 thf(fact_142_fun__diff__def,axiom,
% 0.29/0.46 ( minus_minus_a_o
% 0.29/0.46 = ( ^ [A4: a > $o,B5: a > $o,X2: a] : ( minus_minus_o @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fun_diff_def
% 0.29/0.46 thf(fact_143_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y2 ) )
% 0.29/0.46 = ( sup_sup_a_o @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_144_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y2 ) )
% 0.29/0.46 = ( sup_sup_int @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_145_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_146_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) )
% 0.29/0.46 = ( sup_sup_a_o @ Y2 @ ( sup_sup_a_o @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_147_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [X3: int,Y2: int,Z2: int] :
% 0.29/0.46 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) )
% 0.29/0.46 = ( sup_sup_int @ Y2 @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_148_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_149_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
% 0.29/0.46 = ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_150_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [X3: int,Y2: int,Z2: int] :
% 0.29/0.46 ( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
% 0.29/0.46 = ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_151_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
% 0.29/0.46 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_152_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ( sup_sup_a_o
% 0.29/0.46 = ( ^ [X2: a > $o,Y: a > $o] : ( sup_sup_a_o @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_153_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ( sup_sup_int
% 0.29/0.46 = ( ^ [X2: int,Y: int] : ( sup_sup_int @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_154_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_155_sup__fun__def,axiom,
% 0.29/0.46 ( sup_sup_a_o
% 0.29/0.46 = ( ^ [F: a > $o,G: a > $o,X2: a] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_fun_def
% 0.29/0.46 thf(fact_156_boolean__algebra__cancel_Osup1,axiom,
% 0.29/0.46 ! [A3: a > $o,K: a > $o,A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( sup_sup_a_o @ K @ A ) )
% 0.29/0.46 => ( ( sup_sup_a_o @ A3 @ B )
% 0.29/0.46 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup1
% 0.29/0.46 thf(fact_157_boolean__algebra__cancel_Osup1,axiom,
% 0.29/0.46 ! [A3: int,K: int,A: int,B: int] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( sup_sup_int @ K @ A ) )
% 0.29/0.46 => ( ( sup_sup_int @ A3 @ B )
% 0.29/0.46 = ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup1
% 0.29/0.46 thf(fact_158_boolean__algebra__cancel_Osup1,axiom,
% 0.29/0.46 ! [A3: set_a,K: set_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( sup_sup_set_a @ K @ A ) )
% 0.29/0.46 => ( ( sup_sup_set_a @ A3 @ B )
% 0.29/0.46 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup1
% 0.29/0.46 thf(fact_159_boolean__algebra__cancel_Osup2,axiom,
% 0.29/0.46 ! [B3: a > $o,K: a > $o,B: a > $o,A: a > $o] :
% 0.29/0.46 ( ( B3
% 0.29/0.46 = ( sup_sup_a_o @ K @ B ) )
% 0.29/0.46 => ( ( sup_sup_a_o @ A @ B3 )
% 0.29/0.46 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup2
% 0.29/0.46 thf(fact_160_boolean__algebra__cancel_Osup2,axiom,
% 0.29/0.46 ! [B3: int,K: int,B: int,A: int] :
% 0.29/0.46 ( ( B3
% 0.29/0.46 = ( sup_sup_int @ K @ B ) )
% 0.29/0.46 => ( ( sup_sup_int @ A @ B3 )
% 0.29/0.46 = ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup2
% 0.29/0.46 thf(fact_161_boolean__algebra__cancel_Osup2,axiom,
% 0.29/0.46 ! [B3: set_a,K: set_a,B: set_a,A: set_a] :
% 0.29/0.46 ( ( B3
% 0.29/0.46 = ( sup_sup_set_a @ K @ B ) )
% 0.29/0.46 => ( ( sup_sup_set_a @ A @ B3 )
% 0.29/0.46 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup2
% 0.29/0.46 thf(fact_162_sup_Oassoc,axiom,
% 0.29/0.46 ! [A: a > $o,B: a > $o,C2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B ) @ C2 )
% 0.29/0.46 = ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_163_sup_Oassoc,axiom,
% 0.29/0.46 ! [A: int,B: int,C2: int] :
% 0.29/0.46 ( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ C2 )
% 0.29/0.46 = ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_164_sup_Oassoc,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
% 0.29/0.46 = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_165_sup__assoc,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
% 0.29/0.46 = ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_166_sup__assoc,axiom,
% 0.29/0.46 ! [X3: int,Y2: int,Z2: int] :
% 0.29/0.46 ( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
% 0.29/0.46 = ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_167_sup__assoc,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
% 0.29/0.46 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_168_sup_Ocommute,axiom,
% 0.29/0.46 ( sup_sup_a_o
% 0.29/0.46 = ( ^ [A2: a > $o,B2: a > $o] : ( sup_sup_a_o @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_169_sup_Ocommute,axiom,
% 0.29/0.46 ( sup_sup_int
% 0.29/0.46 = ( ^ [A2: int,B2: int] : ( sup_sup_int @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_170_sup_Ocommute,axiom,
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 = ( ^ [A2: set_a,B2: set_a] : ( sup_sup_set_a @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_171_sup__commute,axiom,
% 0.29/0.46 ( sup_sup_a_o
% 0.29/0.46 = ( ^ [X2: a > $o,Y: a > $o] : ( sup_sup_a_o @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_172_sup__commute,axiom,
% 0.29/0.46 ( sup_sup_int
% 0.29/0.46 = ( ^ [X2: int,Y: int] : ( sup_sup_int @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_173_sup__commute,axiom,
% 0.29/0.46 ( sup_sup_set_a
% 0.29/0.46 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_174_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [B: a > $o,A: a > $o,C2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ B @ ( sup_sup_a_o @ A @ C2 ) )
% 0.29/0.46 = ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_175_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [B: int,A: int,C2: int] :
% 0.29/0.46 ( ( sup_sup_int @ B @ ( sup_sup_int @ A @ C2 ) )
% 0.29/0.46 = ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_176_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [B: set_a,A: set_a,C2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_177_sup__left__commute,axiom,
% 0.29/0.46 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 0.29/0.46 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) )
% 0.29/0.46 = ( sup_sup_a_o @ Y2 @ ( sup_sup_a_o @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_commute
% 0.29/0.46 thf(fact_178_sup__left__commute,axiom,
% 0.29/0.46 ! [X3: int,Y2: int,Z2: int] :
% 0.29/0.46 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) )
% 0.29/0.46 = ( sup_sup_int @ Y2 @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_commute
% 0.29/0.46 thf(fact_179_sup__left__commute,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.46 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
% 0.29/0.46 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_commute
% 0.29/0.46 thf(fact_180_less__supI1,axiom,
% 0.29/0.46 ! [X3: a > $o,A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( ord_less_a_o @ X3 @ A )
% 0.29/0.46 => ( ord_less_a_o @ X3 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI1
% 0.29/0.46 thf(fact_181_less__supI1,axiom,
% 0.29/0.46 ! [X3: set_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ A )
% 0.29/0.46 => ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI1
% 0.29/0.46 thf(fact_182_less__supI1,axiom,
% 0.29/0.46 ! [X3: int,A: int,B: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ A )
% 0.29/0.46 => ( ord_less_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI1
% 0.29/0.46 thf(fact_183_less__supI2,axiom,
% 0.29/0.46 ! [X3: a > $o,B: a > $o,A: a > $o] :
% 0.29/0.46 ( ( ord_less_a_o @ X3 @ B )
% 0.29/0.46 => ( ord_less_a_o @ X3 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI2
% 0.29/0.46 thf(fact_184_less__supI2,axiom,
% 0.29/0.46 ! [X3: set_a,B: set_a,A: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ B )
% 0.29/0.46 => ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI2
% 0.29/0.46 thf(fact_185_less__supI2,axiom,
% 0.29/0.46 ! [X3: int,B: int,A: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ B )
% 0.29/0.46 => ( ord_less_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI2
% 0.29/0.46 thf(fact_186_sup_Ostrict__boundedE,axiom,
% 0.29/0.46 ! [B: a > $o,C2: a > $o,A: a > $o] :
% 0.29/0.46 ( ( ord_less_a_o @ ( sup_sup_a_o @ B @ C2 ) @ A )
% 0.29/0.46 => ~ ( ( ord_less_a_o @ B @ A )
% 0.29/0.46 => ~ ( ord_less_a_o @ C2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_boundedE
% 0.29/0.46 thf(fact_187_sup_Ostrict__boundedE,axiom,
% 0.29/0.46 ! [B: set_a,C2: set_a,A: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
% 0.29/0.46 => ~ ( ( ord_less_set_a @ B @ A )
% 0.29/0.46 => ~ ( ord_less_set_a @ C2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_boundedE
% 0.29/0.46 thf(fact_188_sup_Ostrict__boundedE,axiom,
% 0.29/0.46 ! [B: int,C2: int,A: int] :
% 0.29/0.46 ( ( ord_less_int @ ( sup_sup_int @ B @ C2 ) @ A )
% 0.29/0.46 => ~ ( ( ord_less_int @ B @ A )
% 0.29/0.46 => ~ ( ord_less_int @ C2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_boundedE
% 0.29/0.46 thf(fact_189_sup_Ostrict__order__iff,axiom,
% 0.29/0.46 ( ord_less_a_o
% 0.29/0.46 = ( ^ [B2: a > $o,A2: a > $o] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( sup_sup_a_o @ A2 @ B2 ) )
% 0.29/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_order_iff
% 0.29/0.46 thf(fact_190_sup_Ostrict__order__iff,axiom,
% 0.29/0.46 ( ord_less_set_a
% 0.29/0.46 = ( ^ [B2: set_a,A2: set_a] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( sup_sup_set_a @ A2 @ B2 ) )
% 0.29/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_order_iff
% 0.29/0.46 thf(fact_191_sup_Ostrict__order__iff,axiom,
% 0.29/0.46 ( ord_less_int
% 0.29/0.46 = ( ^ [B2: int,A2: int] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( sup_sup_int @ A2 @ B2 ) )
% 0.29/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_order_iff
% 0.29/0.46 thf(fact_192_sup_Ostrict__coboundedI1,axiom,
% 0.29/0.46 ! [C2: a > $o,A: a > $o,B: a > $o] :
% 0.29/0.46 ( ( ord_less_a_o @ C2 @ A )
% 0.29/0.46 => ( ord_less_a_o @ C2 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI1
% 0.29/0.46 thf(fact_193_sup_Ostrict__coboundedI1,axiom,
% 0.29/0.46 ! [C2: set_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ C2 @ A )
% 0.29/0.46 => ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI1
% 0.29/0.46 thf(fact_194_sup_Ostrict__coboundedI1,axiom,
% 0.29/0.46 ! [C2: int,A: int,B: int] :
% 0.29/0.46 ( ( ord_less_int @ C2 @ A )
% 0.29/0.46 => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI1
% 0.29/0.46 thf(fact_195_sup_Ostrict__coboundedI2,axiom,
% 0.29/0.46 ! [C2: a > $o,B: a > $o,A: a > $o] :
% 0.29/0.46 ( ( ord_less_a_o @ C2 @ B )
% 0.29/0.46 => ( ord_less_a_o @ C2 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI2
% 0.29/0.46 thf(fact_196_sup_Ostrict__coboundedI2,axiom,
% 0.29/0.46 ! [C2: set_a,B: set_a,A: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ C2 @ B )
% 0.29/0.46 => ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI2
% 0.29/0.46 thf(fact_197_sup_Ostrict__coboundedI2,axiom,
% 0.29/0.46 ! [C2: int,B: int,A: int] :
% 0.29/0.46 ( ( ord_less_int @ C2 @ B )
% 0.29/0.46 => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI2
% 0.29/0.46 thf(fact_198_the__elem__eq,axiom,
% 0.29/0.46 ! [X3: a] :
% 0.29/0.46 ( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
% 0.29/0.46 = X3 ) ).
% 0.29/0.46
% 0.29/0.46 % the_elem_eq
% 0.29/0.46 thf(fact_199_bot__apply,axiom,
% 0.29/0.46 ( bot_bot_a_o
% 0.29/0.46 = ( ^ [X2: a] : bot_bot_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_apply
% 0.29/0.46 thf(fact_200_is__singletonI,axiom,
% 0.29/0.46 ! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI
% 0.29/0.46 thf(fact_201_diff__strict__mono,axiom,
% 0.29/0.46 ! [A: int,B: int,D: int,C2: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( ( ord_less_int @ D @ C2 )
% 0.29/0.46 => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_strict_mono
% 0.29/0.46 thf(fact_202_sup1CI,axiom,
% 0.29/0.46 ! [B3: a > $o,X3: a,A3: a > $o] :
% 0.29/0.46 ( ( ~ ( B3 @ X3 )
% 0.29/0.46 => ( A3 @ X3 ) )
% 0.29/0.46 => ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup1CI
% 0.29/0.46 thf(fact_203_sup1E,axiom,
% 0.29/0.46 ! [A3: a > $o,B3: a > $o,X3: a] :
% 0.29/0.46 ( ( sup_sup_a_o @ A3 @ B3 @ X3 )
% 0.29/0.46 => ( ~ ( A3 @ X3 )
% 0.29/0.46 => ( B3 @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup1E
% 0.29/0.46 thf(fact_204_sup1I1,axiom,
% 0.29/0.46 ! [A3: a > $o,X3: a,B3: a > $o] :
% 0.29/0.46 ( ( A3 @ X3 )
% 0.29/0.46 => ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup1I1
% 0.29/0.46 thf(fact_205_sup1I2,axiom,
% 0.29/0.46 ! [B3: a > $o,X3: a,A3: a > $o] :
% 0.29/0.46 ( ( B3 @ X3 )
% 0.29/0.46 => ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup1I2
% 0.29/0.46 thf(fact_206_not__psubset__empty,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % not_psubset_empty
% 0.29/0.46 thf(fact_207_psubset__imp__ex__mem,axiom,
% 0.29/0.46 ! [A3: set_a,B3: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A3 @ B3 )
% 0.29/0.46 => ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_imp_ex_mem
% 0.29/0.46 thf(fact_208_minus__set__def,axiom,
% 0.29/0.46 ( minus_minus_set_a
% 0.29/0.46 = ( ^ [A4: set_a,B5: set_a] :
% 0.29/0.46 ( collect_a
% 0.29/0.46 @ ( minus_minus_a_o
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ A4 )
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_set_def
% 0.29/0.46 thf(fact_209_is__singleton__the__elem,axiom,
% 0.29/0.46 ( is_singleton_a
% 0.29/0.46 = ( ^ [A4: set_a] :
% 0.29/0.46 ( A4
% 0.29/0.46 = ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singleton_the_elem
% 0.29/0.46 thf(fact_210_is__singletonI_H,axiom,
% 0.29/0.46 ! [A3: set_a] :
% 0.29/0.46 ( ( A3 != bot_bot_set_a )
% 0.29/0.46 => ( ! [X4: a,Y4: a] :
% 0.29/0.46 ( ( member_a @ X4 @ A3 )
% 0.29/0.46 => ( ( member_a @ Y4 @ A3 )
% 0.29/0.46 => ( X4 = Y4 ) ) )
% 0.29/0.46 => ( is_singleton_a @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI'
% 0.29/0.46 thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [B: set_a,A: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ B @ A )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_not_eq
% 0.29/0.46 thf(fact_212_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [B: int,A: int] :
% 0.29/0.46 ( ( ord_less_int @ B @ A )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_not_eq
% 0.29/0.46 thf(fact_213_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_not_eq
% 0.29/0.46 thf(fact_214_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [A: int,B: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_not_eq
% 0.29/0.46 thf(fact_215_not__less__iff__gr__or__eq,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 0.29/0.46 = ( ( ord_less_int @ Y2 @ X3 )
% 0.29/0.46 | ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_iff_gr_or_eq
% 0.29/0.46 thf(fact_216_dual__order_Ostrict__trans,axiom,
% 0.29/0.46 ! [B: set_a,A: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ B @ A )
% 0.29/0.46 => ( ( ord_less_set_a @ C2 @ B )
% 0.29/0.46 => ( ord_less_set_a @ C2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans
% 0.29/0.46 thf(fact_217_dual__order_Ostrict__trans,axiom,
% 0.29/0.46 ! [B: int,A: int,C2: int] :
% 0.29/0.46 ( ( ord_less_int @ B @ A )
% 0.29/0.46 => ( ( ord_less_int @ C2 @ B )
% 0.29/0.46 => ( ord_less_int @ C2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans
% 0.29/0.46 thf(fact_218_linorder__less__wlog,axiom,
% 0.29/0.46 ! [P: int > int > $o,A: int,B: int] :
% 0.29/0.46 ( ! [A5: int,B6: int] :
% 0.29/0.46 ( ( ord_less_int @ A5 @ B6 )
% 0.29/0.46 => ( P @ A5 @ B6 ) )
% 0.29/0.46 => ( ! [A5: int] : ( P @ A5 @ A5 )
% 0.29/0.46 => ( ! [A5: int,B6: int] :
% 0.29/0.46 ( ( P @ B6 @ A5 )
% 0.29/0.46 => ( P @ A5 @ B6 ) )
% 0.29/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_less_wlog
% 0.29/0.46 thf(fact_219_less__imp__not__less,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_less
% 0.29/0.46 thf(fact_220_less__imp__not__less,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_less
% 0.29/0.46 thf(fact_221_order_Ostrict__trans,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.46 => ( ord_less_set_a @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans
% 0.29/0.46 thf(fact_222_order_Ostrict__trans,axiom,
% 0.29/0.46 ! [A: int,B: int,C2: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.46 => ( ord_less_int @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans
% 0.29/0.46 thf(fact_223_dual__order_Oirrefl,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ~ ( ord_less_set_a @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.irrefl
% 0.29/0.46 thf(fact_224_dual__order_Oirrefl,axiom,
% 0.29/0.46 ! [A: int] :
% 0.29/0.46 ~ ( ord_less_int @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.irrefl
% 0.29/0.46 thf(fact_225_linorder__cases,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ~ ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( ( X3 != Y2 )
% 0.29/0.46 => ( ord_less_int @ Y2 @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_cases
% 0.29/0.46 thf(fact_226_less__imp__triv,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,P: $o] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_set_a @ Y2 @ X3 )
% 0.29/0.46 => P ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_triv
% 0.29/0.46 thf(fact_227_less__imp__triv,axiom,
% 0.29/0.46 ! [X3: int,Y2: int,P: $o] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_int @ Y2 @ X3 )
% 0.29/0.46 => P ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_triv
% 0.29/0.46 thf(fact_228_less__imp__not__eq2,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( Y2 != X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq2
% 0.29/0.46 thf(fact_229_less__imp__not__eq2,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( Y2 != X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq2
% 0.29/0.46 thf(fact_230_antisym__conv3,axiom,
% 0.29/0.46 ! [Y2: int,X3: int] :
% 0.29/0.46 ( ~ ( ord_less_int @ Y2 @ X3 )
% 0.29/0.46 => ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 0.29/0.46 = ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv3
% 0.29/0.46 thf(fact_231_less__not__sym,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_sym
% 0.29/0.46 thf(fact_232_less__not__sym,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_sym
% 0.29/0.46 thf(fact_233_less__imp__not__eq,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( X3 != Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq
% 0.29/0.46 thf(fact_234_less__imp__not__eq,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( X3 != Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq
% 0.29/0.46 thf(fact_235_dual__order_Oasym,axiom,
% 0.29/0.46 ! [B: set_a,A: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ B @ A )
% 0.29/0.46 => ~ ( ord_less_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.asym
% 0.29/0.46 thf(fact_236_dual__order_Oasym,axiom,
% 0.29/0.46 ! [B: int,A: int] :
% 0.29/0.46 ( ( ord_less_int @ B @ A )
% 0.29/0.46 => ~ ( ord_less_int @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.asym
% 0.29/0.46 thf(fact_237_ord__less__eq__trans,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ( ( B = C2 )
% 0.29/0.46 => ( ord_less_set_a @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_trans
% 0.29/0.46 thf(fact_238_ord__less__eq__trans,axiom,
% 0.29/0.46 ! [A: int,B: int,C2: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( ( B = C2 )
% 0.29/0.46 => ( ord_less_int @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_trans
% 0.29/0.46 thf(fact_239_ord__eq__less__trans,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.46 => ( ord_less_set_a @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_trans
% 0.29/0.46 thf(fact_240_ord__eq__less__trans,axiom,
% 0.29/0.46 ! [A: int,B: int,C2: int] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.46 => ( ord_less_int @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_trans
% 0.29/0.46 thf(fact_241_less__irrefl,axiom,
% 0.29/0.46 ! [X3: set_a] :
% 0.29/0.46 ~ ( ord_less_set_a @ X3 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl
% 0.29/0.46 thf(fact_242_less__irrefl,axiom,
% 0.29/0.46 ! [X3: int] :
% 0.29/0.46 ~ ( ord_less_int @ X3 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl
% 0.29/0.46 thf(fact_243_less__linear,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 | ( X3 = Y2 )
% 0.29/0.46 | ( ord_less_int @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_linear
% 0.29/0.46 thf(fact_244_less__trans,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_set_a @ Y2 @ Z2 )
% 0.29/0.46 => ( ord_less_set_a @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_trans
% 0.29/0.46 thf(fact_245_less__trans,axiom,
% 0.29/0.46 ! [X3: int,Y2: int,Z2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_int @ Y2 @ Z2 )
% 0.29/0.46 => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_trans
% 0.29/0.46 thf(fact_246_less__asym_H,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ~ ( ord_less_set_a @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym'
% 0.29/0.46 thf(fact_247_less__asym_H,axiom,
% 0.29/0.46 ! [A: int,B: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym'
% 0.29/0.46 thf(fact_248_less__asym,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym
% 0.29/0.46 thf(fact_249_less__asym,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym
% 0.29/0.46 thf(fact_250_less__imp__neq,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( X3 != Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_neq
% 0.29/0.46 thf(fact_251_less__imp__neq,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( X3 != Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_neq
% 0.29/0.46 thf(fact_252_order_Oasym,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ~ ( ord_less_set_a @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.asym
% 0.29/0.46 thf(fact_253_order_Oasym,axiom,
% 0.29/0.46 ! [A: int,B: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.asym
% 0.29/0.46 thf(fact_254_neq__iff,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( X3 != Y2 )
% 0.29/0.46 = ( ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 | ( ord_less_int @ Y2 @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq_iff
% 0.29/0.46 thf(fact_255_neqE,axiom,
% 0.29/0.46 ! [X3: int,Y2: int] :
% 0.29/0.46 ( ( X3 != Y2 )
% 0.29/0.46 => ( ~ ( ord_less_int @ X3 @ Y2 )
% 0.29/0.46 => ( ord_less_int @ Y2 @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neqE
% 0.29/0.46 thf(fact_256_gt__ex,axiom,
% 0.29/0.46 ! [X3: int] :
% 0.29/0.46 ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% 0.29/0.46
% 0.29/0.46 % gt_ex
% 0.29/0.46 thf(fact_257_lt__ex,axiom,
% 0.29/0.46 ! [X3: int] :
% 0.29/0.46 ? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % lt_ex
% 0.29/0.46 thf(fact_258_order__less__subst2,axiom,
% 0.29/0.46 ! [A: int,B: int,F2: int > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 0.29/0.46 => ( ! [X4: int,Y4: int] :
% 0.29/0.46 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_259_order__less__subst2,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 0.29/0.46 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_260_order__less__subst2,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 0.29/0.46 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_261_order__less__subst2,axiom,
% 0.29/0.46 ! [A: int,B: int,F2: int > int,C2: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 0.29/0.46 => ( ! [X4: int,Y4: int] :
% 0.29/0.46 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_262_order__less__subst1,axiom,
% 0.29/0.46 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 0.29/0.46 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.46 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_263_order__less__subst1,axiom,
% 0.29/0.46 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ ( F2 @ B ) )
% 0.29/0.46 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.46 => ( ! [X4: int,Y4: int] :
% 0.29/0.46 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_264_order__less__subst1,axiom,
% 0.29/0.46 ! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ ( F2 @ B ) )
% 0.29/0.46 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.46 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_265_order__less__subst1,axiom,
% 0.29/0.46 ! [A: int,F2: int > int,B: int,C2: int] :
% 0.29/0.46 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 0.29/0.46 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.46 => ( ! [X4: int,Y4: int] :
% 0.29/0.46 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_266_ord__less__eq__subst,axiom,
% 0.29/0.46 ! [A: int,B: int,F2: int > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_int @ A @ B )
% 0.29/0.46 => ( ( ( F2 @ B )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X4: int,Y4: int] :
% 0.29/0.46 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_subst
% 0.29/0.46 thf(fact_267_ord__less__eq__subst,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
% 0.29/0.46 ( ( ord_less_set_a @ A @ B )
% 0.29/0.46 => ( ( ( F2 @ B )
% 0.29/0.47 = C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_less_eq_subst
% 0.29/0.47 thf(fact_268_ord__less__eq__subst,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ A @ B )
% 0.29/0.47 => ( ( ( F2 @ B )
% 0.29/0.47 = C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_less_eq_subst
% 0.29/0.47 thf(fact_269_ord__less__eq__subst,axiom,
% 0.29/0.47 ! [A: int,B: int,F2: int > int,C2: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ B )
% 0.29/0.47 => ( ( ( F2 @ B )
% 0.29/0.47 = C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_less_eq_subst
% 0.29/0.47 thf(fact_270_ord__eq__less__subst,axiom,
% 0.29/0.47 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_less_subst
% 0.29/0.47 thf(fact_271_ord__eq__less__subst,axiom,
% 0.29/0.47 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_less_subst
% 0.29/0.47 thf(fact_272_ord__eq__less__subst,axiom,
% 0.29/0.47 ! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_less_subst
% 0.29/0.47 thf(fact_273_ord__eq__less__subst,axiom,
% 0.29/0.47 ! [A: int,F2: int > int,B: int,C2: int] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_less_subst
% 0.29/0.47 thf(fact_274_bot__fun__def,axiom,
% 0.29/0.47 ( bot_bot_a_o
% 0.29/0.47 = ( ^ [X2: a] : bot_bot_o ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_fun_def
% 0.29/0.47 thf(fact_275_diff__right__commute,axiom,
% 0.29/0.47 ! [A: int,C2: int,B: int] :
% 0.29/0.47 ( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
% 0.29/0.47 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % diff_right_commute
% 0.29/0.47 thf(fact_276_diff__eq__diff__eq,axiom,
% 0.29/0.47 ! [A: int,B: int,C2: int,D: int] :
% 0.29/0.47 ( ( ( minus_minus_int @ A @ B )
% 0.29/0.47 = ( minus_minus_int @ C2 @ D ) )
% 0.29/0.47 => ( ( A = B )
% 0.29/0.47 = ( C2 = D ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % diff_eq_diff_eq
% 0.29/0.47 thf(fact_277_is__singleton__def,axiom,
% 0.29/0.47 ( is_singleton_a
% 0.29/0.47 = ( ^ [A4: set_a] :
% 0.29/0.47 ? [X2: a] :
% 0.29/0.47 ( A4
% 0.29/0.47 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_singleton_def
% 0.29/0.47 thf(fact_278_is__singletonE,axiom,
% 0.29/0.47 ! [A3: set_a] :
% 0.29/0.47 ( ( is_singleton_a @ A3 )
% 0.29/0.47 => ~ ! [X4: a] :
% 0.29/0.47 ( A3
% 0.29/0.47 != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_singletonE
% 0.29/0.47 thf(fact_279_bot_Onot__eq__extremum,axiom,
% 0.29/0.47 ! [A: a > $o] :
% 0.29/0.47 ( ( A != bot_bot_a_o )
% 0.29/0.47 = ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot.not_eq_extremum
% 0.29/0.47 thf(fact_280_bot_Onot__eq__extremum,axiom,
% 0.29/0.47 ! [A: set_a] :
% 0.29/0.47 ( ( A != bot_bot_set_a )
% 0.29/0.47 = ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot.not_eq_extremum
% 0.29/0.47 thf(fact_281_bot_Oextremum__strict,axiom,
% 0.29/0.47 ! [A: a > $o] :
% 0.29/0.47 ~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% 0.29/0.47
% 0.29/0.47 % bot.extremum_strict
% 0.29/0.47 thf(fact_282_bot_Oextremum__strict,axiom,
% 0.29/0.47 ! [A: set_a] :
% 0.29/0.47 ~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% 0.29/0.47
% 0.29/0.47 % bot.extremum_strict
% 0.29/0.47 thf(fact_283_diff__strict__right__mono,axiom,
% 0.29/0.47 ! [A: int,B: int,C2: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ B )
% 0.29/0.47 => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % diff_strict_right_mono
% 0.29/0.47 thf(fact_284_diff__strict__left__mono,axiom,
% 0.29/0.47 ! [B: int,A: int,C2: int] :
% 0.29/0.47 ( ( ord_less_int @ B @ A )
% 0.29/0.47 => ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % diff_strict_left_mono
% 0.29/0.47 thf(fact_285_diff__eq__diff__less,axiom,
% 0.29/0.47 ! [A: int,B: int,C2: int,D: int] :
% 0.29/0.47 ( ( ( minus_minus_int @ A @ B )
% 0.29/0.47 = ( minus_minus_int @ C2 @ D ) )
% 0.29/0.47 => ( ( ord_less_int @ A @ B )
% 0.29/0.47 = ( ord_less_int @ C2 @ D ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % diff_eq_diff_less
% 0.29/0.47 thf(fact_286_sup__Un__eq,axiom,
% 0.29/0.47 ! [R: set_a,S: set_a] :
% 0.29/0.47 ( ( sup_sup_a_o
% 0.29/0.47 @ ^ [X2: a] : ( member_a @ X2 @ R )
% 0.29/0.47 @ ^ [X2: a] : ( member_a @ X2 @ S ) )
% 0.29/0.47 = ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sup_Un_eq
% 0.29/0.47 thf(fact_287_bot__empty__eq,axiom,
% 0.29/0.47 ( bot_bot_a_o
% 0.29/0.47 = ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_empty_eq
% 0.29/0.47 thf(fact_288_Collect__empty__eq__bot,axiom,
% 0.29/0.47 ! [P: a > $o] :
% 0.29/0.47 ( ( ( collect_a @ P )
% 0.29/0.47 = bot_bot_set_a )
% 0.29/0.47 = ( P = bot_bot_a_o ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_empty_eq_bot
% 0.29/0.47 thf(fact_289_psubset__trans,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a,C: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.29/0.47 => ( ( ord_less_set_a @ B3 @ C )
% 0.29/0.47 => ( ord_less_set_a @ A3 @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psubset_trans
% 0.29/0.47 thf(fact_290_less__set__def,axiom,
% 0.29/0.47 ( ord_less_set_a
% 0.29/0.47 = ( ^ [A4: set_a,B5: set_a] :
% 0.29/0.47 ( ord_less_a_o
% 0.29/0.47 @ ^ [X2: a] : ( member_a @ X2 @ A4 )
% 0.29/0.47 @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_set_def
% 0.29/0.47 thf(fact_291_psubsetD,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a,C2: a] :
% 0.29/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.29/0.47 => ( ( member_a @ C2 @ A3 )
% 0.29/0.47 => ( member_a @ C2 @ B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psubsetD
% 0.29/0.47 thf(fact_292_remove__def,axiom,
% 0.29/0.47 ( remove_a
% 0.29/0.47 = ( ^ [X2: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % remove_def
% 0.29/0.47 thf(fact_293_Tree_Osimps_I15_J,axiom,
% 0.29/0.47 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 0.29/0.47 ( ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) )
% 0.29/0.47 = ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary256242811Tree_a @ X21 ) @ ( insert_a @ X22 @ bot_bot_set_a ) ) @ ( binary256242811Tree_a @ X23 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Tree.simps(15)
% 0.29/0.47 thf(fact_294_member__remove,axiom,
% 0.29/0.47 ! [X3: a,Y2: a,A3: set_a] :
% 0.29/0.47 ( ( member_a @ X3 @ ( remove_a @ Y2 @ A3 ) )
% 0.29/0.47 = ( ( member_a @ X3 @ A3 )
% 0.29/0.47 & ( X3 != Y2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % member_remove
% 0.29/0.47 thf(fact_295_Tree_Oset__intros_I3_J,axiom,
% 0.29/0.47 ! [Ya: a,X23: binary1439146945Tree_a,X21: binary1439146945Tree_a,X22: a] :
% 0.29/0.47 ( ( member_a @ Ya @ ( binary256242811Tree_a @ X23 ) )
% 0.29/0.47 => ( member_a @ Ya @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Tree.set_intros(3)
% 0.29/0.47 thf(fact_296_Tree_Oset__intros_I2_J,axiom,
% 0.29/0.47 ! [X22: a,X21: binary1439146945Tree_a,X23: binary1439146945Tree_a] : ( member_a @ X22 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Tree.set_intros(2)
% 0.29/0.47 thf(fact_297_Tree_Oset__intros_I1_J,axiom,
% 0.29/0.47 ! [Y2: a,X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 0.29/0.47 ( ( member_a @ Y2 @ ( binary256242811Tree_a @ X21 ) )
% 0.29/0.47 => ( member_a @ Y2 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Tree.set_intros(1)
% 0.29/0.47 thf(fact_298_Tree_Oset__cases,axiom,
% 0.29/0.47 ! [E: a,A: binary1439146945Tree_a] :
% 0.29/0.47 ( ( member_a @ E @ ( binary256242811Tree_a @ A ) )
% 0.29/0.47 => ( ! [Z1: binary1439146945Tree_a] :
% 0.29/0.47 ( ? [Z22: a,Z3: binary1439146945Tree_a] :
% 0.29/0.47 ( A
% 0.29/0.47 = ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
% 0.29/0.47 => ~ ( member_a @ E @ ( binary256242811Tree_a @ Z1 ) ) )
% 0.29/0.47 => ( ! [Z1: binary1439146945Tree_a,Z3: binary1439146945Tree_a] :
% 0.29/0.47 ( A
% 0.29/0.47 != ( binary717961607le_T_a @ Z1 @ E @ Z3 ) )
% 0.29/0.47 => ~ ! [Z1: binary1439146945Tree_a,Z22: a,Z3: binary1439146945Tree_a] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
% 0.29/0.47 => ~ ( member_a @ E @ ( binary256242811Tree_a @ Z3 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Tree.set_cases
% 0.29/0.47 thf(fact_299_Tree_Osimps_I14_J,axiom,
% 0.29/0.47 ( ( binary256242811Tree_a @ binary476621312_Tip_a )
% 0.29/0.47 = bot_bot_set_a ) ).
% 0.29/0.47
% 0.29/0.47 % Tree.simps(14)
% 0.29/0.47 thf(fact_300_Set_Ois__empty__def,axiom,
% 0.29/0.47 ( is_empty_a
% 0.29/0.47 = ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Set.is_empty_def
% 0.29/0.47 thf(fact_301_the__elem__def,axiom,
% 0.29/0.47 ( the_elem_a
% 0.29/0.47 = ( ^ [X5: set_a] :
% 0.29/0.47 ( the_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( X5
% 0.29/0.47 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % the_elem_def
% 0.29/0.47 thf(fact_302_the__sym__eq__trivial,axiom,
% 0.29/0.47 ! [X3: a] :
% 0.29/0.47 ( ( the_a
% 0.29/0.47 @ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
% 0.29/0.47 @ X3 ) )
% 0.29/0.47 = X3 ) ).
% 0.29/0.47
% 0.29/0.47 % the_sym_eq_trivial
% 0.29/0.47 thf(fact_303_the__eq__trivial,axiom,
% 0.29/0.47 ! [A: a] :
% 0.29/0.47 ( ( the_a
% 0.29/0.47 @ ^ [X2: a] : ( X2 = A ) )
% 0.29/0.47 = A ) ).
% 0.29/0.47
% 0.29/0.47 % the_eq_trivial
% 0.29/0.47 thf(fact_304_the__equality,axiom,
% 0.29/0.47 ! [P: a > $o,A: a] :
% 0.29/0.47 ( ( P @ A )
% 0.29/0.47 => ( ! [X4: a] :
% 0.29/0.47 ( ( P @ X4 )
% 0.29/0.47 => ( X4 = A ) )
% 0.29/0.47 => ( ( the_a @ P )
% 0.29/0.47 = A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % the_equality
% 0.29/0.47 thf(fact_305_theI,axiom,
% 0.29/0.47 ! [P: a > $o,A: a] :
% 0.29/0.47 ( ( P @ A )
% 0.29/0.47 => ( ! [X4: a] :
% 0.29/0.47 ( ( P @ X4 )
% 0.29/0.47 => ( X4 = A ) )
% 0.29/0.47 => ( P @ ( the_a @ P ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % theI
% 0.29/0.47 thf(fact_306_theI_H,axiom,
% 0.29/0.47 ! [P: a > $o] :
% 0.29/0.47 ( ? [X: a] :
% 0.29/0.47 ( ( P @ X )
% 0.29/0.47 & ! [Y4: a] :
% 0.29/0.47 ( ( P @ Y4 )
% 0.29/0.47 => ( Y4 = X ) ) )
% 0.29/0.47 => ( P @ ( the_a @ P ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % theI'
% 0.29/0.47 thf(fact_307_theI2,axiom,
% 0.29/0.47 ! [P: a > $o,A: a,Q: a > $o] :
% 0.29/0.47 ( ( P @ A )
% 0.29/0.47 => ( ! [X4: a] :
% 0.29/0.47 ( ( P @ X4 )
% 0.29/0.47 => ( X4 = A ) )
% 0.29/0.47 => ( ! [X4: a] :
% 0.29/0.47 ( ( P @ X4 )
% 0.29/0.47 => ( Q @ X4 ) )
% 0.29/0.47 => ( Q @ ( the_a @ P ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % theI2
% 0.29/0.47 thf(fact_308_If__def,axiom,
% 0.29/0.47 ( if_a
% 0.29/0.47 = ( ^ [P2: $o,X2: a,Y: a] :
% 0.29/0.47 ( the_a
% 0.29/0.47 @ ^ [Z4: a] :
% 0.29/0.47 ( ( P2
% 0.29/0.47 => ( Z4 = X2 ) )
% 0.29/0.47 & ( ~ P2
% 0.29/0.47 => ( Z4 = Y ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % If_def
% 0.29/0.47 thf(fact_309_the1I2,axiom,
% 0.29/0.47 ! [P: a > $o,Q: a > $o] :
% 0.29/0.47 ( ? [X: a] :
% 0.29/0.47 ( ( P @ X )
% 0.29/0.47 & ! [Y4: a] :
% 0.29/0.47 ( ( P @ Y4 )
% 0.29/0.47 => ( Y4 = X ) ) )
% 0.29/0.47 => ( ! [X4: a] :
% 0.29/0.47 ( ( P @ X4 )
% 0.29/0.47 => ( Q @ X4 ) )
% 0.29/0.47 => ( Q @ ( the_a @ P ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % the1I2
% 0.29/0.47 thf(fact_310_the1__equality,axiom,
% 0.29/0.47 ! [P: a > $o,A: a] :
% 0.29/0.47 ( ? [X: a] :
% 0.29/0.47 ( ( P @ X )
% 0.29/0.47 & ! [Y4: a] :
% 0.29/0.47 ( ( P @ Y4 )
% 0.29/0.47 => ( Y4 = X ) ) )
% 0.29/0.47 => ( ( P @ A )
% 0.29/0.47 => ( ( the_a @ P )
% 0.29/0.47 = A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % the1_equality
% 0.29/0.47 thf(fact_311_psubset__insert__iff,axiom,
% 0.29/0.47 ! [A3: set_a,X3: a,B3: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ A3 @ ( insert_a @ X3 @ B3 ) )
% 0.29/0.47 = ( ( ( member_a @ X3 @ B3 )
% 0.29/0.47 => ( ord_less_set_a @ A3 @ B3 ) )
% 0.29/0.47 & ( ~ ( member_a @ X3 @ B3 )
% 0.29/0.47 => ( ( ( member_a @ X3 @ A3 )
% 0.29/0.47 => ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B3 ) )
% 0.29/0.47 & ( ~ ( member_a @ X3 @ A3 )
% 0.29/0.47 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psubset_insert_iff
% 0.29/0.47 thf(fact_312_pairwise__alt,axiom,
% 0.29/0.47 ( pairwise_a
% 0.29/0.47 = ( ^ [R2: a > a > $o,S2: set_a] :
% 0.29/0.47 ! [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ S2 )
% 0.29/0.47 => ! [Y: a] :
% 0.29/0.47 ( ( member_a @ Y @ ( minus_minus_set_a @ S2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
% 0.29/0.47 => ( R2 @ X2 @ Y ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pairwise_alt
% 0.29/0.47 thf(fact_313_order__refl,axiom,
% 0.29/0.47 ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% 0.29/0.47
% 0.29/0.47 % order_refl
% 0.29/0.47 thf(fact_314_order__refl,axiom,
% 0.29/0.47 ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% 0.29/0.47
% 0.29/0.47 % order_refl
% 0.29/0.47 thf(fact_315_subset__antisym,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ B3 @ A3 )
% 0.29/0.47 => ( A3 = B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_antisym
% 0.29/0.47 thf(fact_316_subsetI,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a] :
% 0.29/0.47 ( ! [X4: a] :
% 0.29/0.47 ( ( member_a @ X4 @ A3 )
% 0.29/0.47 => ( member_a @ X4 @ B3 ) )
% 0.29/0.47 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetI
% 0.29/0.47 thf(fact_317_sup_Obounded__iff,axiom,
% 0.29/0.47 ! [B: a > $o,C2: a > $o,A: a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B @ C2 ) @ A )
% 0.29/0.47 = ( ( ord_less_eq_a_o @ B @ A )
% 0.29/0.47 & ( ord_less_eq_a_o @ C2 @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sup.bounded_iff
% 0.29/0.47 thf(fact_318_sup_Obounded__iff,axiom,
% 0.29/0.47 ! [B: int,C2: int,A: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
% 0.29/0.47 = ( ( ord_less_eq_int @ B @ A )
% 0.29/0.47 & ( ord_less_eq_int @ C2 @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sup.bounded_iff
% 0.29/0.47 thf(fact_319_sup_Obounded__iff,axiom,
% 0.29/0.47 ! [B: set_a,C2: set_a,A: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.47 & ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sup.bounded_iff
% 0.29/0.47 thf(fact_320_le__sup__iff,axiom,
% 0.29/0.47 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
% 0.29/0.47 = ( ( ord_less_eq_a_o @ X3 @ Z2 )
% 0.29/0.47 & ( ord_less_eq_a_o @ Y2 @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_sup_iff
% 0.29/0.47 thf(fact_321_le__sup__iff,axiom,
% 0.29/0.47 ! [X3: int,Y2: int,Z2: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
% 0.29/0.47 = ( ( ord_less_eq_int @ X3 @ Z2 )
% 0.29/0.47 & ( ord_less_eq_int @ Y2 @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_sup_iff
% 0.29/0.47 thf(fact_322_le__sup__iff,axiom,
% 0.29/0.47 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ X3 @ Z2 )
% 0.29/0.47 & ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_sup_iff
% 0.29/0.47 thf(fact_323_empty__subsetI,axiom,
% 0.29/0.47 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_324_subset__empty,axiom,
% 0.29/0.47 ! [A3: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 0.29/0.47 = ( A3 = bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_empty
% 0.29/0.47 thf(fact_325_insert__subset,axiom,
% 0.29/0.47 ! [X3: a,A3: set_a,B3: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 0.29/0.47 = ( ( member_a @ X3 @ B3 )
% 0.29/0.47 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % insert_subset
% 0.29/0.47 thf(fact_326_Un__subset__iff,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a,C: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ A3 @ C )
% 0.29/0.47 & ( ord_less_eq_set_a @ B3 @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_subset_iff
% 0.29/0.47 thf(fact_327_psubsetI,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.29/0.47 => ( ( A3 != B3 )
% 0.29/0.47 => ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psubsetI
% 0.29/0.47 thf(fact_328_singleton__insert__inj__eq_H,axiom,
% 0.29/0.47 ! [A: a,A3: set_a,B: a] :
% 0.29/0.47 ( ( ( insert_a @ A @ A3 )
% 0.29/0.47 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.29/0.47 = ( ( A = B )
% 0.29/0.47 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % singleton_insert_inj_eq'
% 0.29/0.47 thf(fact_329_singleton__insert__inj__eq,axiom,
% 0.29/0.47 ! [B: a,A: a,A3: set_a] :
% 0.29/0.47 ( ( ( insert_a @ B @ bot_bot_set_a )
% 0.29/0.47 = ( insert_a @ A @ A3 ) )
% 0.29/0.47 = ( ( A = B )
% 0.29/0.47 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % singleton_insert_inj_eq
% 0.29/0.47 thf(fact_330_Diff__eq__empty__iff,axiom,
% 0.29/0.47 ! [A3: set_a,B3: set_a] :
% 0.29/0.47 ( ( ( minus_minus_set_a @ A3 @ B3 )
% 0.29/0.47 = bot_bot_set_a )
% 0.29/0.47 = ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Diff_eq_empty_iff
% 0.29/0.47 thf(fact_331_leD,axiom,
% 0.29/0.47 ! [Y2: set_a,X3: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ Y2 @ X3 )
% 0.29/0.47 => ~ ( ord_less_set_a @ X3 @ Y2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % leD
% 0.29/0.47 thf(fact_332_leD,axiom,
% 0.29/0.47 ! [Y2: int,X3: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ Y2 @ X3 )
% 0.29/0.47 => ~ ( ord_less_int @ X3 @ Y2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % leD
% 0.29/0.47 thf(fact_333_leI,axiom,
% 0.29/0.47 ! [X3: int,Y2: int] :
% 0.29/0.47 ( ~ ( ord_less_int @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % leI
% 0.29/0.47 thf(fact_334_le__less,axiom,
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 = ( ^ [X2: set_a,Y: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X2 @ Y )
% 0.29/0.47 | ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_less
% 0.29/0.47 thf(fact_335_le__less,axiom,
% 0.29/0.47 ( ord_less_eq_int
% 0.29/0.47 = ( ^ [X2: int,Y: int] :
% 0.29/0.47 ( ( ord_less_int @ X2 @ Y )
% 0.29/0.47 | ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_less
% 0.29/0.47 thf(fact_336_less__le,axiom,
% 0.29/0.47 ( ord_less_set_a
% 0.29/0.47 = ( ^ [X2: set_a,Y: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.29/0.47 & ( X2 != Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_le
% 0.29/0.47 thf(fact_337_less__le,axiom,
% 0.29/0.47 ( ord_less_int
% 0.29/0.47 = ( ^ [X2: int,Y: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ X2 @ Y )
% 0.29/0.47 & ( X2 != Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_le
% 0.29/0.47 thf(fact_338_order__le__less__subst1,axiom,
% 0.29/0.47 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst1
% 0.29/0.47 thf(fact_339_order__le__less__subst1,axiom,
% 0.29/0.47 ! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst1
% 0.29/0.47 thf(fact_340_order__le__less__subst1,axiom,
% 0.29/0.47 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_set_a @ B @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst1
% 0.29/0.47 thf(fact_341_order__le__less__subst1,axiom,
% 0.29/0.47 ! [A: int,F2: int > int,B: int,C2: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_int @ B @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst1
% 0.29/0.47 thf(fact_342_order__le__less__subst2,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.47 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst2
% 0.29/0.47 thf(fact_343_order__le__less__subst2,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.47 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst2
% 0.29/0.47 thf(fact_344_order__le__less__subst2,axiom,
% 0.29/0.47 ! [A: int,B: int,F2: int > set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_int @ A @ B )
% 0.29/0.47 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst2
% 0.29/0.47 thf(fact_345_order__le__less__subst2,axiom,
% 0.29/0.47 ! [A: int,B: int,F2: int > int,C2: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A @ B )
% 0.29/0.47 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst2
% 0.29/0.47 thf(fact_346_order__less__le__subst1,axiom,
% 0.29/0.47 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.29/0.47 => ( ! [X4: set_a,Y4: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_le_subst1
% 0.29/0.47 thf(fact_347_order__less__le__subst1,axiom,
% 0.29/0.47 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 0.29/0.47 ( ( ord_less_set_a @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_eq_int @ B @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_le_subst1
% 0.29/0.47 thf(fact_348_order__less__le__subst1,axiom,
% 0.29/0.47 ! [A: int,F2: int > int,B: int,C2: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 0.29/0.47 => ( ( ord_less_eq_int @ B @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_le_subst1
% 0.29/0.47 thf(fact_349_order__less__le__subst2,axiom,
% 0.29/0.47 ! [A: int,B: int,F2: int > int,C2: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ B )
% 0.29/0.47 => ( ( ord_less_eq_int @ ( F2 @ B ) @ C2 )
% 0.29/0.47 => ( ! [X4: int,Y4: int] :
% 0.29/0.47 ( ( ord_less_int @ X4 @ Y4 )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_le_subst2
% 0.29/0.47 thf(fact_350_not__le,axiom,
% 0.29/0.47 ! [X3: int,Y2: int] :
% 0.29/0.47 ( ( ~ ( ord_less_eq_int @ X3 @ Y2 ) )
% 0.29/0.47 = ( ord_less_int @ Y2 @ X3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_le
% 0.29/0.47 thf(fact_351_not__less,axiom,
% 0.29/0.47 ! [X3: int,Y2: int] :
% 0.29/0.47 ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 0.29/0.47 = ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less
% 0.29/0.47 thf(fact_352_le__neq__trans,axiom,
% 0.29/0.47 ! [A: int,B: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A @ B )
% 0.29/0.47 => ( ( A != B )
% 0.29/0.47 => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_neq_trans
% 0.29/0.47 thf(fact_353_antisym__conv1,axiom,
% 0.29/0.47 ! [X3: int,Y2: int] :
% 0.29/0.47 ( ~ ( ord_less_int @ X3 @ Y2 )
% 0.29/0.47 => ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.29/0.47 = ( X3 = Y2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv1
% 0.29/0.47 thf(fact_354_antisym__conv2,axiom,
% 0.29/0.47 ! [X3: int,Y2: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.29/0.47 => ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 0.29/0.47 = ( X3 = Y2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv2
% 0.29/0.47
% 0.29/0.47 % Helper facts (3)
% 0.29/0.47 thf(help_If_3_1_If_001tf__a_T,axiom,
% 0.29/0.47 ! [P: $o] :
% 0.29/0.47 ( ( P = $true )
% 0.29/0.47 | ( P = $false ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_2_1_If_001tf__a_T,axiom,
% 0.29/0.47 ! [X3: a,Y2: a] :
% 0.29/0.47 ( ( if_a @ $false @ X3 @ Y2 )
% 0.29/0.47 = Y2 ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_1_1_If_001tf__a_T,axiom,
% 0.29/0.47 ! [X3: a,Y2: a] :
% 0.29/0.47 ( ( if_a @ $true @ X3 @ Y2 )
% 0.29/0.47 = X3 ) ).
% 0.29/0.47
% 0.29/0.47 % Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 ! [X4: a] :
% 0.29/0.47 ( ( member_a @ X4 @ ( binary504661350_eqs_a @ h @ e ) )
% 0.29/0.47 => ( ( h @ X4 )
% 0.29/0.47 != ( h @ x ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.43/1.48 ---- Embedded in THF ---
% 2.43/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.43/1.48 %%% Generated on Tue Feb 24 08:24:36 EST 2026
% 2.43/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.43/1.48 %%% Logic specification used:
% 2.43/1.48 %%% thf(spec, logic, $$dhol).
% 2.43/1.48
% 2.43/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2DTF4570.p
% See solution above
% 2.43/1.48 ------------------------
% 2.67/1.51 ---- Cleaned THF ---
% 2.67/1.51 thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
% 2.67/1.51 binary1439146945Tree_a: $tType ).
% 2.67/1.51
% 2.67/1.51 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.67/1.51 set_a: $tType ).
% 2.67/1.51
% 2.67/1.51 thf(ty_n_t__Int__Oint,type,
% 2.67/1.51 int: $tType ).
% 2.67/1.51
% 2.67/1.51 thf(ty_n_tf__a,type,
% 2.67/1.51 a: $tType ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001tf__a,type,
% 2.67/1.51 binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001tf__a,type,
% 2.67/1.51 binary476621312_Tip_a: binary1439146945Tree_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Oset__Tree_001tf__a,type,
% 2.67/1.51 binary256242811Tree_a: binary1439146945Tree_a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Obinsert_001tf__a,type,
% 2.67/1.51 binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oeqs_001tf__a,type,
% 2.67/1.51 binary504661350_eqs_a: ( a > int ) > a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Omemb_001tf__a,type,
% 2.67/1.51 binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001tf__a,type,
% 2.67/1.51 binary945792244etOf_a: binary1439146945Tree_a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001tf__a,type,
% 2.67/1.51 binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001tf__a,type,
% 2.67/1.51 binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
% 2.67/1.51 minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
% 2.67/1.51 minus_minus_o: $o > $o > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 2.67/1.51 minus_minus_int: int > int > int ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 2.67/1.51 minus_minus_set_a: set_a > set_a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_HOL_OThe_001tf__a,type,
% 2.67/1.51 the_a: ( a > $o ) > a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_If_001tf__a,type,
% 2.67/1.51 if_a: $o > a > a > a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
% 2.67/1.51 sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
% 2.67/1.51 sup_sup_o: $o > $o > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 2.67/1.51 sup_sup_int: int > int > int ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
% 2.67/1.51 sup_sup_set_a: set_a > set_a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 2.67/1.51 bot_bot_a_o: a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 2.67/1.51 bot_bot_o: $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 2.67/1.51 bot_bot_set_a: set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
% 2.67/1.51 ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 2.67/1.51 ord_less_int: int > int > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 2.67/1.51 ord_less_set_a: set_a > set_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
% 2.67/1.51 ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.67/1.51 ord_less_eq_int: int > int > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.67/1.51 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.67/1.51 collect_a: ( a > $o ) > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_Oinsert_001tf__a,type,
% 2.67/1.51 insert_a: a > set_a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 2.67/1.51 is_empty_a: set_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_Ois__singleton_001tf__a,type,
% 2.67/1.51 is_singleton_a: set_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_Opairwise_001tf__a,type,
% 2.67/1.51 pairwise_a: ( a > a > $o ) > set_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_Oremove_001tf__a,type,
% 2.67/1.51 remove_a: a > set_a > set_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 2.67/1.51 the_elem_a: set_a > a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_c_member_001tf__a,type,
% 2.67/1.51 member_a: a > set_a > $o ).
% 2.67/1.51
% 2.67/1.51 thf(sy_v_e,type,
% 2.67/1.51 e: a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_v_h,type,
% 2.67/1.51 h: a > int ).
% 2.67/1.51
% 2.67/1.51 thf(sy_v_t1____,type,
% 2.67/1.51 t1: binary1439146945Tree_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_v_t2____,type,
% 2.67/1.51 t2: binary1439146945Tree_a ).
% 2.67/1.51
% 2.67/1.51 thf(sy_v_x____,type,
% 2.67/1.51 x: a ).
% 2.67/1.51
% 2.67/1.51 thf(fact_0_eqsLessX,axiom,
% 2.67/1.51 ! [X: a] :
% 2.67/1.51 ( ( member_a @ X @ ( binary504661350_eqs_a @ h @ e ) )
% 2.67/1.51 => ( ord_less_int @ ( h @ X ) @ ( h @ x ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_1_eLess,axiom,
% 2.67/1.51 ord_less_int @ ( h @ e ) @ ( h @ x ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_2_res,axiom,
% 2.67/1.51 ( ( binary1226383794sert_a @ h @ e @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
% 2.67/1.51 = ( binary717961607le_T_a @ ( binary1226383794sert_a @ h @ e @ t1 ) @ x @ t2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_3_s1,axiom,
% 2.67/1.51 binary1721989714Tree_a @ h @ t1 ).
% 2.67/1.51
% 2.67/1.51 thf(fact_4_s2,axiom,
% 2.67/1.51 binary1721989714Tree_a @ h @ t2 ).
% 2.67/1.51
% 2.67/1.51 thf(fact_5_s,axiom,
% 2.67/1.51 binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_6_sorted__distinct,axiom,
% 2.67/1.51 ! [H: a > int,A: a,B: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A @ B @ T ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_7_c1,axiom,
% 2.67/1.51 ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t1 ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t1 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_8_c2,axiom,
% 2.67/1.51 ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t2 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_9_eqs__def,axiom,
% 2.67/1.51 ( binary504661350_eqs_a
% 2.67/1.51 = ( ^ [H2: a > int,X2: a] :
% 2.67/1.51 ( collect_a
% 2.67/1.51 @ ^ [Y: a] :
% 2.67/1.51 ( ( H2 @ Y )
% 2.67/1.51 = ( H2 @ X2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_10_h1,axiom,
% 2.67/1.51 ( ( binary1721989714Tree_a @ h @ t1 )
% 2.67/1.51 => ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t1 ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t1 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_11_h2,axiom,
% 2.67/1.51 ( ( binary1721989714Tree_a @ h @ t2 )
% 2.67/1.51 => ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t2 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_12__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062,axiom,
% 2.67/1.51 ( ( binary1721989714Tree_a @ h @ binary476621312_Tip_a )
% 2.67/1.51 => ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ binary476621312_Tip_a ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ binary476621312_Tip_a ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_13_Tree_Oinject,axiom,
% 2.67/1.51 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a,Y21: binary1439146945Tree_a,Y22: a,Y23: binary1439146945Tree_a] :
% 2.67/1.51 ( ( ( binary717961607le_T_a @ X21 @ X22 @ X23 )
% 2.67/1.51 = ( binary717961607le_T_a @ Y21 @ Y22 @ Y23 ) )
% 2.67/1.51 = ( ( X21 = Y21 )
% 2.67/1.51 & ( X22 = Y22 )
% 2.67/1.51 & ( X23 = Y23 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_14_setOf_Osimps_I2_J,axiom,
% 2.67/1.51 ! [T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 2.67/1.51 ( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_15_setOf_Osimps_I1_J,axiom,
% 2.67/1.51 ( ( binary945792244etOf_a @ binary476621312_Tip_a )
% 2.67/1.51 = bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_16_Tree_Odistinct_I1_J,axiom,
% 2.67/1.51 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 2.67/1.51 ( binary476621312_Tip_a
% 2.67/1.51 != ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_17_binsert_Osimps_I2_J,axiom,
% 2.67/1.51 ! [H: a > int,E: a,X3: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
% 2.67/1.51 ( ( ( ord_less_int @ ( H @ E ) @ ( H @ X3 ) )
% 2.67/1.51 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 = ( binary717961607le_T_a @ ( binary1226383794sert_a @ H @ E @ T1 ) @ X3 @ T2 ) ) )
% 2.67/1.51 & ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X3 ) )
% 2.67/1.51 => ( ( ( ord_less_int @ ( H @ X3 ) @ ( H @ E ) )
% 2.67/1.51 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 = ( binary717961607le_T_a @ T1 @ X3 @ ( binary1226383794sert_a @ H @ E @ T2 ) ) ) )
% 2.67/1.51 & ( ~ ( ord_less_int @ ( H @ X3 ) @ ( H @ E ) )
% 2.67/1.51 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 = ( binary717961607le_T_a @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_18_binsert_Osimps_I1_J,axiom,
% 2.67/1.51 ! [H: a > int,E: a] :
% 2.67/1.51 ( ( binary1226383794sert_a @ H @ E @ binary476621312_Tip_a )
% 2.67/1.51 = ( binary717961607le_T_a @ binary476621312_Tip_a @ E @ binary476621312_Tip_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_19_sortedTree_Osimps_I2_J,axiom,
% 2.67/1.51 ! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 2.67/1.51 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 = ( ( binary1721989714Tree_a @ H @ T1 )
% 2.67/1.51 & ! [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ ( binary945792244etOf_a @ T1 ) )
% 2.67/1.51 => ( ord_less_int @ ( H @ X2 ) @ ( H @ X3 ) ) )
% 2.67/1.51 & ! [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ ( binary945792244etOf_a @ T2 ) )
% 2.67/1.51 => ( ord_less_int @ ( H @ X3 ) @ ( H @ X2 ) ) )
% 2.67/1.51 & ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_20_sortedTree_Osimps_I1_J,axiom,
% 2.67/1.51 ! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_21_sortLemmaL,axiom,
% 2.67/1.51 ! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 2.67/1.51 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 => ( binary1721989714Tree_a @ H @ T1 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_22_sortLemmaR,axiom,
% 2.67/1.51 ! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
% 2.67/1.51 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
% 2.67/1.51 => ( binary1721989714Tree_a @ H @ T2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_23_Tree_Oinduct,axiom,
% 2.67/1.51 ! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
% 2.67/1.51 ( ( P @ binary476621312_Tip_a )
% 2.67/1.51 => ( ! [X1: binary1439146945Tree_a,X24: a,X32: binary1439146945Tree_a] :
% 2.67/1.51 ( ( P @ X1 )
% 2.67/1.51 => ( ( P @ X32 )
% 2.67/1.51 => ( P @ ( binary717961607le_T_a @ X1 @ X24 @ X32 ) ) ) )
% 2.67/1.51 => ( P @ Tree ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_24_Tree_Oexhaust,axiom,
% 2.67/1.51 ! [Y2: binary1439146945Tree_a] :
% 2.67/1.51 ( ( Y2 != binary476621312_Tip_a )
% 2.67/1.51 => ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
% 2.67/1.51 ( Y2
% 2.67/1.51 != ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_25_sorted__distinct__pred__def,axiom,
% 2.67/1.51 ( binary670562003pred_a
% 2.67/1.51 = ( ^ [H2: a > int,A2: a,B2: a,T3: binary1439146945Tree_a] :
% 2.67/1.51 ( ( ( binary1721989714Tree_a @ H2 @ T3 )
% 2.67/1.51 & ( member_a @ A2 @ ( binary945792244etOf_a @ T3 ) )
% 2.67/1.51 & ( member_a @ B2 @ ( binary945792244etOf_a @ T3 ) )
% 2.67/1.51 & ( ( H2 @ A2 )
% 2.67/1.51 = ( H2 @ B2 ) ) )
% 2.67/1.51 => ( A2 = B2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_26_insert__Diff__single,axiom,
% 2.67/1.51 ! [A: a,A3: set_a] :
% 2.67/1.51 ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
% 2.67/1.51 = ( insert_a @ A @ A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_27_singleton__conv,axiom,
% 2.67/1.51 ! [A: a] :
% 2.67/1.51 ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] : ( X2 = A ) )
% 2.67/1.51 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_28_singleton__conv2,axiom,
% 2.67/1.51 ! [A: a] :
% 2.67/1.51 ( ( collect_a
% 2.67/1.51 @ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
% 2.67/1.51 @ A ) )
% 2.67/1.51 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_29_Un__Diff__cancel,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
% 2.67/1.51 = ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_30_Un__Diff__cancel2,axiom,
% 2.67/1.51 ! [B3: set_a,A3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( minus_minus_set_a @ B3 @ A3 ) @ A3 )
% 2.67/1.51 = ( sup_sup_set_a @ B3 @ A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_31_Diff__insert0,axiom,
% 2.67/1.51 ! [X3: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ~ ( member_a @ X3 @ A3 )
% 2.67/1.51 => ( ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ B3 ) )
% 2.67/1.51 = ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_32_insert__Diff1,axiom,
% 2.67/1.51 ! [X3: a,B3: set_a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ X3 @ B3 )
% 2.67/1.51 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 2.67/1.51 = ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_33_Un__insert__left,axiom,
% 2.67/1.51 ! [A: a,B3: set_a,C: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( insert_a @ A @ B3 ) @ C )
% 2.67/1.51 = ( insert_a @ A @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_34_Un__insert__right,axiom,
% 2.67/1.51 ! [A3: set_a,A: a,B3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B3 ) )
% 2.67/1.51 = ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_35_Diff__empty,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
% 2.67/1.51 = A3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_36_empty__Diff,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
% 2.67/1.51 = bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_37_empty__Collect__eq,axiom,
% 2.67/1.51 ! [P: a > $o] :
% 2.67/1.51 ( ( bot_bot_set_a
% 2.67/1.51 = ( collect_a @ P ) )
% 2.67/1.51 = ( ! [X2: a] :
% 2.67/1.51 ~ ( P @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_38_Collect__empty__eq,axiom,
% 2.67/1.51 ! [P: a > $o] :
% 2.67/1.51 ( ( ( collect_a @ P )
% 2.67/1.51 = bot_bot_set_a )
% 2.67/1.51 = ( ! [X2: a] :
% 2.67/1.51 ~ ( P @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_39_all__not__in__conv,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( ! [X2: a] :
% 2.67/1.51 ~ ( member_a @ X2 @ A3 ) )
% 2.67/1.51 = ( A3 = bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_40_empty__iff,axiom,
% 2.67/1.51 ! [C2: a] :
% 2.67/1.51 ~ ( member_a @ C2 @ bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_41_insert__absorb2,axiom,
% 2.67/1.51 ! [X3: a,A3: set_a] :
% 2.67/1.51 ( ( insert_a @ X3 @ ( insert_a @ X3 @ A3 ) )
% 2.67/1.51 = ( insert_a @ X3 @ A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_42_insert__iff,axiom,
% 2.67/1.51 ! [A: a,B: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 2.67/1.51 = ( ( A = B )
% 2.67/1.51 | ( member_a @ A @ A3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_43_insertCI,axiom,
% 2.67/1.51 ! [A: a,B3: set_a,B: a] :
% 2.67/1.51 ( ( ~ ( member_a @ A @ B3 )
% 2.67/1.51 => ( A = B ) )
% 2.67/1.51 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_44_Un__iff,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.67/1.51 = ( ( member_a @ C2 @ A3 )
% 2.67/1.51 | ( member_a @ C2 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_45_mem__Collect__eq,axiom,
% 2.67/1.51 ! [A: a,P: a > $o] :
% 2.67/1.51 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.67/1.51 = ( P @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_46_Collect__mem__eq,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
% 2.67/1.51 = A3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_47_Collect__cong,axiom,
% 2.67/1.51 ! [P: a > $o,Q: a > $o] :
% 2.67/1.51 ( ! [X4: a] :
% 2.67/1.51 ( ( P @ X4 )
% 2.67/1.51 = ( Q @ X4 ) )
% 2.67/1.51 => ( ( collect_a @ P )
% 2.67/1.51 = ( collect_a @ Q ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_48_UnCI,axiom,
% 2.67/1.51 ! [C2: a,B3: set_a,A3: set_a] :
% 2.67/1.51 ( ( ~ ( member_a @ C2 @ B3 )
% 2.67/1.51 => ( member_a @ C2 @ A3 ) )
% 2.67/1.51 => ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_49_Diff__idemp,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ B3 )
% 2.67/1.51 = ( minus_minus_set_a @ A3 @ B3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_50_Diff__iff,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 2.67/1.51 = ( ( member_a @ C2 @ A3 )
% 2.67/1.51 & ~ ( member_a @ C2 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_51_DiffI,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ A3 )
% 2.67/1.51 => ( ~ ( member_a @ C2 @ B3 )
% 2.67/1.51 => ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_52_singletonI,axiom,
% 2.67/1.51 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_53_Un__empty,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 2.67/1.51 = bot_bot_set_a )
% 2.67/1.51 = ( ( A3 = bot_bot_set_a )
% 2.67/1.51 & ( B3 = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_54_Diff__cancel,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ A3 @ A3 )
% 2.67/1.51 = bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_55_ex__in__conv,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
% 2.67/1.51 = ( A3 != bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_56_equals0I,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ! [Y4: a] :
% 2.67/1.51 ~ ( member_a @ Y4 @ A3 )
% 2.67/1.51 => ( A3 = bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_57_equals0D,axiom,
% 2.67/1.51 ! [A3: set_a,A: a] :
% 2.67/1.51 ( ( A3 = bot_bot_set_a )
% 2.67/1.51 => ~ ( member_a @ A @ A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_58_emptyE,axiom,
% 2.67/1.51 ! [A: a] :
% 2.67/1.51 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_59_mk__disjoint__insert,axiom,
% 2.67/1.51 ! [A: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ A @ A3 )
% 2.67/1.51 => ? [B4: set_a] :
% 2.67/1.51 ( ( A3
% 2.67/1.51 = ( insert_a @ A @ B4 ) )
% 2.67/1.51 & ~ ( member_a @ A @ B4 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_60_insert__commute,axiom,
% 2.67/1.51 ! [X3: a,Y2: a,A3: set_a] :
% 2.67/1.51 ( ( insert_a @ X3 @ ( insert_a @ Y2 @ A3 ) )
% 2.67/1.51 = ( insert_a @ Y2 @ ( insert_a @ X3 @ A3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_61_insert__eq__iff,axiom,
% 2.67/1.51 ! [A: a,A3: set_a,B: a,B3: set_a] :
% 2.67/1.51 ( ~ ( member_a @ A @ A3 )
% 2.67/1.51 => ( ~ ( member_a @ B @ B3 )
% 2.67/1.51 => ( ( ( insert_a @ A @ A3 )
% 2.67/1.51 = ( insert_a @ B @ B3 ) )
% 2.67/1.51 = ( ( ( A = B )
% 2.67/1.51 => ( A3 = B3 ) )
% 2.67/1.51 & ( ( A != B )
% 2.67/1.51 => ? [C3: set_a] :
% 2.67/1.51 ( ( A3
% 2.67/1.51 = ( insert_a @ B @ C3 ) )
% 2.67/1.51 & ~ ( member_a @ B @ C3 )
% 2.67/1.51 & ( B3
% 2.67/1.51 = ( insert_a @ A @ C3 ) )
% 2.67/1.51 & ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_62_insert__absorb,axiom,
% 2.67/1.51 ! [A: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ A @ A3 )
% 2.67/1.51 => ( ( insert_a @ A @ A3 )
% 2.67/1.51 = A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_63_insert__ident,axiom,
% 2.67/1.51 ! [X3: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ~ ( member_a @ X3 @ A3 )
% 2.67/1.51 => ( ~ ( member_a @ X3 @ B3 )
% 2.67/1.51 => ( ( ( insert_a @ X3 @ A3 )
% 2.67/1.51 = ( insert_a @ X3 @ B3 ) )
% 2.67/1.51 = ( A3 = B3 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_64_Set_Oset__insert,axiom,
% 2.67/1.51 ! [X3: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ X3 @ A3 )
% 2.67/1.51 => ~ ! [B4: set_a] :
% 2.67/1.51 ( ( A3
% 2.67/1.51 = ( insert_a @ X3 @ B4 ) )
% 2.67/1.51 => ( member_a @ X3 @ B4 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_65_insertI2,axiom,
% 2.67/1.51 ! [A: a,B3: set_a,B: a] :
% 2.67/1.51 ( ( member_a @ A @ B3 )
% 2.67/1.51 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_66_insertI1,axiom,
% 2.67/1.51 ! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_67_insertE,axiom,
% 2.67/1.51 ! [A: a,B: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 2.67/1.51 => ( ( A != B )
% 2.67/1.51 => ( member_a @ A @ A3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_68_Un__left__commute,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,C: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C ) )
% 2.67/1.51 = ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_69_Un__left__absorb,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.67/1.51 = ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_70_Un__commute,axiom,
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 = ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_71_Un__absorb,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A3 @ A3 )
% 2.67/1.51 = A3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_72_Un__assoc,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,C: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
% 2.67/1.51 = ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_73_ball__Un,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,P: a > $o] :
% 2.67/1.51 ( ( ! [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.67/1.51 => ( P @ X2 ) ) )
% 2.67/1.51 = ( ! [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ A3 )
% 2.67/1.51 => ( P @ X2 ) )
% 2.67/1.51 & ! [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ B3 )
% 2.67/1.51 => ( P @ X2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_74_bex__Un,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,P: a > $o] :
% 2.67/1.51 ( ( ? [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.67/1.51 & ( P @ X2 ) ) )
% 2.67/1.51 = ( ? [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ A3 )
% 2.67/1.51 & ( P @ X2 ) )
% 2.67/1.51 | ? [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ B3 )
% 2.67/1.51 & ( P @ X2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_75_UnI2,axiom,
% 2.67/1.51 ! [C2: a,B3: set_a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ B3 )
% 2.67/1.51 => ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_76_UnI1,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ A3 )
% 2.67/1.51 => ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_77_UnE,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.67/1.51 => ( ~ ( member_a @ C2 @ A3 )
% 2.67/1.51 => ( member_a @ C2 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_78_DiffD2,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 2.67/1.51 => ~ ( member_a @ C2 @ B3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_79_DiffD1,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 2.67/1.51 => ( member_a @ C2 @ A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_80_DiffE,axiom,
% 2.67/1.51 ! [C2: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
% 2.67/1.51 => ~ ( ( member_a @ C2 @ A3 )
% 2.67/1.51 => ( member_a @ C2 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_81_empty__def,axiom,
% 2.67/1.51 ( bot_bot_set_a
% 2.67/1.51 = ( collect_a
% 2.67/1.51 @ ^ [X2: a] : $false ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_82_insert__Collect,axiom,
% 2.67/1.51 ! [A: a,P: a > $o] :
% 2.67/1.51 ( ( insert_a @ A @ ( collect_a @ P ) )
% 2.67/1.51 = ( collect_a
% 2.67/1.51 @ ^ [U: a] :
% 2.67/1.51 ( ( U != A )
% 2.67/1.51 => ( P @ U ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_83_insert__compr,axiom,
% 2.67/1.51 ( insert_a
% 2.67/1.51 = ( ^ [A2: a,B5: set_a] :
% 2.67/1.51 ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( X2 = A2 )
% 2.67/1.51 | ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_84_Collect__disj__eq,axiom,
% 2.67/1.51 ! [P: a > $o,Q: a > $o] :
% 2.67/1.51 ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( P @ X2 )
% 2.67/1.51 | ( Q @ X2 ) ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_85_Un__def,axiom,
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 = ( ^ [A4: set_a,B5: set_a] :
% 2.67/1.51 ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ A4 )
% 2.67/1.51 | ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_86_set__diff__eq,axiom,
% 2.67/1.51 ( minus_minus_set_a
% 2.67/1.51 = ( ^ [A4: set_a,B5: set_a] :
% 2.67/1.51 ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( member_a @ X2 @ A4 )
% 2.67/1.51 & ~ ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_87_singleton__inject,axiom,
% 2.67/1.51 ! [A: a,B: a] :
% 2.67/1.51 ( ( ( insert_a @ A @ bot_bot_set_a )
% 2.67/1.51 = ( insert_a @ B @ bot_bot_set_a ) )
% 2.67/1.51 => ( A = B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_88_insert__not__empty,axiom,
% 2.67/1.51 ! [A: a,A3: set_a] :
% 2.67/1.51 ( ( insert_a @ A @ A3 )
% 2.67/1.51 != bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_89_doubleton__eq__iff,axiom,
% 2.67/1.51 ! [A: a,B: a,C2: a,D: a] :
% 2.67/1.51 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 2.67/1.51 = ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 2.67/1.51 = ( ( ( A = C2 )
% 2.67/1.51 & ( B = D ) )
% 2.67/1.51 | ( ( A = D )
% 2.67/1.51 & ( B = C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_90_singleton__iff,axiom,
% 2.67/1.51 ! [B: a,A: a] :
% 2.67/1.51 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 2.67/1.51 = ( B = A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_91_singletonD,axiom,
% 2.67/1.51 ! [B: a,A: a] :
% 2.67/1.51 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 2.67/1.51 => ( B = A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_92_Un__empty__right,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
% 2.67/1.51 = A3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_93_Un__empty__left,axiom,
% 2.67/1.51 ! [B3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
% 2.67/1.51 = B3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_94_insert__Diff__if,axiom,
% 2.67/1.51 ! [X3: a,B3: set_a,A3: set_a] :
% 2.67/1.51 ( ( ( member_a @ X3 @ B3 )
% 2.67/1.51 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 2.67/1.51 = ( minus_minus_set_a @ A3 @ B3 ) ) )
% 2.67/1.51 & ( ~ ( member_a @ X3 @ B3 )
% 2.67/1.51 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 2.67/1.51 = ( insert_a @ X3 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_95_Un__Diff,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,C: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
% 2.67/1.51 = ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C ) @ ( minus_minus_set_a @ B3 @ C ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_96_Collect__conv__if2,axiom,
% 2.67/1.51 ! [P: a > $o,A: a] :
% 2.67/1.51 ( ( ( P @ A )
% 2.67/1.51 => ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( A = X2 )
% 2.67/1.51 & ( P @ X2 ) ) )
% 2.67/1.51 = ( insert_a @ A @ bot_bot_set_a ) ) )
% 2.67/1.51 & ( ~ ( P @ A )
% 2.67/1.51 => ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( A = X2 )
% 2.67/1.51 & ( P @ X2 ) ) )
% 2.67/1.51 = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_97_Collect__conv__if,axiom,
% 2.67/1.51 ! [P: a > $o,A: a] :
% 2.67/1.51 ( ( ( P @ A )
% 2.67/1.51 => ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( X2 = A )
% 2.67/1.51 & ( P @ X2 ) ) )
% 2.67/1.51 = ( insert_a @ A @ bot_bot_set_a ) ) )
% 2.67/1.51 & ( ~ ( P @ A )
% 2.67/1.51 => ( ( collect_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( ( X2 = A )
% 2.67/1.51 & ( P @ X2 ) ) )
% 2.67/1.51 = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_98_insert__def,axiom,
% 2.67/1.51 ( insert_a
% 2.67/1.51 = ( ^ [A2: a] :
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 @ ( collect_a
% 2.67/1.51 @ ^ [X2: a] : ( X2 = A2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_99_singleton__Un__iff,axiom,
% 2.67/1.51 ! [X3: a,A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( ( insert_a @ X3 @ bot_bot_set_a )
% 2.67/1.51 = ( sup_sup_set_a @ A3 @ B3 ) )
% 2.67/1.51 = ( ( ( A3 = bot_bot_set_a )
% 2.67/1.51 & ( B3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) ) )
% 2.67/1.51 | ( ( A3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 & ( B3 = bot_bot_set_a ) )
% 2.67/1.51 | ( ( A3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 & ( B3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_100_Un__singleton__iff,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,X3: a] :
% 2.67/1.51 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 = ( ( ( A3 = bot_bot_set_a )
% 2.67/1.51 & ( B3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) ) )
% 2.67/1.51 | ( ( A3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 & ( B3 = bot_bot_set_a ) )
% 2.67/1.51 | ( ( A3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 & ( B3
% 2.67/1.51 = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_101_insert__is__Un,axiom,
% 2.67/1.51 ( insert_a
% 2.67/1.51 = ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_102_Diff__insert__absorb,axiom,
% 2.67/1.51 ! [X3: a,A3: set_a] :
% 2.67/1.51 ( ~ ( member_a @ X3 @ A3 )
% 2.67/1.51 => ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 = A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_103_Diff__insert2,axiom,
% 2.67/1.51 ! [A3: set_a,A: a,B3: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
% 2.67/1.51 = ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_104_insert__Diff,axiom,
% 2.67/1.51 ! [A: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ A @ A3 )
% 2.67/1.51 => ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
% 2.67/1.51 = A3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_105_Diff__insert,axiom,
% 2.67/1.51 ! [A3: set_a,A: a,B3: set_a] :
% 2.67/1.51 ( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
% 2.67/1.51 = ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_106_sup__bot_Oright__neutral,axiom,
% 2.67/1.51 ! [A: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ A @ bot_bot_a_o )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_107_sup__bot_Oright__neutral,axiom,
% 2.67/1.51 ! [A: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A @ bot_bot_set_a )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_108_sup__bot_Oneutr__eq__iff,axiom,
% 2.67/1.51 ! [A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( bot_bot_a_o
% 2.67/1.51 = ( sup_sup_a_o @ A @ B ) )
% 2.67/1.51 = ( ( A = bot_bot_a_o )
% 2.67/1.51 & ( B = bot_bot_a_o ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_109_sup__bot_Oneutr__eq__iff,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( bot_bot_set_a
% 2.67/1.51 = ( sup_sup_set_a @ A @ B ) )
% 2.67/1.51 = ( ( A = bot_bot_set_a )
% 2.67/1.51 & ( B = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_110_sup__bot_Oleft__neutral,axiom,
% 2.67/1.51 ! [A: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ bot_bot_a_o @ A )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_111_sup__bot_Oleft__neutral,axiom,
% 2.67/1.51 ! [A: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ bot_bot_set_a @ A )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_112_sup__bot_Oeq__neutr__iff,axiom,
% 2.67/1.51 ! [A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( ( sup_sup_a_o @ A @ B )
% 2.67/1.51 = bot_bot_a_o )
% 2.67/1.51 = ( ( A = bot_bot_a_o )
% 2.67/1.51 & ( B = bot_bot_a_o ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_113_sup__bot_Oeq__neutr__iff,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( ( sup_sup_set_a @ A @ B )
% 2.67/1.51 = bot_bot_set_a )
% 2.67/1.51 = ( ( A = bot_bot_set_a )
% 2.67/1.51 & ( B = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_114_sup__eq__bot__iff,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o] :
% 2.67/1.51 ( ( ( sup_sup_a_o @ X3 @ Y2 )
% 2.67/1.51 = bot_bot_a_o )
% 2.67/1.51 = ( ( X3 = bot_bot_a_o )
% 2.67/1.51 & ( Y2 = bot_bot_a_o ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_115_sup__eq__bot__iff,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ( sup_sup_set_a @ X3 @ Y2 )
% 2.67/1.51 = bot_bot_set_a )
% 2.67/1.51 = ( ( X3 = bot_bot_set_a )
% 2.67/1.51 & ( Y2 = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_116_bot__eq__sup__iff,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o] :
% 2.67/1.51 ( ( bot_bot_a_o
% 2.67/1.51 = ( sup_sup_a_o @ X3 @ Y2 ) )
% 2.67/1.51 = ( ( X3 = bot_bot_a_o )
% 2.67/1.51 & ( Y2 = bot_bot_a_o ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_117_bot__eq__sup__iff,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( bot_bot_set_a
% 2.67/1.51 = ( sup_sup_set_a @ X3 @ Y2 ) )
% 2.67/1.51 = ( ( X3 = bot_bot_set_a )
% 2.67/1.51 & ( Y2 = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_118_sup__bot__right,axiom,
% 2.67/1.51 ! [X3: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ X3 @ bot_bot_a_o )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_119_sup__bot__right,axiom,
% 2.67/1.51 ! [X3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_120_sup__bot__left,axiom,
% 2.67/1.51 ! [X3: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ bot_bot_a_o @ X3 )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_121_sup__bot__left,axiom,
% 2.67/1.51 ! [X3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_122_memb__spec,axiom,
% 2.67/1.51 ! [H: a > int,T: binary1439146945Tree_a,X3: a] :
% 2.67/1.51 ( ( binary1721989714Tree_a @ H @ T )
% 2.67/1.51 => ( ( binary2053421120memb_a @ H @ X3 @ T )
% 2.67/1.51 = ( member_a @ X3 @ ( binary945792244etOf_a @ T ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_123_minus__apply,axiom,
% 2.67/1.51 ( minus_minus_a_o
% 2.67/1.51 = ( ^ [A4: a > $o,B5: a > $o,X2: a] : ( minus_minus_o @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_124_sup__apply,axiom,
% 2.67/1.51 ( sup_sup_a_o
% 2.67/1.51 = ( ^ [F: a > $o,G: a > $o,X2: a] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_125_sup_Oidem,axiom,
% 2.67/1.51 ! [A: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ A @ A )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_126_sup_Oidem,axiom,
% 2.67/1.51 ! [A: int] :
% 2.67/1.51 ( ( sup_sup_int @ A @ A )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_127_sup_Oidem,axiom,
% 2.67/1.51 ! [A: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A @ A )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_128_sup__idem,axiom,
% 2.67/1.51 ! [X3: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ X3 @ X3 )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_129_sup__idem,axiom,
% 2.67/1.51 ! [X3: int] :
% 2.67/1.51 ( ( sup_sup_int @ X3 @ X3 )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_130_sup__idem,axiom,
% 2.67/1.51 ! [X3: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ X3 @ X3 )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_131_sup_Oleft__idem,axiom,
% 2.67/1.51 ! [A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ A @ ( sup_sup_a_o @ A @ B ) )
% 2.67/1.51 = ( sup_sup_a_o @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_132_sup_Oleft__idem,axiom,
% 2.67/1.51 ! [A: int,B: int] :
% 2.67/1.51 ( ( sup_sup_int @ A @ ( sup_sup_int @ A @ B ) )
% 2.67/1.51 = ( sup_sup_int @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_133_sup_Oleft__idem,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
% 2.67/1.51 = ( sup_sup_set_a @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_134_sup__left__idem,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y2 ) )
% 2.67/1.51 = ( sup_sup_a_o @ X3 @ Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_135_sup__left__idem,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y2 ) )
% 2.67/1.51 = ( sup_sup_int @ X3 @ Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_136_sup__left__idem,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_137_sup_Oright__idem,axiom,
% 2.67/1.51 ! [A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B ) @ B )
% 2.67/1.51 = ( sup_sup_a_o @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_138_sup_Oright__idem,axiom,
% 2.67/1.51 ! [A: int,B: int] :
% 2.67/1.51 ( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ B )
% 2.67/1.51 = ( sup_sup_int @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_139_sup_Oright__idem,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
% 2.67/1.51 = ( sup_sup_set_a @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_140_bot__set__def,axiom,
% 2.67/1.51 ( bot_bot_set_a
% 2.67/1.51 = ( collect_a @ bot_bot_a_o ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_141_sup__set__def,axiom,
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 = ( ^ [A4: set_a,B5: set_a] :
% 2.67/1.51 ( collect_a
% 2.67/1.51 @ ( sup_sup_a_o
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ A4 )
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_142_fun__diff__def,axiom,
% 2.67/1.51 ( minus_minus_a_o
% 2.67/1.51 = ( ^ [A4: a > $o,B5: a > $o,X2: a] : ( minus_minus_o @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_143_inf__sup__aci_I8_J,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y2 ) )
% 2.67/1.51 = ( sup_sup_a_o @ X3 @ Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_144_inf__sup__aci_I8_J,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y2 ) )
% 2.67/1.51 = ( sup_sup_int @ X3 @ Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_145_inf__sup__aci_I8_J,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_146_inf__sup__aci_I7_J,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) )
% 2.67/1.51 = ( sup_sup_a_o @ Y2 @ ( sup_sup_a_o @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_147_inf__sup__aci_I7_J,axiom,
% 2.67/1.51 ! [X3: int,Y2: int,Z2: int] :
% 2.67/1.51 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) )
% 2.67/1.51 = ( sup_sup_int @ Y2 @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_148_inf__sup__aci_I7_J,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_149_inf__sup__aci_I6_J,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
% 2.67/1.51 = ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_150_inf__sup__aci_I6_J,axiom,
% 2.67/1.51 ! [X3: int,Y2: int,Z2: int] :
% 2.67/1.51 ( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
% 2.67/1.51 = ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_151_inf__sup__aci_I6_J,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
% 2.67/1.51 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_152_inf__sup__aci_I5_J,axiom,
% 2.67/1.51 ( sup_sup_a_o
% 2.67/1.51 = ( ^ [X2: a > $o,Y: a > $o] : ( sup_sup_a_o @ Y @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_153_inf__sup__aci_I5_J,axiom,
% 2.67/1.51 ( sup_sup_int
% 2.67/1.51 = ( ^ [X2: int,Y: int] : ( sup_sup_int @ Y @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_154_inf__sup__aci_I5_J,axiom,
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_155_sup__fun__def,axiom,
% 2.67/1.51 ( sup_sup_a_o
% 2.67/1.51 = ( ^ [F: a > $o,G: a > $o,X2: a] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_156_boolean__algebra__cancel_Osup1,axiom,
% 2.67/1.51 ! [A3: a > $o,K: a > $o,A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( A3
% 2.67/1.51 = ( sup_sup_a_o @ K @ A ) )
% 2.67/1.51 => ( ( sup_sup_a_o @ A3 @ B )
% 2.67/1.51 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_157_boolean__algebra__cancel_Osup1,axiom,
% 2.67/1.51 ! [A3: int,K: int,A: int,B: int] :
% 2.67/1.51 ( ( A3
% 2.67/1.51 = ( sup_sup_int @ K @ A ) )
% 2.67/1.51 => ( ( sup_sup_int @ A3 @ B )
% 2.67/1.51 = ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_158_boolean__algebra__cancel_Osup1,axiom,
% 2.67/1.51 ! [A3: set_a,K: set_a,A: set_a,B: set_a] :
% 2.67/1.51 ( ( A3
% 2.67/1.51 = ( sup_sup_set_a @ K @ A ) )
% 2.67/1.51 => ( ( sup_sup_set_a @ A3 @ B )
% 2.67/1.51 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_159_boolean__algebra__cancel_Osup2,axiom,
% 2.67/1.51 ! [B3: a > $o,K: a > $o,B: a > $o,A: a > $o] :
% 2.67/1.51 ( ( B3
% 2.67/1.51 = ( sup_sup_a_o @ K @ B ) )
% 2.67/1.51 => ( ( sup_sup_a_o @ A @ B3 )
% 2.67/1.51 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_160_boolean__algebra__cancel_Osup2,axiom,
% 2.67/1.51 ! [B3: int,K: int,B: int,A: int] :
% 2.67/1.51 ( ( B3
% 2.67/1.51 = ( sup_sup_int @ K @ B ) )
% 2.67/1.51 => ( ( sup_sup_int @ A @ B3 )
% 2.67/1.51 = ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_161_boolean__algebra__cancel_Osup2,axiom,
% 2.67/1.51 ! [B3: set_a,K: set_a,B: set_a,A: set_a] :
% 2.67/1.51 ( ( B3
% 2.67/1.51 = ( sup_sup_set_a @ K @ B ) )
% 2.67/1.51 => ( ( sup_sup_set_a @ A @ B3 )
% 2.67/1.51 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_162_sup_Oassoc,axiom,
% 2.67/1.51 ! [A: a > $o,B: a > $o,C2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B ) @ C2 )
% 2.67/1.51 = ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_163_sup_Oassoc,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int] :
% 2.67/1.51 ( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ C2 )
% 2.67/1.51 = ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_164_sup_Oassoc,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
% 2.67/1.51 = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_165_sup__assoc,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
% 2.67/1.51 = ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_166_sup__assoc,axiom,
% 2.67/1.51 ! [X3: int,Y2: int,Z2: int] :
% 2.67/1.51 ( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
% 2.67/1.51 = ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_167_sup__assoc,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
% 2.67/1.51 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_168_sup_Ocommute,axiom,
% 2.67/1.51 ( sup_sup_a_o
% 2.67/1.51 = ( ^ [A2: a > $o,B2: a > $o] : ( sup_sup_a_o @ B2 @ A2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_169_sup_Ocommute,axiom,
% 2.67/1.51 ( sup_sup_int
% 2.67/1.51 = ( ^ [A2: int,B2: int] : ( sup_sup_int @ B2 @ A2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_170_sup_Ocommute,axiom,
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 = ( ^ [A2: set_a,B2: set_a] : ( sup_sup_set_a @ B2 @ A2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_171_sup__commute,axiom,
% 2.67/1.51 ( sup_sup_a_o
% 2.67/1.51 = ( ^ [X2: a > $o,Y: a > $o] : ( sup_sup_a_o @ Y @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_172_sup__commute,axiom,
% 2.67/1.51 ( sup_sup_int
% 2.67/1.51 = ( ^ [X2: int,Y: int] : ( sup_sup_int @ Y @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_173_sup__commute,axiom,
% 2.67/1.51 ( sup_sup_set_a
% 2.67/1.51 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_174_sup_Oleft__commute,axiom,
% 2.67/1.51 ! [B: a > $o,A: a > $o,C2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ B @ ( sup_sup_a_o @ A @ C2 ) )
% 2.67/1.51 = ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_175_sup_Oleft__commute,axiom,
% 2.67/1.51 ! [B: int,A: int,C2: int] :
% 2.67/1.51 ( ( sup_sup_int @ B @ ( sup_sup_int @ A @ C2 ) )
% 2.67/1.51 = ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_176_sup_Oleft__commute,axiom,
% 2.67/1.51 ! [B: set_a,A: set_a,C2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_177_sup__left__commute,axiom,
% 2.67/1.51 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 2.67/1.51 ( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) )
% 2.67/1.51 = ( sup_sup_a_o @ Y2 @ ( sup_sup_a_o @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_178_sup__left__commute,axiom,
% 2.67/1.51 ! [X3: int,Y2: int,Z2: int] :
% 2.67/1.51 ( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) )
% 2.67/1.51 = ( sup_sup_int @ Y2 @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_179_sup__left__commute,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 2.67/1.51 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
% 2.67/1.51 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_180_less__supI1,axiom,
% 2.67/1.51 ! [X3: a > $o,A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( ord_less_a_o @ X3 @ A )
% 2.67/1.51 => ( ord_less_a_o @ X3 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_181_less__supI1,axiom,
% 2.67/1.51 ! [X3: set_a,A: set_a,B: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ A )
% 2.67/1.51 => ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_182_less__supI1,axiom,
% 2.67/1.51 ! [X3: int,A: int,B: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ A )
% 2.67/1.51 => ( ord_less_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_183_less__supI2,axiom,
% 2.67/1.51 ! [X3: a > $o,B: a > $o,A: a > $o] :
% 2.67/1.51 ( ( ord_less_a_o @ X3 @ B )
% 2.67/1.51 => ( ord_less_a_o @ X3 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_184_less__supI2,axiom,
% 2.67/1.51 ! [X3: set_a,B: set_a,A: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ B )
% 2.67/1.51 => ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_185_less__supI2,axiom,
% 2.67/1.51 ! [X3: int,B: int,A: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ B )
% 2.67/1.51 => ( ord_less_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_186_sup_Ostrict__boundedE,axiom,
% 2.67/1.51 ! [B: a > $o,C2: a > $o,A: a > $o] :
% 2.67/1.51 ( ( ord_less_a_o @ ( sup_sup_a_o @ B @ C2 ) @ A )
% 2.67/1.51 => ~ ( ( ord_less_a_o @ B @ A )
% 2.67/1.51 => ~ ( ord_less_a_o @ C2 @ A ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_187_sup_Ostrict__boundedE,axiom,
% 2.67/1.51 ! [B: set_a,C2: set_a,A: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
% 2.67/1.51 => ~ ( ( ord_less_set_a @ B @ A )
% 2.67/1.51 => ~ ( ord_less_set_a @ C2 @ A ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_188_sup_Ostrict__boundedE,axiom,
% 2.67/1.51 ! [B: int,C2: int,A: int] :
% 2.67/1.51 ( ( ord_less_int @ ( sup_sup_int @ B @ C2 ) @ A )
% 2.67/1.51 => ~ ( ( ord_less_int @ B @ A )
% 2.67/1.51 => ~ ( ord_less_int @ C2 @ A ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_189_sup_Ostrict__order__iff,axiom,
% 2.67/1.51 ( ord_less_a_o
% 2.67/1.51 = ( ^ [B2: a > $o,A2: a > $o] :
% 2.67/1.51 ( ( A2
% 2.67/1.51 = ( sup_sup_a_o @ A2 @ B2 ) )
% 2.67/1.51 & ( A2 != B2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_190_sup_Ostrict__order__iff,axiom,
% 2.67/1.51 ( ord_less_set_a
% 2.67/1.51 = ( ^ [B2: set_a,A2: set_a] :
% 2.67/1.51 ( ( A2
% 2.67/1.51 = ( sup_sup_set_a @ A2 @ B2 ) )
% 2.67/1.51 & ( A2 != B2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_191_sup_Ostrict__order__iff,axiom,
% 2.67/1.51 ( ord_less_int
% 2.67/1.51 = ( ^ [B2: int,A2: int] :
% 2.67/1.51 ( ( A2
% 2.67/1.51 = ( sup_sup_int @ A2 @ B2 ) )
% 2.67/1.51 & ( A2 != B2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_192_sup_Ostrict__coboundedI1,axiom,
% 2.67/1.51 ! [C2: a > $o,A: a > $o,B: a > $o] :
% 2.67/1.51 ( ( ord_less_a_o @ C2 @ A )
% 2.67/1.51 => ( ord_less_a_o @ C2 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_193_sup_Ostrict__coboundedI1,axiom,
% 2.67/1.51 ! [C2: set_a,A: set_a,B: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ C2 @ A )
% 2.67/1.51 => ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_194_sup_Ostrict__coboundedI1,axiom,
% 2.67/1.51 ! [C2: int,A: int,B: int] :
% 2.67/1.51 ( ( ord_less_int @ C2 @ A )
% 2.67/1.51 => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_195_sup_Ostrict__coboundedI2,axiom,
% 2.67/1.51 ! [C2: a > $o,B: a > $o,A: a > $o] :
% 2.67/1.51 ( ( ord_less_a_o @ C2 @ B )
% 2.67/1.51 => ( ord_less_a_o @ C2 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_196_sup_Ostrict__coboundedI2,axiom,
% 2.67/1.51 ! [C2: set_a,B: set_a,A: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ C2 @ B )
% 2.67/1.51 => ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_197_sup_Ostrict__coboundedI2,axiom,
% 2.67/1.51 ! [C2: int,B: int,A: int] :
% 2.67/1.51 ( ( ord_less_int @ C2 @ B )
% 2.67/1.51 => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_198_the__elem__eq,axiom,
% 2.67/1.51 ! [X3: a] :
% 2.67/1.51 ( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_199_bot__apply,axiom,
% 2.67/1.51 ( bot_bot_a_o
% 2.67/1.51 = ( ^ [X2: a] : bot_bot_o ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_200_is__singletonI,axiom,
% 2.67/1.51 ! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_201_diff__strict__mono,axiom,
% 2.67/1.51 ! [A: int,B: int,D: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( ord_less_int @ D @ C2 )
% 2.67/1.51 => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_202_sup1CI,axiom,
% 2.67/1.51 ! [B3: a > $o,X3: a,A3: a > $o] :
% 2.67/1.51 ( ( ~ ( B3 @ X3 )
% 2.67/1.51 => ( A3 @ X3 ) )
% 2.67/1.51 => ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_203_sup1E,axiom,
% 2.67/1.51 ! [A3: a > $o,B3: a > $o,X3: a] :
% 2.67/1.51 ( ( sup_sup_a_o @ A3 @ B3 @ X3 )
% 2.67/1.51 => ( ~ ( A3 @ X3 )
% 2.67/1.51 => ( B3 @ X3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_204_sup1I1,axiom,
% 2.67/1.51 ! [A3: a > $o,X3: a,B3: a > $o] :
% 2.67/1.51 ( ( A3 @ X3 )
% 2.67/1.51 => ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_205_sup1I2,axiom,
% 2.67/1.51 ! [B3: a > $o,X3: a,A3: a > $o] :
% 2.67/1.51 ( ( B3 @ X3 )
% 2.67/1.51 => ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_206_not__psubset__empty,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_207_psubset__imp__ex__mem,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A3 @ B3 )
% 2.67/1.51 => ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_208_minus__set__def,axiom,
% 2.67/1.51 ( minus_minus_set_a
% 2.67/1.51 = ( ^ [A4: set_a,B5: set_a] :
% 2.67/1.51 ( collect_a
% 2.67/1.51 @ ( minus_minus_a_o
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ A4 )
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_209_is__singleton__the__elem,axiom,
% 2.67/1.51 ( is_singleton_a
% 2.67/1.51 = ( ^ [A4: set_a] :
% 2.67/1.51 ( A4
% 2.67/1.51 = ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_210_is__singletonI_H,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( A3 != bot_bot_set_a )
% 2.67/1.51 => ( ! [X4: a,Y4: a] :
% 2.67/1.51 ( ( member_a @ X4 @ A3 )
% 2.67/1.51 => ( ( member_a @ Y4 @ A3 )
% 2.67/1.51 => ( X4 = Y4 ) ) )
% 2.67/1.51 => ( is_singleton_a @ A3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
% 2.67/1.51 ! [B: set_a,A: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ B @ A )
% 2.67/1.51 => ( A != B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_212_dual__order_Ostrict__implies__not__eq,axiom,
% 2.67/1.51 ! [B: int,A: int] :
% 2.67/1.51 ( ( ord_less_int @ B @ A )
% 2.67/1.51 => ( A != B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_213_order_Ostrict__implies__not__eq,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( A != B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_214_order_Ostrict__implies__not__eq,axiom,
% 2.67/1.51 ! [A: int,B: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( A != B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_215_not__less__iff__gr__or__eq,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 2.67/1.51 = ( ( ord_less_int @ Y2 @ X3 )
% 2.67/1.51 | ( X3 = Y2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_216_dual__order_Ostrict__trans,axiom,
% 2.67/1.51 ! [B: set_a,A: set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ B @ A )
% 2.67/1.51 => ( ( ord_less_set_a @ C2 @ B )
% 2.67/1.51 => ( ord_less_set_a @ C2 @ A ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_217_dual__order_Ostrict__trans,axiom,
% 2.67/1.51 ! [B: int,A: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ B @ A )
% 2.67/1.51 => ( ( ord_less_int @ C2 @ B )
% 2.67/1.51 => ( ord_less_int @ C2 @ A ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_218_linorder__less__wlog,axiom,
% 2.67/1.51 ! [P: int > int > $o,A: int,B: int] :
% 2.67/1.51 ( ! [A5: int,B6: int] :
% 2.67/1.51 ( ( ord_less_int @ A5 @ B6 )
% 2.67/1.51 => ( P @ A5 @ B6 ) )
% 2.67/1.51 => ( ! [A5: int] : ( P @ A5 @ A5 )
% 2.67/1.51 => ( ! [A5: int,B6: int] :
% 2.67/1.51 ( ( P @ B6 @ A5 )
% 2.67/1.51 => ( P @ A5 @ B6 ) )
% 2.67/1.51 => ( P @ A @ B ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_219_less__imp__not__less,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_220_less__imp__not__less,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_221_order_Ostrict__trans,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.51 => ( ord_less_set_a @ A @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_222_order_Ostrict__trans,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.51 => ( ord_less_int @ A @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_223_dual__order_Oirrefl,axiom,
% 2.67/1.51 ! [A: set_a] :
% 2.67/1.51 ~ ( ord_less_set_a @ A @ A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_224_dual__order_Oirrefl,axiom,
% 2.67/1.51 ! [A: int] :
% 2.67/1.51 ~ ( ord_less_int @ A @ A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_225_linorder__cases,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ~ ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( ( X3 != Y2 )
% 2.67/1.51 => ( ord_less_int @ Y2 @ X3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_226_less__imp__triv,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a,P: $o] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ( ( ord_less_set_a @ Y2 @ X3 )
% 2.67/1.51 => P ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_227_less__imp__triv,axiom,
% 2.67/1.51 ! [X3: int,Y2: int,P: $o] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( ( ord_less_int @ Y2 @ X3 )
% 2.67/1.51 => P ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_228_less__imp__not__eq2,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ( Y2 != X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_229_less__imp__not__eq2,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( Y2 != X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_230_antisym__conv3,axiom,
% 2.67/1.51 ! [Y2: int,X3: int] :
% 2.67/1.51 ( ~ ( ord_less_int @ Y2 @ X3 )
% 2.67/1.51 => ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 2.67/1.51 = ( X3 = Y2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_231_less__not__sym,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_232_less__not__sym,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_233_less__imp__not__eq,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ( X3 != Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_234_less__imp__not__eq,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( X3 != Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_235_dual__order_Oasym,axiom,
% 2.67/1.51 ! [B: set_a,A: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ B @ A )
% 2.67/1.51 => ~ ( ord_less_set_a @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_236_dual__order_Oasym,axiom,
% 2.67/1.51 ! [B: int,A: int] :
% 2.67/1.51 ( ( ord_less_int @ B @ A )
% 2.67/1.51 => ~ ( ord_less_int @ A @ B ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_237_ord__less__eq__trans,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( ( B = C2 )
% 2.67/1.51 => ( ord_less_set_a @ A @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_238_ord__less__eq__trans,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( B = C2 )
% 2.67/1.51 => ( ord_less_int @ A @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_239_ord__eq__less__trans,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( A = B )
% 2.67/1.51 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.51 => ( ord_less_set_a @ A @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_240_ord__eq__less__trans,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int] :
% 2.67/1.51 ( ( A = B )
% 2.67/1.51 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.51 => ( ord_less_int @ A @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_241_less__irrefl,axiom,
% 2.67/1.51 ! [X3: set_a] :
% 2.67/1.51 ~ ( ord_less_set_a @ X3 @ X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_242_less__irrefl,axiom,
% 2.67/1.51 ! [X3: int] :
% 2.67/1.51 ~ ( ord_less_int @ X3 @ X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_243_less__linear,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 | ( X3 = Y2 )
% 2.67/1.51 | ( ord_less_int @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_244_less__trans,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ( ( ord_less_set_a @ Y2 @ Z2 )
% 2.67/1.51 => ( ord_less_set_a @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_245_less__trans,axiom,
% 2.67/1.51 ! [X3: int,Y2: int,Z2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( ( ord_less_int @ Y2 @ Z2 )
% 2.67/1.51 => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_246_less__asym_H,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ~ ( ord_less_set_a @ B @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_247_less__asym_H,axiom,
% 2.67/1.51 ! [A: int,B: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ~ ( ord_less_int @ B @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_248_less__asym,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_249_less__asym,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_250_less__imp__neq,axiom,
% 2.67/1.51 ! [X3: set_a,Y2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X3 @ Y2 )
% 2.67/1.51 => ( X3 != Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_251_less__imp__neq,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( X3 != Y2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_252_order_Oasym,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ~ ( ord_less_set_a @ B @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_253_order_Oasym,axiom,
% 2.67/1.51 ! [A: int,B: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ~ ( ord_less_int @ B @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_254_neq__iff,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( X3 != Y2 )
% 2.67/1.51 = ( ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 | ( ord_less_int @ Y2 @ X3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_255_neqE,axiom,
% 2.67/1.51 ! [X3: int,Y2: int] :
% 2.67/1.51 ( ( X3 != Y2 )
% 2.67/1.51 => ( ~ ( ord_less_int @ X3 @ Y2 )
% 2.67/1.51 => ( ord_less_int @ Y2 @ X3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_256_gt__ex,axiom,
% 2.67/1.51 ! [X3: int] :
% 2.67/1.51 ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_257_lt__ex,axiom,
% 2.67/1.51 ! [X3: int] :
% 2.67/1.51 ? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_258_order__less__subst2,axiom,
% 2.67/1.51 ! [A: int,B: int,F2: int > set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_259_order__less__subst2,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_260_order__less__subst2,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_261_order__less__subst2,axiom,
% 2.67/1.51 ! [A: int,B: int,F2: int > int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_262_order__less__subst1,axiom,
% 2.67/1.51 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_263_order__less__subst1,axiom,
% 2.67/1.51 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_264_order__less__subst1,axiom,
% 2.67/1.51 ! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_265_order__less__subst1,axiom,
% 2.67/1.51 ! [A: int,F2: int > int,B: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_266_ord__less__eq__subst,axiom,
% 2.67/1.51 ! [A: int,B: int,F2: int > set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( ( F2 @ B )
% 2.67/1.51 = C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_267_ord__less__eq__subst,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( ( ( F2 @ B )
% 2.67/1.51 = C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_268_ord__less__eq__subst,axiom,
% 2.67/1.51 ! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A @ B )
% 2.67/1.51 => ( ( ( F2 @ B )
% 2.67/1.51 = C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_269_ord__less__eq__subst,axiom,
% 2.67/1.51 ! [A: int,B: int,F2: int > int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ( ( F2 @ B )
% 2.67/1.51 = C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_270_ord__eq__less__subst,axiom,
% 2.67/1.51 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 2.67/1.51 ( ( A
% 2.67/1.51 = ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_271_ord__eq__less__subst,axiom,
% 2.67/1.51 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( A
% 2.67/1.51 = ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_272_ord__eq__less__subst,axiom,
% 2.67/1.51 ! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
% 2.67/1.51 ( ( A
% 2.67/1.51 = ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.51 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_273_ord__eq__less__subst,axiom,
% 2.67/1.51 ! [A: int,F2: int > int,B: int,C2: int] :
% 2.67/1.51 ( ( A
% 2.67/1.51 = ( F2 @ B ) )
% 2.67/1.51 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.51 => ( ! [X4: int,Y4: int] :
% 2.67/1.51 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.51 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.51 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_274_bot__fun__def,axiom,
% 2.67/1.51 ( bot_bot_a_o
% 2.67/1.51 = ( ^ [X2: a] : bot_bot_o ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_275_diff__right__commute,axiom,
% 2.67/1.51 ! [A: int,C2: int,B: int] :
% 2.67/1.51 ( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
% 2.67/1.51 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_276_diff__eq__diff__eq,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int,D: int] :
% 2.67/1.51 ( ( ( minus_minus_int @ A @ B )
% 2.67/1.51 = ( minus_minus_int @ C2 @ D ) )
% 2.67/1.51 => ( ( A = B )
% 2.67/1.51 = ( C2 = D ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_277_is__singleton__def,axiom,
% 2.67/1.51 ( is_singleton_a
% 2.67/1.51 = ( ^ [A4: set_a] :
% 2.67/1.51 ? [X2: a] :
% 2.67/1.51 ( A4
% 2.67/1.51 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_278_is__singletonE,axiom,
% 2.67/1.51 ! [A3: set_a] :
% 2.67/1.51 ( ( is_singleton_a @ A3 )
% 2.67/1.51 => ~ ! [X4: a] :
% 2.67/1.51 ( A3
% 2.67/1.51 != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_279_bot_Onot__eq__extremum,axiom,
% 2.67/1.51 ! [A: a > $o] :
% 2.67/1.51 ( ( A != bot_bot_a_o )
% 2.67/1.51 = ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_280_bot_Onot__eq__extremum,axiom,
% 2.67/1.51 ! [A: set_a] :
% 2.67/1.51 ( ( A != bot_bot_set_a )
% 2.67/1.51 = ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_281_bot_Oextremum__strict,axiom,
% 2.67/1.51 ! [A: a > $o] :
% 2.67/1.51 ~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_282_bot_Oextremum__strict,axiom,
% 2.67/1.51 ! [A: set_a] :
% 2.67/1.51 ~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_283_diff__strict__right__mono,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ A @ B )
% 2.67/1.51 => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_284_diff__strict__left__mono,axiom,
% 2.67/1.51 ! [B: int,A: int,C2: int] :
% 2.67/1.51 ( ( ord_less_int @ B @ A )
% 2.67/1.51 => ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_285_diff__eq__diff__less,axiom,
% 2.67/1.51 ! [A: int,B: int,C2: int,D: int] :
% 2.67/1.51 ( ( ( minus_minus_int @ A @ B )
% 2.67/1.51 = ( minus_minus_int @ C2 @ D ) )
% 2.67/1.51 => ( ( ord_less_int @ A @ B )
% 2.67/1.51 = ( ord_less_int @ C2 @ D ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_286_sup__Un__eq,axiom,
% 2.67/1.51 ! [R: set_a,S: set_a] :
% 2.67/1.51 ( ( sup_sup_a_o
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ R )
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ S ) )
% 2.67/1.51 = ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_287_bot__empty__eq,axiom,
% 2.67/1.51 ( bot_bot_a_o
% 2.67/1.51 = ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_288_Collect__empty__eq__bot,axiom,
% 2.67/1.51 ! [P: a > $o] :
% 2.67/1.51 ( ( ( collect_a @ P )
% 2.67/1.51 = bot_bot_set_a )
% 2.67/1.51 = ( P = bot_bot_a_o ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_289_psubset__trans,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,C: set_a] :
% 2.67/1.51 ( ( ord_less_set_a @ A3 @ B3 )
% 2.67/1.51 => ( ( ord_less_set_a @ B3 @ C )
% 2.67/1.51 => ( ord_less_set_a @ A3 @ C ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_290_less__set__def,axiom,
% 2.67/1.51 ( ord_less_set_a
% 2.67/1.51 = ( ^ [A4: set_a,B5: set_a] :
% 2.67/1.51 ( ord_less_a_o
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ A4 )
% 2.67/1.51 @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_291_psubsetD,axiom,
% 2.67/1.51 ! [A3: set_a,B3: set_a,C2: a] :
% 2.67/1.51 ( ( ord_less_set_a @ A3 @ B3 )
% 2.67/1.51 => ( ( member_a @ C2 @ A3 )
% 2.67/1.51 => ( member_a @ C2 @ B3 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_292_remove__def,axiom,
% 2.67/1.51 ( remove_a
% 2.67/1.51 = ( ^ [X2: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_293_Tree_Osimps_I15_J,axiom,
% 2.67/1.51 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 2.67/1.51 ( ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) )
% 2.67/1.51 = ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary256242811Tree_a @ X21 ) @ ( insert_a @ X22 @ bot_bot_set_a ) ) @ ( binary256242811Tree_a @ X23 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_294_member__remove,axiom,
% 2.67/1.51 ! [X3: a,Y2: a,A3: set_a] :
% 2.67/1.51 ( ( member_a @ X3 @ ( remove_a @ Y2 @ A3 ) )
% 2.67/1.51 = ( ( member_a @ X3 @ A3 )
% 2.67/1.51 & ( X3 != Y2 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_295_Tree_Oset__intros_I3_J,axiom,
% 2.67/1.51 ! [Ya: a,X23: binary1439146945Tree_a,X21: binary1439146945Tree_a,X22: a] :
% 2.67/1.51 ( ( member_a @ Ya @ ( binary256242811Tree_a @ X23 ) )
% 2.67/1.51 => ( member_a @ Ya @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_296_Tree_Oset__intros_I2_J,axiom,
% 2.67/1.51 ! [X22: a,X21: binary1439146945Tree_a,X23: binary1439146945Tree_a] : ( member_a @ X22 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_297_Tree_Oset__intros_I1_J,axiom,
% 2.67/1.51 ! [Y2: a,X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 2.67/1.51 ( ( member_a @ Y2 @ ( binary256242811Tree_a @ X21 ) )
% 2.67/1.51 => ( member_a @ Y2 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_298_Tree_Oset__cases,axiom,
% 2.67/1.51 ! [E: a,A: binary1439146945Tree_a] :
% 2.67/1.51 ( ( member_a @ E @ ( binary256242811Tree_a @ A ) )
% 2.67/1.51 => ( ! [Z1: binary1439146945Tree_a] :
% 2.67/1.51 ( ? [Z22: a,Z3: binary1439146945Tree_a] :
% 2.67/1.51 ( A
% 2.67/1.51 = ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
% 2.67/1.51 => ~ ( member_a @ E @ ( binary256242811Tree_a @ Z1 ) ) )
% 2.67/1.51 => ( ! [Z1: binary1439146945Tree_a,Z3: binary1439146945Tree_a] :
% 2.67/1.51 ( A
% 2.67/1.51 != ( binary717961607le_T_a @ Z1 @ E @ Z3 ) )
% 2.67/1.51 => ~ ! [Z1: binary1439146945Tree_a,Z22: a,Z3: binary1439146945Tree_a] :
% 2.67/1.51 ( ( A
% 2.67/1.51 = ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
% 2.67/1.51 => ~ ( member_a @ E @ ( binary256242811Tree_a @ Z3 ) ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_299_Tree_Osimps_I14_J,axiom,
% 2.67/1.51 ( ( binary256242811Tree_a @ binary476621312_Tip_a )
% 2.67/1.51 = bot_bot_set_a ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_300_Set_Ois__empty__def,axiom,
% 2.67/1.51 ( is_empty_a
% 2.67/1.51 = ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_301_the__elem__def,axiom,
% 2.67/1.51 ( the_elem_a
% 2.67/1.51 = ( ^ [X5: set_a] :
% 2.67/1.51 ( the_a
% 2.67/1.51 @ ^ [X2: a] :
% 2.67/1.51 ( X5
% 2.67/1.51 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_302_the__sym__eq__trivial,axiom,
% 2.67/1.51 ! [X3: a] :
% 2.67/1.51 ( ( the_a
% 2.67/1.51 @ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
% 2.67/1.51 @ X3 ) )
% 2.67/1.51 = X3 ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_303_the__eq__trivial,axiom,
% 2.67/1.51 ! [A: a] :
% 2.67/1.51 ( ( the_a
% 2.67/1.51 @ ^ [X2: a] : ( X2 = A ) )
% 2.67/1.51 = A ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_304_the__equality,axiom,
% 2.67/1.51 ! [P: a > $o,A: a] :
% 2.67/1.51 ( ( P @ A )
% 2.67/1.51 => ( ! [X4: a] :
% 2.67/1.51 ( ( P @ X4 )
% 2.67/1.51 => ( X4 = A ) )
% 2.67/1.51 => ( ( the_a @ P )
% 2.67/1.51 = A ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_305_theI,axiom,
% 2.67/1.51 ! [P: a > $o,A: a] :
% 2.67/1.51 ( ( P @ A )
% 2.67/1.51 => ( ! [X4: a] :
% 2.67/1.51 ( ( P @ X4 )
% 2.67/1.51 => ( X4 = A ) )
% 2.67/1.51 => ( P @ ( the_a @ P ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_306_theI_H,axiom,
% 2.67/1.51 ! [P: a > $o] :
% 2.67/1.51 ( ? [X: a] :
% 2.67/1.51 ( ( P @ X )
% 2.67/1.51 & ! [Y4: a] :
% 2.67/1.51 ( ( P @ Y4 )
% 2.67/1.51 => ( Y4 = X ) ) )
% 2.67/1.51 => ( P @ ( the_a @ P ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_307_theI2,axiom,
% 2.67/1.51 ! [P: a > $o,A: a,Q: a > $o] :
% 2.67/1.51 ( ( P @ A )
% 2.67/1.51 => ( ! [X4: a] :
% 2.67/1.51 ( ( P @ X4 )
% 2.67/1.51 => ( X4 = A ) )
% 2.67/1.51 => ( ! [X4: a] :
% 2.67/1.51 ( ( P @ X4 )
% 2.67/1.51 => ( Q @ X4 ) )
% 2.67/1.51 => ( Q @ ( the_a @ P ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_308_If__def,axiom,
% 2.67/1.51 ( if_a
% 2.67/1.51 = ( ^ [P2: $o,X2: a,Y: a] :
% 2.67/1.51 ( the_a
% 2.67/1.51 @ ^ [Z4: a] :
% 2.67/1.51 ( ( P2
% 2.67/1.51 => ( Z4 = X2 ) )
% 2.67/1.51 & ( ~ P2
% 2.67/1.51 => ( Z4 = Y ) ) ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_309_the1I2,axiom,
% 2.67/1.51 ! [P: a > $o,Q: a > $o] :
% 2.67/1.51 ( ? [X: a] :
% 2.67/1.51 ( ( P @ X )
% 2.67/1.51 & ! [Y4: a] :
% 2.67/1.51 ( ( P @ Y4 )
% 2.67/1.51 => ( Y4 = X ) ) )
% 2.67/1.51 => ( ! [X4: a] :
% 2.67/1.51 ( ( P @ X4 )
% 2.67/1.51 => ( Q @ X4 ) )
% 2.67/1.51 => ( Q @ ( the_a @ P ) ) ) ) ).
% 2.67/1.51
% 2.67/1.51 thf(fact_310_the1__equality,axiom,
% 2.67/1.51 ! [P: a > $o,A: a] :
% 2.67/1.51 ( ? [X: a] :
% 2.67/1.51 ( ( P @ X )
% 2.67/1.51 & ! [Y4: a] :
% 2.67/1.51 ( ( P @ Y4 )
% 2.67/1.51 => ( Y4 = X ) ) )
% 2.67/1.51 => ( ( P @ A )
% 2.67/1.52 => ( ( the_a @ P )
% 2.67/1.52 = A ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_311_psubset__insert__iff,axiom,
% 2.67/1.52 ! [A3: set_a,X3: a,B3: set_a] :
% 2.67/1.52 ( ( ord_less_set_a @ A3 @ ( insert_a @ X3 @ B3 ) )
% 2.67/1.52 = ( ( ( member_a @ X3 @ B3 )
% 2.67/1.52 => ( ord_less_set_a @ A3 @ B3 ) )
% 2.67/1.52 & ( ~ ( member_a @ X3 @ B3 )
% 2.67/1.52 => ( ( ( member_a @ X3 @ A3 )
% 2.67/1.52 => ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B3 ) )
% 2.67/1.52 & ( ~ ( member_a @ X3 @ A3 )
% 2.67/1.52 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_312_pairwise__alt,axiom,
% 2.67/1.52 ( pairwise_a
% 2.67/1.52 = ( ^ [R2: a > a > $o,S2: set_a] :
% 2.67/1.52 ! [X2: a] :
% 2.67/1.52 ( ( member_a @ X2 @ S2 )
% 2.67/1.52 => ! [Y: a] :
% 2.67/1.52 ( ( member_a @ Y @ ( minus_minus_set_a @ S2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
% 2.67/1.52 => ( R2 @ X2 @ Y ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_313_order__refl,axiom,
% 2.67/1.52 ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_314_order__refl,axiom,
% 2.67/1.52 ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_315_subset__antisym,axiom,
% 2.67/1.52 ! [A3: set_a,B3: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.67/1.52 => ( ( ord_less_eq_set_a @ B3 @ A3 )
% 2.67/1.52 => ( A3 = B3 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_316_subsetI,axiom,
% 2.67/1.52 ! [A3: set_a,B3: set_a] :
% 2.67/1.52 ( ! [X4: a] :
% 2.67/1.52 ( ( member_a @ X4 @ A3 )
% 2.67/1.52 => ( member_a @ X4 @ B3 ) )
% 2.67/1.52 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_317_sup_Obounded__iff,axiom,
% 2.67/1.52 ! [B: a > $o,C2: a > $o,A: a > $o] :
% 2.67/1.52 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B @ C2 ) @ A )
% 2.67/1.52 = ( ( ord_less_eq_a_o @ B @ A )
% 2.67/1.52 & ( ord_less_eq_a_o @ C2 @ A ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_318_sup_Obounded__iff,axiom,
% 2.67/1.52 ! [B: int,C2: int,A: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
% 2.67/1.52 = ( ( ord_less_eq_int @ B @ A )
% 2.67/1.52 & ( ord_less_eq_int @ C2 @ A ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_319_sup_Obounded__iff,axiom,
% 2.67/1.52 ! [B: set_a,C2: set_a,A: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
% 2.67/1.52 = ( ( ord_less_eq_set_a @ B @ A )
% 2.67/1.52 & ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_320_le__sup__iff,axiom,
% 2.67/1.52 ! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
% 2.67/1.52 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
% 2.67/1.52 = ( ( ord_less_eq_a_o @ X3 @ Z2 )
% 2.67/1.52 & ( ord_less_eq_a_o @ Y2 @ Z2 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_321_le__sup__iff,axiom,
% 2.67/1.52 ! [X3: int,Y2: int,Z2: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
% 2.67/1.52 = ( ( ord_less_eq_int @ X3 @ Z2 )
% 2.67/1.52 & ( ord_less_eq_int @ Y2 @ Z2 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_322_le__sup__iff,axiom,
% 2.67/1.52 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
% 2.67/1.52 = ( ( ord_less_eq_set_a @ X3 @ Z2 )
% 2.67/1.52 & ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_323_empty__subsetI,axiom,
% 2.67/1.52 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_324_subset__empty,axiom,
% 2.67/1.52 ! [A3: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 2.67/1.52 = ( A3 = bot_bot_set_a ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_325_insert__subset,axiom,
% 2.67/1.52 ! [X3: a,A3: set_a,B3: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
% 2.67/1.52 = ( ( member_a @ X3 @ B3 )
% 2.67/1.52 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_326_Un__subset__iff,axiom,
% 2.67/1.52 ! [A3: set_a,B3: set_a,C: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
% 2.67/1.52 = ( ( ord_less_eq_set_a @ A3 @ C )
% 2.67/1.52 & ( ord_less_eq_set_a @ B3 @ C ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_327_psubsetI,axiom,
% 2.67/1.52 ! [A3: set_a,B3: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.67/1.52 => ( ( A3 != B3 )
% 2.67/1.52 => ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_328_singleton__insert__inj__eq_H,axiom,
% 2.67/1.52 ! [A: a,A3: set_a,B: a] :
% 2.67/1.52 ( ( ( insert_a @ A @ A3 )
% 2.67/1.52 = ( insert_a @ B @ bot_bot_set_a ) )
% 2.67/1.52 = ( ( A = B )
% 2.67/1.52 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_329_singleton__insert__inj__eq,axiom,
% 2.67/1.52 ! [B: a,A: a,A3: set_a] :
% 2.67/1.52 ( ( ( insert_a @ B @ bot_bot_set_a )
% 2.67/1.52 = ( insert_a @ A @ A3 ) )
% 2.67/1.52 = ( ( A = B )
% 2.67/1.52 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_330_Diff__eq__empty__iff,axiom,
% 2.67/1.52 ! [A3: set_a,B3: set_a] :
% 2.67/1.52 ( ( ( minus_minus_set_a @ A3 @ B3 )
% 2.67/1.52 = bot_bot_set_a )
% 2.67/1.52 = ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_331_leD,axiom,
% 2.67/1.52 ! [Y2: set_a,X3: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ Y2 @ X3 )
% 2.67/1.52 => ~ ( ord_less_set_a @ X3 @ Y2 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_332_leD,axiom,
% 2.67/1.52 ! [Y2: int,X3: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ Y2 @ X3 )
% 2.67/1.52 => ~ ( ord_less_int @ X3 @ Y2 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_333_leI,axiom,
% 2.67/1.52 ! [X3: int,Y2: int] :
% 2.67/1.52 ( ~ ( ord_less_int @ X3 @ Y2 )
% 2.67/1.52 => ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_334_le__less,axiom,
% 2.67/1.52 ( ord_less_eq_set_a
% 2.67/1.52 = ( ^ [X2: set_a,Y: set_a] :
% 2.67/1.52 ( ( ord_less_set_a @ X2 @ Y )
% 2.67/1.52 | ( X2 = Y ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_335_le__less,axiom,
% 2.67/1.52 ( ord_less_eq_int
% 2.67/1.52 = ( ^ [X2: int,Y: int] :
% 2.67/1.52 ( ( ord_less_int @ X2 @ Y )
% 2.67/1.52 | ( X2 = Y ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_336_less__le,axiom,
% 2.67/1.52 ( ord_less_set_a
% 2.67/1.52 = ( ^ [X2: set_a,Y: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.67/1.52 & ( X2 != Y ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_337_less__le,axiom,
% 2.67/1.52 ( ord_less_int
% 2.67/1.52 = ( ^ [X2: int,Y: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ X2 @ Y )
% 2.67/1.52 & ( X2 != Y ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_338_order__le__less__subst1,axiom,
% 2.67/1.52 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_339_order__le__less__subst1,axiom,
% 2.67/1.52 ! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.52 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.52 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_340_order__le__less__subst1,axiom,
% 2.67/1.52 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 2.67/1.52 ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_set_a @ B @ C2 )
% 2.67/1.52 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.52 ( ( ord_less_set_a @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_341_order__le__less__subst1,axiom,
% 2.67/1.52 ! [A: int,F2: int > int,B: int,C2: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_int @ B @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_342_order__le__less__subst2,axiom,
% 2.67/1.52 ! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A @ B )
% 2.67/1.52 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 2.67/1.52 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_343_order__le__less__subst2,axiom,
% 2.67/1.52 ! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ A @ B )
% 2.67/1.52 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 2.67/1.52 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_344_order__le__less__subst2,axiom,
% 2.67/1.52 ! [A: int,B: int,F2: int > set_a,C2: set_a] :
% 2.67/1.52 ( ( ord_less_eq_int @ A @ B )
% 2.67/1.52 => ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_345_order__le__less__subst2,axiom,
% 2.67/1.52 ! [A: int,B: int,F2: int > int,C2: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ A @ B )
% 2.67/1.52 => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_346_order__less__le__subst1,axiom,
% 2.67/1.52 ! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
% 2.67/1.52 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_eq_set_a @ B @ C2 )
% 2.67/1.52 => ( ! [X4: set_a,Y4: set_a] :
% 2.67/1.52 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_347_order__less__le__subst1,axiom,
% 2.67/1.52 ! [A: set_a,F2: int > set_a,B: int,C2: int] :
% 2.67/1.52 ( ( ord_less_set_a @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_eq_int @ B @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_348_order__less__le__subst1,axiom,
% 2.67/1.52 ! [A: int,F2: int > int,B: int,C2: int] :
% 2.67/1.52 ( ( ord_less_int @ A @ ( F2 @ B ) )
% 2.67/1.52 => ( ( ord_less_eq_int @ B @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_349_order__less__le__subst2,axiom,
% 2.67/1.52 ! [A: int,B: int,F2: int > int,C2: int] :
% 2.67/1.52 ( ( ord_less_int @ A @ B )
% 2.67/1.52 => ( ( ord_less_eq_int @ ( F2 @ B ) @ C2 )
% 2.67/1.52 => ( ! [X4: int,Y4: int] :
% 2.67/1.52 ( ( ord_less_int @ X4 @ Y4 )
% 2.67/1.52 => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
% 2.67/1.52 => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_350_not__le,axiom,
% 2.67/1.52 ! [X3: int,Y2: int] :
% 2.67/1.52 ( ( ~ ( ord_less_eq_int @ X3 @ Y2 ) )
% 2.67/1.52 = ( ord_less_int @ Y2 @ X3 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_351_not__less,axiom,
% 2.67/1.52 ! [X3: int,Y2: int] :
% 2.67/1.52 ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 2.67/1.52 = ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_352_le__neq__trans,axiom,
% 2.67/1.52 ! [A: int,B: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ A @ B )
% 2.67/1.52 => ( ( A != B )
% 2.67/1.52 => ( ord_less_int @ A @ B ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_353_antisym__conv1,axiom,
% 2.67/1.52 ! [X3: int,Y2: int] :
% 2.67/1.52 ( ~ ( ord_less_int @ X3 @ Y2 )
% 2.67/1.52 => ( ( ord_less_eq_int @ X3 @ Y2 )
% 2.67/1.52 = ( X3 = Y2 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(fact_354_antisym__conv2,axiom,
% 2.67/1.52 ! [X3: int,Y2: int] :
% 2.67/1.52 ( ( ord_less_eq_int @ X3 @ Y2 )
% 2.67/1.52 => ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
% 2.67/1.52 = ( X3 = Y2 ) ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(help_If_3_1_If_001tf__a_T,axiom,
% 2.67/1.52 ! [P: $o] :
% 2.67/1.52 ( ( P = $true )
% 2.67/1.52 | ( P = $false ) ) ).
% 2.67/1.52
% 2.67/1.52 thf(help_If_2_1_If_001tf__a_T,axiom,
% 2.67/1.52 ! [X3: a,Y2: a] :
% 2.67/1.52 ( ( if_a @ $false @ X3 @ Y2 )
% 2.67/1.52 = Y2 ) ).
% 2.67/1.52
% 2.67/1.52 thf(help_If_1_1_If_001tf__a_T,axiom,
% 2.67/1.52 ! [X3: a,Y2: a] :
% 2.67/1.52 ( ( if_a @ $true @ X3 @ Y2 )
% 2.67/1.52 = X3 ) ).
% 2.67/1.52
% 2.67/1.52 thf(conj_0,conjecture,
% 2.67/1.52 ! [X4: a] :
% 2.67/1.52 ( ( member_a @ X4 @ ( binary504661350_eqs_a @ h @ e ) )
% 2.67/1.52 => ( ( h @ X4 )
% 2.67/1.52 != ( h @ x ) ) ) ).
% 2.67/1.52 ------------------------
% 2.67/1.54 % (4863)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/2Mi)
% 2.67/1.54 % (4861)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_4570 for (2999ds/4Mi)
% 2.67/1.54 % (4863)Instruction limit reached!
% 2.67/1.54 % (4863)------------------------------
% 2.67/1.54 % (4863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.54 % (4863)Termination reason: Unknown
% 2.67/1.54 % (4863)Termination phase: shuffling
% 2.67/1.54
% 2.67/1.54 % (4863)Memory used [KB]: 1407
% 2.67/1.54 % (4863)Time elapsed: 0.002 s
% 2.67/1.54 % (4863)Instructions burned: 2 (million)
% 2.67/1.54 % (4863)------------------------------
% 2.67/1.54 % (4863)------------------------------
% 2.67/1.55 % (4861)Instruction limit reached!
% 2.67/1.55 % (4861)------------------------------
% 2.67/1.55 % (4861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.55 % (4861)Termination reason: Unknown
% 2.67/1.55 % (4861)Termination phase: shuffling
% 2.67/1.55
% 2.67/1.55 % (4861)Memory used [KB]: 1535
% 2.67/1.55 % (4861)Time elapsed: 0.003 s
% 2.67/1.55 % (4861)Instructions burned: 6 (million)
% 2.67/1.55 % (4861)------------------------------
% 2.67/1.55 % (4861)------------------------------
% 2.67/1.55 % (4860)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/183Mi)
% 2.67/1.55 % (4862)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_4570 for (2999ds/27Mi)
% 2.67/1.55 % (4864)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_4570 for (2999ds/2Mi)
% 2.67/1.55 % (4865)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_4570 for (2999ds/275Mi)
% 2.67/1.55 % (4866)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/18Mi)
% 2.67/1.55 % (4864)Instruction limit reached!
% 2.67/1.55 % (4864)------------------------------
% 2.67/1.55 % (4864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.55 % (4864)Termination reason: Unknown
% 2.67/1.55 % (4864)Termination phase: shuffling
% 2.67/1.55
% 2.67/1.55 % (4864)Memory used [KB]: 1407
% 2.67/1.55 % (4864)Time elapsed: 0.004 s
% 2.67/1.55 % (4864)Instructions burned: 3 (million)
% 2.67/1.55 % (4864)------------------------------
% 2.67/1.55 % (4864)------------------------------
% 2.67/1.55 % (4867)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/3Mi)
% 2.67/1.55 % (4867)Instruction limit reached!
% 2.67/1.55 % (4867)------------------------------
% 2.67/1.55 % (4867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.55 % (4867)Termination reason: Unknown
% 2.67/1.55 % (4867)Termination phase: shuffling
% 2.67/1.55
% 2.67/1.55 % (4867)Memory used [KB]: 1407
% 2.67/1.55 % (4867)Time elapsed: 0.002 s
% 2.67/1.55 % (4867)Instructions burned: 3 (million)
% 2.67/1.55 % (4867)------------------------------
% 2.67/1.55 % (4867)------------------------------
% 2.67/1.56 % (4868)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_4570 for (2999ds/37Mi)
% 2.67/1.56 % (4866)Instruction limit reached!
% 2.67/1.56 % (4866)------------------------------
% 2.67/1.56 % (4866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.56 % (4866)Termination reason: Unknown
% 2.67/1.56 % (4866)Termination phase: shuffling
% 2.67/1.56
% 2.67/1.56 % (4866)Memory used [KB]: 1791
% 2.67/1.56 % (4866)Time elapsed: 0.014 s
% 2.67/1.56 % (4866)Instructions burned: 19 (million)
% 2.67/1.56 % (4866)------------------------------
% 2.67/1.56 % (4866)------------------------------
% 2.67/1.56 % (4870)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_4570 for (2999ds/3Mi)
% 2.67/1.57 % (4870)Instruction limit reached!
% 2.67/1.57 % (4870)------------------------------
% 2.67/1.57 % (4870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (4870)Termination reason: Unknown
% 2.67/1.57 % (4870)Termination phase: shuffling
% 2.67/1.57
% 2.67/1.57 % (4870)Memory used [KB]: 1535
% 2.67/1.57 % (4870)Time elapsed: 0.003 s
% 2.67/1.57 % (4870)Instructions burned: 5 (million)
% 2.67/1.57 % (4870)------------------------------
% 2.67/1.57 % (4870)------------------------------
% 2.67/1.57 % (4862)Instruction limit reached!
% 2.67/1.57 % (4862)------------------------------
% 2.67/1.57 % (4862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (4862)Termination reason: Unknown
% 2.67/1.57 % (4862)Termination phase: Property scanning
% 2.67/1.57
% 2.67/1.57 % (4862)Memory used [KB]: 1918
% 2.67/1.57 % (4862)Time elapsed: 0.018 s
% 2.67/1.57 % (4862)Instructions burned: 27 (million)
% 2.67/1.57 % (4862)------------------------------
% 2.67/1.57 % (4862)------------------------------
% 2.67/1.57 % (4868)Instruction limit reached!
% 2.67/1.57 % (4868)------------------------------
% 2.67/1.57 % (4868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (4868)Termination reason: Unknown
% 2.67/1.57 % (4868)Termination phase: shuffling
% 2.67/1.57
% 2.67/1.57 % (4868)Memory used [KB]: 2302
% 2.67/1.57 % (4868)Time elapsed: 0.013 s
% 2.67/1.57 % (4868)Instructions burned: 40 (million)
% 2.67/1.57 % (4868)------------------------------
% 2.67/1.57 % (4868)------------------------------
% 2.67/1.57 % (4869)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_4570 for (2999ds/15Mi)
% 2.67/1.57 % (4869)Instruction limit reached!
% 2.67/1.57 % (4869)------------------------------
% 2.67/1.57 % (4869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (4869)Termination reason: Unknown
% 2.67/1.57 % (4869)Termination phase: shuffling
% 2.67/1.57
% 2.67/1.57 % (4869)Memory used [KB]: 1663
% 2.67/1.57 % (4869)Time elapsed: 0.008 s
% 2.67/1.57 % (4869)Instructions burned: 17 (million)
% 2.67/1.57 % (4869)------------------------------
% 2.67/1.57 % (4869)------------------------------
% 2.67/1.57 % (4872)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/7Mi)
% 2.67/1.58 % (4874)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_4570 for (2999ds/3Mi)
% 2.67/1.58 % (4872)Instruction limit reached!
% 2.67/1.58 % (4872)------------------------------
% 2.67/1.58 % (4872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.58 % (4872)Termination reason: Unknown
% 2.67/1.58 % (4872)Termination phase: shuffling
% 2.67/1.58
% 2.67/1.58 % (4872)Memory used [KB]: 1535
% 2.67/1.58 % (4872)Time elapsed: 0.004 s
% 2.67/1.58 % (4872)Instructions burned: 9 (million)
% 2.67/1.58 % (4872)------------------------------
% 2.67/1.58 % (4872)------------------------------
% 2.67/1.58 % (4874)Instruction limit reached!
% 2.67/1.58 % (4874)------------------------------
% 2.67/1.58 % (4874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.58 % (4874)Termination reason: Unknown
% 2.67/1.58 % (4874)Termination phase: shuffling
% 2.67/1.58
% 2.67/1.58 % (4874)Memory used [KB]: 1535
% 2.67/1.58 % (4874)Time elapsed: 0.003 s
% 2.67/1.58 % (4874)Instructions burned: 6 (million)
% 2.67/1.58 % (4874)------------------------------
% 2.67/1.58 % (4874)------------------------------
% 2.67/1.58 % (4871)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_4570 for (2999ds/1041Mi)
% 2.67/1.58 % (4873)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/16Mi)
% 2.67/1.58 % (4875)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_4570 for (2999ds/3Mi)
% 2.67/1.59 % (4875)Instruction limit reached!
% 2.67/1.59 % (4875)------------------------------
% 2.67/1.59 % (4875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.59 % (4875)Termination reason: Unknown
% 2.67/1.59 % (4875)Termination phase: shuffling
% 2.67/1.59
% 2.67/1.59 % (4875)Memory used [KB]: 1407
% 2.67/1.59 % (4875)Time elapsed: 0.003 s
% 2.67/1.59 % (4875)Instructions burned: 4 (million)
% 2.67/1.59 % (4875)------------------------------
% 2.67/1.59 % (4875)------------------------------
% 2.67/1.59 % (4876)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/7Mi)
% 2.67/1.59 % (4877)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/3Mi)
% 2.67/1.59 % (4877)Instruction limit reached!
% 2.67/1.59 % (4877)------------------------------
% 2.67/1.59 % (4877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.59 % (4877)Termination reason: Unknown
% 2.67/1.59 % (4877)Termination phase: shuffling
% 2.67/1.59
% 2.67/1.59 % (4877)Memory used [KB]: 1407
% 2.67/1.59 % (4877)Time elapsed: 0.002 s
% 2.67/1.59 % (4877)Instructions burned: 3 (million)
% 2.67/1.59 % (4877)------------------------------
% 2.67/1.59 % (4877)------------------------------
% 2.67/1.59 % (4876)Instruction limit reached!
% 2.67/1.59 % (4876)------------------------------
% 2.67/1.59 % (4876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.59 % (4876)Termination reason: Unknown
% 2.67/1.59 % (4876)Termination phase: shuffling
% 2.67/1.59
% 2.67/1.59 % (4876)Memory used [KB]: 1535
% 2.67/1.59 % (4876)Time elapsed: 0.004 s
% 2.67/1.59 % (4876)Instructions burned: 10 (million)
% 2.67/1.59 % (4876)------------------------------
% 2.67/1.59 % (4876)------------------------------
% 2.67/1.59 % (4873)Instruction limit reached!
% 2.67/1.59 % (4873)------------------------------
% 2.67/1.59 % (4873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.59 % (4873)Termination reason: Unknown
% 2.67/1.59 % (4873)Termination phase: shuffling
% 2.67/1.59
% 2.67/1.59 % (4873)Memory used [KB]: 1791
% 2.67/1.59 % (4873)Time elapsed: 0.009 s
% 2.67/1.59 % (4873)Instructions burned: 18 (million)
% 2.67/1.59 % (4873)------------------------------
% 2.67/1.59 % (4873)------------------------------
% 2.67/1.60 % (4878)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/4Mi)
% 2.70/1.60 % (4878)Instruction limit reached!
% 2.70/1.60 % (4878)------------------------------
% 2.70/1.60 % (4878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (4878)Termination reason: Unknown
% 2.70/1.60 % (4878)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (4878)Memory used [KB]: 1407
% 2.70/1.60 % (4878)Time elapsed: 0.003 s
% 2.70/1.60 % (4878)Instructions burned: 4 (million)
% 2.70/1.60 % (4878)------------------------------
% 2.70/1.60 % (4878)------------------------------
% 2.70/1.60 % (4879)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/18Mi)
% 2.70/1.60 % (4880)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/710Mi)
% 2.70/1.60 % (4881)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/6Mi)
% 2.70/1.60 % (4881)Instruction limit reached!
% 2.70/1.60 % (4881)------------------------------
% 2.70/1.60 % (4881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (4881)Termination reason: Unknown
% 2.70/1.60 % (4881)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (4881)Memory used [KB]: 1535
% 2.70/1.60 % (4881)Time elapsed: 0.004 s
% 2.70/1.60 % (4881)Instructions burned: 6 (million)
% 2.70/1.60 % (4881)------------------------------
% 2.70/1.60 % (4881)------------------------------
% 2.70/1.60 % (4879)Instruction limit reached!
% 2.70/1.60 % (4879)------------------------------
% 2.70/1.60 % (4879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (4879)Termination reason: Unknown
% 2.70/1.60 % (4879)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (4879)Memory used [KB]: 1791
% 2.70/1.60 % (4879)Time elapsed: 0.007 s
% 2.70/1.60 % (4879)Instructions burned: 19 (million)
% 2.70/1.60 % (4879)------------------------------
% 2.70/1.60 % (4879)------------------------------
% 2.70/1.61 % (4882)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_4570 for (2999ds/902Mi)
% 2.70/1.61 % (4884)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4570 for (2999ds/5Mi)
% 2.70/1.62 % (4884)Instruction limit reached!
% 2.70/1.62 % (4884)------------------------------
% 2.70/1.62 % (4884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (4884)Termination reason: Unknown
% 2.70/1.62 % (4884)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (4884)Memory used [KB]: 1535
% 2.70/1.62 % (4884)Time elapsed: 0.003 s
% 2.70/1.62 % (4884)Instructions burned: 7 (million)
% 2.70/1.62 % (4884)------------------------------
% 2.70/1.62 % (4884)------------------------------
% 2.70/1.62 % (4883)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_4570 for (2999ds/21Mi)
% 2.70/1.62 % (4865)First to succeed.
% 2.70/1.62 % (4885)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_4570 for (2999ds/6Mi)
% 2.70/1.63 % (4865)Refutation found. Thanks to Tanya!
% 2.70/1.63 % SZS status Theorem for DTF2THF_4570
% 2.70/1.63 % SZS output start Proof for DTF2THF_4570
% 2.70/1.63 thf(type_def_5, type, binary1439146945Tree_a: $tType).
% 2.70/1.63 thf(type_def_6, type, a: $tType).
% 2.70/1.63 thf(type_def_8, type, set_a: $tType).
% 2.70/1.63 thf(type_def_9, type, int: $tType).
% 2.70/1.63 thf(func_def_0, type, binary1439146945Tree_a: $tType).
% 2.70/1.63 thf(func_def_1, type, set_a: $tType).
% 2.70/1.63 thf(func_def_2, type, int: $tType).
% 2.70/1.63 thf(func_def_3, type, a: $tType).
% 2.70/1.63 thf(func_def_4, type, binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_5, type, binary476621312_Tip_a: binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_6, type, binary256242811Tree_a: binary1439146945Tree_a > set_a).
% 2.70/1.63 thf(func_def_7, type, binary1226383794sert_a: (a > int) > a > binary1439146945Tree_a > binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_8, type, binary504661350_eqs_a: (a > int) > a > set_a).
% 2.70/1.63 thf(func_def_9, type, binary2053421120memb_a: (a > int) > a > binary1439146945Tree_a > $o).
% 2.70/1.63 thf(func_def_10, type, binary945792244etOf_a: binary1439146945Tree_a > set_a).
% 2.70/1.63 thf(func_def_11, type, binary1721989714Tree_a: (a > int) > binary1439146945Tree_a > $o).
% 2.70/1.63 thf(func_def_12, type, binary670562003pred_a: (a > int) > a > a > binary1439146945Tree_a > $o).
% 2.70/1.63 thf(func_def_13, type, minus_minus_a_o: (a > $o) > (a > $o) > a > $o).
% 2.70/1.63 thf(func_def_14, type, minus_minus_o: $o > $o > $o).
% 2.70/1.63 thf(func_def_15, type, minus_minus_int: int > int > int).
% 2.70/1.63 thf(func_def_16, type, minus_minus_set_a: set_a > set_a > set_a).
% 2.70/1.63 thf(func_def_17, type, the_a: (a > $o) > a).
% 2.70/1.63 thf(func_def_18, type, if_a: $o > a > a > a).
% 2.70/1.63 thf(func_def_19, type, sup_sup_a_o: (a > $o) > (a > $o) > a > $o).
% 2.70/1.63 thf(func_def_20, type, sup_sup_o: $o > $o > $o).
% 2.70/1.63 thf(func_def_21, type, sup_sup_int: int > int > int).
% 2.70/1.63 thf(func_def_22, type, sup_sup_set_a: set_a > set_a > set_a).
% 2.70/1.63 thf(func_def_23, type, bot_bot_a_o: a > $o).
% 2.70/1.63 thf(func_def_25, type, bot_bot_set_a: set_a).
% 2.70/1.63 thf(func_def_26, type, ord_less_a_o: (a > $o) > (a > $o) > $o).
% 2.70/1.63 thf(func_def_27, type, ord_less_int: int > int > $o).
% 2.70/1.63 thf(func_def_28, type, ord_less_set_a: set_a > set_a > $o).
% 2.70/1.63 thf(func_def_29, type, ord_less_eq_a_o: (a > $o) > (a > $o) > $o).
% 2.70/1.63 thf(func_def_30, type, ord_less_eq_int: int > int > $o).
% 2.70/1.63 thf(func_def_31, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.70/1.63 thf(func_def_32, type, collect_a: (a > $o) > set_a).
% 2.70/1.63 thf(func_def_33, type, insert_a: a > set_a > set_a).
% 2.70/1.63 thf(func_def_34, type, is_empty_a: set_a > $o).
% 2.70/1.63 thf(func_def_35, type, is_singleton_a: set_a > $o).
% 2.70/1.63 thf(func_def_36, type, pairwise_a: (a > a > $o) > set_a > $o).
% 2.70/1.63 thf(func_def_37, type, remove_a: a > set_a > set_a).
% 2.70/1.63 thf(func_def_38, type, the_elem_a: set_a > a).
% 2.70/1.63 thf(func_def_39, type, member_a: a > set_a > $o).
% 2.70/1.63 thf(func_def_40, type, e: a).
% 2.70/1.63 thf(func_def_41, type, h: a > int).
% 2.70/1.63 thf(func_def_42, type, t1: binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_43, type, t2: binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_44, type, x: a).
% 2.70/1.63 thf(func_def_60, type, sP0: set_a > a > set_a > $o).
% 2.70/1.63 thf(func_def_61, type, sP1: set_a > a > a > set_a > $o).
% 2.70/1.63 thf(func_def_62, type, sP2: a > set_a > a > set_a > $o).
% 2.70/1.63 thf(func_def_63, type, sP3: set_a > a > set_a > $o).
% 2.70/1.63 thf(func_def_64, type, sK4: (a > $o) > (a > $o) > a).
% 2.70/1.63 thf(func_def_65, type, sK5: set_a > a).
% 2.70/1.63 thf(func_def_66, type, sK6: (a > $o) > a).
% 2.70/1.63 thf(func_def_67, type, sK7: int > int).
% 2.70/1.63 thf(func_def_68, type, sK8: (a > $o) > set_a > a).
% 2.70/1.63 thf(func_def_69, type, sK9: (a > $o) > set_a > a).
% 2.70/1.63 thf(func_def_70, type, sK10: (a > $o) > set_a > set_a > a).
% 2.70/1.63 thf(func_def_71, type, sK11: int > int).
% 2.70/1.63 thf(func_def_72, type, sK12: set_a > set_a > a).
% 2.70/1.63 thf(func_def_73, type, sK13: set_a > a > set_a).
% 2.70/1.63 thf(func_def_74, type, sK14: (int > int > $o) > int).
% 2.70/1.63 thf(func_def_75, type, sK15: (int > int > $o) > int).
% 2.70/1.63 thf(func_def_76, type, sK16: (int > int > $o) > int).
% 2.70/1.63 thf(func_def_77, type, sK17: (int > int > $o) > int).
% 2.70/1.63 thf(func_def_78, type, sK18: (int > int > $o) > int).
% 2.70/1.63 thf(func_def_79, type, sK19: set_a > a).
% 2.70/1.63 thf(func_def_80, type, sK20: a > set_a > set_a).
% 2.70/1.63 thf(func_def_81, type, sK21: (a > $o) > a).
% 2.70/1.63 thf(func_def_82, type, sK22: (binary1439146945Tree_a > $o) > binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_83, type, sK23: (binary1439146945Tree_a > $o) > a).
% 2.70/1.63 thf(func_def_84, type, sK24: (binary1439146945Tree_a > $o) > binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_85, type, sK25: set_a > a > a > set_a > set_a).
% 2.70/1.63 thf(func_def_86, type, sK26: (a > int) > binary1439146945Tree_a > a > a).
% 2.70/1.63 thf(func_def_87, type, sK27: (a > int) > binary1439146945Tree_a > a > a).
% 2.70/1.63 thf(func_def_88, type, sK28: set_a > a).
% 2.70/1.63 thf(func_def_89, type, sK29: binary1439146945Tree_a > a).
% 2.70/1.63 thf(func_def_90, type, sK30: binary1439146945Tree_a > binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_91, type, sK31: binary1439146945Tree_a > binary1439146945Tree_a).
% 2.70/1.63 thf(func_def_92, type, sK32: (int > int) > int).
% 2.70/1.63 thf(func_def_93, type, sK33: (int > int) > int).
% 2.70/1.63 thf(func_def_94, type, sK34: (int > int) > int).
% 2.70/1.63 thf(func_def_95, type, sK35: (int > int) > int).
% 2.70/1.63 thf(func_def_96, type, sK36: (a > $o) > set_a > set_a > a).
% 2.70/1.63 thf(func_def_97, type, sK37: (a > $o) > set_a > a).
% 2.70/1.63 thf(func_def_98, type, sK38: (a > $o) > set_a > a).
% 2.70/1.63 thf(func_def_99, type, sK39: (int > int) > int).
% 2.70/1.63 thf(func_def_100, type, sK40: (int > int) > int).
% 2.70/1.63 thf(func_def_101, type, sK41: a).
% 2.70/1.63 thf(func_def_102, type, sK42: (int > int) > int).
% 2.70/1.63 thf(func_def_103, type, sK43: (int > int) > int).
% 2.70/1.63 thf(func_def_105, type, ph45: !>[X0: $tType]:(X0)).
% 2.70/1.63 thf(f1904,plain,(
% 2.70/1.63 $false),
% 2.70/1.63 inference(subsumption_resolution,[status(thm)],[f1900,f1353])).
% 2.70/1.63 thf(f1353,plain,(
% 2.70/1.63 ( ! [X0 : int] : (((ord_less_int @ X0 @ X0) != $true)) )),
% 2.70/1.63 inference(cnf_transformation,[status(thm)],[f972])).
% 2.70/1.63 thf(f972,plain,(
% 2.70/1.63 ! [X0 : int] : ((ord_less_int @ X0 @ X0) != $true)),
% 2.70/1.63 inference(flattening,[status(thm)],[f841])).
% 2.70/1.63 thf(f841,plain,(
% 2.70/1.63 ! [X0 : int] : ~((ord_less_int @ X0 @ X0) = $true)),
% 2.70/1.63 inference(fool_elimination,[status(thm)],[f840])).
% 2.70/1.63 thf(f840,plain,(
% 2.70/1.63 ! [X0 : int] : ~(ord_less_int @ X0 @ X0)),
% 2.70/1.63 inference(rectify,[status(thm)],[f243])).
% 2.70/1.63 thf(f243,axiom,(
% 2.70/1.63 ! [X15 : int] : ~(ord_less_int @ X15 @ X15)),
% 2.70/1.63 file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_242_less__irrefl)).
% 2.70/1.63 thf(f1900,plain,(
% 2.70/1.63 ($true = (ord_less_int @ (h @ e) @ (h @ e)))),
% 2.70/1.63 inference(superposition,[status(thm)],[f1211,f1861])).
% 2.70/1.63 thf(f1861,plain,(
% 2.70/1.63 ((h @ x) = (h @ e))),
% 2.70/1.63 inference(superposition,[status(thm)],[f1855,f1429])).
% 2.70/1.63 thf(f1429,plain,(
% 2.70/1.63 ((h @ x) = (h @ sK41))),
% 2.70/1.63 inference(cnf_transformation,[status(thm)],[f1177])).
% 2.70/1.63 thf(f1177,plain,(
% 2.70/1.63 ((h @ x) = (h @ sK41)) & ($true = (member_a @ sK41 @ (binary504661350_eqs_a @ h @ e)))),
% 2.70/1.63 inference(skolemisation,[status(esa),new_symbols(skolem,[sK41])],[f990,f1176])).
% 2.70/1.63 thf(f1176,plain,(
% 2.70/1.63 ? [X0 : a] : (((h @ X0) = (h @ x)) & ((member_a @ X0 @ (binary504661350_eqs_a @ h @ e)) = $true)) => (((h @ x) = (h @ sK41)) & ($true = (member_a @ sK41 @ (binary504661350_eqs_a @ h @ e))))),
% 2.70/1.63 introduced(definition,[],[choice_axiom])).
% 2.70/1.63 thf(f990,plain,(
% 2.70/1.63 ? [X0 : a] : (((h @ X0) = (h @ x)) & ((member_a @ X0 @ (binary504661350_eqs_a @ h @ e)) = $true))),
% 2.70/1.63 inference(ennf_transformation,[status(thm)],[f715])).
% 2.70/1.63 thf(f715,plain,(
% 2.70/1.63 ~! [X0 : a] : (((member_a @ X0 @ (binary504661350_eqs_a @ h @ e)) = $true) => ((h @ X0) != (h @ x)))),
% 2.70/1.63 inference(fool_elimination,[status(thm)],[f714])).
% 2.70/1.63 thf(f714,plain,(
% 2.70/1.63 ~! [X0 : a] : ((member_a @ X0 @ (binary504661350_eqs_a @ h @ e)) => ((h @ X0) != (h @ x)))),
% 2.70/1.63 inference(rectify,[status(thm)],[f360])).
% 2.70/1.63 thf(f360,negated_conjecture,(
% 2.70/1.63 ~! [X37 : a] : ((member_a @ X37 @ (binary504661350_eqs_a @ h @ e)) => ((h @ x) != (h @ X37)))),
% 2.70/1.63 inference(negated_conjecture,[status(cth)],[f359])).
% 2.70/1.63 thf(f359,conjecture,(
% 2.70/1.63 ! [X37 : a] : ((member_a @ X37 @ (binary504661350_eqs_a @ h @ e)) => ((h @ x) != (h @ X37)))),
% 2.70/1.63 file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',conj_0)).
% 2.70/1.63 thf(f1855,plain,(
% 2.70/1.63 ((h @ e) = (h @ sK41))),
% 2.70/1.63 inference(equality_proxy_clausification,[status(thm)],[f1854])).
% 2.70/1.63 thf(f1854,plain,(
% 2.70/1.63 (((h @ e) = (h @ sK41)) = $true)),
% 2.70/1.63 inference(beta_eta_normalization,[status(thm)],[f1838])).
% 2.70/1.63 thf(f1838,plain,(
% 2.70/1.63 (((^[Y0 : a]: ((h @ e) = (h @ Y0))) @ sK41) = $true)),
% 2.70/1.63 inference(superposition,[status(thm)],[f1832,f1404])).
% 2.70/1.63 thf(f1404,plain,(
% 2.70/1.63 ( ! [X0 : a > $o,X1 : a] : (((X0 @ X1) = (member_a @ X1 @ (collect_a @ X0)))) )),
% 2.70/1.63 inference(cnf_transformation,[status(thm)],[f916])).
% 2.70/1.63 thf(f916,plain,(
% 2.70/1.63 ! [X0 : a > $o,X1 : a] : ((X0 @ X1) = (member_a @ X1 @ (collect_a @ X0)))),
% 2.70/1.63 inference(rectify,[status(thm)],[f46])).
% 2.70/1.63 thf(f46,axiom,(
% 2.70/1.63 ! [X18 : a > $o,X2 : a] : ((member_a @ X2 @ (collect_a @ X18)) = (X18 @ X2))),
% 2.70/1.63 file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_45_mem__Collect__eq)).
% 2.70/1.63 thf(f1832,plain,(
% 2.70/1.63 ((member_a @ sK41 @ (collect_a @ (^[Y0 : a]: ((h @ e) = (h @ Y0))))) = $true)),
% 2.70/1.63 inference(beta_eta_normalization,[status(thm)],[f1605])).
% 2.70/1.63 thf(f1605,plain,(
% 2.70/1.63 ((member_a @ sK41 @ ((^[Y0 : a > int]: ((^[Y1 : a]: (collect_a @ (^[Y2 : a]: ((Y0 @ Y1) = (Y0 @ Y2))))))) @ h @ e)) = $true)),
% 2.70/1.63 inference(definition_unfolding,[status(thm)],[f1428,f1326])).
% 2.70/1.63 thf(f1326,plain,(
% 2.70/1.63 (binary504661350_eqs_a = (^[Y0 : a > int]: ((^[Y1 : a]: (collect_a @ (^[Y2 : a]: ((Y0 @ Y1) = (Y0 @ Y2))))))))),
% 2.70/1.63 inference(cnf_transformation,[status(thm)],[f567])).
% 2.70/1.63 thf(f567,plain,(
% 2.70/1.63 (binary504661350_eqs_a = (^[Y0 : a > int]: ((^[Y1 : a]: (collect_a @ (^[Y2 : a]: ((Y0 @ Y1) = (Y0 @ Y2))))))))),
% 2.70/1.63 inference(fool_elimination,[status(thm)],[f566])).
% 2.70/1.63 thf(f566,plain,(
% 2.70/1.63 ((^[X0 : a > int, X1 : a] : (collect_a @ (^[X2 : a] : (((X0 @ X1) = (X0 @ X2)))))) = binary504661350_eqs_a)),
% 2.70/1.63 inference(rectify,[status(thm)],[f10])).
% 2.70/1.63 thf(f10,axiom,(
% 2.70/1.63 ((^[X5 : a > int, X6 : a] : (collect_a @ (^[X7 : a] : (((X5 @ X7) = (X5 @ X6)))))) = binary504661350_eqs_a)),
% 2.70/1.63 file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_9_eqs__def)).
% 2.70/1.63 thf(f1428,plain,(
% 2.70/1.63 ($true = (member_a @ sK41 @ (binary504661350_eqs_a @ h @ e)))),
% 2.70/1.63 inference(cnf_transformation,[status(thm)],[f1177])).
% 2.70/1.63 thf(f1211,plain,(
% 2.70/1.63 ((ord_less_int @ (h @ e) @ (h @ x)) = $true)),
% 2.70/1.63 inference(cnf_transformation,[status(thm)],[f471])).
% 2.70/1.63 thf(f471,plain,(
% 2.70/1.63 ((ord_less_int @ (h @ e) @ (h @ x)) = $true)),
% 2.70/1.63 inference(fool_elimination,[status(thm)],[f470])).
% 2.70/1.63 thf(f470,plain,(
% 2.70/1.63 (ord_less_int @ (h @ e) @ (h @ x))),
% 2.70/1.63 inference(rectify,[status(thm)],[f2])).
% 2.70/1.63 thf(f2,axiom,(
% 2.70/1.63 (ord_less_int @ (h @ e) @ (h @ x))),
% 2.70/1.63 file('/export/starexec/sandbox2/tmp/tmp.RuC2w6X7YM/DTF2THF_4570.p',fact_1_eLess)).
% 2.70/1.63 % SZS output end Proof for DTF2THF_4570
% 2.70/1.63 % (4865)------------------------------
% 2.70/1.63 % (4865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (4865)Termination reason: Refutation
% 2.70/1.63
% 2.70/1.63 % (4865)Memory used [KB]: 7291
% 2.70/1.63 % (4865)Time elapsed: 0.075 s
% 2.70/1.63 % (4865)Instructions burned: 185 (million)
% 2.70/1.63 % (4865)------------------------------
% 2.70/1.63 % (4865)------------------------------
% 2.70/1.63 % (4859)Success in time 0.103 s
%------------------------------------------------------------------------------