%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP134^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:49 AM UTC 2026
% Result : Theorem 3.24s 1.85s
% Output : Refutation 3.24s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 4
% Syntax : Number of formulae : 25 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 157 ( 39 equ; 0 cnn)
% Maximal formula atoms : 5 ( 6 avg)
% Number of connectives : 282 ( 23 ~; 9 |; 11 &; 206 @)
% ( 0 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 23 ( 23 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 81 usr; 6 con; 0-5 aty)
% ( 14 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 67 ( 37 ^; 30 !; 0 ?; 67 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_nat_nat_nat_nat3: $tType ).
thf(type_def_6,type,
set_se1657353at_nat: $tType ).
thf(type_def_8,type,
set_nat_nat_nat2: $tType ).
thf(type_def_9,type,
set_set_nat_nat_nat2: $tType ).
thf(type_def_10,type,
set_nat_nat_nat: $tType ).
thf(type_def_11,type,
set_set_nat_nat_nat: $tType ).
thf(type_def_12,type,
set_nat_nat: $tType ).
thf(type_def_13,type,
set_set_nat_nat: $tType ).
thf(type_def_14,type,
set_nat: $tType ).
thf(type_def_15,type,
set_set_nat: $tType ).
thf(type_def_16,type,
set_nat_nat_nat_nat5: $tType ).
thf(type_def_17,type,
set_nat_nat_nat_nat4: $tType ).
thf(type_def_18,type,
set_na1201295426at_nat: $tType ).
thf(type_def_19,type,
set_na1064332610at_nat: $tType ).
thf(type_def_20,type,
set_nat_nat_nat_nat2: $tType ).
thf(type_def_21,type,
set_nat_nat_nat_nat: $tType ).
thf(type_def_22,type,
nat: $tType ).
thf(func_def_0,type,
set_na1064332610at_nat: $tType ).
thf(func_def_1,type,
set_na1201295426at_nat: $tType ).
thf(func_def_2,type,
set_se1657353at_nat: $tType ).
thf(func_def_3,type,
set_nat_nat_nat_nat: $tType ).
thf(func_def_4,type,
set_nat_nat_nat_nat2: $tType ).
thf(func_def_5,type,
set_nat_nat_nat_nat3: $tType ).
thf(func_def_6,type,
set_nat_nat_nat_nat4: $tType ).
thf(func_def_7,type,
set_nat_nat_nat_nat5: $tType ).
thf(func_def_8,type,
set_set_nat_nat_nat: $tType ).
thf(func_def_9,type,
set_set_nat_nat_nat2: $tType ).
thf(func_def_10,type,
set_nat_nat_nat: $tType ).
thf(func_def_11,type,
set_nat_nat_nat2: $tType ).
thf(func_def_12,type,
set_set_nat_nat: $tType ).
thf(func_def_13,type,
set_nat_nat: $tType ).
thf(func_def_14,type,
set_set_nat: $tType ).
thf(func_def_15,type,
set_nat: $tType ).
thf(func_def_16,type,
nat: $tType ).
thf(func_def_17,type,
finite252967095at_nat: set_nat_nat_nat_nat3 > set_se1657353at_nat ).
thf(func_def_18,type,
finite1633036872at_nat: set_nat_nat_nat2 > set_set_nat_nat_nat2 ).
thf(func_def_19,type,
finite90785608at_nat: set_nat_nat_nat > set_set_nat_nat_nat ).
thf(func_def_20,type,
finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
thf(func_def_21,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(func_def_22,type,
finite1526199988at_nat: set_nat_nat_nat_nat5 > $o ).
thf(func_def_23,type,
finite2081887156at_nat: set_nat_nat_nat_nat4 > $o ).
thf(func_def_24,type,
finite1090277411at_nat: set_na1201295426at_nat > $o ).
thf(func_def_25,type,
finite1397812515at_nat: set_na1064332610at_nat > $o ).
thf(func_def_26,type,
finite1064868788at_nat: set_nat_nat_nat_nat3 > $o ).
thf(func_def_27,type,
finite1440337093at_nat: set_nat_nat_nat2 > $o ).
thf(func_def_28,type,
finite584233140at_nat: set_nat_nat_nat_nat2 > $o ).
thf(func_def_29,type,
finite891768244at_nat: set_nat_nat_nat_nat > $o ).
thf(func_def_30,type,
finite2045569477at_nat: set_nat_nat_nat > $o ).
thf(func_def_31,type,
finite570312790at_nat: set_nat_nat > $o ).
thf(func_def_32,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_33,type,
finite547180394at_nat: set_se1657353at_nat > $o ).
thf(func_def_34,type,
finite1977496699at_nat: set_set_nat_nat_nat2 > $o ).
thf(func_def_35,type,
finite137548155at_nat: set_set_nat_nat_nat > $o ).
thf(func_def_36,type,
finite604103692at_nat: set_set_nat_nat > $o ).
thf(func_def_37,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(func_def_38,type,
one_one_nat: nat ).
thf(func_def_39,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_40,type,
zero_zero_nat: nat ).
thf(func_def_41,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(func_def_42,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_43,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(func_def_44,type,
ord_le2059018749at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(func_def_45,type,
ord_le2026740592_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(func_def_46,type,
ord_le809907342at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(func_def_47,type,
ord_le1415139726at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(func_def_48,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_49,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_50,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_51,type,
ord_le2040082867at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(func_def_52,type,
ord_le633272388at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(func_def_53,type,
ord_le940807492at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(func_def_54,type,
ord_le1415039317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(func_def_55,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_56,type,
collec2071977458at_nat: ( ( ( ( nat > nat ) > nat ) > nat ) > $o ) > set_nat_nat_nat_nat5 ).
thf(func_def_57,type,
collec480180978at_nat: ( ( ( nat > nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat4 ).
thf(func_def_58,type,
collec6783585at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na1201295426at_nat ).
thf(func_def_59,type,
collec314318689at_nat: ( ( ( nat > nat ) > nat > nat > nat ) > $o ) > set_na1064332610at_nat ).
thf(func_def_60,type,
collec1610646258at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
thf(func_def_61,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(func_def_62,type,
collec1130010610at_nat: ( ( nat > ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat2 ).
thf(func_def_63,type,
collec1437545714at_nat: ( ( nat > nat > nat > nat ) > $o ) > set_nat_nat_nat_nat ).
thf(func_def_64,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(func_def_65,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(func_def_66,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_67,type,
collec1702444712at_nat: ( set_nat_nat_nat_nat3 > $o ) > set_se1657353at_nat ).
thf(func_def_68,type,
collec309745849at_nat: ( set_nat_nat_nat2 > $o ) > set_set_nat_nat_nat2 ).
thf(func_def_69,type,
collec617280953at_nat: ( set_nat_nat_nat > $o ) > set_set_nat_nat_nat ).
thf(func_def_70,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(func_def_71,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_72,type,
member1128122036at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(func_def_73,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(func_def_74,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(func_def_75,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(func_def_76,type,
member_nat: nat > set_nat > $o ).
thf(func_def_77,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(func_def_78,type,
n: nat ).
thf(func_def_93,type,
sK0: set_nat_nat > ( nat > nat ) > nat > nat ).
thf(func_def_94,type,
sK1: ( ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_95,type,
sK2: set_nat_nat > set_nat_nat > ( ( nat > nat ) > ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_96,type,
sK3: set_nat_nat > set_nat_nat > ( ( nat > nat ) > ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_97,type,
sK4: set_nat_nat > ( ( nat > nat ) > ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 > nat > nat ).
thf(func_def_98,type,
sK5: set_nat_nat > ( ( nat > nat ) > ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 > ( nat > nat ) > nat ).
thf(func_def_99,type,
sK6: set_nat > ( ( nat > nat ) > nat > $o ) > set_nat_nat > nat > nat ).
thf(func_def_100,type,
sK7: set_nat > ( ( nat > nat ) > nat > $o ) > set_nat_nat > nat ).
thf(func_def_101,type,
sK8: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_102,type,
sK9: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_103,type,
sK10: ( nat > nat ) > set_nat_nat > nat > nat ).
thf(func_def_104,type,
sK11: ( ( nat > nat ) > ( nat > nat > nat ) > $o ) > set_nat_nat > set_nat_nat_nat > nat > nat > nat ).
thf(func_def_105,type,
sK12: ( ( nat > nat ) > ( nat > nat > nat ) > $o ) > set_nat_nat > set_nat_nat_nat > nat > nat ).
thf(func_def_106,type,
sK13: set_nat > set_nat_nat > ( nat > ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_107,type,
sK14: set_nat > set_nat_nat > ( nat > ( nat > nat ) > $o ) > nat ).
thf(func_def_108,type,
sK15: set_nat_nat > ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_110,type,
sK17: ( ( nat > nat ) > $o ) > set_nat_nat > nat > nat ).
thf(func_def_111,type,
sK18: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > nat > nat ).
thf(func_def_112,type,
ph19:
!>[X0: $tType] : X0 ).
thf(f1283,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1282,f1114]) ).
thf(f1114,plain,
! [X0: nat,X1: nat] :
( $true
= ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ X1 @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ X0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f834]) ).
thf(f834,plain,
! [X0: nat,X1: nat] :
( $true
= ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ X1 @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ X0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f833]) ).
thf(f833,plain,
! [X0: nat,X1: nat] :
( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ! [X3: nat] : ( ord_less_eq_nat @ ( X2 @ X3 ) @ X0 )
& ! [X4: nat] :
( ( ord_less_eq_nat @ X1 @ X4 )
=> ( zero_zero_nat
= ( X2 @ X4 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f45]) ).
thf(f45,axiom,
! [X7: nat,X18: nat] :
( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X0: nat > nat] :
( ! [X1: nat] : ( ord_less_eq_nat @ ( X0 @ X1 ) @ X7 )
& ! [X1: nat] :
( ( ord_less_eq_nat @ X18 @ X1 )
=> ( ( X0 @ X1 )
= zero_zero_nat ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',fact_44_bound__domain__and__range__impl__finitely__many__functions) ).
thf(f1282,plain,
( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ n ) ) ) ) )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f1281,f1090]) ).
thf(f1090,plain,
( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f971]) ).
thf(f971,plain,
( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) ) )
!= $true ),
inference(flattening,[status(thm)],[f768]) ).
thf(f768,plain,
( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f767]) ).
thf(f767,plain,
~ ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X0: nat > nat] : ( number1551313001itions @ X0 @ n ) ) ),
inference(rectify,[status(thm)],[f351]) ).
thf(f351,negated_conjecture,
~ ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] : ( number1551313001itions @ X2 @ n ) ) ),
inference(negated_conjecture,[status(cth)],[f350]) ).
thf(f350,conjecture,
( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] : ( number1551313001itions @ X2 @ n ) ) ),
file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',conj_0) ).
thf(f1281,plain,
( ( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) ) )
= $true )
| ( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ n ) ) ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1280]) ).
thf(f1280,plain,
( ( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ n ) ) ) ) )
!= $true )
| ( $true != $true )
| ( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) ) )
= $true ) ),
inference(superposition,[status(thm)],[f1057,f1067]) ).
thf(f1067,plain,
( ( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) )
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ n ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f650]) ).
thf(f650,plain,
( ( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] : ( number1551313001itions @ Y0 @ n ) )
@ ( collect_nat_nat
@ ^ [Y0: nat > nat] :
( ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ Y1 )
=> ( zero_zero_nat
= ( Y0 @ Y1 ) ) ) )
& ( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( Y0 @ Y1 ) @ n ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f649]) ).
thf(f649,plain,
( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [X0: nat > nat] : ( number1551313001itions @ X0 @ n ) )
@ ( collect_nat_nat
@ ^ [X1: nat > nat] :
( ! [X2: nat] : ( ord_less_eq_nat @ ( X1 @ X2 ) @ n )
& ! [X3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ X3 )
=> ( zero_zero_nat
= ( X1 @ X3 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] : ( number1551313001itions @ X2 @ n ) )
@ ( collect_nat_nat
@ ^ [X0: nat > nat] :
( ! [X1: nat] : ( ord_less_eq_nat @ ( X0 @ X1 ) @ n )
& ! [X1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ X1 )
=> ( ( X0 @ X1 )
= zero_zero_nat ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062) ).
thf(f1057,plain,
! [X0: set_nat_nat,X1: set_nat_nat] :
( ( ( ord_le1415039317at_nat @ X1 @ X0 )
!= $true )
| ( ( finite570312790at_nat @ X1 )
= $true )
| ( $true
!= ( finite570312790at_nat @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1003]) ).
thf(f1003,plain,
! [X0: set_nat_nat,X1: set_nat_nat] :
( ( $true
!= ( finite570312790at_nat @ X0 ) )
| ( ( ord_le1415039317at_nat @ X1 @ X0 )
!= $true )
| ( ( finite570312790at_nat @ X1 )
= $true ) ),
inference(flattening,[status(thm)],[f1002]) ).
thf(f1002,plain,
! [X0: set_nat_nat,X1: set_nat_nat] :
( ( $true
!= ( finite570312790at_nat @ X0 ) )
| ( ( finite570312790at_nat @ X1 )
= $true )
| ( ( ord_le1415039317at_nat @ X1 @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f977]) ).
thf(f977,plain,
! [X0: set_nat_nat,X1: set_nat_nat] :
( ( ( ord_le1415039317at_nat @ X1 @ X0 )
= $true )
=> ( ( ( finite570312790at_nat @ X1 )
!= $true )
=> ( $true
!= ( finite570312790at_nat @ X0 ) ) ) ),
inference(flattening,[status(thm)],[f612]) ).
thf(f612,plain,
! [X0: set_nat_nat,X1: set_nat_nat] :
( ( ( ord_le1415039317at_nat @ X1 @ X0 )
= $true )
=> ( ( ( finite570312790at_nat @ X1 )
!= $true )
=> ( $true
!= ( finite570312790at_nat @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f611]) ).
thf(f611,plain,
! [X0: set_nat_nat,X1: set_nat_nat] :
( ( ord_le1415039317at_nat @ X1 @ X0 )
=> ( ~ ( finite570312790at_nat @ X1 )
=> ~ ( finite570312790at_nat @ X0 ) ) ),
inference(rectify,[status(thm)],[f68]) ).
thf(f68,axiom,
! [X23: set_nat_nat,X22: set_nat_nat] :
( ( ord_le1415039317at_nat @ X22 @ X23 )
=> ( ~ ( finite570312790at_nat @ X22 )
=> ~ ( finite570312790at_nat @ X23 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',fact_67_infinite__super) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP134^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n015.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 08:27:46 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.21/0.47 ---- Original DTF file ---
% 0.21/0.47 thf(spec,logic,$$dhol).
% 0.21/0.47 %------------------------------------------------------------------------------
% 0.21/0.47 % File : ITP134^1 : TPTP v9.2.1. Released v7.5.0.
% 0.21/0.47 % Domain : Interactive Theorem Proving
% 0.21/0.47 % Problem : Sledgehammer Number_Partition problem prob_94__5324562_1
% 0.21/0.47 % Version : Especial.
% 0.21/0.47 % English :
% 0.21/0.47
% 0.21/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.21/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.21/0.47 % Source : [Des21]
% 0.21/0.47 % Names : Number_Partition/prob_94__5324562_1 [Des21]
% 0.21/0.47
% 0.21/0.47 % Status : Theorem
% 0.21/0.47 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.21/0.47 % Syntax : Number of formulae : 429 ( 135 unt; 79 typ; 0 def)
% 0.21/0.47 % Number of atoms : 1178 ( 279 equ; 0 cnn)
% 0.21/0.47 % Maximal formula atoms : 12 ( 3 avg)
% 0.21/0.47 % Number of connectives : 3087 ( 93 ~; 21 |; 162 &;2318 @)
% 0.21/0.47 % ( 0 <=>; 493 =>; 0 <=; 0 <~>)
% 0.21/0.47 % Maximal formula depth : 16 ( 7 avg)
% 0.21/0.47 % Number of types : 18 ( 17 usr)
% 0.21/0.47 % Number of type conns : 820 ( 820 >; 0 *; 0 +; 0 <<)
% 0.21/0.47 % Number of symbols : 63 ( 62 usr; 3 con; 0-2 aty)
% 0.21/0.47 % Number of variables : 1092 ( 187 ^; 862 !; 43 ?;1092 :)
% 0.21/0.47 % SPC : TH0_THM_EQU_NAR
% 0.21/0.47
% 0.21/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:42:27.163
% 0.21/0.47 %------------------------------------------------------------------------------
% 0.21/0.47 % Could-be-implicit typings (17)
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.21/0.47 set_na1064332610at_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 set_na1201295426at_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.21/0.47 set_se1657353at_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.21/0.47 set_nat_nat_nat_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 set_nat_nat_nat_nat2: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 set_nat_nat_nat_nat3: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 set_nat_nat_nat_nat4: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 set_nat_nat_nat_nat5: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.21/0.47 set_set_nat_nat_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 set_set_nat_nat_nat2: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 set_nat_nat_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 set_nat_nat_nat2: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 set_set_nat_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 set_nat_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.21/0.47 set_set_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.47 set_nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.21/0.47 nat: $tType ).
% 0.21/0.47
% 0.21/0.47 % Explicit typings (62)
% 0.21/0.47 thf(sy_c_Finite__Set_OFpow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite252967095at_nat: set_nat_nat_nat_nat3 > set_se1657353at_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_OFpow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 finite1633036872at_nat: set_nat_nat_nat2 > set_set_nat_nat_nat2 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite90785608at_nat: set_nat_nat_nat > set_set_nat_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.21/0.47 finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
% 0.21/0.47 finite_Fpow_nat: set_nat > set_set_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 finite1526199988at_nat: set_nat_nat_nat_nat5 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 finite2081887156at_nat: set_nat_nat_nat_nat4 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite1090277411at_nat: set_na1201295426at_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 finite1397812515at_nat: set_na1064332610at_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite1064868788at_nat: set_nat_nat_nat_nat3 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 finite1440337093at_nat: set_nat_nat_nat2 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite584233140at_nat: set_nat_nat_nat_nat2 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 finite891768244at_nat: set_nat_nat_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite2045569477at_nat: set_nat_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.21/0.47 finite570312790at_nat: set_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.21/0.47 finite_finite_nat: set_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 finite547180394at_nat: set_se1657353at_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite1977496699at_nat: set_set_nat_nat_nat2 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 finite137548155at_nat: set_set_nat_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 finite604103692at_nat: set_set_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.47 finite2012248349et_nat: set_set_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.21/0.47 one_one_nat: nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.21/0.47 plus_plus_nat: nat > nat > nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.21/0.47 zero_zero_nat: nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 0.21/0.47 number1551313001itions: ( nat > nat ) > nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.21/0.47 ord_less_nat: nat > nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 ord_le2059018749at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.21/0.47 ord_le2026740592_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 ord_le809907342at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 ord_le1415139726at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.21/0.47 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.21/0.47 ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.21/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 ord_le2040082867at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 ord_le633272388at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 ord_le940807492at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 ord_le1415039317at_nat: set_nat_nat > set_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.47 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 collec2071977458at_nat: ( ( ( ( nat > nat ) > nat ) > nat ) > $o ) > set_nat_nat_nat_nat5 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 collec480180978at_nat: ( ( ( nat > nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat4 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 collec6783585at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na1201295426at_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 collec314318689at_nat: ( ( ( nat > nat ) > nat > nat > nat ) > $o ) > set_na1064332610at_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 collec1610646258at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 collec1130010610at_nat: ( ( nat > ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat2 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 collec1437545714at_nat: ( ( nat > nat > nat > nat ) > $o ) > set_nat_nat_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.21/0.47 collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.21/0.47 collect_nat: ( nat > $o ) > set_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 collec1702444712at_nat: ( set_nat_nat_nat_nat3 > $o ) > set_se1657353at_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 collec309745849at_nat: ( set_nat_nat_nat2 > $o ) > set_set_nat_nat_nat2 ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.21/0.47 collec617280953at_nat: ( set_nat_nat_nat > $o ) > set_set_nat_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.47 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 member1128122036at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.21/0.47 member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.21/0.47 member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_member_001t__Nat__Onat,type,
% 0.21/0.47 member_nat: nat > set_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.21/0.47 member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_v_n,type,
% 0.21/0.47 n: nat ).
% 0.21/0.47
% 0.21/0.47 % Relevant facts (349)
% 0.21/0.47 thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 0.21/0.47 ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [F: nat > nat] :
% 0.21/0.47 ( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ n )
% 0.21/0.47 & ! [I: nat] :
% 0.21/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ I )
% 0.21/0.47 => ( ( F @ I )
% 0.21/0.47 = zero_zero_nat ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % \<open>finite {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
% 0.21/0.47 thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 0.21/0.47 ( ord_le1415039317at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [P: nat > nat] : ( number1551313001itions @ P @ n ) )
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [F: nat > nat] :
% 0.21/0.47 ( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ n )
% 0.21/0.47 & ! [I: nat] :
% 0.21/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ I )
% 0.21/0.47 => ( ( F @ I )
% 0.21/0.47 = zero_zero_nat ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % \<open>{p. p partitions n} \<subseteq> {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
% 0.21/0.47 thf(fact_2_finite__Collect__conjI,axiom,
% 0.21/0.47 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 0.21/0.47 ( ( ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
% 0.21/0.47 | ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ Q ) ) )
% 0.21/0.47 => ( finite2045569477at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat2
% 0.21/0.47 @ ^ [X: nat > nat > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_conjI
% 0.21/0.47 thf(fact_3_finite__Collect__conjI,axiom,
% 0.21/0.47 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 0.21/0.47 ( ( ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
% 0.21/0.47 | ( finite1440337093at_nat @ ( collect_nat_nat_nat @ Q ) ) )
% 0.21/0.47 => ( finite1440337093at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat
% 0.21/0.47 @ ^ [X: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_conjI
% 0.21/0.47 thf(fact_4_finite__Collect__conjI,axiom,
% 0.21/0.47 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.21/0.47 ( ( ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
% 0.21/0.47 | ( finite1064868788at_nat @ ( collec1610646258at_nat @ Q ) ) )
% 0.21/0.47 => ( finite1064868788at_nat
% 0.21/0.47 @ ( collec1610646258at_nat
% 0.21/0.47 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_conjI
% 0.21/0.47 thf(fact_5_finite__Collect__conjI,axiom,
% 0.21/0.47 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 0.21/0.47 ( ( ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
% 0.21/0.47 | ( finite570312790at_nat @ ( collect_nat_nat @ Q ) ) )
% 0.21/0.47 => ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [X: nat > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_conjI
% 0.21/0.47 thf(fact_6_finite__Collect__conjI,axiom,
% 0.21/0.47 ! [P2: nat > $o,Q: nat > $o] :
% 0.21/0.47 ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 0.21/0.47 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.21/0.47 => ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [X: nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_conjI
% 0.21/0.47 thf(fact_7_finite__Collect__disjI,axiom,
% 0.21/0.47 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 0.21/0.47 ( ( finite2045569477at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat2
% 0.21/0.47 @ ^ [X: nat > nat > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 | ( Q @ X ) ) ) )
% 0.21/0.47 = ( ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
% 0.21/0.47 & ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_disjI
% 0.21/0.47 thf(fact_8_finite__Collect__disjI,axiom,
% 0.21/0.47 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 0.21/0.47 ( ( finite1440337093at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat
% 0.21/0.47 @ ^ [X: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 | ( Q @ X ) ) ) )
% 0.21/0.47 = ( ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
% 0.21/0.47 & ( finite1440337093at_nat @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_disjI
% 0.21/0.47 thf(fact_9_finite__Collect__disjI,axiom,
% 0.21/0.47 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.21/0.47 ( ( finite1064868788at_nat
% 0.21/0.47 @ ( collec1610646258at_nat
% 0.21/0.47 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 | ( Q @ X ) ) ) )
% 0.21/0.47 = ( ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
% 0.21/0.47 & ( finite1064868788at_nat @ ( collec1610646258at_nat @ Q ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_disjI
% 0.21/0.47 thf(fact_10_finite__Collect__disjI,axiom,
% 0.21/0.47 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 0.21/0.47 ( ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [X: nat > nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 | ( Q @ X ) ) ) )
% 0.21/0.47 = ( ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
% 0.21/0.47 & ( finite570312790at_nat @ ( collect_nat_nat @ Q ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_disjI
% 0.21/0.47 thf(fact_11_finite__Collect__disjI,axiom,
% 0.21/0.47 ! [P2: nat > $o,Q: nat > $o] :
% 0.21/0.47 ( ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [X: nat] :
% 0.21/0.47 ( ( P2 @ X )
% 0.21/0.47 | ( Q @ X ) ) ) )
% 0.21/0.47 = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 0.21/0.47 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_disjI
% 0.21/0.47 thf(fact_12_partitions__zero,axiom,
% 0.21/0.47 ! [P3: nat > nat] :
% 0.21/0.47 ( ( number1551313001itions @ P3 @ zero_zero_nat )
% 0.21/0.47 = ( P3
% 0.21/0.47 = ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % partitions_zero
% 0.21/0.47 thf(fact_13_partitions__bounds,axiom,
% 0.21/0.47 ! [P3: nat > nat,N: nat,I2: nat] :
% 0.21/0.47 ( ( number1551313001itions @ P3 @ N )
% 0.21/0.47 => ( ord_less_eq_nat @ ( P3 @ I2 ) @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % partitions_bounds
% 0.21/0.47 thf(fact_14_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat,D: nat] :
% 0.21/0.47 ( ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [F: nat > nat] :
% 0.21/0.47 ! [X: nat] :
% 0.21/0.47 ( ( ( member_nat @ X @ A )
% 0.21/0.47 => ( member_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_15_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat,D: nat] :
% 0.21/0.47 ( ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( finite1440337093at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat
% 0.21/0.47 @ ^ [F: ( nat > nat ) > nat] :
% 0.21/0.47 ! [X: nat > nat] :
% 0.21/0.47 ( ( ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( member_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_16_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat_nat,D: nat > nat] :
% 0.21/0.47 ( ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite570312790at_nat @ B )
% 0.21/0.47 => ( finite2045569477at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat2
% 0.21/0.47 @ ^ [F: nat > nat > nat] :
% 0.21/0.47 ! [X: nat] :
% 0.21/0.47 ( ( ( member_nat @ X @ A )
% 0.21/0.47 => ( member_nat_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_17_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat_nat,D: nat > nat] :
% 0.21/0.47 ( ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite570312790at_nat @ B )
% 0.21/0.47 => ( finite1064868788at_nat
% 0.21/0.47 @ ( collec1610646258at_nat
% 0.21/0.47 @ ^ [F: ( nat > nat ) > nat > nat] :
% 0.21/0.47 ! [X: nat > nat] :
% 0.21/0.47 ( ( ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( member_nat_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_18_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat_nat_nat,D: nat > nat > nat] :
% 0.21/0.47 ( ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite2045569477at_nat @ B )
% 0.21/0.47 => ( finite891768244at_nat
% 0.21/0.47 @ ( collec1437545714at_nat
% 0.21/0.47 @ ^ [F: nat > nat > nat > nat] :
% 0.21/0.47 ! [X: nat] :
% 0.21/0.47 ( ( ( member_nat @ X @ A )
% 0.21/0.47 => ( member_nat_nat_nat2 @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_19_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat_nat_nat2,D: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite1440337093at_nat @ B )
% 0.21/0.47 => ( finite584233140at_nat
% 0.21/0.47 @ ( collec1130010610at_nat
% 0.21/0.47 @ ^ [F: nat > ( nat > nat ) > nat] :
% 0.21/0.47 ! [X: nat] :
% 0.21/0.47 ( ( ( member_nat @ X @ A )
% 0.21/0.47 => ( member_nat_nat_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_20_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat,B: set_nat,D: nat] :
% 0.21/0.47 ( ( finite2045569477at_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( finite2081887156at_nat
% 0.21/0.47 @ ( collec480180978at_nat
% 0.21/0.47 @ ^ [F: ( nat > nat > nat ) > nat] :
% 0.21/0.47 ! [X: nat > nat > nat] :
% 0.21/0.47 ( ( ( member_nat_nat_nat2 @ X @ A )
% 0.21/0.47 => ( member_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat_nat_nat2 @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_21_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat2,B: set_nat,D: nat] :
% 0.21/0.47 ( ( finite1440337093at_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( finite1526199988at_nat
% 0.21/0.47 @ ( collec2071977458at_nat
% 0.21/0.47 @ ^ [F: ( ( nat > nat ) > nat ) > nat] :
% 0.21/0.47 ! [X: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( ( member_nat_nat_nat @ X @ A )
% 0.21/0.47 => ( member_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat_nat_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_22_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat_nat_nat,D: nat > nat > nat] :
% 0.21/0.47 ( ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite2045569477at_nat @ B )
% 0.21/0.47 => ( finite1397812515at_nat
% 0.21/0.47 @ ( collec314318689at_nat
% 0.21/0.47 @ ^ [F: ( nat > nat ) > nat > nat > nat] :
% 0.21/0.47 ! [X: nat > nat] :
% 0.21/0.47 ( ( ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( member_nat_nat_nat2 @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_23_finite__set__of__finite__funs,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat_nat_nat2,D: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite1440337093at_nat @ B )
% 0.21/0.47 => ( finite1090277411at_nat
% 0.21/0.47 @ ( collec6783585at_nat
% 0.21/0.47 @ ^ [F: ( nat > nat ) > ( nat > nat ) > nat] :
% 0.21/0.47 ! [X: nat > nat] :
% 0.21/0.47 ( ( ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( member_nat_nat_nat @ ( F @ X ) @ B ) )
% 0.21/0.47 & ( ~ ( member_nat_nat @ X @ A )
% 0.21/0.47 => ( ( F @ X )
% 0.21/0.47 = D ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_set_of_finite_funs
% 0.21/0.47 thf(fact_24_not__finite__existsD,axiom,
% 0.21/0.47 ! [P2: ( nat > nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
% 0.21/0.47 => ? [X_1: nat > nat > nat] : ( P2 @ X_1 ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_finite_existsD
% 0.21/0.47 thf(fact_25_not__finite__existsD,axiom,
% 0.21/0.47 ! [P2: ( ( nat > nat ) > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
% 0.21/0.47 => ? [X_1: ( nat > nat ) > nat] : ( P2 @ X_1 ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_finite_existsD
% 0.21/0.47 thf(fact_26_not__finite__existsD,axiom,
% 0.21/0.47 ! [P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
% 0.21/0.47 => ? [X_1: ( nat > nat ) > nat > nat] : ( P2 @ X_1 ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_finite_existsD
% 0.21/0.47 thf(fact_27_not__finite__existsD,axiom,
% 0.21/0.47 ! [P2: ( nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
% 0.21/0.47 => ? [X_1: nat > nat] : ( P2 @ X_1 ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_finite_existsD
% 0.21/0.47 thf(fact_28_not__finite__existsD,axiom,
% 0.21/0.47 ! [P2: nat > $o] :
% 0.21/0.47 ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 0.21/0.47 => ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_finite_existsD
% 0.21/0.47 thf(fact_29_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat,R: nat > nat > $o] :
% 0.21/0.47 ( ~ ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( ! [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat] :
% 0.21/0.47 ( ( member_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [A2: nat] :
% 0.21/0.47 ( ( member_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_30_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat,R: ( nat > nat ) > nat > $o] :
% 0.21/0.47 ( ~ ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( ! [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat] :
% 0.21/0.47 ( ( member_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [A2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_31_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat_nat,R: nat > ( nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite570312790at_nat @ B )
% 0.21/0.47 => ( ! [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [A2: nat] :
% 0.21/0.47 ( ( member_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_32_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat_nat,R: ( nat > nat ) > ( nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite570312790at_nat @ B )
% 0.21/0.47 => ( ! [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [A2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_33_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat_nat_nat,R: nat > ( nat > nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite2045569477at_nat @ B )
% 0.21/0.47 => ( ! [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ X2 @ B )
% 0.21/0.47 & ~ ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [A2: nat] :
% 0.21/0.47 ( ( member_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_34_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat,B: set_nat_nat_nat2,R: nat > ( ( nat > nat ) > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite_finite_nat @ A )
% 0.21/0.47 => ( ( finite1440337093at_nat @ B )
% 0.21/0.47 => ( ! [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [A2: nat] :
% 0.21/0.47 ( ( member_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_35_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat,B: set_nat,R: ( nat > nat > nat ) > nat > $o] :
% 0.21/0.47 ( ~ ( finite2045569477at_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( ! [X2: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat] :
% 0.21/0.47 ( ( member_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite2045569477at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat2
% 0.21/0.47 @ ^ [A2: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_36_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat2,B: set_nat,R: ( ( nat > nat ) > nat ) > nat > $o] :
% 0.21/0.47 ( ~ ( finite1440337093at_nat @ A )
% 0.21/0.47 => ( ( finite_finite_nat @ B )
% 0.21/0.47 => ( ! [X2: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat] :
% 0.21/0.47 ( ( member_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat] :
% 0.21/0.47 ( ( member_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite1440337093at_nat
% 0.21/0.47 @ ( collect_nat_nat_nat
% 0.21/0.47 @ ^ [A2: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_37_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat_nat_nat,R: ( nat > nat ) > ( nat > nat > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite2045569477at_nat @ B )
% 0.21/0.47 => ( ! [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ X2 @ B )
% 0.21/0.47 & ~ ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [A2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_38_pigeonhole__infinite__rel,axiom,
% 0.21/0.47 ! [A: set_nat_nat,B: set_nat_nat_nat2,R: ( nat > nat ) > ( ( nat > nat ) > nat ) > $o] :
% 0.21/0.47 ( ~ ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( finite1440337093at_nat @ B )
% 0.21/0.47 => ( ! [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.21/0.47 => ? [Xa: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ Xa @ B )
% 0.21/0.47 & ( R @ X2 @ Xa ) ) )
% 0.21/0.47 => ? [X2: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ X2 @ B )
% 0.21/0.47 & ~ ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [A2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ A2 @ A )
% 0.21/0.47 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % pigeonhole_infinite_rel
% 0.21/0.47 thf(fact_39_finite__Collect__subsets,axiom,
% 0.21/0.47 ! [A: set_nat] :
% 0.21/0.47 ( ( finite_finite_nat @ A )
% 0.21/0.47 => ( finite2012248349et_nat
% 0.21/0.47 @ ( collect_set_nat
% 0.21/0.47 @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_subsets
% 0.21/0.47 thf(fact_40_finite__Collect__subsets,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat] :
% 0.21/0.47 ( ( finite2045569477at_nat @ A )
% 0.21/0.47 => ( finite137548155at_nat
% 0.21/0.47 @ ( collec617280953at_nat
% 0.21/0.47 @ ^ [B2: set_nat_nat_nat] : ( ord_le940807492at_nat @ B2 @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_subsets
% 0.21/0.47 thf(fact_41_finite__Collect__subsets,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat2] :
% 0.21/0.47 ( ( finite1440337093at_nat @ A )
% 0.21/0.47 => ( finite1977496699at_nat
% 0.21/0.47 @ ( collec309745849at_nat
% 0.21/0.47 @ ^ [B2: set_nat_nat_nat2] : ( ord_le633272388at_nat @ B2 @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_subsets
% 0.21/0.47 thf(fact_42_finite__Collect__subsets,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat_nat3] :
% 0.21/0.47 ( ( finite1064868788at_nat @ A )
% 0.21/0.47 => ( finite547180394at_nat
% 0.21/0.47 @ ( collec1702444712at_nat
% 0.21/0.47 @ ^ [B2: set_nat_nat_nat_nat3] : ( ord_le2040082867at_nat @ B2 @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_subsets
% 0.21/0.47 thf(fact_43_finite__Collect__subsets,axiom,
% 0.21/0.47 ! [A: set_nat_nat] :
% 0.21/0.47 ( ( finite570312790at_nat @ A )
% 0.21/0.47 => ( finite604103692at_nat
% 0.21/0.47 @ ( collect_set_nat_nat
% 0.21/0.47 @ ^ [B2: set_nat_nat] : ( ord_le1415039317at_nat @ B2 @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_subsets
% 0.21/0.47 thf(fact_44_bound__domain__and__range__impl__finitely__many__functions,axiom,
% 0.21/0.47 ! [N: nat,M: nat] :
% 0.21/0.47 ( finite570312790at_nat
% 0.21/0.47 @ ( collect_nat_nat
% 0.21/0.47 @ ^ [F: nat > nat] :
% 0.21/0.47 ( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ N )
% 0.21/0.47 & ! [I: nat] :
% 0.21/0.47 ( ( ord_less_eq_nat @ M @ I )
% 0.21/0.47 => ( ( F @ I )
% 0.21/0.47 = zero_zero_nat ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % bound_domain_and_range_impl_finitely_many_functions
% 0.21/0.47 thf(fact_45_finite__Collect__le__nat,axiom,
% 0.21/0.47 ! [K: nat] :
% 0.21/0.47 ( finite_finite_nat
% 0.21/0.47 @ ( collect_nat
% 0.21/0.47 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_Collect_le_nat
% 0.21/0.47 thf(fact_46_finite__has__minimal2,axiom,
% 0.21/0.47 ! [A: set_nat_nat,A3: nat > nat] :
% 0.21/0.47 ( ( finite570312790at_nat @ A )
% 0.21/0.47 => ( ( member_nat_nat @ A3 @ A )
% 0.21/0.47 => ? [X2: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.21/0.47 & ( ord_less_eq_nat_nat @ X2 @ A3 )
% 0.21/0.47 & ! [Xa: nat > nat] :
% 0.21/0.47 ( ( member_nat_nat @ Xa @ A )
% 0.21/0.47 => ( ( ord_less_eq_nat_nat @ Xa @ X2 )
% 0.21/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_has_minimal2
% 0.21/0.47 thf(fact_47_finite__has__minimal2,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat,A3: nat > nat > nat] :
% 0.21/0.47 ( ( finite2045569477at_nat @ A )
% 0.21/0.47 => ( ( member_nat_nat_nat2 @ A3 @ A )
% 0.21/0.47 => ? [X2: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ X2 @ A )
% 0.21/0.47 & ( ord_le1415139726at_nat @ X2 @ A3 )
% 0.21/0.47 & ! [Xa: nat > nat > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat2 @ Xa @ A )
% 0.21/0.47 => ( ( ord_le1415139726at_nat @ Xa @ X2 )
% 0.21/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_has_minimal2
% 0.21/0.47 thf(fact_48_finite__has__minimal2,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat2,A3: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( finite1440337093at_nat @ A )
% 0.21/0.47 => ( ( member_nat_nat_nat @ A3 @ A )
% 0.21/0.47 => ? [X2: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ X2 @ A )
% 0.21/0.47 & ( ord_le809907342at_nat @ X2 @ A3 )
% 0.21/0.47 & ! [Xa: ( nat > nat ) > nat] :
% 0.21/0.47 ( ( member_nat_nat_nat @ Xa @ A )
% 0.21/0.47 => ( ( ord_le809907342at_nat @ Xa @ X2 )
% 0.21/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_has_minimal2
% 0.21/0.47 thf(fact_49_finite__has__minimal2,axiom,
% 0.21/0.47 ! [A: set_nat_nat_nat_nat3,A3: ( nat > nat ) > nat > nat] :
% 0.21/0.47 ( ( finite1064868788at_nat @ A )
% 0.21/0.47 => ( ( member1128122036at_nat @ A3 @ A )
% 0.21/0.47 => ? [X2: ( nat > nat ) > nat > nat] :
% 0.21/0.47 ( ( member1128122036at_nat @ X2 @ A )
% 0.21/0.47 & ( ord_le2059018749at_nat @ X2 @ A3 )
% 0.21/0.47 & ! [Xa: ( nat > nat ) > nat > nat] :
% 0.21/0.47 ( ( member1128122036at_nat @ Xa @ A )
% 0.21/0.47 => ( ( ord_le2059018749at_nat @ Xa @ X2 )
% 0.21/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % finite_has_minimal2
% 0.32/0.47 thf(fact_50_finite__has__minimal2,axiom,
% 0.32/0.47 ! [A: set_nat,A3: nat] :
% 0.32/0.47 ( ( finite_finite_nat @ A )
% 0.32/0.47 => ( ( member_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: nat] :
% 0.32/0.47 ( ( member_nat @ X2 @ A )
% 0.32/0.47 & ( ord_less_eq_nat @ X2 @ A3 )
% 0.32/0.47 & ! [Xa: nat] :
% 0.32/0.47 ( ( member_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_less_eq_nat @ Xa @ X2 )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_minimal2
% 0.32/0.47 thf(fact_51_finite__has__minimal2,axiom,
% 0.32/0.47 ! [A: set_set_nat_nat,A3: set_nat_nat] :
% 0.32/0.47 ( ( finite604103692at_nat @ A )
% 0.32/0.47 => ( ( member_set_nat_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: set_nat_nat] :
% 0.32/0.47 ( ( member_set_nat_nat @ X2 @ A )
% 0.32/0.47 & ( ord_le1415039317at_nat @ X2 @ A3 )
% 0.32/0.47 & ! [Xa: set_nat_nat] :
% 0.32/0.47 ( ( member_set_nat_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ Xa @ X2 )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_minimal2
% 0.32/0.47 thf(fact_52_finite__has__maximal2,axiom,
% 0.32/0.47 ! [A: set_nat_nat,A3: nat > nat] :
% 0.32/0.47 ( ( finite570312790at_nat @ A )
% 0.32/0.47 => ( ( member_nat_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: nat > nat] :
% 0.32/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.32/0.47 & ( ord_less_eq_nat_nat @ A3 @ X2 )
% 0.32/0.47 & ! [Xa: nat > nat] :
% 0.32/0.47 ( ( member_nat_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_less_eq_nat_nat @ X2 @ Xa )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_maximal2
% 0.32/0.47 thf(fact_53_finite__has__maximal2,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat,A3: nat > nat > nat] :
% 0.32/0.47 ( ( finite2045569477at_nat @ A )
% 0.32/0.47 => ( ( member_nat_nat_nat2 @ A3 @ A )
% 0.32/0.47 => ? [X2: nat > nat > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat2 @ X2 @ A )
% 0.32/0.47 & ( ord_le1415139726at_nat @ A3 @ X2 )
% 0.32/0.47 & ! [Xa: nat > nat > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat2 @ Xa @ A )
% 0.32/0.47 => ( ( ord_le1415139726at_nat @ X2 @ Xa )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_maximal2
% 0.32/0.47 thf(fact_54_finite__has__maximal2,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat2,A3: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( finite1440337093at_nat @ A )
% 0.32/0.47 => ( ( member_nat_nat_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat @ X2 @ A )
% 0.32/0.47 & ( ord_le809907342at_nat @ A3 @ X2 )
% 0.32/0.47 & ! [Xa: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_le809907342at_nat @ X2 @ Xa )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_maximal2
% 0.32/0.47 thf(fact_55_finite__has__maximal2,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat_nat3,A3: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( finite1064868788at_nat @ A )
% 0.32/0.47 => ( ( member1128122036at_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( member1128122036at_nat @ X2 @ A )
% 0.32/0.47 & ( ord_le2059018749at_nat @ A3 @ X2 )
% 0.32/0.47 & ! [Xa: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( member1128122036at_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_le2059018749at_nat @ X2 @ Xa )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_maximal2
% 0.32/0.47 thf(fact_56_finite__has__maximal2,axiom,
% 0.32/0.47 ! [A: set_nat,A3: nat] :
% 0.32/0.47 ( ( finite_finite_nat @ A )
% 0.32/0.47 => ( ( member_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: nat] :
% 0.32/0.47 ( ( member_nat @ X2 @ A )
% 0.32/0.47 & ( ord_less_eq_nat @ A3 @ X2 )
% 0.32/0.47 & ! [Xa: nat] :
% 0.32/0.47 ( ( member_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X2 @ Xa )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_maximal2
% 0.32/0.47 thf(fact_57_finite__has__maximal2,axiom,
% 0.32/0.47 ! [A: set_set_nat_nat,A3: set_nat_nat] :
% 0.32/0.47 ( ( finite604103692at_nat @ A )
% 0.32/0.47 => ( ( member_set_nat_nat @ A3 @ A )
% 0.32/0.47 => ? [X2: set_nat_nat] :
% 0.32/0.47 ( ( member_set_nat_nat @ X2 @ A )
% 0.32/0.47 & ( ord_le1415039317at_nat @ A3 @ X2 )
% 0.32/0.47 & ! [Xa: set_nat_nat] :
% 0.32/0.47 ( ( member_set_nat_nat @ Xa @ A )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ X2 @ Xa )
% 0.32/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_has_maximal2
% 0.32/0.47 thf(fact_58_rev__finite__subset,axiom,
% 0.32/0.47 ! [B: set_nat,A: set_nat] :
% 0.32/0.47 ( ( finite_finite_nat @ B )
% 0.32/0.47 => ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.47 => ( finite_finite_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_finite_subset
% 0.32/0.47 thf(fact_59_rev__finite__subset,axiom,
% 0.32/0.47 ! [B: set_nat_nat_nat,A: set_nat_nat_nat] :
% 0.32/0.47 ( ( finite2045569477at_nat @ B )
% 0.32/0.47 => ( ( ord_le940807492at_nat @ A @ B )
% 0.32/0.47 => ( finite2045569477at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_finite_subset
% 0.32/0.47 thf(fact_60_rev__finite__subset,axiom,
% 0.32/0.47 ! [B: set_nat_nat_nat2,A: set_nat_nat_nat2] :
% 0.32/0.47 ( ( finite1440337093at_nat @ B )
% 0.32/0.47 => ( ( ord_le633272388at_nat @ A @ B )
% 0.32/0.47 => ( finite1440337093at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_finite_subset
% 0.32/0.47 thf(fact_61_rev__finite__subset,axiom,
% 0.32/0.47 ! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3] :
% 0.32/0.47 ( ( finite1064868788at_nat @ B )
% 0.32/0.47 => ( ( ord_le2040082867at_nat @ A @ B )
% 0.32/0.47 => ( finite1064868788at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_finite_subset
% 0.32/0.47 thf(fact_62_rev__finite__subset,axiom,
% 0.32/0.47 ! [B: set_nat_nat,A: set_nat_nat] :
% 0.32/0.47 ( ( finite570312790at_nat @ B )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.47 => ( finite570312790at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_finite_subset
% 0.32/0.47 thf(fact_63_infinite__super,axiom,
% 0.32/0.47 ! [S: set_nat,T: set_nat] :
% 0.32/0.47 ( ( ord_less_eq_set_nat @ S @ T )
% 0.32/0.47 => ( ~ ( finite_finite_nat @ S )
% 0.32/0.47 => ~ ( finite_finite_nat @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_super
% 0.32/0.47 thf(fact_64_infinite__super,axiom,
% 0.32/0.47 ! [S: set_nat_nat_nat,T: set_nat_nat_nat] :
% 0.32/0.47 ( ( ord_le940807492at_nat @ S @ T )
% 0.32/0.47 => ( ~ ( finite2045569477at_nat @ S )
% 0.32/0.47 => ~ ( finite2045569477at_nat @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_super
% 0.32/0.47 thf(fact_65_infinite__super,axiom,
% 0.32/0.47 ! [S: set_nat_nat_nat2,T: set_nat_nat_nat2] :
% 0.32/0.47 ( ( ord_le633272388at_nat @ S @ T )
% 0.32/0.47 => ( ~ ( finite1440337093at_nat @ S )
% 0.32/0.47 => ~ ( finite1440337093at_nat @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_super
% 0.32/0.47 thf(fact_66_infinite__super,axiom,
% 0.32/0.47 ! [S: set_nat_nat_nat_nat3,T: set_nat_nat_nat_nat3] :
% 0.32/0.47 ( ( ord_le2040082867at_nat @ S @ T )
% 0.32/0.47 => ( ~ ( finite1064868788at_nat @ S )
% 0.32/0.47 => ~ ( finite1064868788at_nat @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_super
% 0.32/0.47 thf(fact_67_infinite__super,axiom,
% 0.32/0.47 ! [S: set_nat_nat,T: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ S @ T )
% 0.32/0.47 => ( ~ ( finite570312790at_nat @ S )
% 0.32/0.47 => ~ ( finite570312790at_nat @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_super
% 0.32/0.47 thf(fact_68_finite__subset,axiom,
% 0.32/0.47 ! [A: set_nat,B: set_nat] :
% 0.32/0.47 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.47 => ( ( finite_finite_nat @ B )
% 0.32/0.47 => ( finite_finite_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_subset
% 0.32/0.47 thf(fact_69_finite__subset,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
% 0.32/0.47 ( ( ord_le940807492at_nat @ A @ B )
% 0.32/0.47 => ( ( finite2045569477at_nat @ B )
% 0.32/0.47 => ( finite2045569477at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_subset
% 0.32/0.47 thf(fact_70_finite__subset,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
% 0.32/0.47 ( ( ord_le633272388at_nat @ A @ B )
% 0.32/0.47 => ( ( finite1440337093at_nat @ B )
% 0.32/0.47 => ( finite1440337093at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_subset
% 0.32/0.47 thf(fact_71_finite__subset,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat_nat3,B: set_nat_nat_nat_nat3] :
% 0.32/0.47 ( ( ord_le2040082867at_nat @ A @ B )
% 0.32/0.47 => ( ( finite1064868788at_nat @ B )
% 0.32/0.47 => ( finite1064868788at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_subset
% 0.32/0.47 thf(fact_72_finite__subset,axiom,
% 0.32/0.47 ! [A: set_nat_nat,B: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.47 => ( ( finite570312790at_nat @ B )
% 0.32/0.47 => ( finite570312790at_nat @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_subset
% 0.32/0.47 thf(fact_73_add__le__same__cancel1,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
% 0.32/0.47 = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_same_cancel1
% 0.32/0.47 thf(fact_74_add__le__same__cancel2,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
% 0.32/0.47 = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_same_cancel2
% 0.32/0.47 thf(fact_75_le__add__same__cancel1,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
% 0.32/0.47 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_add_same_cancel1
% 0.32/0.47 thf(fact_76_le__add__same__cancel2,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
% 0.32/0.47 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_add_same_cancel2
% 0.32/0.47 thf(fact_77_nat__add__left__cancel__le,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_add_left_cancel_le
% 0.32/0.47 thf(fact_78_add__is__0,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.47 = zero_zero_nat )
% 0.32/0.47 = ( ( M = zero_zero_nat )
% 0.32/0.47 & ( N = zero_zero_nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_is_0
% 0.32/0.47 thf(fact_79_Nat_Oadd__0__right,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.32/0.47 = M ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.add_0_right
% 0.32/0.47 thf(fact_80_le0,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le0
% 0.32/0.47 thf(fact_81_bot__nat__0_Oextremum,axiom,
% 0.32/0.47 ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum
% 0.32/0.47 thf(fact_82_add__right__cancel,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat,C: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ B3 @ A3 )
% 0.32/0.47 = ( plus_plus_nat @ C @ A3 ) )
% 0.32/0.47 = ( B3 = C ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_cancel
% 0.32/0.47 thf(fact_83_add__left__cancel,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ A3 @ B3 )
% 0.32/0.47 = ( plus_plus_nat @ A3 @ C ) )
% 0.32/0.47 = ( B3 = C ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_cancel
% 0.32/0.47 thf(fact_84_le__zero__eq,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.47 = ( N = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47 thf(fact_85_add__le__cancel__right,axiom,
% 0.32/0.47 ! [A3: nat,C: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.32/0.47 = ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_cancel_right
% 0.32/0.47 thf(fact_86_add__le__cancel__left,axiom,
% 0.32/0.47 ! [C: nat,A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.32/0.47 = ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_cancel_left
% 0.32/0.47 thf(fact_87_zero__eq__add__iff__both__eq__0,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( zero_zero_nat
% 0.32/0.47 = ( plus_plus_nat @ X3 @ Y ) )
% 0.32/0.47 = ( ( X3 = zero_zero_nat )
% 0.32/0.47 & ( Y = zero_zero_nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_eq_add_iff_both_eq_0
% 0.32/0.47 thf(fact_88_add__eq__0__iff__both__eq__0,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ X3 @ Y )
% 0.32/0.47 = zero_zero_nat )
% 0.32/0.47 = ( ( X3 = zero_zero_nat )
% 0.32/0.47 & ( Y = zero_zero_nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_eq_0_iff_both_eq_0
% 0.32/0.47 thf(fact_89_add__cancel__right__right,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( plus_plus_nat @ A3 @ B3 ) )
% 0.32/0.47 = ( B3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_cancel_right_right
% 0.32/0.47 thf(fact_90_add__cancel__right__left,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( plus_plus_nat @ B3 @ A3 ) )
% 0.32/0.47 = ( B3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_cancel_right_left
% 0.32/0.47 thf(fact_91_add__cancel__left__right,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ A3 @ B3 )
% 0.32/0.47 = A3 )
% 0.32/0.47 = ( B3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_cancel_left_right
% 0.32/0.47 thf(fact_92_add__cancel__left__left,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ B3 @ A3 )
% 0.32/0.47 = A3 )
% 0.32/0.47 = ( B3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_cancel_left_left
% 0.32/0.47 thf(fact_93_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A3: nat > nat,P2: ( nat > nat ) > $o] :
% 0.32/0.47 ( ( member_nat_nat @ A3 @ ( collect_nat_nat @ P2 ) )
% 0.32/0.47 = ( P2 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_94_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A3: nat,P2: nat > $o] :
% 0.32/0.47 ( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
% 0.32/0.47 = ( P2 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_95_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A3: nat > nat > nat,P2: ( nat > nat > nat ) > $o] :
% 0.32/0.47 ( ( member_nat_nat_nat2 @ A3 @ ( collect_nat_nat_nat2 @ P2 ) )
% 0.32/0.47 = ( P2 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_96_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A3: ( nat > nat ) > nat,P2: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.47 ( ( member_nat_nat_nat @ A3 @ ( collect_nat_nat_nat @ P2 ) )
% 0.32/0.47 = ( P2 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_97_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A3: ( nat > nat ) > nat > nat,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.47 ( ( member1128122036at_nat @ A3 @ ( collec1610646258at_nat @ P2 ) )
% 0.32/0.47 = ( P2 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_98_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: set_nat_nat] :
% 0.32/0.47 ( ( collect_nat_nat
% 0.32/0.47 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_99_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: set_nat] :
% 0.32/0.47 ( ( collect_nat
% 0.32/0.47 @ ^ [X: nat] : ( member_nat @ X @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_100_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat] :
% 0.32/0.47 ( ( collect_nat_nat_nat2
% 0.32/0.47 @ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_101_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat2] :
% 0.32/0.47 ( ( collect_nat_nat_nat
% 0.32/0.47 @ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_102_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: set_nat_nat_nat_nat3] :
% 0.32/0.47 ( ( collec1610646258at_nat
% 0.32/0.47 @ ^ [X: ( nat > nat ) > nat > nat] : ( member1128122036at_nat @ X @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_103_Collect__cong,axiom,
% 0.32/0.47 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 0.32/0.47 ( ! [X2: nat > nat] :
% 0.32/0.47 ( ( P2 @ X2 )
% 0.32/0.47 = ( Q @ X2 ) )
% 0.32/0.47 => ( ( collect_nat_nat @ P2 )
% 0.32/0.47 = ( collect_nat_nat @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_104_Collect__cong,axiom,
% 0.32/0.47 ! [P2: nat > $o,Q: nat > $o] :
% 0.32/0.47 ( ! [X2: nat] :
% 0.32/0.47 ( ( P2 @ X2 )
% 0.32/0.47 = ( Q @ X2 ) )
% 0.32/0.47 => ( ( collect_nat @ P2 )
% 0.32/0.47 = ( collect_nat @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_105_Collect__cong,axiom,
% 0.32/0.47 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 0.32/0.47 ( ! [X2: nat > nat > nat] :
% 0.32/0.47 ( ( P2 @ X2 )
% 0.32/0.47 = ( Q @ X2 ) )
% 0.32/0.47 => ( ( collect_nat_nat_nat2 @ P2 )
% 0.32/0.47 = ( collect_nat_nat_nat2 @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_106_Collect__cong,axiom,
% 0.32/0.47 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.47 ( ! [X2: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( P2 @ X2 )
% 0.32/0.47 = ( Q @ X2 ) )
% 0.32/0.47 => ( ( collect_nat_nat_nat @ P2 )
% 0.32/0.47 = ( collect_nat_nat_nat @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_107_Collect__cong,axiom,
% 0.32/0.47 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.47 ( ! [X2: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( P2 @ X2 )
% 0.32/0.47 = ( Q @ X2 ) )
% 0.32/0.47 => ( ( collec1610646258at_nat @ P2 )
% 0.32/0.47 = ( collec1610646258at_nat @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_108_add_Oright__neutral,axiom,
% 0.32/0.47 ! [A3: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ A3 @ zero_zero_nat )
% 0.32/0.47 = A3 ) ).
% 0.32/0.47
% 0.32/0.47 % add.right_neutral
% 0.32/0.47 thf(fact_109_add_Oleft__neutral,axiom,
% 0.32/0.47 ! [A3: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ zero_zero_nat @ A3 )
% 0.32/0.47 = A3 ) ).
% 0.32/0.47
% 0.32/0.47 % add.left_neutral
% 0.32/0.47 thf(fact_110_zero__reorient,axiom,
% 0.32/0.47 ! [X3: nat] :
% 0.32/0.47 ( ( zero_zero_nat = X3 )
% 0.32/0.47 = ( X3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_reorient
% 0.32/0.47 thf(fact_111_add__right__imp__eq,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat,C: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ B3 @ A3 )
% 0.32/0.47 = ( plus_plus_nat @ C @ A3 ) )
% 0.32/0.47 => ( B3 = C ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_imp_eq
% 0.32/0.47 thf(fact_112_add__left__imp__eq,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ A3 @ B3 )
% 0.32/0.47 = ( plus_plus_nat @ A3 @ C ) )
% 0.32/0.47 => ( B3 = C ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_imp_eq
% 0.32/0.47 thf(fact_113_add_Oleft__commute,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat,C: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) )
% 0.32/0.47 = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.left_commute
% 0.32/0.47 thf(fact_114_add_Ocommute,axiom,
% 0.32/0.47 ( plus_plus_nat
% 0.32/0.47 = ( ^ [A2: nat,B4: nat] : ( plus_plus_nat @ B4 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.commute
% 0.32/0.47 thf(fact_115_add_Oassoc,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
% 0.32/0.47 = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.assoc
% 0.32/0.47 thf(fact_116_group__cancel_Oadd2,axiom,
% 0.32/0.47 ! [B: nat,K: nat,B3: nat,A3: nat] :
% 0.32/0.47 ( ( B
% 0.32/0.47 = ( plus_plus_nat @ K @ B3 ) )
% 0.32/0.47 => ( ( plus_plus_nat @ A3 @ B )
% 0.32/0.47 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % group_cancel.add2
% 0.32/0.47 thf(fact_117_group__cancel_Oadd1,axiom,
% 0.32/0.47 ! [A: nat,K: nat,A3: nat,B3: nat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( plus_plus_nat @ K @ A3 ) )
% 0.32/0.47 => ( ( plus_plus_nat @ A @ B3 )
% 0.32/0.47 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % group_cancel.add1
% 0.32/0.47 thf(fact_118_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.47 ( ( ( I2 = J )
% 0.32/0.47 & ( K = L ) )
% 0.32/0.47 => ( ( plus_plus_nat @ I2 @ K )
% 0.32/0.47 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(4)
% 0.32/0.47 thf(fact_119_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
% 0.32/0.47 = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.47 thf(fact_120_one__reorient,axiom,
% 0.32/0.47 ! [X3: nat] :
% 0.32/0.47 ( ( one_one_nat = X3 )
% 0.32/0.47 = ( X3 = one_one_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % one_reorient
% 0.32/0.47 thf(fact_121_Nat_Oex__has__greatest__nat,axiom,
% 0.32/0.47 ! [P2: nat > $o,K: nat,B3: nat] :
% 0.32/0.47 ( ( P2 @ K )
% 0.32/0.47 => ( ! [Y2: nat] :
% 0.32/0.47 ( ( P2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ Y2 @ B3 ) )
% 0.32/0.47 => ? [X2: nat] :
% 0.32/0.47 ( ( P2 @ X2 )
% 0.32/0.47 & ! [Y3: nat] :
% 0.32/0.47 ( ( P2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.ex_has_greatest_nat
% 0.32/0.47 thf(fact_122_nat__le__linear,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_le_linear
% 0.32/0.47 thf(fact_123_le__antisym,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_antisym
% 0.32/0.47 thf(fact_124_eq__imp__le,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( M = N )
% 0.32/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_imp_le
% 0.32/0.47 thf(fact_125_le__trans,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 => ( ( ord_less_eq_nat @ J @ K )
% 0.32/0.47 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_trans
% 0.32/0.47 thf(fact_126_le__refl,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le_refl
% 0.32/0.47 thf(fact_127_zero__le,axiom,
% 0.32/0.47 ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le
% 0.32/0.47 thf(fact_128_add__le__imp__le__right,axiom,
% 0.32/0.47 ! [A3: nat,C: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_imp_le_right
% 0.32/0.47 thf(fact_129_add__le__imp__le__left,axiom,
% 0.32/0.47 ! [C: nat,A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_imp_le_left
% 0.32/0.47 thf(fact_130_le__iff__add,axiom,
% 0.32/0.47 ( ord_less_eq_nat
% 0.32/0.47 = ( ^ [A2: nat,B4: nat] :
% 0.32/0.47 ? [C2: nat] :
% 0.32/0.47 ( B4
% 0.32/0.47 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_add
% 0.32/0.47 thf(fact_131_add__right__mono,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_mono
% 0.32/0.47 thf(fact_132_less__eqE,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ~ ! [C3: nat] :
% 0.32/0.47 ( B3
% 0.32/0.47 != ( plus_plus_nat @ A3 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eqE
% 0.32/0.47 thf(fact_133_add__left__mono,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_mono
% 0.32/0.47 thf(fact_134_add__mono,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat,D: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono
% 0.32/0.47 thf(fact_135_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.47 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(1)
% 0.32/0.47 thf(fact_136_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.47 ( ( ( I2 = J )
% 0.32/0.47 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(2)
% 0.32/0.47 thf(fact_137_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.47 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 & ( K = L ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(3)
% 0.32/0.47 thf(fact_138_add_Ocomm__neutral,axiom,
% 0.32/0.47 ! [A3: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ A3 @ zero_zero_nat )
% 0.32/0.47 = A3 ) ).
% 0.32/0.47
% 0.32/0.47 % add.comm_neutral
% 0.32/0.47 thf(fact_139_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.47 ! [A3: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ zero_zero_nat @ A3 )
% 0.32/0.47 = A3 ) ).
% 0.32/0.47
% 0.32/0.47 % comm_monoid_add_class.add_0
% 0.32/0.47 thf(fact_140_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 0.32/0.47 => ( A3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_uniqueI
% 0.32/0.47 thf(fact_141_bot__nat__0_Oextremum__unique,axiom,
% 0.32/0.47 ! [A3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 0.32/0.47 = ( A3 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_unique
% 0.32/0.47 thf(fact_142_le__0__eq,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.47 = ( N = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_0_eq
% 0.32/0.47 thf(fact_143_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_nat.simps(1)
% 0.32/0.47 thf(fact_144_add__eq__self__zero,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.47 = M )
% 0.32/0.47 => ( N = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_eq_self_zero
% 0.32/0.47 thf(fact_145_plus__nat_Oadd__0,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.32/0.47 = N ) ).
% 0.32/0.47
% 0.32/0.47 % plus_nat.add_0
% 0.32/0.47 thf(fact_146_nat__le__iff__add,axiom,
% 0.32/0.47 ( ord_less_eq_nat
% 0.32/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.32/0.47 ? [K2: nat] :
% 0.32/0.47 ( N2
% 0.32/0.47 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_le_iff_add
% 0.32/0.47 thf(fact_147_trans__le__add2,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % trans_le_add2
% 0.32/0.47 thf(fact_148_trans__le__add1,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % trans_le_add1
% 0.32/0.47 thf(fact_149_add__le__mono1,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_mono1
% 0.32/0.47 thf(fact_150_add__le__mono,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.47 => ( ( ord_less_eq_nat @ K @ L )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_mono
% 0.32/0.47 thf(fact_151_le__Suc__ex,axiom,
% 0.32/0.47 ! [K: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ K @ L )
% 0.32/0.47 => ? [N3: nat] :
% 0.32/0.47 ( L
% 0.32/0.47 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_Suc_ex
% 0.32/0.47 thf(fact_152_add__leD2,axiom,
% 0.32/0.47 ! [M: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.47 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_leD2
% 0.32/0.47 thf(fact_153_add__leD1,axiom,
% 0.32/0.47 ! [M: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_leD1
% 0.32/0.47 thf(fact_154_le__add2,axiom,
% 0.32/0.47 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_add2
% 0.32/0.47 thf(fact_155_le__add1,axiom,
% 0.32/0.47 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_add1
% 0.32/0.47 thf(fact_156_add__leE,axiom,
% 0.32/0.47 ! [M: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.47 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_leE
% 0.32/0.47 thf(fact_157_add__nonpos__eq__0__iff,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
% 0.32/0.47 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.32/0.47 => ( ( ( plus_plus_nat @ X3 @ Y )
% 0.32/0.47 = zero_zero_nat )
% 0.32/0.47 = ( ( X3 = zero_zero_nat )
% 0.32/0.47 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_nonpos_eq_0_iff
% 0.32/0.47 thf(fact_158_add__nonneg__eq__0__iff,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.47 => ( ( ( plus_plus_nat @ X3 @ Y )
% 0.32/0.47 = zero_zero_nat )
% 0.32/0.47 = ( ( X3 = zero_zero_nat )
% 0.32/0.47 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_nonneg_eq_0_iff
% 0.32/0.47 thf(fact_159_add__nonpos__nonpos,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_nonpos_nonpos
% 0.32/0.47 thf(fact_160_add__nonneg__nonneg,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.32/0.47 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_nonneg_nonneg
% 0.32/0.47 thf(fact_161_add__increasing2,axiom,
% 0.32/0.47 ! [C: nat,B3: nat,A3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.32/0.47 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_increasing2
% 0.32/0.47 thf(fact_162_add__decreasing2,axiom,
% 0.32/0.47 ! [C: nat,A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.32/0.47 => ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_decreasing2
% 0.32/0.47 thf(fact_163_add__increasing,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_increasing
% 0.32/0.47 thf(fact_164_add__decreasing,axiom,
% 0.32/0.47 ! [A3: nat,C: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 0.32/0.47 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_decreasing
% 0.32/0.47 thf(fact_165_sum_Ofinite__Collect__op,axiom,
% 0.32/0.47 ! [I3: set_nat_nat,X3: ( nat > nat ) > nat,Y: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( finite570312790at_nat
% 0.32/0.47 @ ( collect_nat_nat
% 0.32/0.47 @ ^ [I: nat > nat] :
% 0.32/0.47 ( ( member_nat_nat @ I @ I3 )
% 0.32/0.47 & ( ( X3 @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( ( finite570312790at_nat
% 0.32/0.47 @ ( collect_nat_nat
% 0.32/0.47 @ ^ [I: nat > nat] :
% 0.32/0.47 ( ( member_nat_nat @ I @ I3 )
% 0.32/0.47 & ( ( Y @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( finite570312790at_nat
% 0.32/0.47 @ ( collect_nat_nat
% 0.32/0.47 @ ^ [I: nat > nat] :
% 0.32/0.47 ( ( member_nat_nat @ I @ I3 )
% 0.32/0.47 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 0.32/0.47 != zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sum.finite_Collect_op
% 0.32/0.47 thf(fact_166_sum_Ofinite__Collect__op,axiom,
% 0.32/0.47 ! [I3: set_nat,X3: nat > nat,Y: nat > nat] :
% 0.32/0.47 ( ( finite_finite_nat
% 0.32/0.47 @ ( collect_nat
% 0.32/0.47 @ ^ [I: nat] :
% 0.32/0.47 ( ( member_nat @ I @ I3 )
% 0.32/0.47 & ( ( X3 @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( ( finite_finite_nat
% 0.32/0.47 @ ( collect_nat
% 0.32/0.47 @ ^ [I: nat] :
% 0.32/0.47 ( ( member_nat @ I @ I3 )
% 0.32/0.47 & ( ( Y @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( finite_finite_nat
% 0.32/0.47 @ ( collect_nat
% 0.32/0.47 @ ^ [I: nat] :
% 0.32/0.47 ( ( member_nat @ I @ I3 )
% 0.32/0.47 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 0.32/0.47 != zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sum.finite_Collect_op
% 0.32/0.47 thf(fact_167_sum_Ofinite__Collect__op,axiom,
% 0.32/0.47 ! [I3: set_nat_nat_nat,X3: ( nat > nat > nat ) > nat,Y: ( nat > nat > nat ) > nat] :
% 0.32/0.47 ( ( finite2045569477at_nat
% 0.32/0.47 @ ( collect_nat_nat_nat2
% 0.32/0.47 @ ^ [I: nat > nat > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat2 @ I @ I3 )
% 0.32/0.47 & ( ( X3 @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( ( finite2045569477at_nat
% 0.32/0.47 @ ( collect_nat_nat_nat2
% 0.32/0.47 @ ^ [I: nat > nat > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat2 @ I @ I3 )
% 0.32/0.47 & ( ( Y @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( finite2045569477at_nat
% 0.32/0.47 @ ( collect_nat_nat_nat2
% 0.32/0.47 @ ^ [I: nat > nat > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat2 @ I @ I3 )
% 0.32/0.47 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 0.32/0.47 != zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sum.finite_Collect_op
% 0.32/0.47 thf(fact_168_sum_Ofinite__Collect__op,axiom,
% 0.32/0.47 ! [I3: set_nat_nat_nat2,X3: ( ( nat > nat ) > nat ) > nat,Y: ( ( nat > nat ) > nat ) > nat] :
% 0.32/0.47 ( ( finite1440337093at_nat
% 0.32/0.47 @ ( collect_nat_nat_nat
% 0.32/0.47 @ ^ [I: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat @ I @ I3 )
% 0.32/0.47 & ( ( X3 @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( ( finite1440337093at_nat
% 0.32/0.47 @ ( collect_nat_nat_nat
% 0.32/0.47 @ ^ [I: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat @ I @ I3 )
% 0.32/0.47 & ( ( Y @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( finite1440337093at_nat
% 0.32/0.47 @ ( collect_nat_nat_nat
% 0.32/0.47 @ ^ [I: ( nat > nat ) > nat] :
% 0.32/0.47 ( ( member_nat_nat_nat @ I @ I3 )
% 0.32/0.47 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 0.32/0.47 != zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sum.finite_Collect_op
% 0.32/0.47 thf(fact_169_sum_Ofinite__Collect__op,axiom,
% 0.32/0.47 ! [I3: set_nat_nat_nat_nat3,X3: ( ( nat > nat ) > nat > nat ) > nat,Y: ( ( nat > nat ) > nat > nat ) > nat] :
% 0.32/0.47 ( ( finite1064868788at_nat
% 0.32/0.47 @ ( collec1610646258at_nat
% 0.32/0.47 @ ^ [I: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( member1128122036at_nat @ I @ I3 )
% 0.32/0.47 & ( ( X3 @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( ( finite1064868788at_nat
% 0.32/0.47 @ ( collec1610646258at_nat
% 0.32/0.47 @ ^ [I: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( member1128122036at_nat @ I @ I3 )
% 0.32/0.47 & ( ( Y @ I )
% 0.32/0.47 != zero_zero_nat ) ) ) )
% 0.32/0.47 => ( finite1064868788at_nat
% 0.32/0.47 @ ( collec1610646258at_nat
% 0.32/0.47 @ ^ [I: ( nat > nat ) > nat > nat] :
% 0.32/0.47 ( ( member1128122036at_nat @ I @ I3 )
% 0.32/0.47 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 0.32/0.47 != zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sum.finite_Collect_op
% 0.32/0.47 thf(fact_170_not__one__le__zero,axiom,
% 0.32/0.47 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.32/0.47
% 0.32/0.47 % not_one_le_zero
% 0.32/0.47 thf(fact_171_zero__le__one,axiom,
% 0.32/0.47 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.47
% 0.32/0.47 % zero_le_one
% 0.32/0.47 thf(fact_172_finite__less__ub,axiom,
% 0.32/0.47 ! [F2: nat > nat,U: nat] :
% 0.32/0.47 ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
% 0.32/0.47 => ( finite_finite_nat
% 0.32/0.47 @ ( collect_nat
% 0.32/0.47 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_less_ub
% 0.32/0.47 thf(fact_173_subset__antisym,axiom,
% 0.32/0.47 ! [A: set_nat_nat,B: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_antisym
% 0.32/0.47 thf(fact_174_subsetI,axiom,
% 0.32/0.47 ! [A: set_nat,B: set_nat] :
% 0.32/0.47 ( ! [X2: nat] :
% 0.32/0.47 ( ( member_nat @ X2 @ A )
% 0.32/0.47 => ( member_nat @ X2 @ B ) )
% 0.32/0.47 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetI
% 0.32/0.47 thf(fact_175_subsetI,axiom,
% 0.32/0.47 ! [A: set_nat_nat,B: set_nat_nat] :
% 0.32/0.47 ( ! [X2: nat > nat] :
% 0.32/0.47 ( ( member_nat_nat @ X2 @ A )
% 0.32/0.47 => ( member_nat_nat @ X2 @ B ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetI
% 0.32/0.47 thf(fact_176_order__refl,axiom,
% 0.32/0.47 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_177_order__refl,axiom,
% 0.32/0.47 ! [X3: set_nat_nat] : ( ord_le1415039317at_nat @ X3 @ X3 ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_178_finite__nat__set__iff__bounded__le,axiom,
% 0.32/0.47 ( finite_finite_nat
% 0.32/0.47 = ( ^ [N4: set_nat] :
% 0.32/0.47 ? [M2: nat] :
% 0.32/0.47 ! [X: nat] :
% 0.32/0.47 ( ( member_nat @ X @ N4 )
% 0.32/0.47 => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_nat_set_iff_bounded_le
% 0.32/0.47 thf(fact_179_Euclid__induct,axiom,
% 0.32/0.47 ! [P2: nat > nat > $o,A3: nat,B3: nat] :
% 0.32/0.47 ( ! [A4: nat,B5: nat] :
% 0.32/0.47 ( ( P2 @ A4 @ B5 )
% 0.32/0.47 = ( P2 @ B5 @ A4 ) )
% 0.32/0.47 => ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
% 0.32/0.47 => ( ! [A4: nat,B5: nat] :
% 0.32/0.47 ( ( P2 @ A4 @ B5 )
% 0.32/0.47 => ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
% 0.32/0.47 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Euclid_induct
% 0.32/0.47 thf(fact_180_order__subst1,axiom,
% 0.32/0.47 ! [A3: nat,F2: nat > nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_181_order__subst1,axiom,
% 0.32/0.47 ! [A3: nat,F2: set_nat_nat > nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_182_order__subst1,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,F2: nat > set_nat_nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_183_order__subst1,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,F2: set_nat_nat > set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_184_order__subst2,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,F2: nat > nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_185_order__subst2,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ ( F2 @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_186_order__subst2,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_187_order__subst2,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ ( F2 @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_188_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A3: nat,F2: nat > nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_189_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,F2: nat > set_nat_nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_190_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A3: nat,F2: set_nat_nat > nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_191_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,F2: set_nat_nat > set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F2 @ B3 ) )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_192_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,F2: nat > nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ( F2 @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_193_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ( F2 @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_194_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ( F2 @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_195_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ( F2 @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 0.32/0.47 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_196_eq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [X: nat,Y5: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X @ Y5 )
% 0.32/0.47 & ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_197_eq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [X: set_nat_nat,Y5: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X @ Y5 )
% 0.32/0.47 & ( ord_le1415039317at_nat @ Y5 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_198_antisym,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ( ord_less_eq_nat @ Y @ X3 )
% 0.32/0.47 => ( X3 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_199_antisym,axiom,
% 0.32/0.47 ! [X3: set_nat_nat,Y: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X3 @ Y )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ Y @ X3 )
% 0.32/0.47 => ( X3 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_200_linear,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 | ( ord_less_eq_nat @ Y @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_201_eq__refl,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( X3 = Y )
% 0.32/0.47 => ( ord_less_eq_nat @ X3 @ Y ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_202_eq__refl,axiom,
% 0.32/0.47 ! [X3: set_nat_nat,Y: set_nat_nat] :
% 0.32/0.47 ( ( X3 = Y )
% 0.32/0.47 => ( ord_le1415039317at_nat @ X3 @ Y ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_203_le__cases,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ~ ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_204_order_Otrans,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_205_order_Otrans,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_206_le__cases3,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat,Z2: nat] :
% 0.32/0.47 ( ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ Y @ X3 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Y @ X3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
% 0.32/0.47 => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_207_antisym__conv,axiom,
% 0.32/0.47 ! [Y: nat,X3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ Y @ X3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 = ( X3 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_208_antisym__conv,axiom,
% 0.32/0.47 ! [Y: set_nat_nat,X3: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ Y @ X3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ X3 @ Y )
% 0.32/0.47 = ( X3 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_209_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [A2: nat,B4: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A2 @ B4 )
% 0.32/0.47 & ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_210_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A2 @ B4 )
% 0.32/0.47 & ( ord_le1415039317at_nat @ B4 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_211_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( A3 = B3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_212_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( A3 = B3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_213_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( B3 = C )
% 0.32/0.47 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_214_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( B3 = C )
% 0.32/0.47 => ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_215_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.32/0.47 => ( A3 = B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_216_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A3: set_nat_nat,B3: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 0.32/0.47 => ( A3 = B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_217_order__trans,axiom,
% 0.32/0.47 ! [X3: nat,Y: nat,Z2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.32/0.47 => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_218_order__trans,axiom,
% 0.32/0.47 ! [X3: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ X3 @ Y )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ Y @ Z2 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ X3 @ Z2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_219_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_220_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A3: set_nat_nat] : ( ord_le1415039317at_nat @ A3 @ A3 ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_221_linorder__wlog,axiom,
% 0.32/0.47 ! [P2: nat > nat > $o,A3: nat,B3: nat] :
% 0.32/0.47 ( ! [A4: nat,B5: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A4 @ B5 )
% 0.32/0.47 => ( P2 @ A4 @ B5 ) )
% 0.32/0.47 => ( ! [A4: nat,B5: nat] :
% 0.32/0.47 ( ( P2 @ B5 @ A4 )
% 0.32/0.47 => ( P2 @ A4 @ B5 ) )
% 0.32/0.47 => ( P2 @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_222_dual__order_Otrans,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.32/0.47 => ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_223_dual__order_Otrans,axiom,
% 0.32/0.47 ! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ C @ B3 )
% 0.32/0.47 => ( ord_le1415039317at_nat @ C @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_224_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [A2: nat,B4: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ B4 @ A2 )
% 0.32/0.47 & ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_225_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ B4 @ A2 )
% 0.32/0.47 & ( ord_le1415039317at_nat @ A2 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_226_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [B3: nat,A3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( A3 = B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_227_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [B3: set_nat_nat,A3: set_nat_nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.47 => ( A3 = B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_228_in__mono,axiom,
% 0.32/0.47 ! [A: set_nat,B: set_nat,X3: nat] :
% 0.32/0.47 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.47 => ( ( member_nat @ X3 @ A )
% 0.32/0.47 => ( member_nat @ X3 @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_mono
% 0.32/0.47 thf(fact_229_in__mono,axiom,
% 0.32/0.47 ! [A: set_nat_nat,B: set_nat_nat,X3: nat > nat] :
% 0.32/0.47 ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.47 => ( ( member_nat_nat @ X3 @ A )
% 0.32/0.47 => ( member_nat_nat @ X3 @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_mono
% 0.32/0.47 thf(fact_230_subsetD,axiom,
% 0.32/0.47 ! [A: set_nat,B: set_nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.47 => ( ( member_nat @ C @ A )
% 0.32/0.48 => ( member_nat @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subsetD
% 0.32/0.48 thf(fact_231_subsetD,axiom,
% 0.32/0.48 ! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.48 => ( ( member_nat_nat @ C @ A )
% 0.32/0.48 => ( member_nat_nat @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subsetD
% 0.32/0.48 thf(fact_232_equalityE,axiom,
% 0.32/0.48 ! [A: set_nat_nat,B: set_nat_nat] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ~ ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.48 => ~ ( ord_le1415039317at_nat @ B @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % equalityE
% 0.32/0.48 thf(fact_233_subset__eq,axiom,
% 0.32/0.48 ( ord_less_eq_set_nat
% 0.32/0.48 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.32/0.48 ! [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ A5 )
% 0.32/0.48 => ( member_nat @ X @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_eq
% 0.32/0.48 thf(fact_234_subset__eq,axiom,
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 0.32/0.48 ! [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ A5 )
% 0.32/0.48 => ( member_nat_nat @ X @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_eq
% 0.32/0.48 thf(fact_235_equalityD1,axiom,
% 0.32/0.48 ! [A: set_nat_nat,B: set_nat_nat] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ( ord_le1415039317at_nat @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % equalityD1
% 0.32/0.48 thf(fact_236_equalityD2,axiom,
% 0.32/0.48 ! [A: set_nat_nat,B: set_nat_nat] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ( ord_le1415039317at_nat @ B @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % equalityD2
% 0.32/0.48 thf(fact_237_subset__iff,axiom,
% 0.32/0.48 ( ord_less_eq_set_nat
% 0.32/0.48 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.32/0.48 ! [T2: nat] :
% 0.32/0.48 ( ( member_nat @ T2 @ A5 )
% 0.32/0.48 => ( member_nat @ T2 @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_iff
% 0.32/0.48 thf(fact_238_subset__iff,axiom,
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 0.32/0.48 ! [T2: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ T2 @ A5 )
% 0.32/0.48 => ( member_nat_nat @ T2 @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_iff
% 0.32/0.48 thf(fact_239_subset__refl,axiom,
% 0.32/0.48 ! [A: set_nat_nat] : ( ord_le1415039317at_nat @ A @ A ) ).
% 0.32/0.48
% 0.32/0.48 % subset_refl
% 0.32/0.48 thf(fact_240_Collect__mono,axiom,
% 0.32/0.48 ! [P2: nat > $o,Q: nat > $o] :
% 0.32/0.48 ( ! [X2: nat] :
% 0.32/0.48 ( ( P2 @ X2 )
% 0.32/0.48 => ( Q @ X2 ) )
% 0.32/0.48 => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono
% 0.32/0.48 thf(fact_241_Collect__mono,axiom,
% 0.32/0.48 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ! [X2: nat > nat > nat] :
% 0.32/0.48 ( ( P2 @ X2 )
% 0.32/0.48 => ( Q @ X2 ) )
% 0.32/0.48 => ( ord_le940807492at_nat @ ( collect_nat_nat_nat2 @ P2 ) @ ( collect_nat_nat_nat2 @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono
% 0.32/0.48 thf(fact_242_Collect__mono,axiom,
% 0.32/0.48 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ! [X2: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( P2 @ X2 )
% 0.32/0.48 => ( Q @ X2 ) )
% 0.32/0.48 => ( ord_le633272388at_nat @ ( collect_nat_nat_nat @ P2 ) @ ( collect_nat_nat_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono
% 0.32/0.48 thf(fact_243_Collect__mono,axiom,
% 0.32/0.48 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ! [X2: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( P2 @ X2 )
% 0.32/0.48 => ( Q @ X2 ) )
% 0.32/0.48 => ( ord_le2040082867at_nat @ ( collec1610646258at_nat @ P2 ) @ ( collec1610646258at_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono
% 0.32/0.48 thf(fact_244_Collect__mono,axiom,
% 0.32/0.48 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 0.32/0.48 ( ! [X2: nat > nat] :
% 0.32/0.48 ( ( P2 @ X2 )
% 0.32/0.48 => ( Q @ X2 ) )
% 0.32/0.48 => ( ord_le1415039317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono
% 0.32/0.48 thf(fact_245_subset__trans,axiom,
% 0.32/0.48 ! [A: set_nat_nat,B: set_nat_nat,C4: set_nat_nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ A @ B )
% 0.32/0.48 => ( ( ord_le1415039317at_nat @ B @ C4 )
% 0.32/0.48 => ( ord_le1415039317at_nat @ A @ C4 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_trans
% 0.32/0.48 thf(fact_246_set__eq__subset,axiom,
% 0.32/0.48 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 0.32/0.48 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ A5 @ B2 )
% 0.32/0.48 & ( ord_le1415039317at_nat @ B2 @ A5 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % set_eq_subset
% 0.32/0.48 thf(fact_247_Collect__mono__iff,axiom,
% 0.32/0.48 ! [P2: nat > $o,Q: nat > $o] :
% 0.32/0.48 ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
% 0.32/0.48 = ( ! [X: nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 => ( Q @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono_iff
% 0.32/0.48 thf(fact_248_Collect__mono__iff,axiom,
% 0.32/0.48 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le940807492at_nat @ ( collect_nat_nat_nat2 @ P2 ) @ ( collect_nat_nat_nat2 @ Q ) )
% 0.32/0.48 = ( ! [X: nat > nat > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 => ( Q @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono_iff
% 0.32/0.48 thf(fact_249_Collect__mono__iff,axiom,
% 0.32/0.48 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ( ord_le633272388at_nat @ ( collect_nat_nat_nat @ P2 ) @ ( collect_nat_nat_nat @ Q ) )
% 0.32/0.48 = ( ! [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 => ( Q @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono_iff
% 0.32/0.48 thf(fact_250_Collect__mono__iff,axiom,
% 0.32/0.48 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le2040082867at_nat @ ( collec1610646258at_nat @ P2 ) @ ( collec1610646258at_nat @ Q ) )
% 0.32/0.48 = ( ! [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 => ( Q @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono_iff
% 0.32/0.48 thf(fact_251_Collect__mono__iff,axiom,
% 0.32/0.48 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) )
% 0.32/0.48 = ( ! [X: nat > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 => ( Q @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mono_iff
% 0.32/0.48 thf(fact_252_bounded__Max__nat,axiom,
% 0.32/0.48 ! [P2: nat > $o,X3: nat,M3: nat] :
% 0.32/0.48 ( ( P2 @ X3 )
% 0.32/0.48 => ( ! [X2: nat] :
% 0.32/0.48 ( ( P2 @ X2 )
% 0.32/0.48 => ( ord_less_eq_nat @ X2 @ M3 ) )
% 0.32/0.48 => ~ ! [M4: nat] :
% 0.32/0.48 ( ( P2 @ M4 )
% 0.32/0.48 => ~ ! [X4: nat] :
% 0.32/0.48 ( ( P2 @ X4 )
% 0.32/0.48 => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % bounded_Max_nat
% 0.32/0.48 thf(fact_253_Collect__subset,axiom,
% 0.32/0.48 ! [A: set_nat,P2: nat > $o] :
% 0.32/0.48 ( ord_less_eq_set_nat
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ A ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_subset
% 0.32/0.48 thf(fact_254_Collect__subset,axiom,
% 0.32/0.48 ! [A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ord_le940807492at_nat
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ A ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_subset
% 0.32/0.48 thf(fact_255_Collect__subset,axiom,
% 0.32/0.48 ! [A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ord_le633272388at_nat
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ A ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_subset
% 0.32/0.48 thf(fact_256_Collect__subset,axiom,
% 0.32/0.48 ! [A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ord_le2040082867at_nat
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ A ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_subset
% 0.32/0.48 thf(fact_257_Collect__subset,axiom,
% 0.32/0.48 ! [A: set_nat_nat,P2: ( nat > nat ) > $o] :
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ A ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_subset
% 0.32/0.48 thf(fact_258_less__eq__set__def,axiom,
% 0.32/0.48 ( ord_less_eq_set_nat
% 0.32/0.48 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.32/0.48 ( ord_less_eq_nat_o
% 0.32/0.48 @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 0.32/0.48 @ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_eq_set_def
% 0.32/0.48 thf(fact_259_less__eq__set__def,axiom,
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 0.32/0.48 ( ord_le2026740592_nat_o
% 0.32/0.48 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ A5 )
% 0.32/0.48 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_eq_set_def
% 0.32/0.48 thf(fact_260_zero__neq__one,axiom,
% 0.32/0.48 zero_zero_nat != one_one_nat ).
% 0.32/0.48
% 0.32/0.48 % zero_neq_one
% 0.32/0.48 thf(fact_261_le__numeral__extra_I4_J,axiom,
% 0.32/0.48 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.32/0.48
% 0.32/0.48 % le_numeral_extra(4)
% 0.32/0.48 thf(fact_262_verit__sum__simplify,axiom,
% 0.32/0.48 ! [A3: nat] :
% 0.32/0.48 ( ( plus_plus_nat @ A3 @ zero_zero_nat )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % verit_sum_simplify
% 0.32/0.48 thf(fact_263_add__0__iff,axiom,
% 0.32/0.48 ! [B3: nat,A3: nat] :
% 0.32/0.48 ( ( B3
% 0.32/0.48 = ( plus_plus_nat @ B3 @ A3 ) )
% 0.32/0.48 = ( A3 = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_0_iff
% 0.32/0.48 thf(fact_264_le__numeral__extra_I3_J,axiom,
% 0.32/0.48 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.32/0.48
% 0.32/0.48 % le_numeral_extra(3)
% 0.32/0.48 thf(fact_265_verit__la__disequality,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( A3 = B3 )
% 0.32/0.48 | ~ ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.48 | ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % verit_la_disequality
% 0.32/0.48 thf(fact_266_pred__subset__eq,axiom,
% 0.32/0.48 ! [R: set_nat,S: set_nat] :
% 0.32/0.48 ( ( ord_less_eq_nat_o
% 0.32/0.48 @ ^ [X: nat] : ( member_nat @ X @ R )
% 0.32/0.48 @ ^ [X: nat] : ( member_nat @ X @ S ) )
% 0.32/0.48 = ( ord_less_eq_set_nat @ R @ S ) ) ).
% 0.32/0.48
% 0.32/0.48 % pred_subset_eq
% 0.32/0.48 thf(fact_267_pred__subset__eq,axiom,
% 0.32/0.48 ! [R: set_nat_nat,S: set_nat_nat] :
% 0.32/0.48 ( ( ord_le2026740592_nat_o
% 0.32/0.48 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
% 0.32/0.48 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
% 0.32/0.48 = ( ord_le1415039317at_nat @ R @ S ) ) ).
% 0.32/0.48
% 0.32/0.48 % pred_subset_eq
% 0.32/0.48 thf(fact_268_conj__subset__def,axiom,
% 0.32/0.48 ! [A: set_nat,P2: nat > $o,Q: nat > $o] :
% 0.32/0.48 ( ( ord_less_eq_set_nat @ A
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 & ( Q @ X ) ) ) )
% 0.32/0.48 = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P2 ) )
% 0.32/0.48 & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % conj_subset_def
% 0.32/0.48 thf(fact_269_conj__subset__def,axiom,
% 0.32/0.48 ! [A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le940807492at_nat @ A
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 & ( Q @ X ) ) ) )
% 0.32/0.48 = ( ( ord_le940807492at_nat @ A @ ( collect_nat_nat_nat2 @ P2 ) )
% 0.32/0.48 & ( ord_le940807492at_nat @ A @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % conj_subset_def
% 0.32/0.48 thf(fact_270_conj__subset__def,axiom,
% 0.32/0.48 ! [A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ( ord_le633272388at_nat @ A
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 & ( Q @ X ) ) ) )
% 0.32/0.48 = ( ( ord_le633272388at_nat @ A @ ( collect_nat_nat_nat @ P2 ) )
% 0.32/0.48 & ( ord_le633272388at_nat @ A @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % conj_subset_def
% 0.32/0.48 thf(fact_271_conj__subset__def,axiom,
% 0.32/0.48 ! [A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le2040082867at_nat @ A
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 & ( Q @ X ) ) ) )
% 0.32/0.48 = ( ( ord_le2040082867at_nat @ A @ ( collec1610646258at_nat @ P2 ) )
% 0.32/0.48 & ( ord_le2040082867at_nat @ A @ ( collec1610646258at_nat @ Q ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % conj_subset_def
% 0.32/0.48 thf(fact_272_conj__subset__def,axiom,
% 0.32/0.48 ! [A: set_nat_nat,P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ A
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( P2 @ X )
% 0.32/0.48 & ( Q @ X ) ) ) )
% 0.32/0.48 = ( ( ord_le1415039317at_nat @ A @ ( collect_nat_nat @ P2 ) )
% 0.32/0.48 & ( ord_le1415039317at_nat @ A @ ( collect_nat_nat @ Q ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % conj_subset_def
% 0.32/0.48 thf(fact_273_subset__Collect__iff,axiom,
% 0.32/0.48 ! [B: set_nat,A: set_nat,P2: nat > $o] :
% 0.32/0.48 ( ( ord_less_eq_set_nat @ B @ A )
% 0.32/0.48 => ( ( ord_less_eq_set_nat @ B
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 = ( ! [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ B )
% 0.32/0.48 => ( P2 @ X ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_Collect_iff
% 0.32/0.48 thf(fact_274_subset__Collect__iff,axiom,
% 0.32/0.48 ! [B: set_nat_nat_nat,A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le940807492at_nat @ B @ A )
% 0.32/0.48 => ( ( ord_le940807492at_nat @ B
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 = ( ! [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ B )
% 0.32/0.48 => ( P2 @ X ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_Collect_iff
% 0.32/0.48 thf(fact_275_subset__Collect__iff,axiom,
% 0.32/0.48 ! [B: set_nat_nat_nat2,A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ( ord_le633272388at_nat @ B @ A )
% 0.32/0.48 => ( ( ord_le633272388at_nat @ B
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 = ( ! [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ B )
% 0.32/0.48 => ( P2 @ X ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_Collect_iff
% 0.32/0.48 thf(fact_276_subset__Collect__iff,axiom,
% 0.32/0.48 ! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le2040082867at_nat @ B @ A )
% 0.32/0.48 => ( ( ord_le2040082867at_nat @ B
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 = ( ! [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ B )
% 0.32/0.48 => ( P2 @ X ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_Collect_iff
% 0.32/0.48 thf(fact_277_subset__Collect__iff,axiom,
% 0.32/0.48 ! [B: set_nat_nat,A: set_nat_nat,P2: ( nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ B @ A )
% 0.32/0.48 => ( ( ord_le1415039317at_nat @ B
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 = ( ! [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ B )
% 0.32/0.48 => ( P2 @ X ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_Collect_iff
% 0.32/0.48 thf(fact_278_subset__CollectI,axiom,
% 0.32/0.48 ! [B: set_nat,A: set_nat,Q: nat > $o,P2: nat > $o] :
% 0.32/0.48 ( ( ord_less_eq_set_nat @ B @ A )
% 0.32/0.48 => ( ! [X2: nat] :
% 0.32/0.48 ( ( member_nat @ X2 @ B )
% 0.32/0.48 => ( ( Q @ X2 )
% 0.32/0.48 => ( P2 @ X2 ) ) )
% 0.32/0.48 => ( ord_less_eq_set_nat
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ B )
% 0.32/0.48 & ( Q @ X ) ) )
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_CollectI
% 0.32/0.48 thf(fact_279_subset__CollectI,axiom,
% 0.32/0.48 ! [B: set_nat_nat_nat,A: set_nat_nat_nat,Q: ( nat > nat > nat ) > $o,P2: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le940807492at_nat @ B @ A )
% 0.32/0.48 => ( ! [X2: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X2 @ B )
% 0.32/0.48 => ( ( Q @ X2 )
% 0.32/0.48 => ( P2 @ X2 ) ) )
% 0.32/0.48 => ( ord_le940807492at_nat
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ B )
% 0.32/0.48 & ( Q @ X ) ) )
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_CollectI
% 0.32/0.48 thf(fact_280_subset__CollectI,axiom,
% 0.32/0.48 ! [B: set_nat_nat_nat2,A: set_nat_nat_nat2,Q: ( ( nat > nat ) > nat ) > $o,P2: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ( ord_le633272388at_nat @ B @ A )
% 0.32/0.48 => ( ! [X2: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X2 @ B )
% 0.32/0.48 => ( ( Q @ X2 )
% 0.32/0.48 => ( P2 @ X2 ) ) )
% 0.32/0.48 => ( ord_le633272388at_nat
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ B )
% 0.32/0.48 & ( Q @ X ) ) )
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_CollectI
% 0.32/0.48 thf(fact_281_subset__CollectI,axiom,
% 0.32/0.48 ! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3,Q: ( ( nat > nat ) > nat > nat ) > $o,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le2040082867at_nat @ B @ A )
% 0.32/0.48 => ( ! [X2: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X2 @ B )
% 0.32/0.48 => ( ( Q @ X2 )
% 0.32/0.48 => ( P2 @ X2 ) ) )
% 0.32/0.48 => ( ord_le2040082867at_nat
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ B )
% 0.32/0.48 & ( Q @ X ) ) )
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_CollectI
% 0.32/0.48 thf(fact_282_subset__CollectI,axiom,
% 0.32/0.48 ! [B: set_nat_nat,A: set_nat_nat,Q: ( nat > nat ) > $o,P2: ( nat > nat ) > $o] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ B @ A )
% 0.32/0.48 => ( ! [X2: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X2 @ B )
% 0.32/0.48 => ( ( Q @ X2 )
% 0.32/0.48 => ( P2 @ X2 ) ) )
% 0.32/0.48 => ( ord_le1415039317at_nat
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ B )
% 0.32/0.48 & ( Q @ X ) ) )
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ A )
% 0.32/0.48 & ( P2 @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_CollectI
% 0.32/0.48 thf(fact_283_Collect__restrict,axiom,
% 0.32/0.48 ! [X5: set_nat,P2: nat > $o] :
% 0.32/0.48 ( ord_less_eq_set_nat
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ X5 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_restrict
% 0.32/0.48 thf(fact_284_Collect__restrict,axiom,
% 0.32/0.48 ! [X5: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ord_le940807492at_nat
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ X5 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_restrict
% 0.32/0.48 thf(fact_285_Collect__restrict,axiom,
% 0.32/0.48 ! [X5: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ord_le633272388at_nat
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ X5 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_restrict
% 0.32/0.48 thf(fact_286_Collect__restrict,axiom,
% 0.32/0.48 ! [X5: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ord_le2040082867at_nat
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ X5 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_restrict
% 0.32/0.48 thf(fact_287_Collect__restrict,axiom,
% 0.32/0.48 ! [X5: set_nat_nat,P2: ( nat > nat ) > $o] :
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) )
% 0.32/0.48 @ X5 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_restrict
% 0.32/0.48 thf(fact_288_prop__restrict,axiom,
% 0.32/0.48 ! [X3: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
% 0.32/0.48 ( ( member_nat @ X3 @ Z3 )
% 0.32/0.48 => ( ( ord_less_eq_set_nat @ Z3
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 => ( P2 @ X3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prop_restrict
% 0.32/0.48 thf(fact_289_prop__restrict,axiom,
% 0.32/0.48 ! [X3: nat > nat > nat,Z3: set_nat_nat_nat,X5: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X3 @ Z3 )
% 0.32/0.48 => ( ( ord_le940807492at_nat @ Z3
% 0.32/0.48 @ ( collect_nat_nat_nat2
% 0.32/0.48 @ ^ [X: nat > nat > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat2 @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 => ( P2 @ X3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prop_restrict
% 0.32/0.48 thf(fact_290_prop__restrict,axiom,
% 0.32/0.48 ! [X3: ( nat > nat ) > nat,Z3: set_nat_nat_nat2,X5: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X3 @ Z3 )
% 0.32/0.48 => ( ( ord_le633272388at_nat @ Z3
% 0.32/0.48 @ ( collect_nat_nat_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat] :
% 0.32/0.48 ( ( member_nat_nat_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 => ( P2 @ X3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prop_restrict
% 0.32/0.48 thf(fact_291_prop__restrict,axiom,
% 0.32/0.48 ! [X3: ( nat > nat ) > nat > nat,Z3: set_nat_nat_nat_nat3,X5: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 0.32/0.48 ( ( member1128122036at_nat @ X3 @ Z3 )
% 0.32/0.48 => ( ( ord_le2040082867at_nat @ Z3
% 0.32/0.48 @ ( collec1610646258at_nat
% 0.32/0.48 @ ^ [X: ( nat > nat ) > nat > nat] :
% 0.32/0.48 ( ( member1128122036at_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 => ( P2 @ X3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prop_restrict
% 0.32/0.48 thf(fact_292_prop__restrict,axiom,
% 0.32/0.48 ! [X3: nat > nat,Z3: set_nat_nat,X5: set_nat_nat,P2: ( nat > nat ) > $o] :
% 0.32/0.48 ( ( member_nat_nat @ X3 @ Z3 )
% 0.32/0.48 => ( ( ord_le1415039317at_nat @ Z3
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [X: nat > nat] :
% 0.32/0.48 ( ( member_nat_nat @ X @ X5 )
% 0.32/0.48 & ( P2 @ X ) ) ) )
% 0.32/0.48 => ( P2 @ X3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prop_restrict
% 0.32/0.48 thf(fact_293_Fpow__def,axiom,
% 0.32/0.48 ( finite_Fpow_nat
% 0.32/0.48 = ( ^ [A5: set_nat] :
% 0.32/0.48 ( collect_set_nat
% 0.32/0.48 @ ^ [X6: set_nat] :
% 0.32/0.48 ( ( ord_less_eq_set_nat @ X6 @ A5 )
% 0.32/0.48 & ( finite_finite_nat @ X6 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Fpow_def
% 0.32/0.48 thf(fact_294_Fpow__def,axiom,
% 0.32/0.48 ( finite90785608at_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat_nat] :
% 0.32/0.48 ( collec617280953at_nat
% 0.32/0.48 @ ^ [X6: set_nat_nat_nat] :
% 0.32/0.48 ( ( ord_le940807492at_nat @ X6 @ A5 )
% 0.32/0.48 & ( finite2045569477at_nat @ X6 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Fpow_def
% 0.32/0.48 thf(fact_295_Fpow__def,axiom,
% 0.32/0.48 ( finite1633036872at_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat_nat2] :
% 0.32/0.48 ( collec309745849at_nat
% 0.32/0.48 @ ^ [X6: set_nat_nat_nat2] :
% 0.32/0.48 ( ( ord_le633272388at_nat @ X6 @ A5 )
% 0.32/0.48 & ( finite1440337093at_nat @ X6 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Fpow_def
% 0.32/0.48 thf(fact_296_Fpow__def,axiom,
% 0.32/0.48 ( finite252967095at_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat_nat_nat3] :
% 0.32/0.48 ( collec1702444712at_nat
% 0.32/0.48 @ ^ [X6: set_nat_nat_nat_nat3] :
% 0.32/0.48 ( ( ord_le2040082867at_nat @ X6 @ A5 )
% 0.32/0.48 & ( finite1064868788at_nat @ X6 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Fpow_def
% 0.32/0.48 thf(fact_297_Fpow__def,axiom,
% 0.32/0.48 ( finite_Fpow_nat_nat
% 0.32/0.48 = ( ^ [A5: set_nat_nat] :
% 0.32/0.48 ( collect_set_nat_nat
% 0.32/0.48 @ ^ [X6: set_nat_nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ X6 @ A5 )
% 0.32/0.48 & ( finite570312790at_nat @ X6 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Fpow_def
% 0.32/0.48 thf(fact_298_partitions__imp__finite__elements,axiom,
% 0.32/0.48 ! [P3: nat > nat,N: nat] :
% 0.32/0.48 ( ( number1551313001itions @ P3 @ N )
% 0.32/0.48 => ( finite_finite_nat
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [I: nat] : ( ord_less_nat @ zero_zero_nat @ ( P3 @ I ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % partitions_imp_finite_elements
% 0.32/0.48 thf(fact_299_not__gr__zero,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/0.48 = ( N = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % not_gr_zero
% 0.32/0.48 thf(fact_300_add__less__cancel__right,axiom,
% 0.32/0.48 ! [A3: nat,C: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.32/0.48 = ( ord_less_nat @ A3 @ B3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_cancel_right
% 0.32/0.48 thf(fact_301_add__less__cancel__left,axiom,
% 0.32/0.48 ! [C: nat,A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.32/0.48 = ( ord_less_nat @ A3 @ B3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_cancel_left
% 0.32/0.48 thf(fact_302_bot__nat__0_Onot__eq__extremum,axiom,
% 0.32/0.48 ! [A3: nat] :
% 0.32/0.48 ( ( A3 != zero_zero_nat )
% 0.32/0.48 = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % bot_nat_0.not_eq_extremum
% 0.32/0.48 thf(fact_303_less__nat__zero__code,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % less_nat_zero_code
% 0.32/0.48 thf(fact_304_neq0__conv,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( N != zero_zero_nat )
% 0.32/0.48 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % neq0_conv
% 0.32/0.48 thf(fact_305_nat__add__left__cancel__less,axiom,
% 0.32/0.48 ! [K: nat,M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.48 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_add_left_cancel_less
% 0.32/0.48 thf(fact_306_finite__Collect__less__nat,axiom,
% 0.32/0.48 ! [K: nat] :
% 0.32/0.48 ( finite_finite_nat
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % finite_Collect_less_nat
% 0.32/0.48 thf(fact_307_add__less__same__cancel1,axiom,
% 0.32/0.48 ! [B3: nat,A3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
% 0.32/0.48 = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_same_cancel1
% 0.32/0.48 thf(fact_308_add__less__same__cancel2,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
% 0.32/0.48 = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_same_cancel2
% 0.32/0.48 thf(fact_309_less__add__same__cancel1,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
% 0.32/0.48 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_add_same_cancel1
% 0.32/0.48 thf(fact_310_less__add__same__cancel2,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
% 0.32/0.48 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_add_same_cancel2
% 0.32/0.48 thf(fact_311_add__gr__0,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.48 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.48 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_gr_0
% 0.32/0.48 thf(fact_312_less__one,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( ord_less_nat @ N @ one_one_nat )
% 0.32/0.48 = ( N = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_one
% 0.32/0.48 thf(fact_313_add__less__le__mono,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat,C: nat,D: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A3 @ B3 )
% 0.32/0.48 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_le_mono
% 0.32/0.48 thf(fact_314_add__le__less__mono,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat,C: nat,D: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.48 => ( ( ord_less_nat @ C @ D )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_less_mono
% 0.32/0.48 thf(fact_315_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( ord_less_nat @ I2 @ J )
% 0.32/0.48 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(3)
% 0.32/0.48 thf(fact_316_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.48 & ( ord_less_nat @ K @ L ) )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(4)
% 0.32/0.48 thf(fact_317_pos__add__strict,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_nat @ zero_zero_nat @ A3 )
% 0.32/0.48 => ( ( ord_less_nat @ B3 @ C )
% 0.32/0.48 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pos_add_strict
% 0.32/0.48 thf(fact_318_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A3 @ B3 )
% 0.32/0.48 => ~ ! [C3: nat] :
% 0.32/0.48 ( ( B3
% 0.32/0.48 = ( plus_plus_nat @ A3 @ C3 ) )
% 0.32/0.48 => ( C3 = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % canonically_ordered_monoid_add_class.lessE
% 0.32/0.48 thf(fact_319_add__pos__pos,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ zero_zero_nat @ A3 )
% 0.32/0.48 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.32/0.48 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_pos_pos
% 0.32/0.48 thf(fact_320_add__neg__neg,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A3 @ zero_zero_nat )
% 0.32/0.48 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_neg_neg
% 0.32/0.48 thf(fact_321_order_Onot__eq__order__implies__strict,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( A3 != B3 )
% 0.32/0.48 => ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.48 => ( ord_less_nat @ A3 @ B3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.not_eq_order_implies_strict
% 0.32/0.48 thf(fact_322_order_Onot__eq__order__implies__strict,axiom,
% 0.32/0.48 ! [A3: set_nat_nat,B3: set_nat_nat] :
% 0.32/0.48 ( ( A3 != B3 )
% 0.32/0.48 => ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 0.32/0.48 => ( ord_less_set_nat_nat @ A3 @ B3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.not_eq_order_implies_strict
% 0.32/0.48 thf(fact_323_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [B3: nat,A3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ B3 @ A3 )
% 0.32/0.48 => ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_implies_order
% 0.32/0.48 thf(fact_324_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [B3: set_nat_nat,A3: set_nat_nat] :
% 0.32/0.48 ( ( ord_less_set_nat_nat @ B3 @ A3 )
% 0.32/0.48 => ( ord_le1415039317at_nat @ B3 @ A3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_implies_order
% 0.32/0.48 thf(fact_325_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.48 ( ord_less_nat
% 0.32/0.48 = ( ^ [B4: nat,A2: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ B4 @ A2 )
% 0.32/0.48 & ( A2 != B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_iff_order
% 0.32/0.48 thf(fact_326_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.48 ( ord_less_set_nat_nat
% 0.32/0.48 = ( ^ [B4: set_nat_nat,A2: set_nat_nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ B4 @ A2 )
% 0.32/0.48 & ( A2 != B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_iff_order
% 0.32/0.48 thf(fact_327_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.48 ( ord_less_eq_nat
% 0.32/0.48 = ( ^ [B4: nat,A2: nat] :
% 0.32/0.48 ( ( ord_less_nat @ B4 @ A2 )
% 0.32/0.48 | ( A2 = B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.order_iff_strict
% 0.32/0.48 thf(fact_328_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 = ( ^ [B4: set_nat_nat,A2: set_nat_nat] :
% 0.32/0.48 ( ( ord_less_set_nat_nat @ B4 @ A2 )
% 0.32/0.48 | ( A2 = B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.order_iff_strict
% 0.32/0.48 thf(fact_329_order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A3: nat,B3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A3 @ B3 )
% 0.32/0.48 => ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_implies_order
% 0.32/0.48 thf(fact_330_order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A3: set_nat_nat,B3: set_nat_nat] :
% 0.32/0.48 ( ( ord_less_set_nat_nat @ A3 @ B3 )
% 0.32/0.48 => ( ord_le1415039317at_nat @ A3 @ B3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_implies_order
% 0.32/0.48 thf(fact_331_dual__order_Ostrict__trans2,axiom,
% 0.32/0.48 ! [B3: nat,A3: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_nat @ B3 @ A3 )
% 0.32/0.48 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.32/0.48 => ( ord_less_nat @ C @ A3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_trans2
% 0.32/0.48 thf(fact_332_dual__order_Ostrict__trans2,axiom,
% 0.32/0.48 ! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.48 ( ( ord_less_set_nat_nat @ B3 @ A3 )
% 0.32/0.48 => ( ( ord_le1415039317at_nat @ C @ B3 )
% 0.32/0.48 => ( ord_less_set_nat_nat @ C @ A3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_trans2
% 0.32/0.48 thf(fact_333_dual__order_Ostrict__trans1,axiom,
% 0.32/0.48 ! [B3: nat,A3: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.32/0.48 => ( ( ord_less_nat @ C @ B3 )
% 0.32/0.48 => ( ord_less_nat @ C @ A3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_trans1
% 0.32/0.48 thf(fact_334_dual__order_Ostrict__trans1,axiom,
% 0.32/0.48 ! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 0.32/0.48 => ( ( ord_less_set_nat_nat @ C @ B3 )
% 0.32/0.48 => ( ord_less_set_nat_nat @ C @ A3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_trans1
% 0.32/0.48 thf(fact_335_order_Ostrict__iff__order,axiom,
% 0.32/0.48 ( ord_less_nat
% 0.32/0.48 = ( ^ [A2: nat,B4: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ A2 @ B4 )
% 0.32/0.48 & ( A2 != B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_iff_order
% 0.32/0.48 thf(fact_336_order_Ostrict__iff__order,axiom,
% 0.32/0.48 ( ord_less_set_nat_nat
% 0.32/0.48 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 0.32/0.48 ( ( ord_le1415039317at_nat @ A2 @ B4 )
% 0.32/0.48 & ( A2 != B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_iff_order
% 0.32/0.48 thf(fact_337_order_Oorder__iff__strict,axiom,
% 0.32/0.48 ( ord_less_eq_nat
% 0.32/0.48 = ( ^ [A2: nat,B4: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A2 @ B4 )
% 0.32/0.48 | ( A2 = B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.order_iff_strict
% 0.32/0.48 thf(fact_338_order_Oorder__iff__strict,axiom,
% 0.32/0.48 ( ord_le1415039317at_nat
% 0.32/0.48 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 0.32/0.48 ( ( ord_less_set_nat_nat @ A2 @ B4 )
% 0.32/0.48 | ( A2 = B4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.order_iff_strict
% 0.32/0.48 thf(fact_339_order_Ostrict__trans2,axiom,
% 0.32/0.48 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 0.32/0.48 ( ( ord_less_set_nat_nat @ A3 @ B3 )
% 0.32/0.48 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 0.32/0.48 => ( ord_less_set_nat_nat @ A3 @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_trans2
% 0.32/0.48 thf(fact_340_nat__less__le,axiom,
% 0.32/0.48 ( ord_less_nat
% 0.32/0.48 = ( ^ [M2: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/0.48 & ( M2 != N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_less_le
% 0.32/0.48 thf(fact_341_less__imp__le__nat,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_nat @ M @ N )
% 0.32/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_le_nat
% 0.32/0.48 thf(fact_342_le__eq__less__or__eq,axiom,
% 0.32/0.48 ( ord_less_eq_nat
% 0.32/0.48 = ( ^ [M2: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.32/0.48 | ( M2 = N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_eq_less_or_eq
% 0.32/0.48 thf(fact_343_less__or__eq__imp__le,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( ord_less_nat @ M @ N )
% 0.32/0.48 | ( M = N ) )
% 0.32/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_or_eq_imp_le
% 0.32/0.48 thf(fact_344_le__neq__implies__less,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.48 => ( ( M != N )
% 0.32/0.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_neq_implies_less
% 0.32/0.48 thf(fact_345_less__mono__imp__le__mono,axiom,
% 0.32/0.48 ! [F2: nat > nat,I2: nat,J: nat] :
% 0.32/0.48 ( ! [I4: nat,J2: nat] :
% 0.32/0.48 ( ( ord_less_nat @ I4 @ J2 )
% 0.32/0.48 => ( ord_less_nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) )
% 0.32/0.48 => ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/0.48 => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_mono_imp_le_mono
% 0.32/0.48 thf(fact_346_bounded__nat__set__is__finite,axiom,
% 0.32/0.48 ! [N5: set_nat,N: nat] :
% 0.32/0.48 ( ! [X2: nat] :
% 0.32/0.48 ( ( member_nat @ X2 @ N5 )
% 0.32/0.48 => ( ord_less_nat @ X2 @ N ) )
% 0.32/0.48 => ( finite_finite_nat @ N5 ) ) ).
% 0.32/0.48
% 0.32/0.48 % bounded_nat_set_is_finite
% 0.32/0.48 thf(fact_347_finite__nat__set__iff__bounded,axiom,
% 0.32/0.48 ( finite_finite_nat
% 0.32/0.48 = ( ^ [N4: set_nat] :
% 0.32/0.48 ? [M2: nat] :
% 0.32/0.48 ! [X: nat] :
% 0.32/0.48 ( ( member_nat @ X @ N4 )
% 0.32/0.48 => ( ord_less_nat @ X @ M2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % finite_nat_set_iff_bounded
% 0.32/0.48 thf(fact_348_finite__M__bounded__by__nat,axiom,
% 0.32/0.48 ! [P2: nat > $o,I2: nat] :
% 0.32/0.48 ( finite_finite_nat
% 0.32/0.48 @ ( collect_nat
% 0.32/0.48 @ ^ [K2: nat] :
% 0.32/0.48 ( ( P2 @ K2 )
% 0.32/0.48 & ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % finite_M_bounded_by_nat
% 0.32/0.48
% 0.32/0.48 % Conjectures (1)
% 0.32/0.48 thf(conj_0,conjecture,
% 0.32/0.48 ( finite570312790at_nat
% 0.32/0.48 @ ( collect_nat_nat
% 0.32/0.48 @ ^ [P: nat > nat] : ( number1551313001itions @ P @ n ) ) ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 3.18/1.61 ---- Embedded in THF ---
% 3.18/1.61 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.18/1.61 %%% Generated on Tue Feb 24 08:27:47 EST 2026
% 3.18/1.61 %%% using '$$dhol' embedding, version 1.3.0.
% 3.18/1.61 %%% Logic specification used:
% 3.18/1.61 %%% thf(spec, logic, $$dhol).
% 3.18/1.61
% 3.18/1.61 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2DTF31226.p
% See solution above
% 3.18/1.61 ------------------------
% 3.18/1.64 ---- Cleaned THF ---
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 3.18/1.64 set_na1064332610at_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 set_na1201295426at_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 3.18/1.64 set_se1657353at_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 3.18/1.64 set_nat_nat_nat_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 set_nat_nat_nat_nat2: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 set_nat_nat_nat_nat3: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 set_nat_nat_nat_nat4: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 set_nat_nat_nat_nat5: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 3.18/1.64 set_set_nat_nat_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 set_set_nat_nat_nat2: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 set_nat_nat_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 set_nat_nat_nat2: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 set_set_nat_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 set_nat_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 3.18/1.64 set_set_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 3.18/1.64 set_nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(ty_n_t__Nat__Onat,type,
% 3.18/1.64 nat: $tType ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_OFpow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite252967095at_nat: set_nat_nat_nat_nat3 > set_se1657353at_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_OFpow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 finite1633036872at_nat: set_nat_nat_nat2 > set_set_nat_nat_nat2 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite90785608at_nat: set_nat_nat_nat > set_set_nat_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.18/1.64 finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
% 3.18/1.64 finite_Fpow_nat: set_nat > set_set_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 finite1526199988at_nat: set_nat_nat_nat_nat5 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 finite2081887156at_nat: set_nat_nat_nat_nat4 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite1090277411at_nat: set_na1201295426at_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 finite1397812515at_nat: set_na1064332610at_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite1064868788at_nat: set_nat_nat_nat_nat3 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 finite1440337093at_nat: set_nat_nat_nat2 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite584233140at_nat: set_nat_nat_nat_nat2 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 finite891768244at_nat: set_nat_nat_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite2045569477at_nat: set_nat_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.18/1.64 finite570312790at_nat: set_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 3.18/1.64 finite_finite_nat: set_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 finite547180394at_nat: set_se1657353at_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite1977496699at_nat: set_set_nat_nat_nat2 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 finite137548155at_nat: set_set_nat_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 finite604103692at_nat: set_set_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.18/1.64 finite2012248349et_nat: set_set_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 3.18/1.64 one_one_nat: nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 3.18/1.64 plus_plus_nat: nat > nat > nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 3.18/1.64 zero_zero_nat: nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 3.18/1.64 number1551313001itions: ( nat > nat ) > nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 3.18/1.64 ord_less_nat: nat > nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 ord_le2059018749at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 3.18/1.64 ord_le2026740592_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 ord_le809907342at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 ord_le1415139726at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 3.18/1.64 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.18/1.64 ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.18/1.64 ord_less_eq_nat: nat > nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 ord_le2040082867at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 ord_le633272388at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 ord_le940807492at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 ord_le1415039317at_nat: set_nat_nat > set_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.18/1.64 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 collec2071977458at_nat: ( ( ( ( nat > nat ) > nat ) > nat ) > $o ) > set_nat_nat_nat_nat5 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 collec480180978at_nat: ( ( ( nat > nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat4 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 collec6783585at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na1201295426at_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 collec314318689at_nat: ( ( ( nat > nat ) > nat > nat > nat ) > $o ) > set_na1064332610at_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 collec1610646258at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 collec1130010610at_nat: ( ( nat > ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat2 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 collec1437545714at_nat: ( ( nat > nat > nat > nat ) > $o ) > set_nat_nat_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.18/1.64 collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 3.18/1.64 collect_nat: ( nat > $o ) > set_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 collec1702444712at_nat: ( set_nat_nat_nat_nat3 > $o ) > set_se1657353at_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 collec309745849at_nat: ( set_nat_nat_nat2 > $o ) > set_set_nat_nat_nat2 ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.18/1.64 collec617280953at_nat: ( set_nat_nat_nat > $o ) > set_set_nat_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.18/1.64 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 member1128122036at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 3.18/1.64 member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.18/1.64 member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_member_001t__Nat__Onat,type,
% 3.18/1.64 member_nat: nat > set_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.18/1.64 member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
% 3.18/1.64
% 3.18/1.64 thf(sy_v_n,type,
% 3.18/1.64 n: nat ).
% 3.18/1.64
% 3.18/1.64 thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 3.18/1.64 ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [F: nat > nat] :
% 3.18/1.64 ( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ n )
% 3.18/1.64 & ! [I: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ I )
% 3.18/1.64 => ( ( F @ I )
% 3.18/1.64 = zero_zero_nat ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 3.18/1.64 ( ord_le1415039317at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [P: nat > nat] : ( number1551313001itions @ P @ n ) )
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [F: nat > nat] :
% 3.18/1.64 ( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ n )
% 3.18/1.64 & ! [I: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ I )
% 3.18/1.64 => ( ( F @ I )
% 3.18/1.64 = zero_zero_nat ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_2_finite__Collect__conjI,axiom,
% 3.18/1.64 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ( ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
% 3.18/1.64 | ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ Q ) ) )
% 3.18/1.64 => ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [X: nat > nat > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 & ( Q @ X ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_3_finite__Collect__conjI,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ( ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
% 3.18/1.64 | ( finite1440337093at_nat @ ( collect_nat_nat_nat @ Q ) ) )
% 3.18/1.64 => ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 & ( Q @ X ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_4_finite__Collect__conjI,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.64 ( ( ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
% 3.18/1.64 | ( finite1064868788at_nat @ ( collec1610646258at_nat @ Q ) ) )
% 3.18/1.64 => ( finite1064868788at_nat
% 3.18/1.64 @ ( collec1610646258at_nat
% 3.18/1.64 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 & ( Q @ X ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_5_finite__Collect__conjI,axiom,
% 3.18/1.64 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 3.18/1.64 ( ( ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
% 3.18/1.64 | ( finite570312790at_nat @ ( collect_nat_nat @ Q ) ) )
% 3.18/1.64 => ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [X: nat > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 & ( Q @ X ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_6_finite__Collect__conjI,axiom,
% 3.18/1.64 ! [P2: nat > $o,Q: nat > $o] :
% 3.18/1.64 ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 3.18/1.64 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 3.18/1.64 => ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [X: nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 & ( Q @ X ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_7_finite__Collect__disjI,axiom,
% 3.18/1.64 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [X: nat > nat > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 | ( Q @ X ) ) ) )
% 3.18/1.64 = ( ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
% 3.18/1.64 & ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_8_finite__Collect__disjI,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 | ( Q @ X ) ) ) )
% 3.18/1.64 = ( ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
% 3.18/1.64 & ( finite1440337093at_nat @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_9_finite__Collect__disjI,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.64 ( ( finite1064868788at_nat
% 3.18/1.64 @ ( collec1610646258at_nat
% 3.18/1.64 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 | ( Q @ X ) ) ) )
% 3.18/1.64 = ( ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
% 3.18/1.64 & ( finite1064868788at_nat @ ( collec1610646258at_nat @ Q ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_10_finite__Collect__disjI,axiom,
% 3.18/1.64 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 3.18/1.64 ( ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [X: nat > nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 | ( Q @ X ) ) ) )
% 3.18/1.64 = ( ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
% 3.18/1.64 & ( finite570312790at_nat @ ( collect_nat_nat @ Q ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_11_finite__Collect__disjI,axiom,
% 3.18/1.64 ! [P2: nat > $o,Q: nat > $o] :
% 3.18/1.64 ( ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [X: nat] :
% 3.18/1.64 ( ( P2 @ X )
% 3.18/1.64 | ( Q @ X ) ) ) )
% 3.18/1.64 = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 3.18/1.64 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_12_partitions__zero,axiom,
% 3.18/1.64 ! [P3: nat > nat] :
% 3.18/1.64 ( ( number1551313001itions @ P3 @ zero_zero_nat )
% 3.18/1.64 = ( P3
% 3.18/1.64 = ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_13_partitions__bounds,axiom,
% 3.18/1.64 ! [P3: nat > nat,N: nat,I2: nat] :
% 3.18/1.64 ( ( number1551313001itions @ P3 @ N )
% 3.18/1.64 => ( ord_less_eq_nat @ ( P3 @ I2 ) @ N ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_14_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat,D: nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [F: nat > nat] :
% 3.18/1.64 ! [X: nat] :
% 3.18/1.64 ( ( ( member_nat @ X @ A )
% 3.18/1.64 => ( member_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_15_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat,D: nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [F: ( nat > nat ) > nat] :
% 3.18/1.64 ! [X: nat > nat] :
% 3.18/1.64 ( ( ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( member_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_16_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat_nat,D: nat > nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite570312790at_nat @ B )
% 3.18/1.64 => ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [F: nat > nat > nat] :
% 3.18/1.64 ! [X: nat] :
% 3.18/1.64 ( ( ( member_nat @ X @ A )
% 3.18/1.64 => ( member_nat_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_17_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat,D: nat > nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite570312790at_nat @ B )
% 3.18/1.64 => ( finite1064868788at_nat
% 3.18/1.64 @ ( collec1610646258at_nat
% 3.18/1.64 @ ^ [F: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ! [X: nat > nat] :
% 3.18/1.64 ( ( ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( member_nat_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_18_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat_nat_nat,D: nat > nat > nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite2045569477at_nat @ B )
% 3.18/1.64 => ( finite891768244at_nat
% 3.18/1.64 @ ( collec1437545714at_nat
% 3.18/1.64 @ ^ [F: nat > nat > nat > nat] :
% 3.18/1.64 ! [X: nat] :
% 3.18/1.64 ( ( ( member_nat @ X @ A )
% 3.18/1.64 => ( member_nat_nat_nat2 @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_19_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat_nat_nat2,D: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite1440337093at_nat @ B )
% 3.18/1.64 => ( finite584233140at_nat
% 3.18/1.64 @ ( collec1130010610at_nat
% 3.18/1.64 @ ^ [F: nat > ( nat > nat ) > nat] :
% 3.18/1.64 ! [X: nat] :
% 3.18/1.64 ( ( ( member_nat @ X @ A )
% 3.18/1.64 => ( member_nat_nat_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_20_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat,B: set_nat,D: nat] :
% 3.18/1.64 ( ( finite2045569477at_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( finite2081887156at_nat
% 3.18/1.64 @ ( collec480180978at_nat
% 3.18/1.64 @ ^ [F: ( nat > nat > nat ) > nat] :
% 3.18/1.64 ! [X: nat > nat > nat] :
% 3.18/1.64 ( ( ( member_nat_nat_nat2 @ X @ A )
% 3.18/1.64 => ( member_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat_nat_nat2 @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_21_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2,B: set_nat,D: nat] :
% 3.18/1.64 ( ( finite1440337093at_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( finite1526199988at_nat
% 3.18/1.64 @ ( collec2071977458at_nat
% 3.18/1.64 @ ^ [F: ( ( nat > nat ) > nat ) > nat] :
% 3.18/1.64 ! [X: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( ( member_nat_nat_nat @ X @ A )
% 3.18/1.64 => ( member_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat_nat_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_22_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat_nat,D: nat > nat > nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite2045569477at_nat @ B )
% 3.18/1.64 => ( finite1397812515at_nat
% 3.18/1.64 @ ( collec314318689at_nat
% 3.18/1.64 @ ^ [F: ( nat > nat ) > nat > nat > nat] :
% 3.18/1.64 ! [X: nat > nat] :
% 3.18/1.64 ( ( ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( member_nat_nat_nat2 @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_23_finite__set__of__finite__funs,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat_nat2,D: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite1440337093at_nat @ B )
% 3.18/1.64 => ( finite1090277411at_nat
% 3.18/1.64 @ ( collec6783585at_nat
% 3.18/1.64 @ ^ [F: ( nat > nat ) > ( nat > nat ) > nat] :
% 3.18/1.64 ! [X: nat > nat] :
% 3.18/1.64 ( ( ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( member_nat_nat_nat @ ( F @ X ) @ B ) )
% 3.18/1.64 & ( ~ ( member_nat_nat @ X @ A )
% 3.18/1.64 => ( ( F @ X )
% 3.18/1.64 = D ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_24_not__finite__existsD,axiom,
% 3.18/1.64 ! [P2: ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
% 3.18/1.64 => ? [X_1: nat > nat > nat] : ( P2 @ X_1 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_25_not__finite__existsD,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
% 3.18/1.64 => ? [X_1: ( nat > nat ) > nat] : ( P2 @ X_1 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_26_not__finite__existsD,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
% 3.18/1.64 => ? [X_1: ( nat > nat ) > nat > nat] : ( P2 @ X_1 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_27_not__finite__existsD,axiom,
% 3.18/1.64 ! [P2: ( nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
% 3.18/1.64 => ? [X_1: nat > nat] : ( P2 @ X_1 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_28_not__finite__existsD,axiom,
% 3.18/1.64 ! [P2: nat > $o] :
% 3.18/1.64 ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 3.18/1.64 => ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_29_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat,R: nat > nat > $o] :
% 3.18/1.64 ( ~ ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( ! [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat] :
% 3.18/1.64 ( ( member_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [A2: nat] :
% 3.18/1.64 ( ( member_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_30_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat,R: ( nat > nat ) > nat > $o] :
% 3.18/1.64 ( ~ ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( ! [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat] :
% 3.18/1.64 ( ( member_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [A2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_31_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat_nat,R: nat > ( nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite570312790at_nat @ B )
% 3.18/1.64 => ( ! [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [A2: nat] :
% 3.18/1.64 ( ( member_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_32_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat,R: ( nat > nat ) > ( nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite570312790at_nat @ B )
% 3.18/1.64 => ( ! [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [A2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_33_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat_nat_nat,R: nat > ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite2045569477at_nat @ B )
% 3.18/1.64 => ( ! [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ X2 @ B )
% 3.18/1.64 & ~ ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [A2: nat] :
% 3.18/1.64 ( ( member_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_34_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat_nat_nat2,R: nat > ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( finite1440337093at_nat @ B )
% 3.18/1.64 => ( ! [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [A2: nat] :
% 3.18/1.64 ( ( member_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_35_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat,B: set_nat,R: ( nat > nat > nat ) > nat > $o] :
% 3.18/1.64 ( ~ ( finite2045569477at_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( ! [X2: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat] :
% 3.18/1.64 ( ( member_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [A2: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_36_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2,B: set_nat,R: ( ( nat > nat ) > nat ) > nat > $o] :
% 3.18/1.64 ( ~ ( finite1440337093at_nat @ A )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( ! [X2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat] :
% 3.18/1.64 ( ( member_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [A2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_37_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat_nat,R: ( nat > nat ) > ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite2045569477at_nat @ B )
% 3.18/1.64 => ( ! [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ X2 @ B )
% 3.18/1.64 & ~ ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [A2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_38_pigeonhole__infinite__rel,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat_nat2,R: ( nat > nat ) > ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ~ ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( finite1440337093at_nat @ B )
% 3.18/1.64 => ( ! [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 => ? [Xa: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ Xa @ B )
% 3.18/1.64 & ( R @ X2 @ Xa ) ) )
% 3.18/1.64 => ? [X2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ X2 @ B )
% 3.18/1.64 & ~ ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [A2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ A2 @ A )
% 3.18/1.64 & ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_39_finite__Collect__subsets,axiom,
% 3.18/1.64 ! [A: set_nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( finite2012248349et_nat
% 3.18/1.64 @ ( collect_set_nat
% 3.18/1.64 @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_40_finite__Collect__subsets,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat] :
% 3.18/1.64 ( ( finite2045569477at_nat @ A )
% 3.18/1.64 => ( finite137548155at_nat
% 3.18/1.64 @ ( collec617280953at_nat
% 3.18/1.64 @ ^ [B2: set_nat_nat_nat] : ( ord_le940807492at_nat @ B2 @ A ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_41_finite__Collect__subsets,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2] :
% 3.18/1.64 ( ( finite1440337093at_nat @ A )
% 3.18/1.64 => ( finite1977496699at_nat
% 3.18/1.64 @ ( collec309745849at_nat
% 3.18/1.64 @ ^ [B2: set_nat_nat_nat2] : ( ord_le633272388at_nat @ B2 @ A ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_42_finite__Collect__subsets,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat_nat3] :
% 3.18/1.64 ( ( finite1064868788at_nat @ A )
% 3.18/1.64 => ( finite547180394at_nat
% 3.18/1.64 @ ( collec1702444712at_nat
% 3.18/1.64 @ ^ [B2: set_nat_nat_nat_nat3] : ( ord_le2040082867at_nat @ B2 @ A ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_43_finite__Collect__subsets,axiom,
% 3.18/1.64 ! [A: set_nat_nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( finite604103692at_nat
% 3.18/1.64 @ ( collect_set_nat_nat
% 3.18/1.64 @ ^ [B2: set_nat_nat] : ( ord_le1415039317at_nat @ B2 @ A ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_44_bound__domain__and__range__impl__finitely__many__functions,axiom,
% 3.18/1.64 ! [N: nat,M: nat] :
% 3.18/1.64 ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [F: nat > nat] :
% 3.18/1.64 ( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ N )
% 3.18/1.64 & ! [I: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ M @ I )
% 3.18/1.64 => ( ( F @ I )
% 3.18/1.64 = zero_zero_nat ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_45_finite__Collect__le__nat,axiom,
% 3.18/1.64 ! [K: nat] :
% 3.18/1.64 ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_46_finite__has__minimal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat,A3: nat > nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( member_nat_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 & ( ord_less_eq_nat_nat @ X2 @ A3 )
% 3.18/1.64 & ! [Xa: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_less_eq_nat_nat @ Xa @ X2 )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_47_finite__has__minimal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat,A3: nat > nat > nat] :
% 3.18/1.64 ( ( finite2045569477at_nat @ A )
% 3.18/1.64 => ( ( member_nat_nat_nat2 @ A3 @ A )
% 3.18/1.64 => ? [X2: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ X2 @ A )
% 3.18/1.64 & ( ord_le1415139726at_nat @ X2 @ A3 )
% 3.18/1.64 & ! [Xa: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ Xa @ A )
% 3.18/1.64 => ( ( ord_le1415139726at_nat @ Xa @ X2 )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_48_finite__has__minimal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2,A3: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( finite1440337093at_nat @ A )
% 3.18/1.64 => ( ( member_nat_nat_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ X2 @ A )
% 3.18/1.64 & ( ord_le809907342at_nat @ X2 @ A3 )
% 3.18/1.64 & ! [Xa: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_le809907342at_nat @ Xa @ X2 )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_49_finite__has__minimal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat_nat3,A3: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( finite1064868788at_nat @ A )
% 3.18/1.64 => ( ( member1128122036at_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ X2 @ A )
% 3.18/1.64 & ( ord_le2059018749at_nat @ X2 @ A3 )
% 3.18/1.64 & ! [Xa: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_le2059018749at_nat @ Xa @ X2 )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_50_finite__has__minimal2,axiom,
% 3.18/1.64 ! [A: set_nat,A3: nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( member_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 & ( ord_less_eq_nat @ X2 @ A3 )
% 3.18/1.64 & ! [Xa: nat] :
% 3.18/1.64 ( ( member_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_less_eq_nat @ Xa @ X2 )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_51_finite__has__minimal2,axiom,
% 3.18/1.64 ! [A: set_set_nat_nat,A3: set_nat_nat] :
% 3.18/1.64 ( ( finite604103692at_nat @ A )
% 3.18/1.64 => ( ( member_set_nat_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: set_nat_nat] :
% 3.18/1.64 ( ( member_set_nat_nat @ X2 @ A )
% 3.18/1.64 & ( ord_le1415039317at_nat @ X2 @ A3 )
% 3.18/1.64 & ! [Xa: set_nat_nat] :
% 3.18/1.64 ( ( member_set_nat_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ Xa @ X2 )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_52_finite__has__maximal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat,A3: nat > nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ A )
% 3.18/1.64 => ( ( member_nat_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 & ( ord_less_eq_nat_nat @ A3 @ X2 )
% 3.18/1.64 & ! [Xa: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_less_eq_nat_nat @ X2 @ Xa )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_53_finite__has__maximal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat,A3: nat > nat > nat] :
% 3.18/1.64 ( ( finite2045569477at_nat @ A )
% 3.18/1.64 => ( ( member_nat_nat_nat2 @ A3 @ A )
% 3.18/1.64 => ? [X2: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ X2 @ A )
% 3.18/1.64 & ( ord_le1415139726at_nat @ A3 @ X2 )
% 3.18/1.64 & ! [Xa: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ Xa @ A )
% 3.18/1.64 => ( ( ord_le1415139726at_nat @ X2 @ Xa )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_54_finite__has__maximal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2,A3: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( finite1440337093at_nat @ A )
% 3.18/1.64 => ( ( member_nat_nat_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ X2 @ A )
% 3.18/1.64 & ( ord_le809907342at_nat @ A3 @ X2 )
% 3.18/1.64 & ! [Xa: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_le809907342at_nat @ X2 @ Xa )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_55_finite__has__maximal2,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat_nat3,A3: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( finite1064868788at_nat @ A )
% 3.18/1.64 => ( ( member1128122036at_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ X2 @ A )
% 3.18/1.64 & ( ord_le2059018749at_nat @ A3 @ X2 )
% 3.18/1.64 & ! [Xa: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_le2059018749at_nat @ X2 @ Xa )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_56_finite__has__maximal2,axiom,
% 3.18/1.64 ! [A: set_nat,A3: nat] :
% 3.18/1.64 ( ( finite_finite_nat @ A )
% 3.18/1.64 => ( ( member_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 & ( ord_less_eq_nat @ A3 @ X2 )
% 3.18/1.64 & ! [Xa: nat] :
% 3.18/1.64 ( ( member_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_less_eq_nat @ X2 @ Xa )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_57_finite__has__maximal2,axiom,
% 3.18/1.64 ! [A: set_set_nat_nat,A3: set_nat_nat] :
% 3.18/1.64 ( ( finite604103692at_nat @ A )
% 3.18/1.64 => ( ( member_set_nat_nat @ A3 @ A )
% 3.18/1.64 => ? [X2: set_nat_nat] :
% 3.18/1.64 ( ( member_set_nat_nat @ X2 @ A )
% 3.18/1.64 & ( ord_le1415039317at_nat @ A3 @ X2 )
% 3.18/1.64 & ! [Xa: set_nat_nat] :
% 3.18/1.64 ( ( member_set_nat_nat @ Xa @ A )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ X2 @ Xa )
% 3.18/1.64 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_58_rev__finite__subset,axiom,
% 3.18/1.64 ! [B: set_nat,A: set_nat] :
% 3.18/1.64 ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( ( ord_less_eq_set_nat @ A @ B )
% 3.18/1.64 => ( finite_finite_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_59_rev__finite__subset,axiom,
% 3.18/1.64 ! [B: set_nat_nat_nat,A: set_nat_nat_nat] :
% 3.18/1.64 ( ( finite2045569477at_nat @ B )
% 3.18/1.64 => ( ( ord_le940807492at_nat @ A @ B )
% 3.18/1.64 => ( finite2045569477at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_60_rev__finite__subset,axiom,
% 3.18/1.64 ! [B: set_nat_nat_nat2,A: set_nat_nat_nat2] :
% 3.18/1.64 ( ( finite1440337093at_nat @ B )
% 3.18/1.64 => ( ( ord_le633272388at_nat @ A @ B )
% 3.18/1.64 => ( finite1440337093at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_61_rev__finite__subset,axiom,
% 3.18/1.64 ! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3] :
% 3.18/1.64 ( ( finite1064868788at_nat @ B )
% 3.18/1.64 => ( ( ord_le2040082867at_nat @ A @ B )
% 3.18/1.64 => ( finite1064868788at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_62_rev__finite__subset,axiom,
% 3.18/1.64 ! [B: set_nat_nat,A: set_nat_nat] :
% 3.18/1.64 ( ( finite570312790at_nat @ B )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.64 => ( finite570312790at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_63_infinite__super,axiom,
% 3.18/1.64 ! [S: set_nat,T: set_nat] :
% 3.18/1.64 ( ( ord_less_eq_set_nat @ S @ T )
% 3.18/1.64 => ( ~ ( finite_finite_nat @ S )
% 3.18/1.64 => ~ ( finite_finite_nat @ T ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_64_infinite__super,axiom,
% 3.18/1.64 ! [S: set_nat_nat_nat,T: set_nat_nat_nat] :
% 3.18/1.64 ( ( ord_le940807492at_nat @ S @ T )
% 3.18/1.64 => ( ~ ( finite2045569477at_nat @ S )
% 3.18/1.64 => ~ ( finite2045569477at_nat @ T ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_65_infinite__super,axiom,
% 3.18/1.64 ! [S: set_nat_nat_nat2,T: set_nat_nat_nat2] :
% 3.18/1.64 ( ( ord_le633272388at_nat @ S @ T )
% 3.18/1.64 => ( ~ ( finite1440337093at_nat @ S )
% 3.18/1.64 => ~ ( finite1440337093at_nat @ T ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_66_infinite__super,axiom,
% 3.18/1.64 ! [S: set_nat_nat_nat_nat3,T: set_nat_nat_nat_nat3] :
% 3.18/1.64 ( ( ord_le2040082867at_nat @ S @ T )
% 3.18/1.64 => ( ~ ( finite1064868788at_nat @ S )
% 3.18/1.64 => ~ ( finite1064868788at_nat @ T ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_67_infinite__super,axiom,
% 3.18/1.64 ! [S: set_nat_nat,T: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ S @ T )
% 3.18/1.64 => ( ~ ( finite570312790at_nat @ S )
% 3.18/1.64 => ~ ( finite570312790at_nat @ T ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_68_finite__subset,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat] :
% 3.18/1.64 ( ( ord_less_eq_set_nat @ A @ B )
% 3.18/1.64 => ( ( finite_finite_nat @ B )
% 3.18/1.64 => ( finite_finite_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_69_finite__subset,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
% 3.18/1.64 ( ( ord_le940807492at_nat @ A @ B )
% 3.18/1.64 => ( ( finite2045569477at_nat @ B )
% 3.18/1.64 => ( finite2045569477at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_70_finite__subset,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
% 3.18/1.64 ( ( ord_le633272388at_nat @ A @ B )
% 3.18/1.64 => ( ( finite1440337093at_nat @ B )
% 3.18/1.64 => ( finite1440337093at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_71_finite__subset,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat_nat3,B: set_nat_nat_nat_nat3] :
% 3.18/1.64 ( ( ord_le2040082867at_nat @ A @ B )
% 3.18/1.64 => ( ( finite1064868788at_nat @ B )
% 3.18/1.64 => ( finite1064868788at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_72_finite__subset,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.64 => ( ( finite570312790at_nat @ B )
% 3.18/1.64 => ( finite570312790at_nat @ A ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_73_add__le__same__cancel1,axiom,
% 3.18/1.64 ! [B3: nat,A3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
% 3.18/1.64 = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_74_add__le__same__cancel2,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
% 3.18/1.64 = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_75_le__add__same__cancel1,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
% 3.18/1.64 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_76_le__add__same__cancel2,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
% 3.18/1.64 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_77_nat__add__left__cancel__le,axiom,
% 3.18/1.64 ! [K: nat,M: nat,N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 3.18/1.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_78_add__is__0,axiom,
% 3.18/1.64 ! [M: nat,N: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ M @ N )
% 3.18/1.64 = zero_zero_nat )
% 3.18/1.64 = ( ( M = zero_zero_nat )
% 3.18/1.64 & ( N = zero_zero_nat ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_79_Nat_Oadd__0__right,axiom,
% 3.18/1.64 ! [M: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 3.18/1.64 = M ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_80_le0,axiom,
% 3.18/1.64 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_81_bot__nat__0_Oextremum,axiom,
% 3.18/1.64 ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_82_add__right__cancel,axiom,
% 3.18/1.64 ! [B3: nat,A3: nat,C: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ B3 @ A3 )
% 3.18/1.64 = ( plus_plus_nat @ C @ A3 ) )
% 3.18/1.64 = ( B3 = C ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_83_add__left__cancel,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ A3 @ B3 )
% 3.18/1.64 = ( plus_plus_nat @ A3 @ C ) )
% 3.18/1.64 = ( B3 = C ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_84_le__zero__eq,axiom,
% 3.18/1.64 ! [N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.18/1.64 = ( N = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_85_add__le__cancel__right,axiom,
% 3.18/1.64 ! [A3: nat,C: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 3.18/1.64 = ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_86_add__le__cancel__left,axiom,
% 3.18/1.64 ! [C: nat,A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
% 3.18/1.64 = ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_87_zero__eq__add__iff__both__eq__0,axiom,
% 3.18/1.64 ! [X3: nat,Y: nat] :
% 3.18/1.64 ( ( zero_zero_nat
% 3.18/1.64 = ( plus_plus_nat @ X3 @ Y ) )
% 3.18/1.64 = ( ( X3 = zero_zero_nat )
% 3.18/1.64 & ( Y = zero_zero_nat ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_88_add__eq__0__iff__both__eq__0,axiom,
% 3.18/1.64 ! [X3: nat,Y: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ X3 @ Y )
% 3.18/1.64 = zero_zero_nat )
% 3.18/1.64 = ( ( X3 = zero_zero_nat )
% 3.18/1.64 & ( Y = zero_zero_nat ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_89_add__cancel__right__right,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( A3
% 3.18/1.64 = ( plus_plus_nat @ A3 @ B3 ) )
% 3.18/1.64 = ( B3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_90_add__cancel__right__left,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( A3
% 3.18/1.64 = ( plus_plus_nat @ B3 @ A3 ) )
% 3.18/1.64 = ( B3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_91_add__cancel__left__right,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ A3 @ B3 )
% 3.18/1.64 = A3 )
% 3.18/1.64 = ( B3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_92_add__cancel__left__left,axiom,
% 3.18/1.64 ! [B3: nat,A3: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ B3 @ A3 )
% 3.18/1.64 = A3 )
% 3.18/1.64 = ( B3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_93_mem__Collect__eq,axiom,
% 3.18/1.64 ! [A3: nat > nat,P2: ( nat > nat ) > $o] :
% 3.18/1.64 ( ( member_nat_nat @ A3 @ ( collect_nat_nat @ P2 ) )
% 3.18/1.64 = ( P2 @ A3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_94_mem__Collect__eq,axiom,
% 3.18/1.64 ! [A3: nat,P2: nat > $o] :
% 3.18/1.64 ( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
% 3.18/1.64 = ( P2 @ A3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_95_mem__Collect__eq,axiom,
% 3.18/1.64 ! [A3: nat > nat > nat,P2: ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ A3 @ ( collect_nat_nat_nat2 @ P2 ) )
% 3.18/1.64 = ( P2 @ A3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_96_mem__Collect__eq,axiom,
% 3.18/1.64 ! [A3: ( nat > nat ) > nat,P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ( member_nat_nat_nat @ A3 @ ( collect_nat_nat_nat @ P2 ) )
% 3.18/1.64 = ( P2 @ A3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_97_mem__Collect__eq,axiom,
% 3.18/1.64 ! [A3: ( nat > nat ) > nat > nat,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.64 ( ( member1128122036at_nat @ A3 @ ( collec1610646258at_nat @ P2 ) )
% 3.18/1.64 = ( P2 @ A3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_98_Collect__mem__eq,axiom,
% 3.18/1.64 ! [A: set_nat_nat] :
% 3.18/1.64 ( ( collect_nat_nat
% 3.18/1.64 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
% 3.18/1.64 = A ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_99_Collect__mem__eq,axiom,
% 3.18/1.64 ! [A: set_nat] :
% 3.18/1.64 ( ( collect_nat
% 3.18/1.64 @ ^ [X: nat] : ( member_nat @ X @ A ) )
% 3.18/1.64 = A ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_100_Collect__mem__eq,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat] :
% 3.18/1.64 ( ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A ) )
% 3.18/1.64 = A ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_101_Collect__mem__eq,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat2] :
% 3.18/1.64 ( ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A ) )
% 3.18/1.64 = A ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_102_Collect__mem__eq,axiom,
% 3.18/1.64 ! [A: set_nat_nat_nat_nat3] :
% 3.18/1.64 ( ( collec1610646258at_nat
% 3.18/1.64 @ ^ [X: ( nat > nat ) > nat > nat] : ( member1128122036at_nat @ X @ A ) )
% 3.18/1.64 = A ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_103_Collect__cong,axiom,
% 3.18/1.64 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 3.18/1.64 ( ! [X2: nat > nat] :
% 3.18/1.64 ( ( P2 @ X2 )
% 3.18/1.64 = ( Q @ X2 ) )
% 3.18/1.64 => ( ( collect_nat_nat @ P2 )
% 3.18/1.64 = ( collect_nat_nat @ Q ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_104_Collect__cong,axiom,
% 3.18/1.64 ! [P2: nat > $o,Q: nat > $o] :
% 3.18/1.64 ( ! [X2: nat] :
% 3.18/1.64 ( ( P2 @ X2 )
% 3.18/1.64 = ( Q @ X2 ) )
% 3.18/1.64 => ( ( collect_nat @ P2 )
% 3.18/1.64 = ( collect_nat @ Q ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_105_Collect__cong,axiom,
% 3.18/1.64 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 3.18/1.64 ( ! [X2: nat > nat > nat] :
% 3.18/1.64 ( ( P2 @ X2 )
% 3.18/1.64 = ( Q @ X2 ) )
% 3.18/1.64 => ( ( collect_nat_nat_nat2 @ P2 )
% 3.18/1.64 = ( collect_nat_nat_nat2 @ Q ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_106_Collect__cong,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.64 ( ! [X2: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( P2 @ X2 )
% 3.18/1.64 = ( Q @ X2 ) )
% 3.18/1.64 => ( ( collect_nat_nat_nat @ P2 )
% 3.18/1.64 = ( collect_nat_nat_nat @ Q ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_107_Collect__cong,axiom,
% 3.18/1.64 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.64 ( ! [X2: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( P2 @ X2 )
% 3.18/1.64 = ( Q @ X2 ) )
% 3.18/1.64 => ( ( collec1610646258at_nat @ P2 )
% 3.18/1.64 = ( collec1610646258at_nat @ Q ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_108_add_Oright__neutral,axiom,
% 3.18/1.64 ! [A3: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ A3 @ zero_zero_nat )
% 3.18/1.64 = A3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_109_add_Oleft__neutral,axiom,
% 3.18/1.64 ! [A3: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ zero_zero_nat @ A3 )
% 3.18/1.64 = A3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_110_zero__reorient,axiom,
% 3.18/1.64 ! [X3: nat] :
% 3.18/1.64 ( ( zero_zero_nat = X3 )
% 3.18/1.64 = ( X3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_111_add__right__imp__eq,axiom,
% 3.18/1.64 ! [B3: nat,A3: nat,C: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ B3 @ A3 )
% 3.18/1.64 = ( plus_plus_nat @ C @ A3 ) )
% 3.18/1.64 => ( B3 = C ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_112_add__left__imp__eq,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ A3 @ B3 )
% 3.18/1.64 = ( plus_plus_nat @ A3 @ C ) )
% 3.18/1.64 => ( B3 = C ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_113_add_Oleft__commute,axiom,
% 3.18/1.64 ! [B3: nat,A3: nat,C: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) )
% 3.18/1.64 = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_114_add_Ocommute,axiom,
% 3.18/1.64 ( plus_plus_nat
% 3.18/1.64 = ( ^ [A2: nat,B4: nat] : ( plus_plus_nat @ B4 @ A2 ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_115_add_Oassoc,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
% 3.18/1.64 = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_116_group__cancel_Oadd2,axiom,
% 3.18/1.64 ! [B: nat,K: nat,B3: nat,A3: nat] :
% 3.18/1.64 ( ( B
% 3.18/1.64 = ( plus_plus_nat @ K @ B3 ) )
% 3.18/1.64 => ( ( plus_plus_nat @ A3 @ B )
% 3.18/1.64 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_117_group__cancel_Oadd1,axiom,
% 3.18/1.64 ! [A: nat,K: nat,A3: nat,B3: nat] :
% 3.18/1.64 ( ( A
% 3.18/1.64 = ( plus_plus_nat @ K @ A3 ) )
% 3.18/1.64 => ( ( plus_plus_nat @ A @ B3 )
% 3.18/1.64 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_118_add__mono__thms__linordered__semiring_I4_J,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.64 ( ( ( I2 = J )
% 3.18/1.64 & ( K = L ) )
% 3.18/1.64 => ( ( plus_plus_nat @ I2 @ K )
% 3.18/1.64 = ( plus_plus_nat @ J @ L ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_119_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
% 3.18/1.64 = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_120_one__reorient,axiom,
% 3.18/1.64 ! [X3: nat] :
% 3.18/1.64 ( ( one_one_nat = X3 )
% 3.18/1.64 = ( X3 = one_one_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_121_Nat_Oex__has__greatest__nat,axiom,
% 3.18/1.64 ! [P2: nat > $o,K: nat,B3: nat] :
% 3.18/1.64 ( ( P2 @ K )
% 3.18/1.64 => ( ! [Y2: nat] :
% 3.18/1.64 ( ( P2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ Y2 @ B3 ) )
% 3.18/1.64 => ? [X2: nat] :
% 3.18/1.64 ( ( P2 @ X2 )
% 3.18/1.64 & ! [Y3: nat] :
% 3.18/1.64 ( ( P2 @ Y3 )
% 3.18/1.64 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_122_nat__le__linear,axiom,
% 3.18/1.64 ! [M: nat,N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ M @ N )
% 3.18/1.64 | ( ord_less_eq_nat @ N @ M ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_123_le__antisym,axiom,
% 3.18/1.64 ! [M: nat,N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ M @ N )
% 3.18/1.64 => ( ( ord_less_eq_nat @ N @ M )
% 3.18/1.64 => ( M = N ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_124_eq__imp__le,axiom,
% 3.18/1.64 ! [M: nat,N: nat] :
% 3.18/1.64 ( ( M = N )
% 3.18/1.64 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_125_le__trans,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 => ( ( ord_less_eq_nat @ J @ K )
% 3.18/1.64 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_126_le__refl,axiom,
% 3.18/1.64 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_127_zero__le,axiom,
% 3.18/1.64 ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_128_add__le__imp__le__right,axiom,
% 3.18/1.64 ! [A3: nat,C: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 3.18/1.64 => ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_129_add__le__imp__le__left,axiom,
% 3.18/1.64 ! [C: nat,A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
% 3.18/1.64 => ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_130_le__iff__add,axiom,
% 3.18/1.64 ( ord_less_eq_nat
% 3.18/1.64 = ( ^ [A2: nat,B4: nat] :
% 3.18/1.64 ? [C2: nat] :
% 3.18/1.64 ( B4
% 3.18/1.64 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_131_add__right__mono,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_132_less__eqE,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ~ ! [C3: nat] :
% 3.18/1.64 ( B3
% 3.18/1.64 != ( plus_plus_nat @ A3 @ C3 ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_133_add__left__mono,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_134_add__mono,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat,D: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ord_less_eq_nat @ C @ D )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_135_add__mono__thms__linordered__semiring_I1_J,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.64 ( ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 & ( ord_less_eq_nat @ K @ L ) )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_136_add__mono__thms__linordered__semiring_I2_J,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.64 ( ( ( I2 = J )
% 3.18/1.64 & ( ord_less_eq_nat @ K @ L ) )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_137_add__mono__thms__linordered__semiring_I3_J,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.64 ( ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 & ( K = L ) )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_138_add_Ocomm__neutral,axiom,
% 3.18/1.64 ! [A3: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ A3 @ zero_zero_nat )
% 3.18/1.64 = A3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_139_comm__monoid__add__class_Oadd__0,axiom,
% 3.18/1.64 ! [A3: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ zero_zero_nat @ A3 )
% 3.18/1.64 = A3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_140_bot__nat__0_Oextremum__uniqueI,axiom,
% 3.18/1.64 ! [A3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 3.18/1.64 => ( A3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_141_bot__nat__0_Oextremum__unique,axiom,
% 3.18/1.64 ! [A3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 3.18/1.64 = ( A3 = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_142_le__0__eq,axiom,
% 3.18/1.64 ! [N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.18/1.64 = ( N = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_143_less__eq__nat_Osimps_I1_J,axiom,
% 3.18/1.64 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_144_add__eq__self__zero,axiom,
% 3.18/1.64 ! [M: nat,N: nat] :
% 3.18/1.64 ( ( ( plus_plus_nat @ M @ N )
% 3.18/1.64 = M )
% 3.18/1.64 => ( N = zero_zero_nat ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_145_plus__nat_Oadd__0,axiom,
% 3.18/1.64 ! [N: nat] :
% 3.18/1.64 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 3.18/1.64 = N ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_146_nat__le__iff__add,axiom,
% 3.18/1.64 ( ord_less_eq_nat
% 3.18/1.64 = ( ^ [M2: nat,N2: nat] :
% 3.18/1.64 ? [K2: nat] :
% 3.18/1.64 ( N2
% 3.18/1.64 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_147_trans__le__add2,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,M: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_148_trans__le__add1,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,M: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_149_add__le__mono1,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_150_add__le__mono,axiom,
% 3.18/1.64 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.64 => ( ( ord_less_eq_nat @ K @ L )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_151_le__Suc__ex,axiom,
% 3.18/1.64 ! [K: nat,L: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ K @ L )
% 3.18/1.64 => ? [N3: nat] :
% 3.18/1.64 ( L
% 3.18/1.64 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_152_add__leD2,axiom,
% 3.18/1.64 ! [M: nat,K: nat,N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 3.18/1.64 => ( ord_less_eq_nat @ K @ N ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_153_add__leD1,axiom,
% 3.18/1.64 ! [M: nat,K: nat,N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 3.18/1.64 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_154_le__add2,axiom,
% 3.18/1.64 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_155_le__add1,axiom,
% 3.18/1.64 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_156_add__leE,axiom,
% 3.18/1.64 ! [M: nat,K: nat,N: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 3.18/1.64 => ~ ( ( ord_less_eq_nat @ M @ N )
% 3.18/1.64 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_157_add__nonpos__eq__0__iff,axiom,
% 3.18/1.64 ! [X3: nat,Y: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
% 3.18/1.64 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 3.18/1.64 => ( ( ( plus_plus_nat @ X3 @ Y )
% 3.18/1.64 = zero_zero_nat )
% 3.18/1.64 = ( ( X3 = zero_zero_nat )
% 3.18/1.64 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_158_add__nonneg__eq__0__iff,axiom,
% 3.18/1.64 ! [X3: nat,Y: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 3.18/1.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 3.18/1.64 => ( ( ( plus_plus_nat @ X3 @ Y )
% 3.18/1.64 = zero_zero_nat )
% 3.18/1.64 = ( ( X3 = zero_zero_nat )
% 3.18/1.64 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_159_add__nonpos__nonpos,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_160_add__nonneg__nonneg,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
% 3.18/1.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 3.18/1.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_161_add__increasing2,axiom,
% 3.18/1.64 ! [C: nat,B3: nat,A3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ A3 )
% 3.18/1.64 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_162_add__decreasing2,axiom,
% 3.18/1.64 ! [C: nat,A3: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 3.18/1.64 => ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_163_add__increasing,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.64 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_164_add__decreasing,axiom,
% 3.18/1.64 ! [A3: nat,C: nat,B3: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
% 3.18/1.64 => ( ( ord_less_eq_nat @ C @ B3 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_165_sum_Ofinite__Collect__op,axiom,
% 3.18/1.64 ! [I3: set_nat_nat,X3: ( nat > nat ) > nat,Y: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [I: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ I @ I3 )
% 3.18/1.64 & ( ( X3 @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [I: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ I @ I3 )
% 3.18/1.64 & ( ( Y @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( finite570312790at_nat
% 3.18/1.64 @ ( collect_nat_nat
% 3.18/1.64 @ ^ [I: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ I @ I3 )
% 3.18/1.64 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 3.18/1.64 != zero_zero_nat ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_166_sum_Ofinite__Collect__op,axiom,
% 3.18/1.64 ! [I3: set_nat,X3: nat > nat,Y: nat > nat] :
% 3.18/1.64 ( ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [I: nat] :
% 3.18/1.64 ( ( member_nat @ I @ I3 )
% 3.18/1.64 & ( ( X3 @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [I: nat] :
% 3.18/1.64 ( ( member_nat @ I @ I3 )
% 3.18/1.64 & ( ( Y @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [I: nat] :
% 3.18/1.64 ( ( member_nat @ I @ I3 )
% 3.18/1.64 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 3.18/1.64 != zero_zero_nat ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_167_sum_Ofinite__Collect__op,axiom,
% 3.18/1.64 ! [I3: set_nat_nat_nat,X3: ( nat > nat > nat ) > nat,Y: ( nat > nat > nat ) > nat] :
% 3.18/1.64 ( ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [I: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ I @ I3 )
% 3.18/1.64 & ( ( X3 @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [I: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ I @ I3 )
% 3.18/1.64 & ( ( Y @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( finite2045569477at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat2
% 3.18/1.64 @ ^ [I: nat > nat > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat2 @ I @ I3 )
% 3.18/1.64 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 3.18/1.64 != zero_zero_nat ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_168_sum_Ofinite__Collect__op,axiom,
% 3.18/1.64 ! [I3: set_nat_nat_nat2,X3: ( ( nat > nat ) > nat ) > nat,Y: ( ( nat > nat ) > nat ) > nat] :
% 3.18/1.64 ( ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [I: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ I @ I3 )
% 3.18/1.64 & ( ( X3 @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [I: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ I @ I3 )
% 3.18/1.64 & ( ( Y @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( finite1440337093at_nat
% 3.18/1.64 @ ( collect_nat_nat_nat
% 3.18/1.64 @ ^ [I: ( nat > nat ) > nat] :
% 3.18/1.64 ( ( member_nat_nat_nat @ I @ I3 )
% 3.18/1.64 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 3.18/1.64 != zero_zero_nat ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_169_sum_Ofinite__Collect__op,axiom,
% 3.18/1.64 ! [I3: set_nat_nat_nat_nat3,X3: ( ( nat > nat ) > nat > nat ) > nat,Y: ( ( nat > nat ) > nat > nat ) > nat] :
% 3.18/1.64 ( ( finite1064868788at_nat
% 3.18/1.64 @ ( collec1610646258at_nat
% 3.18/1.64 @ ^ [I: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ I @ I3 )
% 3.18/1.64 & ( ( X3 @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( ( finite1064868788at_nat
% 3.18/1.64 @ ( collec1610646258at_nat
% 3.18/1.64 @ ^ [I: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ I @ I3 )
% 3.18/1.64 & ( ( Y @ I )
% 3.18/1.64 != zero_zero_nat ) ) ) )
% 3.18/1.64 => ( finite1064868788at_nat
% 3.18/1.64 @ ( collec1610646258at_nat
% 3.18/1.64 @ ^ [I: ( nat > nat ) > nat > nat] :
% 3.18/1.64 ( ( member1128122036at_nat @ I @ I3 )
% 3.18/1.64 & ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
% 3.18/1.64 != zero_zero_nat ) ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_170_not__one__le__zero,axiom,
% 3.18/1.64 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_171_zero__le__one,axiom,
% 3.18/1.64 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 3.18/1.64
% 3.18/1.64 thf(fact_172_finite__less__ub,axiom,
% 3.18/1.64 ! [F2: nat > nat,U: nat] :
% 3.18/1.64 ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
% 3.18/1.64 => ( finite_finite_nat
% 3.18/1.64 @ ( collect_nat
% 3.18/1.64 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_173_subset__antisym,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ B @ A )
% 3.18/1.64 => ( A = B ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_174_subsetI,axiom,
% 3.18/1.64 ! [A: set_nat,B: set_nat] :
% 3.18/1.64 ( ! [X2: nat] :
% 3.18/1.64 ( ( member_nat @ X2 @ A )
% 3.18/1.64 => ( member_nat @ X2 @ B ) )
% 3.18/1.64 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_175_subsetI,axiom,
% 3.18/1.64 ! [A: set_nat_nat,B: set_nat_nat] :
% 3.18/1.64 ( ! [X2: nat > nat] :
% 3.18/1.64 ( ( member_nat_nat @ X2 @ A )
% 3.18/1.64 => ( member_nat_nat @ X2 @ B ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ A @ B ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_176_order__refl,axiom,
% 3.18/1.64 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_177_order__refl,axiom,
% 3.18/1.64 ! [X3: set_nat_nat] : ( ord_le1415039317at_nat @ X3 @ X3 ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_178_finite__nat__set__iff__bounded__le,axiom,
% 3.18/1.64 ( finite_finite_nat
% 3.18/1.64 = ( ^ [N4: set_nat] :
% 3.18/1.64 ? [M2: nat] :
% 3.18/1.64 ! [X: nat] :
% 3.18/1.64 ( ( member_nat @ X @ N4 )
% 3.18/1.64 => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_179_Euclid__induct,axiom,
% 3.18/1.64 ! [P2: nat > nat > $o,A3: nat,B3: nat] :
% 3.18/1.64 ( ! [A4: nat,B5: nat] :
% 3.18/1.64 ( ( P2 @ A4 @ B5 )
% 3.18/1.64 = ( P2 @ B5 @ A4 ) )
% 3.18/1.64 => ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
% 3.18/1.64 => ( ! [A4: nat,B5: nat] :
% 3.18/1.64 ( ( P2 @ A4 @ B5 )
% 3.18/1.64 => ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
% 3.18/1.64 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_180_order__subst1,axiom,
% 3.18/1.64 ! [A3: nat,F2: nat > nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_181_order__subst1,axiom,
% 3.18/1.64 ! [A3: nat,F2: set_nat_nat > nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_182_order__subst1,axiom,
% 3.18/1.64 ! [A3: set_nat_nat,F2: nat > set_nat_nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ A3 @ ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_183_order__subst1,axiom,
% 3.18/1.64 ! [A3: set_nat_nat,F2: set_nat_nat > set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ A3 @ ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_184_order__subst2,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,F2: nat > nat,C: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_185_order__subst2,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ ( F2 @ B3 ) @ C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_186_order__subst2,axiom,
% 3.18/1.64 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
% 3.18/1.64 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_187_order__subst2,axiom,
% 3.18/1.64 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ ( F2 @ B3 ) @ C )
% 3.18/1.64 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_188_ord__eq__le__subst,axiom,
% 3.18/1.64 ! [A3: nat,F2: nat > nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( A3
% 3.18/1.64 = ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_189_ord__eq__le__subst,axiom,
% 3.18/1.64 ! [A3: set_nat_nat,F2: nat > set_nat_nat,B3: nat,C: nat] :
% 3.18/1.64 ( ( A3
% 3.18/1.64 = ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_190_ord__eq__le__subst,axiom,
% 3.18/1.64 ! [A3: nat,F2: set_nat_nat > nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( A3
% 3.18/1.64 = ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_191_ord__eq__le__subst,axiom,
% 3.18/1.64 ! [A3: set_nat_nat,F2: set_nat_nat > set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( A3
% 3.18/1.64 = ( F2 @ B3 ) )
% 3.18/1.64 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.64 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.64 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_192_ord__le__eq__subst,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,F2: nat > nat,C: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ( F2 @ B3 )
% 3.18/1.64 = C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.64 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.64
% 3.18/1.64 thf(fact_193_ord__le__eq__subst,axiom,
% 3.18/1.64 ! [A3: nat,B3: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.64 => ( ( ( F2 @ B3 )
% 3.18/1.64 = C )
% 3.18/1.64 => ( ! [X2: nat,Y2: nat] :
% 3.18/1.64 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 3.18/1.64 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.65 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_194_ord__le__eq__subst,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ( F2 @ B3 )
% 3.18/1.65 = C )
% 3.18/1.65 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.65 => ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.65 => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_195_ord__le__eq__subst,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ( F2 @ B3 )
% 3.18/1.65 = C )
% 3.18/1.65 => ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ X2 @ Y2 )
% 3.18/1.65 => ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
% 3.18/1.65 => ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_196_eq__iff,axiom,
% 3.18/1.65 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [X: nat,Y5: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ X @ Y5 )
% 3.18/1.65 & ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_197_eq__iff,axiom,
% 3.18/1.65 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [X: set_nat_nat,Y5: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ X @ Y5 )
% 3.18/1.65 & ( ord_le1415039317at_nat @ Y5 @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_198_antisym,axiom,
% 3.18/1.65 ! [X3: nat,Y: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ X3 @ Y )
% 3.18/1.65 => ( ( ord_less_eq_nat @ Y @ X3 )
% 3.18/1.65 => ( X3 = Y ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_199_antisym,axiom,
% 3.18/1.65 ! [X3: set_nat_nat,Y: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ X3 @ Y )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ Y @ X3 )
% 3.18/1.65 => ( X3 = Y ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_200_linear,axiom,
% 3.18/1.65 ! [X3: nat,Y: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ X3 @ Y )
% 3.18/1.65 | ( ord_less_eq_nat @ Y @ X3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_201_eq__refl,axiom,
% 3.18/1.65 ! [X3: nat,Y: nat] :
% 3.18/1.65 ( ( X3 = Y )
% 3.18/1.65 => ( ord_less_eq_nat @ X3 @ Y ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_202_eq__refl,axiom,
% 3.18/1.65 ! [X3: set_nat_nat,Y: set_nat_nat] :
% 3.18/1.65 ( ( X3 = Y )
% 3.18/1.65 => ( ord_le1415039317at_nat @ X3 @ Y ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_203_le__cases,axiom,
% 3.18/1.65 ! [X3: nat,Y: nat] :
% 3.18/1.65 ( ~ ( ord_less_eq_nat @ X3 @ Y )
% 3.18/1.65 => ( ord_less_eq_nat @ Y @ X3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_204_order_Otrans,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.65 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_205_order_Otrans,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.65 => ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_206_le__cases3,axiom,
% 3.18/1.65 ! [X3: nat,Y: nat,Z2: nat] :
% 3.18/1.65 ( ( ( ord_less_eq_nat @ X3 @ Y )
% 3.18/1.65 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 3.18/1.65 => ( ( ( ord_less_eq_nat @ Y @ X3 )
% 3.18/1.65 => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
% 3.18/1.65 => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
% 3.18/1.65 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 3.18/1.65 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 3.18/1.65 => ~ ( ord_less_eq_nat @ Y @ X3 ) )
% 3.18/1.65 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 3.18/1.65 => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
% 3.18/1.65 => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
% 3.18/1.65 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_207_antisym__conv,axiom,
% 3.18/1.65 ! [Y: nat,X3: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ Y @ X3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ X3 @ Y )
% 3.18/1.65 = ( X3 = Y ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_208_antisym__conv,axiom,
% 3.18/1.65 ! [Y: set_nat_nat,X3: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ Y @ X3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ X3 @ Y )
% 3.18/1.65 = ( X3 = Y ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_209_order__class_Oorder_Oeq__iff,axiom,
% 3.18/1.65 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [A2: nat,B4: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A2 @ B4 )
% 3.18/1.65 & ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_210_order__class_Oorder_Oeq__iff,axiom,
% 3.18/1.65 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A2 @ B4 )
% 3.18/1.65 & ( ord_le1415039317at_nat @ B4 @ A2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_211_ord__eq__le__trans,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.65 ( ( A3 = B3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ B3 @ C )
% 3.18/1.65 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_212_ord__eq__le__trans,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( A3 = B3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.65 => ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_213_ord__le__eq__trans,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( B3 = C )
% 3.18/1.65 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_214_ord__le__eq__trans,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( B3 = C )
% 3.18/1.65 => ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_215_order__class_Oorder_Oantisym,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ B3 @ A3 )
% 3.18/1.65 => ( A3 = B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_216_order__class_Oorder_Oantisym,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 3.18/1.65 => ( A3 = B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_217_order__trans,axiom,
% 3.18/1.65 ! [X3: nat,Y: nat,Z2: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ X3 @ Y )
% 3.18/1.65 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 3.18/1.65 => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_218_order__trans,axiom,
% 3.18/1.65 ! [X3: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ X3 @ Y )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ Y @ Z2 )
% 3.18/1.65 => ( ord_le1415039317at_nat @ X3 @ Z2 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_219_dual__order_Orefl,axiom,
% 3.18/1.65 ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_220_dual__order_Orefl,axiom,
% 3.18/1.65 ! [A3: set_nat_nat] : ( ord_le1415039317at_nat @ A3 @ A3 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_221_linorder__wlog,axiom,
% 3.18/1.65 ! [P2: nat > nat > $o,A3: nat,B3: nat] :
% 3.18/1.65 ( ! [A4: nat,B5: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A4 @ B5 )
% 3.18/1.65 => ( P2 @ A4 @ B5 ) )
% 3.18/1.65 => ( ! [A4: nat,B5: nat] :
% 3.18/1.65 ( ( P2 @ B5 @ A4 )
% 3.18/1.65 => ( P2 @ A4 @ B5 ) )
% 3.18/1.65 => ( P2 @ A3 @ B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_222_dual__order_Otrans,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat,C: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ C @ B3 )
% 3.18/1.65 => ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_223_dual__order_Otrans,axiom,
% 3.18/1.65 ! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ C @ B3 )
% 3.18/1.65 => ( ord_le1415039317at_nat @ C @ A3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_224_dual__order_Oeq__iff,axiom,
% 3.18/1.65 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [A2: nat,B4: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ B4 @ A2 )
% 3.18/1.65 & ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_225_dual__order_Oeq__iff,axiom,
% 3.18/1.65 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B4 @ A2 )
% 3.18/1.65 & ( ord_le1415039317at_nat @ A2 @ B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_226_dual__order_Oantisym,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 => ( A3 = B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_227_dual__order_Oantisym,axiom,
% 3.18/1.65 ! [B3: set_nat_nat,A3: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( A3 = B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_228_in__mono,axiom,
% 3.18/1.65 ! [A: set_nat,B: set_nat,X3: nat] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ A @ B )
% 3.18/1.65 => ( ( member_nat @ X3 @ A )
% 3.18/1.65 => ( member_nat @ X3 @ B ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_229_in__mono,axiom,
% 3.18/1.65 ! [A: set_nat_nat,B: set_nat_nat,X3: nat > nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.65 => ( ( member_nat_nat @ X3 @ A )
% 3.18/1.65 => ( member_nat_nat @ X3 @ B ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_230_subsetD,axiom,
% 3.18/1.65 ! [A: set_nat,B: set_nat,C: nat] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ A @ B )
% 3.18/1.65 => ( ( member_nat @ C @ A )
% 3.18/1.65 => ( member_nat @ C @ B ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_231_subsetD,axiom,
% 3.18/1.65 ! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.65 => ( ( member_nat_nat @ C @ A )
% 3.18/1.65 => ( member_nat_nat @ C @ B ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_232_equalityE,axiom,
% 3.18/1.65 ! [A: set_nat_nat,B: set_nat_nat] :
% 3.18/1.65 ( ( A = B )
% 3.18/1.65 => ~ ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.65 => ~ ( ord_le1415039317at_nat @ B @ A ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_233_subset__eq,axiom,
% 3.18/1.65 ( ord_less_eq_set_nat
% 3.18/1.65 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.18/1.65 ! [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ A5 )
% 3.18/1.65 => ( member_nat @ X @ B2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_234_subset__eq,axiom,
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 3.18/1.65 ! [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ A5 )
% 3.18/1.65 => ( member_nat_nat @ X @ B2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_235_equalityD1,axiom,
% 3.18/1.65 ! [A: set_nat_nat,B: set_nat_nat] :
% 3.18/1.65 ( ( A = B )
% 3.18/1.65 => ( ord_le1415039317at_nat @ A @ B ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_236_equalityD2,axiom,
% 3.18/1.65 ! [A: set_nat_nat,B: set_nat_nat] :
% 3.18/1.65 ( ( A = B )
% 3.18/1.65 => ( ord_le1415039317at_nat @ B @ A ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_237_subset__iff,axiom,
% 3.18/1.65 ( ord_less_eq_set_nat
% 3.18/1.65 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.18/1.65 ! [T2: nat] :
% 3.18/1.65 ( ( member_nat @ T2 @ A5 )
% 3.18/1.65 => ( member_nat @ T2 @ B2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_238_subset__iff,axiom,
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 3.18/1.65 ! [T2: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ T2 @ A5 )
% 3.18/1.65 => ( member_nat_nat @ T2 @ B2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_239_subset__refl,axiom,
% 3.18/1.65 ! [A: set_nat_nat] : ( ord_le1415039317at_nat @ A @ A ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_240_Collect__mono,axiom,
% 3.18/1.65 ! [P2: nat > $o,Q: nat > $o] :
% 3.18/1.65 ( ! [X2: nat] :
% 3.18/1.65 ( ( P2 @ X2 )
% 3.18/1.65 => ( Q @ X2 ) )
% 3.18/1.65 => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_241_Collect__mono,axiom,
% 3.18/1.65 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ! [X2: nat > nat > nat] :
% 3.18/1.65 ( ( P2 @ X2 )
% 3.18/1.65 => ( Q @ X2 ) )
% 3.18/1.65 => ( ord_le940807492at_nat @ ( collect_nat_nat_nat2 @ P2 ) @ ( collect_nat_nat_nat2 @ Q ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_242_Collect__mono,axiom,
% 3.18/1.65 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ! [X2: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( P2 @ X2 )
% 3.18/1.65 => ( Q @ X2 ) )
% 3.18/1.65 => ( ord_le633272388at_nat @ ( collect_nat_nat_nat @ P2 ) @ ( collect_nat_nat_nat @ Q ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_243_Collect__mono,axiom,
% 3.18/1.65 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ! [X2: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( P2 @ X2 )
% 3.18/1.65 => ( Q @ X2 ) )
% 3.18/1.65 => ( ord_le2040082867at_nat @ ( collec1610646258at_nat @ P2 ) @ ( collec1610646258at_nat @ Q ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_244_Collect__mono,axiom,
% 3.18/1.65 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 3.18/1.65 ( ! [X2: nat > nat] :
% 3.18/1.65 ( ( P2 @ X2 )
% 3.18/1.65 => ( Q @ X2 ) )
% 3.18/1.65 => ( ord_le1415039317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_245_subset__trans,axiom,
% 3.18/1.65 ! [A: set_nat_nat,B: set_nat_nat,C4: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A @ B )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ B @ C4 )
% 3.18/1.65 => ( ord_le1415039317at_nat @ A @ C4 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_246_set__eq__subset,axiom,
% 3.18/1.65 ( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
% 3.18/1.65 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A5 @ B2 )
% 3.18/1.65 & ( ord_le1415039317at_nat @ B2 @ A5 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_247_Collect__mono__iff,axiom,
% 3.18/1.65 ! [P2: nat > $o,Q: nat > $o] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
% 3.18/1.65 = ( ! [X: nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 => ( Q @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_248_Collect__mono__iff,axiom,
% 3.18/1.65 ! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le940807492at_nat @ ( collect_nat_nat_nat2 @ P2 ) @ ( collect_nat_nat_nat2 @ Q ) )
% 3.18/1.65 = ( ! [X: nat > nat > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 => ( Q @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_249_Collect__mono__iff,axiom,
% 3.18/1.65 ! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ( ord_le633272388at_nat @ ( collect_nat_nat_nat @ P2 ) @ ( collect_nat_nat_nat @ Q ) )
% 3.18/1.65 = ( ! [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 => ( Q @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_250_Collect__mono__iff,axiom,
% 3.18/1.65 ! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le2040082867at_nat @ ( collec1610646258at_nat @ P2 ) @ ( collec1610646258at_nat @ Q ) )
% 3.18/1.65 = ( ! [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 => ( Q @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_251_Collect__mono__iff,axiom,
% 3.18/1.65 ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) )
% 3.18/1.65 = ( ! [X: nat > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 => ( Q @ X ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_252_bounded__Max__nat,axiom,
% 3.18/1.65 ! [P2: nat > $o,X3: nat,M3: nat] :
% 3.18/1.65 ( ( P2 @ X3 )
% 3.18/1.65 => ( ! [X2: nat] :
% 3.18/1.65 ( ( P2 @ X2 )
% 3.18/1.65 => ( ord_less_eq_nat @ X2 @ M3 ) )
% 3.18/1.65 => ~ ! [M4: nat] :
% 3.18/1.65 ( ( P2 @ M4 )
% 3.18/1.65 => ~ ! [X4: nat] :
% 3.18/1.65 ( ( P2 @ X4 )
% 3.18/1.65 => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_253_Collect__subset,axiom,
% 3.18/1.65 ! [A: set_nat,P2: nat > $o] :
% 3.18/1.65 ( ord_less_eq_set_nat
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ A ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_254_Collect__subset,axiom,
% 3.18/1.65 ! [A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ord_le940807492at_nat
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ A ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_255_Collect__subset,axiom,
% 3.18/1.65 ! [A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ord_le633272388at_nat
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ A ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_256_Collect__subset,axiom,
% 3.18/1.65 ! [A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ord_le2040082867at_nat
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ A ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_257_Collect__subset,axiom,
% 3.18/1.65 ! [A: set_nat_nat,P2: ( nat > nat ) > $o] :
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ A ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_258_less__eq__set__def,axiom,
% 3.18/1.65 ( ord_less_eq_set_nat
% 3.18/1.65 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.18/1.65 ( ord_less_eq_nat_o
% 3.18/1.65 @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 3.18/1.65 @ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_259_less__eq__set__def,axiom,
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
% 3.18/1.65 ( ord_le2026740592_nat_o
% 3.18/1.65 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ A5 )
% 3.18/1.65 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ B2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_260_zero__neq__one,axiom,
% 3.18/1.65 zero_zero_nat != one_one_nat ).
% 3.18/1.65
% 3.18/1.65 thf(fact_261_le__numeral__extra_I4_J,axiom,
% 3.18/1.65 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 3.18/1.65
% 3.18/1.65 thf(fact_262_verit__sum__simplify,axiom,
% 3.18/1.65 ! [A3: nat] :
% 3.18/1.65 ( ( plus_plus_nat @ A3 @ zero_zero_nat )
% 3.18/1.65 = A3 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_263_add__0__iff,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat] :
% 3.18/1.65 ( ( B3
% 3.18/1.65 = ( plus_plus_nat @ B3 @ A3 ) )
% 3.18/1.65 = ( A3 = zero_zero_nat ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_264_le__numeral__extra_I3_J,axiom,
% 3.18/1.65 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 3.18/1.65
% 3.18/1.65 thf(fact_265_verit__la__disequality,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( A3 = B3 )
% 3.18/1.65 | ~ ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 | ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_266_pred__subset__eq,axiom,
% 3.18/1.65 ! [R: set_nat,S: set_nat] :
% 3.18/1.65 ( ( ord_less_eq_nat_o
% 3.18/1.65 @ ^ [X: nat] : ( member_nat @ X @ R )
% 3.18/1.65 @ ^ [X: nat] : ( member_nat @ X @ S ) )
% 3.18/1.65 = ( ord_less_eq_set_nat @ R @ S ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_267_pred__subset__eq,axiom,
% 3.18/1.65 ! [R: set_nat_nat,S: set_nat_nat] :
% 3.18/1.65 ( ( ord_le2026740592_nat_o
% 3.18/1.65 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
% 3.18/1.65 @ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
% 3.18/1.65 = ( ord_le1415039317at_nat @ R @ S ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_268_conj__subset__def,axiom,
% 3.18/1.65 ! [A: set_nat,P2: nat > $o,Q: nat > $o] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ A
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 & ( Q @ X ) ) ) )
% 3.18/1.65 = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P2 ) )
% 3.18/1.65 & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_269_conj__subset__def,axiom,
% 3.18/1.65 ! [A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le940807492at_nat @ A
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 & ( Q @ X ) ) ) )
% 3.18/1.65 = ( ( ord_le940807492at_nat @ A @ ( collect_nat_nat_nat2 @ P2 ) )
% 3.18/1.65 & ( ord_le940807492at_nat @ A @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_270_conj__subset__def,axiom,
% 3.18/1.65 ! [A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ( ord_le633272388at_nat @ A
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 & ( Q @ X ) ) ) )
% 3.18/1.65 = ( ( ord_le633272388at_nat @ A @ ( collect_nat_nat_nat @ P2 ) )
% 3.18/1.65 & ( ord_le633272388at_nat @ A @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_271_conj__subset__def,axiom,
% 3.18/1.65 ! [A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le2040082867at_nat @ A
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 & ( Q @ X ) ) ) )
% 3.18/1.65 = ( ( ord_le2040082867at_nat @ A @ ( collec1610646258at_nat @ P2 ) )
% 3.18/1.65 & ( ord_le2040082867at_nat @ A @ ( collec1610646258at_nat @ Q ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_272_conj__subset__def,axiom,
% 3.18/1.65 ! [A: set_nat_nat,P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( P2 @ X )
% 3.18/1.65 & ( Q @ X ) ) ) )
% 3.18/1.65 = ( ( ord_le1415039317at_nat @ A @ ( collect_nat_nat @ P2 ) )
% 3.18/1.65 & ( ord_le1415039317at_nat @ A @ ( collect_nat_nat @ Q ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_273_subset__Collect__iff,axiom,
% 3.18/1.65 ! [B: set_nat,A: set_nat,P2: nat > $o] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ B @ A )
% 3.18/1.65 => ( ( ord_less_eq_set_nat @ B
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 = ( ! [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ B )
% 3.18/1.65 => ( P2 @ X ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_274_subset__Collect__iff,axiom,
% 3.18/1.65 ! [B: set_nat_nat_nat,A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le940807492at_nat @ B @ A )
% 3.18/1.65 => ( ( ord_le940807492at_nat @ B
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 = ( ! [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ B )
% 3.18/1.65 => ( P2 @ X ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_275_subset__Collect__iff,axiom,
% 3.18/1.65 ! [B: set_nat_nat_nat2,A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ( ord_le633272388at_nat @ B @ A )
% 3.18/1.65 => ( ( ord_le633272388at_nat @ B
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 = ( ! [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ B )
% 3.18/1.65 => ( P2 @ X ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_276_subset__Collect__iff,axiom,
% 3.18/1.65 ! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le2040082867at_nat @ B @ A )
% 3.18/1.65 => ( ( ord_le2040082867at_nat @ B
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 = ( ! [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ B )
% 3.18/1.65 => ( P2 @ X ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_277_subset__Collect__iff,axiom,
% 3.18/1.65 ! [B: set_nat_nat,A: set_nat_nat,P2: ( nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B @ A )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ B
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 = ( ! [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ B )
% 3.18/1.65 => ( P2 @ X ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_278_subset__CollectI,axiom,
% 3.18/1.65 ! [B: set_nat,A: set_nat,Q: nat > $o,P2: nat > $o] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ B @ A )
% 3.18/1.65 => ( ! [X2: nat] :
% 3.18/1.65 ( ( member_nat @ X2 @ B )
% 3.18/1.65 => ( ( Q @ X2 )
% 3.18/1.65 => ( P2 @ X2 ) ) )
% 3.18/1.65 => ( ord_less_eq_set_nat
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ B )
% 3.18/1.65 & ( Q @ X ) ) )
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_279_subset__CollectI,axiom,
% 3.18/1.65 ! [B: set_nat_nat_nat,A: set_nat_nat_nat,Q: ( nat > nat > nat ) > $o,P2: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le940807492at_nat @ B @ A )
% 3.18/1.65 => ( ! [X2: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X2 @ B )
% 3.18/1.65 => ( ( Q @ X2 )
% 3.18/1.65 => ( P2 @ X2 ) ) )
% 3.18/1.65 => ( ord_le940807492at_nat
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ B )
% 3.18/1.65 & ( Q @ X ) ) )
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_280_subset__CollectI,axiom,
% 3.18/1.65 ! [B: set_nat_nat_nat2,A: set_nat_nat_nat2,Q: ( ( nat > nat ) > nat ) > $o,P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ( ord_le633272388at_nat @ B @ A )
% 3.18/1.65 => ( ! [X2: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X2 @ B )
% 3.18/1.65 => ( ( Q @ X2 )
% 3.18/1.65 => ( P2 @ X2 ) ) )
% 3.18/1.65 => ( ord_le633272388at_nat
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ B )
% 3.18/1.65 & ( Q @ X ) ) )
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_281_subset__CollectI,axiom,
% 3.18/1.65 ! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3,Q: ( ( nat > nat ) > nat > nat ) > $o,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le2040082867at_nat @ B @ A )
% 3.18/1.65 => ( ! [X2: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X2 @ B )
% 3.18/1.65 => ( ( Q @ X2 )
% 3.18/1.65 => ( P2 @ X2 ) ) )
% 3.18/1.65 => ( ord_le2040082867at_nat
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ B )
% 3.18/1.65 & ( Q @ X ) ) )
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_282_subset__CollectI,axiom,
% 3.18/1.65 ! [B: set_nat_nat,A: set_nat_nat,Q: ( nat > nat ) > $o,P2: ( nat > nat ) > $o] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B @ A )
% 3.18/1.65 => ( ! [X2: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X2 @ B )
% 3.18/1.65 => ( ( Q @ X2 )
% 3.18/1.65 => ( P2 @ X2 ) ) )
% 3.18/1.65 => ( ord_le1415039317at_nat
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ B )
% 3.18/1.65 & ( Q @ X ) ) )
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ A )
% 3.18/1.65 & ( P2 @ X ) ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_283_Collect__restrict,axiom,
% 3.18/1.65 ! [X5: set_nat,P2: nat > $o] :
% 3.18/1.65 ( ord_less_eq_set_nat
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ X5 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_284_Collect__restrict,axiom,
% 3.18/1.65 ! [X5: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ord_le940807492at_nat
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ X5 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_285_Collect__restrict,axiom,
% 3.18/1.65 ! [X5: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ord_le633272388at_nat
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ X5 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_286_Collect__restrict,axiom,
% 3.18/1.65 ! [X5: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ord_le2040082867at_nat
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ X5 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_287_Collect__restrict,axiom,
% 3.18/1.65 ! [X5: set_nat_nat,P2: ( nat > nat ) > $o] :
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) )
% 3.18/1.65 @ X5 ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_288_prop__restrict,axiom,
% 3.18/1.65 ! [X3: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
% 3.18/1.65 ( ( member_nat @ X3 @ Z3 )
% 3.18/1.65 => ( ( ord_less_eq_set_nat @ Z3
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 => ( P2 @ X3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_289_prop__restrict,axiom,
% 3.18/1.65 ! [X3: nat > nat > nat,Z3: set_nat_nat_nat,X5: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X3 @ Z3 )
% 3.18/1.65 => ( ( ord_le940807492at_nat @ Z3
% 3.18/1.65 @ ( collect_nat_nat_nat2
% 3.18/1.65 @ ^ [X: nat > nat > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat2 @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 => ( P2 @ X3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_290_prop__restrict,axiom,
% 3.18/1.65 ! [X3: ( nat > nat ) > nat,Z3: set_nat_nat_nat2,X5: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X3 @ Z3 )
% 3.18/1.65 => ( ( ord_le633272388at_nat @ Z3
% 3.18/1.65 @ ( collect_nat_nat_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat] :
% 3.18/1.65 ( ( member_nat_nat_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 => ( P2 @ X3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_291_prop__restrict,axiom,
% 3.18/1.65 ! [X3: ( nat > nat ) > nat > nat,Z3: set_nat_nat_nat_nat3,X5: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
% 3.18/1.65 ( ( member1128122036at_nat @ X3 @ Z3 )
% 3.18/1.65 => ( ( ord_le2040082867at_nat @ Z3
% 3.18/1.65 @ ( collec1610646258at_nat
% 3.18/1.65 @ ^ [X: ( nat > nat ) > nat > nat] :
% 3.18/1.65 ( ( member1128122036at_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 => ( P2 @ X3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_292_prop__restrict,axiom,
% 3.18/1.65 ! [X3: nat > nat,Z3: set_nat_nat,X5: set_nat_nat,P2: ( nat > nat ) > $o] :
% 3.18/1.65 ( ( member_nat_nat @ X3 @ Z3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ Z3
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [X: nat > nat] :
% 3.18/1.65 ( ( member_nat_nat @ X @ X5 )
% 3.18/1.65 & ( P2 @ X ) ) ) )
% 3.18/1.65 => ( P2 @ X3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_293_Fpow__def,axiom,
% 3.18/1.65 ( finite_Fpow_nat
% 3.18/1.65 = ( ^ [A5: set_nat] :
% 3.18/1.65 ( collect_set_nat
% 3.18/1.65 @ ^ [X6: set_nat] :
% 3.18/1.65 ( ( ord_less_eq_set_nat @ X6 @ A5 )
% 3.18/1.65 & ( finite_finite_nat @ X6 ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_294_Fpow__def,axiom,
% 3.18/1.65 ( finite90785608at_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat_nat] :
% 3.18/1.65 ( collec617280953at_nat
% 3.18/1.65 @ ^ [X6: set_nat_nat_nat] :
% 3.18/1.65 ( ( ord_le940807492at_nat @ X6 @ A5 )
% 3.18/1.65 & ( finite2045569477at_nat @ X6 ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_295_Fpow__def,axiom,
% 3.18/1.65 ( finite1633036872at_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat_nat2] :
% 3.18/1.65 ( collec309745849at_nat
% 3.18/1.65 @ ^ [X6: set_nat_nat_nat2] :
% 3.18/1.65 ( ( ord_le633272388at_nat @ X6 @ A5 )
% 3.18/1.65 & ( finite1440337093at_nat @ X6 ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_296_Fpow__def,axiom,
% 3.18/1.65 ( finite252967095at_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat_nat_nat3] :
% 3.18/1.65 ( collec1702444712at_nat
% 3.18/1.65 @ ^ [X6: set_nat_nat_nat_nat3] :
% 3.18/1.65 ( ( ord_le2040082867at_nat @ X6 @ A5 )
% 3.18/1.65 & ( finite1064868788at_nat @ X6 ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_297_Fpow__def,axiom,
% 3.18/1.65 ( finite_Fpow_nat_nat
% 3.18/1.65 = ( ^ [A5: set_nat_nat] :
% 3.18/1.65 ( collect_set_nat_nat
% 3.18/1.65 @ ^ [X6: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ X6 @ A5 )
% 3.18/1.65 & ( finite570312790at_nat @ X6 ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_298_partitions__imp__finite__elements,axiom,
% 3.18/1.65 ! [P3: nat > nat,N: nat] :
% 3.18/1.65 ( ( number1551313001itions @ P3 @ N )
% 3.18/1.65 => ( finite_finite_nat
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [I: nat] : ( ord_less_nat @ zero_zero_nat @ ( P3 @ I ) ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_299_not__gr__zero,axiom,
% 3.18/1.65 ! [N: nat] :
% 3.18/1.65 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 3.18/1.65 = ( N = zero_zero_nat ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_300_add__less__cancel__right,axiom,
% 3.18/1.65 ! [A3: nat,C: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 3.18/1.65 = ( ord_less_nat @ A3 @ B3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_301_add__less__cancel__left,axiom,
% 3.18/1.65 ! [C: nat,A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
% 3.18/1.65 = ( ord_less_nat @ A3 @ B3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_302_bot__nat__0_Onot__eq__extremum,axiom,
% 3.18/1.65 ! [A3: nat] :
% 3.18/1.65 ( ( A3 != zero_zero_nat )
% 3.18/1.65 = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_303_less__nat__zero__code,axiom,
% 3.18/1.65 ! [N: nat] :
% 3.18/1.65 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_304_neq0__conv,axiom,
% 3.18/1.65 ! [N: nat] :
% 3.18/1.65 ( ( N != zero_zero_nat )
% 3.18/1.65 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_305_nat__add__left__cancel__less,axiom,
% 3.18/1.65 ! [K: nat,M: nat,N: nat] :
% 3.18/1.65 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 3.18/1.65 = ( ord_less_nat @ M @ N ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_306_finite__Collect__less__nat,axiom,
% 3.18/1.65 ! [K: nat] :
% 3.18/1.65 ( finite_finite_nat
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_307_add__less__same__cancel1,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
% 3.18/1.65 = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_308_add__less__same__cancel2,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
% 3.18/1.65 = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_309_less__add__same__cancel1,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
% 3.18/1.65 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_310_less__add__same__cancel2,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
% 3.18/1.65 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_311_add__gr__0,axiom,
% 3.18/1.65 ! [M: nat,N: nat] :
% 3.18/1.65 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 3.18/1.65 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.18/1.65 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_312_less__one,axiom,
% 3.18/1.65 ! [N: nat] :
% 3.18/1.65 ( ( ord_less_nat @ N @ one_one_nat )
% 3.18/1.65 = ( N = zero_zero_nat ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_313_add__less__le__mono,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat,C: nat,D: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ C @ D )
% 3.18/1.65 => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_314_add__le__less__mono,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat,C: nat,D: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_less_nat @ C @ D )
% 3.18/1.65 => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_315_add__mono__thms__linordered__field_I3_J,axiom,
% 3.18/1.65 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.65 ( ( ( ord_less_nat @ I2 @ J )
% 3.18/1.65 & ( ord_less_eq_nat @ K @ L ) )
% 3.18/1.65 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_316_add__mono__thms__linordered__field_I4_J,axiom,
% 3.18/1.65 ! [I2: nat,J: nat,K: nat,L: nat] :
% 3.18/1.65 ( ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.65 & ( ord_less_nat @ K @ L ) )
% 3.18/1.65 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_317_pos__add__strict,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat,C: nat] :
% 3.18/1.65 ( ( ord_less_nat @ zero_zero_nat @ A3 )
% 3.18/1.65 => ( ( ord_less_nat @ B3 @ C )
% 3.18/1.65 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_318_canonically__ordered__monoid__add__class_OlessE,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A3 @ B3 )
% 3.18/1.65 => ~ ! [C3: nat] :
% 3.18/1.65 ( ( B3
% 3.18/1.65 = ( plus_plus_nat @ A3 @ C3 ) )
% 3.18/1.65 => ( C3 = zero_zero_nat ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_319_add__pos__pos,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ zero_zero_nat @ A3 )
% 3.18/1.65 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 3.18/1.65 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_320_add__neg__neg,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A3 @ zero_zero_nat )
% 3.18/1.65 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 3.18/1.65 => ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_321_order_Onot__eq__order__implies__strict,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( A3 != B3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ A3 @ B3 )
% 3.18/1.65 => ( ord_less_nat @ A3 @ B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_322_order_Onot__eq__order__implies__strict,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat] :
% 3.18/1.65 ( ( A3 != B3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ A3 @ B3 )
% 3.18/1.65 => ( ord_less_set_nat_nat @ A3 @ B3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_323_dual__order_Ostrict__implies__order,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ B3 @ A3 )
% 3.18/1.65 => ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_324_dual__order_Ostrict__implies__order,axiom,
% 3.18/1.65 ! [B3: set_nat_nat,A3: set_nat_nat] :
% 3.18/1.65 ( ( ord_less_set_nat_nat @ B3 @ A3 )
% 3.18/1.65 => ( ord_le1415039317at_nat @ B3 @ A3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_325_dual__order_Ostrict__iff__order,axiom,
% 3.18/1.65 ( ord_less_nat
% 3.18/1.65 = ( ^ [B4: nat,A2: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ B4 @ A2 )
% 3.18/1.65 & ( A2 != B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_326_dual__order_Ostrict__iff__order,axiom,
% 3.18/1.65 ( ord_less_set_nat_nat
% 3.18/1.65 = ( ^ [B4: set_nat_nat,A2: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B4 @ A2 )
% 3.18/1.65 & ( A2 != B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_327_dual__order_Oorder__iff__strict,axiom,
% 3.18/1.65 ( ord_less_eq_nat
% 3.18/1.65 = ( ^ [B4: nat,A2: nat] :
% 3.18/1.65 ( ( ord_less_nat @ B4 @ A2 )
% 3.18/1.65 | ( A2 = B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_328_dual__order_Oorder__iff__strict,axiom,
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 = ( ^ [B4: set_nat_nat,A2: set_nat_nat] :
% 3.18/1.65 ( ( ord_less_set_nat_nat @ B4 @ A2 )
% 3.18/1.65 | ( A2 = B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_329_order_Ostrict__implies__order,axiom,
% 3.18/1.65 ! [A3: nat,B3: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A3 @ B3 )
% 3.18/1.65 => ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_330_order_Ostrict__implies__order,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat] :
% 3.18/1.65 ( ( ord_less_set_nat_nat @ A3 @ B3 )
% 3.18/1.65 => ( ord_le1415039317at_nat @ A3 @ B3 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_331_dual__order_Ostrict__trans2,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat,C: nat] :
% 3.18/1.65 ( ( ord_less_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_less_eq_nat @ C @ B3 )
% 3.18/1.65 => ( ord_less_nat @ C @ A3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_332_dual__order_Ostrict__trans2,axiom,
% 3.18/1.65 ! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_less_set_nat_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ C @ B3 )
% 3.18/1.65 => ( ord_less_set_nat_nat @ C @ A3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_333_dual__order_Ostrict__trans1,axiom,
% 3.18/1.65 ! [B3: nat,A3: nat,C: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_less_nat @ C @ B3 )
% 3.18/1.65 => ( ord_less_nat @ C @ A3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_334_dual__order_Ostrict__trans1,axiom,
% 3.18/1.65 ! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ B3 @ A3 )
% 3.18/1.65 => ( ( ord_less_set_nat_nat @ C @ B3 )
% 3.18/1.65 => ( ord_less_set_nat_nat @ C @ A3 ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_335_order_Ostrict__iff__order,axiom,
% 3.18/1.65 ( ord_less_nat
% 3.18/1.65 = ( ^ [A2: nat,B4: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ A2 @ B4 )
% 3.18/1.65 & ( A2 != B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_336_order_Ostrict__iff__order,axiom,
% 3.18/1.65 ( ord_less_set_nat_nat
% 3.18/1.65 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 3.18/1.65 ( ( ord_le1415039317at_nat @ A2 @ B4 )
% 3.18/1.65 & ( A2 != B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_337_order_Oorder__iff__strict,axiom,
% 3.18/1.65 ( ord_less_eq_nat
% 3.18/1.65 = ( ^ [A2: nat,B4: nat] :
% 3.18/1.65 ( ( ord_less_nat @ A2 @ B4 )
% 3.18/1.65 | ( A2 = B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_338_order_Oorder__iff__strict,axiom,
% 3.18/1.65 ( ord_le1415039317at_nat
% 3.18/1.65 = ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
% 3.18/1.65 ( ( ord_less_set_nat_nat @ A2 @ B4 )
% 3.18/1.65 | ( A2 = B4 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_339_order_Ostrict__trans2,axiom,
% 3.18/1.65 ! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
% 3.18/1.65 ( ( ord_less_set_nat_nat @ A3 @ B3 )
% 3.18/1.65 => ( ( ord_le1415039317at_nat @ B3 @ C )
% 3.18/1.65 => ( ord_less_set_nat_nat @ A3 @ C ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_340_nat__less__le,axiom,
% 3.18/1.65 ( ord_less_nat
% 3.18/1.65 = ( ^ [M2: nat,N2: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.18/1.65 & ( M2 != N2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_341_less__imp__le__nat,axiom,
% 3.18/1.65 ! [M: nat,N: nat] :
% 3.18/1.65 ( ( ord_less_nat @ M @ N )
% 3.18/1.65 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_342_le__eq__less__or__eq,axiom,
% 3.18/1.65 ( ord_less_eq_nat
% 3.18/1.65 = ( ^ [M2: nat,N2: nat] :
% 3.18/1.65 ( ( ord_less_nat @ M2 @ N2 )
% 3.18/1.65 | ( M2 = N2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_343_less__or__eq__imp__le,axiom,
% 3.18/1.65 ! [M: nat,N: nat] :
% 3.18/1.65 ( ( ( ord_less_nat @ M @ N )
% 3.18/1.65 | ( M = N ) )
% 3.18/1.65 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_344_le__neq__implies__less,axiom,
% 3.18/1.65 ! [M: nat,N: nat] :
% 3.18/1.65 ( ( ord_less_eq_nat @ M @ N )
% 3.18/1.65 => ( ( M != N )
% 3.18/1.65 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_345_less__mono__imp__le__mono,axiom,
% 3.18/1.65 ! [F2: nat > nat,I2: nat,J: nat] :
% 3.18/1.65 ( ! [I4: nat,J2: nat] :
% 3.18/1.65 ( ( ord_less_nat @ I4 @ J2 )
% 3.18/1.65 => ( ord_less_nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) )
% 3.18/1.65 => ( ( ord_less_eq_nat @ I2 @ J )
% 3.18/1.65 => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_346_bounded__nat__set__is__finite,axiom,
% 3.18/1.65 ! [N5: set_nat,N: nat] :
% 3.18/1.65 ( ! [X2: nat] :
% 3.18/1.65 ( ( member_nat @ X2 @ N5 )
% 3.18/1.65 => ( ord_less_nat @ X2 @ N ) )
% 3.18/1.65 => ( finite_finite_nat @ N5 ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_347_finite__nat__set__iff__bounded,axiom,
% 3.18/1.65 ( finite_finite_nat
% 3.18/1.65 = ( ^ [N4: set_nat] :
% 3.18/1.65 ? [M2: nat] :
% 3.18/1.65 ! [X: nat] :
% 3.18/1.65 ( ( member_nat @ X @ N4 )
% 3.18/1.65 => ( ord_less_nat @ X @ M2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(fact_348_finite__M__bounded__by__nat,axiom,
% 3.18/1.65 ! [P2: nat > $o,I2: nat] :
% 3.18/1.65 ( finite_finite_nat
% 3.18/1.65 @ ( collect_nat
% 3.18/1.65 @ ^ [K2: nat] :
% 3.18/1.65 ( ( P2 @ K2 )
% 3.18/1.65 & ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% 3.18/1.65
% 3.18/1.65 thf(conj_0,conjecture,
% 3.18/1.65 ( finite570312790at_nat
% 3.18/1.65 @ ( collect_nat_nat
% 3.18/1.65 @ ^ [P: nat > nat] : ( number1551313001itions @ P @ n ) ) ) ).
% 3.18/1.65 ------------------------
% 3.18/1.67 % (31438)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/183Mi)
% 3.18/1.68 % (31444)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/18Mi)
% 3.18/1.68 % (31441)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/2Mi)
% 3.18/1.68 % (31440)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_31226 for (2999ds/27Mi)
% 3.18/1.68 % (31442)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_31226 for (2999ds/2Mi)
% 3.18/1.68 % (31441)Instruction limit reached!
% 3.18/1.68 % (31441)------------------------------
% 3.18/1.68 % (31441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (31441)Termination reason: Unknown
% 3.18/1.68 % (31441)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (31441)Memory used [KB]: 1407
% 3.18/1.68 % (31441)Time elapsed: 0.003 s
% 3.18/1.68 % (31441)Instructions burned: 3 (million)
% 3.18/1.68 % (31441)------------------------------
% 3.18/1.68 % (31441)------------------------------
% 3.18/1.68 % (31439)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_31226 for (2999ds/4Mi)
% 3.18/1.68 % (31443)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_31226 for (2999ds/275Mi)
% 3.18/1.68 % (31442)Instruction limit reached!
% 3.18/1.68 % (31442)------------------------------
% 3.18/1.68 % (31442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (31442)Termination reason: Unknown
% 3.18/1.68 % (31442)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (31442)Memory used [KB]: 1535
% 3.18/1.68 % (31442)Time elapsed: 0.003 s
% 3.18/1.68 % (31442)Instructions burned: 4 (million)
% 3.18/1.68 % (31442)------------------------------
% 3.18/1.68 % (31442)------------------------------
% 3.18/1.68 % (31439)Instruction limit reached!
% 3.18/1.68 % (31439)------------------------------
% 3.18/1.68 % (31439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (31439)Termination reason: Unknown
% 3.18/1.68 % (31439)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (31439)Memory used [KB]: 1535
% 3.18/1.68 % (31439)Time elapsed: 0.005 s
% 3.18/1.68 % (31439)Instructions burned: 6 (million)
% 3.18/1.68 % (31439)------------------------------
% 3.18/1.68 % (31439)------------------------------
% 3.18/1.68 % (31444)Instruction limit reached!
% 3.18/1.68 % (31444)------------------------------
% 3.18/1.68 % (31444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (31444)Termination reason: Unknown
% 3.18/1.68 % (31444)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (31444)Memory used [KB]: 1918
% 3.18/1.68 % (31444)Time elapsed: 0.010 s
% 3.18/1.68 % (31444)Instructions burned: 18 (million)
% 3.18/1.68 % (31444)------------------------------
% 3.18/1.68 % (31444)------------------------------
% 3.18/1.69 % (31440)Instruction limit reached!
% 3.18/1.69 % (31440)------------------------------
% 3.18/1.69 % (31440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.69 % (31440)Termination reason: Unknown
% 3.18/1.69 % (31440)Termination phase: shuffling
% 3.18/1.69
% 3.18/1.69 % (31440)Memory used [KB]: 1918
% 3.18/1.69 % (31440)Time elapsed: 0.015 s
% 3.18/1.69 % (31440)Instructions burned: 27 (million)
% 3.18/1.69 % (31440)------------------------------
% 3.18/1.69 % (31440)------------------------------
% 3.18/1.69 % (31449)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/3Mi)
% 3.18/1.69 % (31451)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_31226 for (2999ds/37Mi)
% 3.18/1.69 % (31455)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_31226 for (2999ds/3Mi)
% 3.18/1.70 % (31449)Instruction limit reached!
% 3.18/1.70 % (31449)------------------------------
% 3.18/1.70 % (31449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.70 % (31449)Termination reason: Unknown
% 3.18/1.70 % (31449)Termination phase: shuffling
% 3.18/1.70
% 3.18/1.70 % (31449)Memory used [KB]: 1535
% 3.18/1.70 % (31449)Time elapsed: 0.004 s
% 3.18/1.70 % (31449)Instructions burned: 4 (million)
% 3.18/1.70 % (31449)------------------------------
% 3.18/1.70 % (31449)------------------------------
% 3.24/1.70 % (31455)Instruction limit reached!
% 3.24/1.70 % (31455)------------------------------
% 3.24/1.70 % (31455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.70 % (31455)Termination reason: Unknown
% 3.24/1.70 % (31455)Termination phase: shuffling
% 3.24/1.70
% 3.24/1.70 % (31455)Memory used [KB]: 1535
% 3.24/1.70 % (31455)Time elapsed: 0.003 s
% 3.24/1.70 % (31455)Instructions burned: 4 (million)
% 3.24/1.70 % (31455)------------------------------
% 3.24/1.70 % (31455)------------------------------
% 3.24/1.70 % (31453)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_31226 for (2999ds/15Mi)
% 3.24/1.70 % (31453)Instruction limit reached!
% 3.24/1.70 % (31453)------------------------------
% 3.24/1.70 % (31453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.70 % (31453)Termination reason: Unknown
% 3.24/1.70 % (31453)Termination phase: shuffling
% 3.24/1.70
% 3.24/1.70 % (31453)Memory used [KB]: 1791
% 3.24/1.70 % (31453)Time elapsed: 0.007 s
% 3.24/1.70 % (31453)Instructions burned: 15 (million)
% 3.24/1.70 % (31453)------------------------------
% 3.24/1.70 % (31453)------------------------------
% 3.24/1.70 % (31457)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_31226 for (2999ds/1041Mi)
% 3.24/1.71 % (31451)Instruction limit reached!
% 3.24/1.71 % (31451)------------------------------
% 3.24/1.71 % (31451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.71 % (31451)Termination reason: Unknown
% 3.24/1.71 % (31451)Termination phase: shuffling
% 3.24/1.71
% 3.24/1.71 % (31451)Memory used [KB]: 2302
% 3.24/1.71 % (31451)Time elapsed: 0.019 s
% 3.24/1.71 % (31451)Instructions burned: 38 (million)
% 3.24/1.71 % (31451)------------------------------
% 3.24/1.71 % (31451)------------------------------
% 3.24/1.71 % (31461)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/16Mi)
% 3.24/1.71 % (31459)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/7Mi)
% 3.24/1.71 % (31463)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_31226 for (2999ds/3Mi)
% 3.24/1.71 % (31463)Instruction limit reached!
% 3.24/1.71 % (31463)------------------------------
% 3.24/1.71 % (31463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.71 % (31463)Termination reason: Unknown
% 3.24/1.71 % (31463)Termination phase: shuffling
% 3.24/1.71
% 3.24/1.71 % (31463)Memory used [KB]: 1407
% 3.24/1.71 % (31463)Time elapsed: 0.002 s
% 3.24/1.71 % (31463)Instructions burned: 3 (million)
% 3.24/1.71 % (31463)------------------------------
% 3.24/1.71 % (31463)------------------------------
% 3.24/1.72 % (31459)Instruction limit reached!
% 3.24/1.72 % (31459)------------------------------
% 3.24/1.72 % (31459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.72 % (31459)Termination reason: Unknown
% 3.24/1.72 % (31459)Termination phase: shuffling
% 3.24/1.72
% 3.24/1.72 % (31459)Memory used [KB]: 1535
% 3.24/1.72 % (31459)Time elapsed: 0.005 s
% 3.24/1.72 % (31459)Instructions burned: 7 (million)
% 3.24/1.72 % (31459)------------------------------
% 3.24/1.72 % (31459)------------------------------
% 3.24/1.72 % (31461)Instruction limit reached!
% 3.24/1.72 % (31461)------------------------------
% 3.24/1.72 % (31461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.72 % (31461)Termination reason: Unknown
% 3.24/1.72 % (31461)Termination phase: shuffling
% 3.24/1.72
% 3.24/1.72 % (31461)Memory used [KB]: 1791
% 3.24/1.72 % (31461)Time elapsed: 0.010 s
% 3.24/1.72 % (31461)Instructions burned: 17 (million)
% 3.24/1.72 % (31461)------------------------------
% 3.24/1.72 % (31461)------------------------------
% 3.24/1.72 % (31467)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/7Mi)
% 3.24/1.73 % (31466)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_31226 for (2999ds/3Mi)
% 3.24/1.73 % (31468)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/3Mi)
% 3.24/1.73 % (31467)Instruction limit reached!
% 3.24/1.73 % (31467)------------------------------
% 3.24/1.73 % (31467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.73 % (31467)Termination reason: Unknown
% 3.24/1.73 % (31467)Termination phase: shuffling
% 3.24/1.73
% 3.24/1.73 % (31467)Memory used [KB]: 1535
% 3.24/1.73 % (31467)Time elapsed: 0.005 s
% 3.24/1.73 % (31467)Instructions burned: 8 (million)
% 3.24/1.73 % (31467)------------------------------
% 3.24/1.73 % (31467)------------------------------
% 3.24/1.73 % (31466)Instruction limit reached!
% 3.24/1.73 % (31466)------------------------------
% 3.24/1.73 % (31466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.73 % (31468)Instruction limit reached!
% 3.24/1.73 % (31468)------------------------------
% 3.24/1.73 % (31468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.73 % (31468)Termination reason: Unknown
% 3.24/1.73 % (31468)Termination phase: shuffling
% 3.24/1.73
% 3.24/1.73 % (31468)Memory used [KB]: 1535
% 3.24/1.73 % (31468)Time elapsed: 0.003 s
% 3.24/1.73 % (31468)Instructions burned: 5 (million)
% 3.24/1.73 % (31468)------------------------------
% 3.24/1.73 % (31468)------------------------------
% 3.24/1.73 % (31466)Termination reason: Unknown
% 3.24/1.73 % (31466)Termination phase: shuffling
% 3.24/1.73
% 3.24/1.73 % (31466)Memory used [KB]: 1535
% 3.24/1.73 % (31466)Time elapsed: 0.003 s
% 3.24/1.73 % (31466)Instructions burned: 3 (million)
% 3.24/1.73 % (31466)------------------------------
% 3.24/1.73 % (31466)------------------------------
% 3.24/1.74 % (31469)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/4Mi)
% 3.24/1.74 % (31474)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/710Mi)
% 3.24/1.74 % (31469)Instruction limit reached!
% 3.24/1.74 % (31469)------------------------------
% 3.24/1.74 % (31469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.74 % (31469)Termination reason: Unknown
% 3.24/1.74 % (31469)Termination phase: shuffling
% 3.24/1.74
% 3.24/1.74 % (31469)Memory used [KB]: 1535
% 3.24/1.74 % (31469)Time elapsed: 0.004 s
% 3.24/1.74 % (31469)Instructions burned: 5 (million)
% 3.24/1.74 % (31469)------------------------------
% 3.24/1.74 % (31469)------------------------------
% 3.24/1.74 % (31473)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/18Mi)
% 3.24/1.75 % (31473)Instruction limit reached!
% 3.24/1.75 % (31473)------------------------------
% 3.24/1.75 % (31473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.75 % (31473)Termination reason: Unknown
% 3.24/1.75 % (31473)Termination phase: shuffling
% 3.24/1.75
% 3.24/1.75 % (31473)Memory used [KB]: 1791
% 3.24/1.75 % (31473)Time elapsed: 0.010 s
% 3.24/1.75 % (31473)Instructions burned: 21 (million)
% 3.24/1.75 % (31473)------------------------------
% 3.24/1.75 % (31473)------------------------------
% 3.24/1.75 % (31475)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31226 for (2999ds/6Mi)
% 3.24/1.76 % (31475)Instruction limit reached!
% 3.24/1.76 % (31475)------------------------------
% 3.24/1.76 % (31475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.76 % (31475)Termination reason: Unknown
% 3.24/1.76 % (31475)Termination phase: shuffling
% 3.24/1.76
% 3.24/1.76 % (31475)Memory used [KB]: 1535
% 3.24/1.76 % (31475)Time elapsed: 0.005 s
% 3.24/1.76 % (31475)Instructions burned: 7 (million)
% 3.24/1.76 % (31475)------------------------------
% 3.24/1.76 % (31475)------------------------------
% 3.24/1.76 % (31479)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_31226 for (2999ds/21Mi)
% 3.24/1.77 % (31477)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_31226 for (2999ds/902Mi)
% 3.24/1.77 % (31438)Instruction limit reached!
% 3.24/1.77 % (31438)------------------------------
% 3.24/1.77 % (31438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.77 % (31438)Termination reason: Unknown
% 3.24/1.77 % (31438)Termination phase: Saturation
% 3.24/1.77
% 3.24/1.77 % (31438)Memory used [KB]: 7419
% 3.24/1.77 % (31438)Time elapsed: 0.100 s
% 3.24/1.77 % (31438)Instructions burned: 184 (million)
% 3.24/1.77 % (31438)------------------------------
% 3.24/1.77 % (31438)------------------------------
% 3.24/1.77 % (31479)Instruction limit reached!
% 3.24/1.77 % (31479)------------------------------
% 3.24/1.77 % (31479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.77 % (31479)Termination reason: Unknown
% 3.24/1.77 % (31479)Termination phase: shuffling
% 3.24/1.77
% 3.24/1.77 % (31479)Memory used [KB]: 1791
% 3.24/1.77 % (31479)Time elapsed: 0.008 s
% 3.24/1.77 % (31479)Instructions burned: 21 (million)
% 3.24/1.77 % (31479)------------------------------
% 3.24/1.77 % (31479)------------------------------
% 3.24/1.77 % (31480)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/5Mi)
% 3.24/1.77 % (31480)Instruction limit reached!
% 3.24/1.77 % (31480)------------------------------
% 3.24/1.77 % (31480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.77 % (31480)Termination reason: Unknown
% 3.24/1.77 % (31480)Termination phase: shuffling
% 3.24/1.77
% 3.24/1.77 % (31480)Memory used [KB]: 1535
% 3.24/1.77 % (31480)Time elapsed: 0.003 s
% 3.24/1.77 % (31480)Instructions burned: 5 (million)
% 3.24/1.77 % (31480)------------------------------
% 3.24/1.77 % (31480)------------------------------
% 3.24/1.78 % (31483)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/377Mi)
% 3.24/1.78 % (31482)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_31226 for (2998ds/6Mi)
% 3.24/1.78 % (31484)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/779Mi)
% 3.24/1.78 % (31482)Instruction limit reached!
% 3.24/1.78 % (31482)------------------------------
% 3.24/1.78 % (31482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.78 % (31482)Termination reason: Unknown
% 3.24/1.78 % (31482)Termination phase: shuffling
% 3.24/1.78
% 3.24/1.78 % (31482)Memory used [KB]: 1535
% 3.24/1.78 % (31482)Time elapsed: 0.003 s
% 3.24/1.78 % (31482)Instructions burned: 7 (million)
% 3.24/1.78 % (31482)------------------------------
% 3.24/1.78 % (31482)------------------------------
% 3.24/1.79 % (31443)Instruction limit reached!
% 3.24/1.79 % (31443)------------------------------
% 3.24/1.79 % (31443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.79 % (31443)Termination reason: Unknown
% 3.24/1.79 % (31443)Termination phase: Saturation
% 3.24/1.79
% 3.24/1.79 % (31443)Memory used [KB]: 7547
% 3.24/1.79 % (31443)Time elapsed: 0.114 s
% 3.24/1.79 % (31443)Instructions burned: 276 (million)
% 3.24/1.79 % (31443)------------------------------
% 3.24/1.79 % (31443)------------------------------
% 3.24/1.79 % (31485)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/19Mi)
% 3.24/1.80 % (31485)Instruction limit reached!
% 3.24/1.80 % (31485)------------------------------
% 3.24/1.80 % (31485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (31485)Termination reason: Unknown
% 3.24/1.80 % (31485)Termination phase: shuffling
% 3.24/1.80
% 3.24/1.80 % (31485)Memory used [KB]: 1918
% 3.24/1.80 % (31485)Time elapsed: 0.008 s
% 3.24/1.80 % (31485)Instructions burned: 21 (million)
% 3.24/1.80 % (31485)------------------------------
% 3.24/1.80 % (31485)------------------------------
% 3.24/1.80 % (31486)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/879Mi)
% 3.24/1.81 % (31487)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/17Mi)
% 3.24/1.81 % (31487)Instruction limit reached!
% 3.24/1.81 % (31487)------------------------------
% 3.24/1.81 % (31487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (31487)Termination reason: Unknown
% 3.24/1.81 % (31487)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (31487)Memory used [KB]: 1791
% 3.24/1.81 % (31487)Time elapsed: 0.008 s
% 3.24/1.81 % (31487)Instructions burned: 18 (million)
% 3.24/1.81 % (31487)------------------------------
% 3.24/1.81 % (31487)------------------------------
% 3.24/1.82 % (31488)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/3Mi)
% 3.24/1.83 % (31488)Instruction limit reached!
% 3.24/1.83 % (31488)------------------------------
% 3.24/1.83 % (31488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.83 % (31488)Termination reason: Unknown
% 3.24/1.83 % (31488)Termination phase: shuffling
% 3.24/1.83
% 3.24/1.83 % (31488)Memory used [KB]: 1535
% 3.24/1.83 % (31488)Time elapsed: 0.003 s
% 3.24/1.83 % (31488)Instructions burned: 3 (million)
% 3.24/1.83 % (31488)------------------------------
% 3.24/1.83 % (31488)------------------------------
% 3.24/1.84 % (31490)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_31226 for (2998ds/30Mi)
% 3.24/1.84 % (31486)First to succeed.
% 3.24/1.85 % (31486)Refutation found. Thanks to Tanya!
% 3.24/1.85 % SZS status Theorem for DTF2THF_31226
% 3.24/1.85 % SZS output start Proof for DTF2THF_31226
% 3.24/1.85 thf(type_def_5, type, set_nat_nat_nat_nat3: $tType).
% 3.24/1.85 thf(type_def_6, type, set_se1657353at_nat: $tType).
% 3.24/1.85 thf(type_def_8, type, set_nat_nat_nat2: $tType).
% 3.24/1.85 thf(type_def_9, type, set_set_nat_nat_nat2: $tType).
% 3.24/1.85 thf(type_def_10, type, set_nat_nat_nat: $tType).
% 3.24/1.85 thf(type_def_11, type, set_set_nat_nat_nat: $tType).
% 3.24/1.85 thf(type_def_12, type, set_nat_nat: $tType).
% 3.24/1.85 thf(type_def_13, type, set_set_nat_nat: $tType).
% 3.24/1.85 thf(type_def_14, type, set_nat: $tType).
% 3.24/1.85 thf(type_def_15, type, set_set_nat: $tType).
% 3.24/1.85 thf(type_def_16, type, set_nat_nat_nat_nat5: $tType).
% 3.24/1.85 thf(type_def_17, type, set_nat_nat_nat_nat4: $tType).
% 3.24/1.85 thf(type_def_18, type, set_na1201295426at_nat: $tType).
% 3.24/1.85 thf(type_def_19, type, set_na1064332610at_nat: $tType).
% 3.24/1.85 thf(type_def_20, type, set_nat_nat_nat_nat2: $tType).
% 3.24/1.85 thf(type_def_21, type, set_nat_nat_nat_nat: $tType).
% 3.24/1.85 thf(type_def_22, type, nat: $tType).
% 3.24/1.85 thf(func_def_0, type, set_na1064332610at_nat: $tType).
% 3.24/1.85 thf(func_def_1, type, set_na1201295426at_nat: $tType).
% 3.24/1.85 thf(func_def_2, type, set_se1657353at_nat: $tType).
% 3.24/1.85 thf(func_def_3, type, set_nat_nat_nat_nat: $tType).
% 3.24/1.85 thf(func_def_4, type, set_nat_nat_nat_nat2: $tType).
% 3.24/1.85 thf(func_def_5, type, set_nat_nat_nat_nat3: $tType).
% 3.24/1.85 thf(func_def_6, type, set_nat_nat_nat_nat4: $tType).
% 3.24/1.85 thf(func_def_7, type, set_nat_nat_nat_nat5: $tType).
% 3.24/1.85 thf(func_def_8, type, set_set_nat_nat_nat: $tType).
% 3.24/1.85 thf(func_def_9, type, set_set_nat_nat_nat2: $tType).
% 3.24/1.85 thf(func_def_10, type, set_nat_nat_nat: $tType).
% 3.24/1.85 thf(func_def_11, type, set_nat_nat_nat2: $tType).
% 3.24/1.85 thf(func_def_12, type, set_set_nat_nat: $tType).
% 3.24/1.85 thf(func_def_13, type, set_nat_nat: $tType).
% 3.24/1.85 thf(func_def_14, type, set_set_nat: $tType).
% 3.24/1.85 thf(func_def_15, type, set_nat: $tType).
% 3.24/1.85 thf(func_def_16, type, nat: $tType).
% 3.24/1.85 thf(func_def_17, type, finite252967095at_nat: set_nat_nat_nat_nat3 > set_se1657353at_nat).
% 3.24/1.85 thf(func_def_18, type, finite1633036872at_nat: set_nat_nat_nat2 > set_set_nat_nat_nat2).
% 3.24/1.85 thf(func_def_19, type, finite90785608at_nat: set_nat_nat_nat > set_set_nat_nat_nat).
% 3.24/1.85 thf(func_def_20, type, finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat).
% 3.24/1.85 thf(func_def_21, type, finite_Fpow_nat: set_nat > set_set_nat).
% 3.24/1.85 thf(func_def_22, type, finite1526199988at_nat: set_nat_nat_nat_nat5 > $o).
% 3.24/1.85 thf(func_def_23, type, finite2081887156at_nat: set_nat_nat_nat_nat4 > $o).
% 3.24/1.85 thf(func_def_24, type, finite1090277411at_nat: set_na1201295426at_nat > $o).
% 3.24/1.85 thf(func_def_25, type, finite1397812515at_nat: set_na1064332610at_nat > $o).
% 3.24/1.85 thf(func_def_26, type, finite1064868788at_nat: set_nat_nat_nat_nat3 > $o).
% 3.24/1.85 thf(func_def_27, type, finite1440337093at_nat: set_nat_nat_nat2 > $o).
% 3.24/1.85 thf(func_def_28, type, finite584233140at_nat: set_nat_nat_nat_nat2 > $o).
% 3.24/1.85 thf(func_def_29, type, finite891768244at_nat: set_nat_nat_nat_nat > $o).
% 3.24/1.85 thf(func_def_30, type, finite2045569477at_nat: set_nat_nat_nat > $o).
% 3.24/1.85 thf(func_def_31, type, finite570312790at_nat: set_nat_nat > $o).
% 3.24/1.85 thf(func_def_32, type, finite_finite_nat: set_nat > $o).
% 3.24/1.85 thf(func_def_33, type, finite547180394at_nat: set_se1657353at_nat > $o).
% 3.24/1.85 thf(func_def_34, type, finite1977496699at_nat: set_set_nat_nat_nat2 > $o).
% 3.24/1.85 thf(func_def_35, type, finite137548155at_nat: set_set_nat_nat_nat > $o).
% 3.24/1.85 thf(func_def_36, type, finite604103692at_nat: set_set_nat_nat > $o).
% 3.24/1.85 thf(func_def_37, type, finite2012248349et_nat: set_set_nat > $o).
% 3.24/1.85 thf(func_def_38, type, one_one_nat: nat).
% 3.24/1.85 thf(func_def_39, type, plus_plus_nat: nat > nat > nat).
% 3.24/1.85 thf(func_def_40, type, zero_zero_nat: nat).
% 3.24/1.85 thf(func_def_41, type, number1551313001itions: (nat > nat) > nat > $o).
% 3.24/1.85 thf(func_def_42, type, ord_less_nat: nat > nat > $o).
% 3.24/1.85 thf(func_def_43, type, ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o).
% 3.24/1.85 thf(func_def_44, type, ord_le2059018749at_nat: ((nat > nat) > nat > nat) > ((nat > nat) > nat > nat) > $o).
% 3.24/1.85 thf(func_def_45, type, ord_le2026740592_nat_o: ((nat > nat) > $o) > ((nat > nat) > $o) > $o).
% 3.24/1.85 thf(func_def_46, type, ord_le809907342at_nat: ((nat > nat) > nat) > ((nat > nat) > nat) > $o).
% 3.24/1.85 thf(func_def_47, type, ord_le1415139726at_nat: (nat > nat > nat) > (nat > nat > nat) > $o).
% 3.24/1.85 thf(func_def_48, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 3.24/1.85 thf(func_def_49, type, ord_less_eq_nat_nat: (nat > nat) > (nat > nat) > $o).
% 3.24/1.85 thf(func_def_50, type, ord_less_eq_nat: nat > nat > $o).
% 3.24/1.85 thf(func_def_51, type, ord_le2040082867at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o).
% 3.24/1.85 thf(func_def_52, type, ord_le633272388at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o).
% 3.24/1.85 thf(func_def_53, type, ord_le940807492at_nat: set_nat_nat_nat > set_nat_nat_nat > $o).
% 3.24/1.85 thf(func_def_54, type, ord_le1415039317at_nat: set_nat_nat > set_nat_nat > $o).
% 3.24/1.85 thf(func_def_55, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 3.24/1.85 thf(func_def_56, type, collec2071977458at_nat: ((((nat > nat) > nat) > nat) > $o) > set_nat_nat_nat_nat5).
% 3.24/1.85 thf(func_def_57, type, collec480180978at_nat: (((nat > nat > nat) > nat) > $o) > set_nat_nat_nat_nat4).
% 3.24/1.85 thf(func_def_58, type, collec6783585at_nat: (((nat > nat) > (nat > nat) > nat) > $o) > set_na1201295426at_nat).
% 3.24/1.85 thf(func_def_59, type, collec314318689at_nat: (((nat > nat) > nat > nat > nat) > $o) > set_na1064332610at_nat).
% 3.24/1.85 thf(func_def_60, type, collec1610646258at_nat: (((nat > nat) > nat > nat) > $o) > set_nat_nat_nat_nat3).
% 3.24/1.85 thf(func_def_61, type, collect_nat_nat_nat: (((nat > nat) > nat) > $o) > set_nat_nat_nat2).
% 3.24/1.85 thf(func_def_62, type, collec1130010610at_nat: ((nat > (nat > nat) > nat) > $o) > set_nat_nat_nat_nat2).
% 3.24/1.85 thf(func_def_63, type, collec1437545714at_nat: ((nat > nat > nat > nat) > $o) > set_nat_nat_nat_nat).
% 3.24/1.85 thf(func_def_64, type, collect_nat_nat_nat2: ((nat > nat > nat) > $o) > set_nat_nat_nat).
% 3.24/1.85 thf(func_def_65, type, collect_nat_nat: ((nat > nat) > $o) > set_nat_nat).
% 3.24/1.85 thf(func_def_66, type, collect_nat: (nat > $o) > set_nat).
% 3.24/1.85 thf(func_def_67, type, collec1702444712at_nat: (set_nat_nat_nat_nat3 > $o) > set_se1657353at_nat).
% 3.24/1.85 thf(func_def_68, type, collec309745849at_nat: (set_nat_nat_nat2 > $o) > set_set_nat_nat_nat2).
% 3.24/1.85 thf(func_def_69, type, collec617280953at_nat: (set_nat_nat_nat > $o) > set_set_nat_nat_nat).
% 3.24/1.85 thf(func_def_70, type, collect_set_nat_nat: (set_nat_nat > $o) > set_set_nat_nat).
% 3.24/1.85 thf(func_def_71, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 3.24/1.85 thf(func_def_72, type, member1128122036at_nat: ((nat > nat) > nat > nat) > set_nat_nat_nat_nat3 > $o).
% 3.24/1.85 thf(func_def_73, type, member_nat_nat_nat: ((nat > nat) > nat) > set_nat_nat_nat2 > $o).
% 3.24/1.85 thf(func_def_74, type, member_nat_nat_nat2: (nat > nat > nat) > set_nat_nat_nat > $o).
% 3.24/1.85 thf(func_def_75, type, member_nat_nat: (nat > nat) > set_nat_nat > $o).
% 3.24/1.85 thf(func_def_76, type, member_nat: nat > set_nat > $o).
% 3.24/1.85 thf(func_def_77, type, member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o).
% 3.24/1.85 thf(func_def_78, type, n: nat).
% 3.24/1.85 thf(func_def_93, type, sK0: set_nat_nat > (nat > nat) > nat > nat).
% 3.24/1.85 thf(func_def_94, type, sK1: ((nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_95, type, sK2: set_nat_nat > set_nat_nat > ((nat > nat) > (nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_96, type, sK3: set_nat_nat > set_nat_nat > ((nat > nat) > (nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_97, type, sK4: set_nat_nat > ((nat > nat) > ((nat > nat) > nat) > $o) > set_nat_nat_nat2 > nat > nat).
% 3.24/1.85 thf(func_def_98, type, sK5: set_nat_nat > ((nat > nat) > ((nat > nat) > nat) > $o) > set_nat_nat_nat2 > (nat > nat) > nat).
% 3.24/1.85 thf(func_def_99, type, sK6: set_nat > ((nat > nat) > nat > $o) > set_nat_nat > nat > nat).
% 3.24/1.85 thf(func_def_100, type, sK7: set_nat > ((nat > nat) > nat > $o) > set_nat_nat > nat).
% 3.24/1.85 thf(func_def_101, type, sK8: ((nat > nat) > $o) > ((nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_102, type, sK9: ((nat > nat) > $o) > ((nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_103, type, sK10: (nat > nat) > set_nat_nat > nat > nat).
% 3.24/1.85 thf(func_def_104, type, sK11: ((nat > nat) > (nat > nat > nat) > $o) > set_nat_nat > set_nat_nat_nat > nat > nat > nat).
% 3.24/1.85 thf(func_def_105, type, sK12: ((nat > nat) > (nat > nat > nat) > $o) > set_nat_nat > set_nat_nat_nat > nat > nat).
% 3.24/1.85 thf(func_def_106, type, sK13: set_nat > set_nat_nat > (nat > (nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_107, type, sK14: set_nat > set_nat_nat > (nat > (nat > nat) > $o) > nat).
% 3.24/1.85 thf(func_def_108, type, sK15: set_nat_nat > ((nat > nat) > $o) > ((nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_110, type, sK17: ((nat > nat) > $o) > set_nat_nat > nat > nat).
% 3.24/1.85 thf(func_def_111, type, sK18: ((nat > nat) > $o) > ((nat > nat) > $o) > nat > nat).
% 3.24/1.85 thf(func_def_112, type, ph19: !>[X0: $tType]:(X0)).
% 3.24/1.85 thf(f1283,plain,(
% 3.24/1.85 $false),
% 3.24/1.85 inference(subsumption_resolution,[status(thm)],[f1282,f1114])).
% 3.24/1.85 thf(f1114,plain,(
% 3.24/1.85 ( ! [X0 : nat,X1 : nat] : (($true = (finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ X1 @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ X0))))))))) )),
% 3.24/1.85 inference(cnf_transformation,[status(thm)],[f834])).
% 3.24/1.85 thf(f834,plain,(
% 3.24/1.85 ! [X0 : nat,X1 : nat] : ($true = (finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ X1 @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ X0))))))))),
% 3.24/1.85 inference(fool_elimination,[status(thm)],[f833])).
% 3.24/1.85 thf(f833,plain,(
% 3.24/1.85 ! [X0 : nat,X1 : nat] : (finite570312790at_nat @ (collect_nat_nat @ (^[X2 : nat > nat] : (! [X3 : nat] : (ord_less_eq_nat @ (X2 @ X3) @ X0) & ! [X4 : nat] : ((ord_less_eq_nat @ X1 @ X4) => (zero_zero_nat = (X2 @ X4)))))))),
% 3.24/1.85 inference(rectify,[status(thm)],[f45])).
% 3.24/1.85 thf(f45,axiom,(
% 3.24/1.85 ! [X7 : nat,X18 : nat] : (finite570312790at_nat @ (collect_nat_nat @ (^[X0 : nat > nat] : (! [X1 : nat] : (ord_less_eq_nat @ (X0 @ X1) @ X7) & ! [X1 : nat] : ((ord_less_eq_nat @ X18 @ X1) => ((X0 @ X1) = zero_zero_nat))))))),
% 3.24/1.85 file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',fact_44_bound__domain__and__range__impl__finitely__many__functions)).
% 3.24/1.85 thf(f1282,plain,(
% 3.24/1.85 ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ n))))))) != $true)),
% 3.24/1.85 inference(subsumption_resolution,[status(thm)],[f1281,f1090])).
% 3.24/1.85 thf(f1090,plain,(
% 3.24/1.85 ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n)))) != $true)),
% 3.24/1.85 inference(cnf_transformation,[status(thm)],[f971])).
% 3.24/1.85 thf(f971,plain,(
% 3.24/1.85 ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n)))) != $true)),
% 3.24/1.85 inference(flattening,[status(thm)],[f768])).
% 3.24/1.85 thf(f768,plain,(
% 3.24/1.85 ~((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n)))) = $true)),
% 3.24/1.85 inference(fool_elimination,[status(thm)],[f767])).
% 3.24/1.85 thf(f767,plain,(
% 3.24/1.85 ~(finite570312790at_nat @ (collect_nat_nat @ (^[X0 : nat > nat] : (number1551313001itions @ X0 @ n))))),
% 3.24/1.85 inference(rectify,[status(thm)],[f351])).
% 3.24/1.85 thf(f351,negated_conjecture,(
% 3.24/1.85 ~(finite570312790at_nat @ (collect_nat_nat @ (^[X2 : nat > nat] : (number1551313001itions @ X2 @ n))))),
% 3.24/1.85 inference(negated_conjecture,[status(cth)],[f350])).
% 3.24/1.85 thf(f350,conjecture,(
% 3.24/1.85 (finite570312790at_nat @ (collect_nat_nat @ (^[X2 : nat > nat] : (number1551313001itions @ X2 @ n))))),
% 3.24/1.85 file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',conj_0)).
% 3.24/1.85 thf(f1281,plain,(
% 3.24/1.85 ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n)))) = $true) | ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ n))))))) != $true)),
% 3.24/1.85 inference(trivial_inequality_removal,[status(thm)],[f1280])).
% 3.24/1.85 thf(f1280,plain,(
% 3.24/1.85 ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ n))))))) != $true) | ($true != $true) | ((finite570312790at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n)))) = $true)),
% 3.24/1.85 inference(superposition,[status(thm)],[f1057,f1067])).
% 3.24/1.85 thf(f1067,plain,(
% 3.24/1.85 ((ord_le1415039317at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n))) @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ n))))))) = $true)),
% 3.24/1.85 inference(cnf_transformation,[status(thm)],[f650])).
% 3.24/1.85 thf(f650,plain,(
% 3.24/1.85 ((ord_le1415039317at_nat @ (collect_nat_nat @ (^[Y0 : nat > nat]: (number1551313001itions @ Y0 @ n))) @ (collect_nat_nat @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ Y1) => (zero_zero_nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (Y0 @ Y1) @ n))))))) = $true)),
% 3.24/1.85 inference(fool_elimination,[status(thm)],[f649])).
% 3.24/1.85 thf(f649,plain,(
% 3.24/1.85 (ord_le1415039317at_nat @ (collect_nat_nat @ (^[X0 : nat > nat] : (number1551313001itions @ X0 @ n))) @ (collect_nat_nat @ (^[X1 : nat > nat] : (! [X2 : nat] : (ord_less_eq_nat @ (X1 @ X2) @ n) & ! [X3 : nat] : ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ X3) => (zero_zero_nat = (X1 @ X3)))))))),
% 3.24/1.85 inference(rectify,[status(thm)],[f2])).
% 3.24/1.85 thf(f2,axiom,(
% 3.24/1.85 (ord_le1415039317at_nat @ (collect_nat_nat @ (^[X2 : nat > nat] : (number1551313001itions @ X2 @ n))) @ (collect_nat_nat @ (^[X0 : nat > nat] : (! [X1 : nat] : (ord_less_eq_nat @ (X0 @ X1) @ n) & ! [X1 : nat] : ((ord_less_eq_nat @ (plus_plus_nat @ n @ one_one_nat) @ X1) => ((X0 @ X1) = zero_zero_nat))))))),
% 3.24/1.85 file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062)).
% 3.24/1.85 thf(f1057,plain,(
% 3.24/1.85 ( ! [X0 : set_nat_nat,X1 : set_nat_nat] : (((ord_le1415039317at_nat @ X1 @ X0) != $true) | ((finite570312790at_nat @ X1) = $true) | ($true != (finite570312790at_nat @ X0))) )),
% 3.24/1.85 inference(cnf_transformation,[status(thm)],[f1003])).
% 3.24/1.85 thf(f1003,plain,(
% 3.24/1.85 ! [X0 : set_nat_nat,X1 : set_nat_nat] : (($true != (finite570312790at_nat @ X0)) | ((ord_le1415039317at_nat @ X1 @ X0) != $true) | ((finite570312790at_nat @ X1) = $true))),
% 3.24/1.85 inference(flattening,[status(thm)],[f1002])).
% 3.24/1.85 thf(f1002,plain,(
% 3.24/1.85 ! [X0 : set_nat_nat,X1 : set_nat_nat] : ((($true != (finite570312790at_nat @ X0)) | ((finite570312790at_nat @ X1) = $true)) | ((ord_le1415039317at_nat @ X1 @ X0) != $true))),
% 3.24/1.85 inference(ennf_transformation,[status(thm)],[f977])).
% 3.24/1.85 thf(f977,plain,(
% 3.24/1.85 ! [X0 : set_nat_nat,X1 : set_nat_nat] : (((ord_le1415039317at_nat @ X1 @ X0) = $true) => (((finite570312790at_nat @ X1) != $true) => ($true != (finite570312790at_nat @ X0))))),
% 3.24/1.85 inference(flattening,[status(thm)],[f612])).
% 3.24/1.85 thf(f612,plain,(
% 3.24/1.85 ! [X0 : set_nat_nat,X1 : set_nat_nat] : (((ord_le1415039317at_nat @ X1 @ X0) = $true) => (~((finite570312790at_nat @ X1) = $true) => ~($true = (finite570312790at_nat @ X0))))),
% 3.24/1.85 inference(fool_elimination,[status(thm)],[f611])).
% 3.24/1.85 thf(f611,plain,(
% 3.24/1.85 ! [X0 : set_nat_nat,X1 : set_nat_nat] : ((ord_le1415039317at_nat @ X1 @ X0) => (~(finite570312790at_nat @ X1) => ~(finite570312790at_nat @ X0)))),
% 3.24/1.85 inference(rectify,[status(thm)],[f68])).
% 3.24/1.85 thf(f68,axiom,(
% 3.24/1.85 ! [X23 : set_nat_nat,X22 : set_nat_nat] : ((ord_le1415039317at_nat @ X22 @ X23) => (~(finite570312790at_nat @ X22) => ~(finite570312790at_nat @ X23)))),
% 3.24/1.85 file('/export/starexec/sandbox/tmp/tmp.RaUGvRk2Y3/DTF2THF_31226.p',fact_67_infinite__super)).
% 3.24/1.85 % SZS output end Proof for DTF2THF_31226
% 3.24/1.85 % (31486)------------------------------
% 3.24/1.85 % (31486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.85 % (31486)Termination reason: Refutation
% 3.24/1.85
% 3.24/1.85 % (31486)Memory used [KB]: 7164
% 3.24/1.85 % (31486)Time elapsed: 0.067 s
% 3.24/1.85 % (31486)Instructions burned: 142 (million)
% 3.24/1.85 % (31486)------------------------------
% 3.24/1.85 % (31486)------------------------------
% 3.24/1.85 % (31431)Success in time 0.206 s
%------------------------------------------------------------------------------