%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SYO868^1 : 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 : n003.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 09:54:57 AM UTC 2026
% Result : Theorem 2.45s 1.56s
% Output : Refutation 2.45s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 6
% Syntax : Number of formulae : 25 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 732 ( 160 equ; 0 cnn)
% Maximal formula atoms : 27 ( 29 avg)
% Number of connectives : 929 ( 72 ~; 55 |; 65 &; 648 @)
% ( 0 <=>; 78 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 11 ( 11 >; 0 *; 0 +; 0 <<)
% Number of symbols : 113 ( 109 usr; 7 con; 0-14 aty)
% ( 11 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 83 ( 17 ^; 38 !; 28 ?; 83 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
c_Eps_i: ( $i > $o ) > $i ).
thf(func_def_3,type,
c_not: $o > $o ).
thf(func_def_4,type,
c_and: $o > $o > $o ).
thf(func_def_5,type,
c_or: $o > $o > $o ).
thf(func_def_6,type,
c_iff: $o > $o > $o ).
thf(func_def_7,type,
c_In: $i > $i > $o ).
thf(func_def_8,type,
c_Subq: $i > $i > $o ).
thf(func_def_10,type,
c_Union: $i > $i ).
thf(func_def_11,type,
c_Power: $i > $i ).
thf(func_def_12,type,
c_Repl: $i > ( $i > $i ) > $i ).
thf(func_def_13,type,
c_TransSet: $i > $o ).
thf(func_def_14,type,
c_atleast2: $i > $o ).
thf(func_def_15,type,
c_atleast3: $i > $o ).
thf(func_def_16,type,
c_atleast4: $i > $o ).
thf(func_def_17,type,
c_atleast5: $i > $o ).
thf(func_def_18,type,
c_atleast6: $i > $o ).
thf(func_def_19,type,
c_exactly2: $i > $o ).
thf(func_def_20,type,
c_exactly3: $i > $o ).
thf(func_def_21,type,
c_exactly4: $i > $o ).
thf(func_def_22,type,
c_exactly5: $i > $o ).
thf(func_def_23,type,
c_exu_i: ( $i > $o ) > $o ).
thf(func_def_24,type,
c_reflexive_i: ( $i > $i > $o ) > $o ).
thf(func_def_25,type,
c_irreflexive_i: ( $i > $i > $o ) > $o ).
thf(func_def_26,type,
c_symmetric_i: ( $i > $i > $o ) > $o ).
thf(func_def_27,type,
c_antisymmetric_i: ( $i > $i > $o ) > $o ).
thf(func_def_28,type,
c_transitive_i: ( $i > $i > $o ) > $o ).
thf(func_def_29,type,
c_eqreln_i: ( $i > $i > $o ) > $o ).
thf(func_def_30,type,
c_per_i: ( $i > $i > $o ) > $o ).
thf(func_def_31,type,
c_linear_i: ( $i > $i > $o ) > $o ).
thf(func_def_32,type,
c_trichotomous_or_i: ( $i > $i > $o ) > $o ).
thf(func_def_33,type,
c_partialorder_i: ( $i > $i > $o ) > $o ).
thf(func_def_34,type,
c_totalorder_i: ( $i > $i > $o ) > $o ).
thf(func_def_35,type,
c_strictpartialorder_i: ( $i > $i > $o ) > $o ).
thf(func_def_36,type,
c_stricttotalorder_i: ( $i > $i > $o ) > $o ).
thf(func_def_37,type,
c_If_i: $o > $i > $i > $i ).
thf(func_def_38,type,
c_exactly1of2: $o > $o > $o ).
thf(func_def_39,type,
c_exactly1of3: $o > $o > $o > $o ).
thf(func_def_40,type,
c_nIn: $i > $i > $o ).
thf(func_def_41,type,
c_nSubq: $i > $i > $o ).
thf(func_def_42,type,
c_UPair: $i > $i > $i ).
thf(func_def_43,type,
c_Sing: $i > $i ).
thf(func_def_44,type,
c_binunion: $i > $i > $i ).
thf(func_def_45,type,
c_SetAdjoin: $i > $i > $i ).
thf(func_def_46,type,
c_famunion: $i > ( $i > $i ) > $i ).
thf(func_def_47,type,
c_Sep: $i > ( $i > $o ) > $i ).
thf(func_def_48,type,
c_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
thf(func_def_49,type,
c_binintersect: $i > $i > $i ).
thf(func_def_50,type,
c_setminus: $i > $i > $i ).
thf(func_def_51,type,
c_inj: $i > $i > ( $i > $i ) > $o ).
thf(func_def_52,type,
c_bij: $i > $i > ( $i > $i ) > $o ).
thf(func_def_53,type,
c_atleastp: $i > $i > $o ).
thf(func_def_54,type,
c_equip: $i > $i > $o ).
thf(func_def_55,type,
c_In_rec_poly_G_i: ( $i > ( $i > $i ) > $i ) > $i > $i > $o ).
thf(func_def_56,type,
c_In_rec_poly_i: ( $i > ( $i > $i ) > $i ) > $i > $i ).
thf(func_def_57,type,
c_ordsucc: $i > $i ).
thf(func_def_58,type,
c_nat_p: $i > $o ).
thf(func_def_59,type,
c_nat_primrec: $i > ( $i > $i > $i ) > $i > $i ).
thf(func_def_60,type,
c_add_nat: $i > $i > $i ).
thf(func_def_61,type,
c_mul_nat: $i > $i > $i ).
thf(func_def_62,type,
c_ordinal: $i > $o ).
thf(func_def_63,type,
c_V_: $i > $i ).
thf(func_def_64,type,
c_Inj1: $i > $i ).
thf(func_def_65,type,
c_Inj0: $i > $i ).
thf(func_def_66,type,
c_Unj: $i > $i ).
thf(func_def_67,type,
c_combine_funcs: $i > $i > ( $i > $i ) > ( $i > $i ) > $i > $i ).
thf(func_def_68,type,
c_setsum: $i > $i > $i ).
thf(func_def_69,type,
c_proj0: $i > $i ).
thf(func_def_70,type,
c_proj1: $i > $i ).
thf(func_def_71,type,
c_binrep: $i > $i > $i ).
thf(func_def_72,type,
c_lam: $i > ( $i > $i ) > $i ).
thf(func_def_73,type,
c_setprod: $i > $i > $i ).
thf(func_def_74,type,
c_ap: $i > $i > $i ).
thf(func_def_75,type,
c_setsum_p: $i > $o ).
thf(func_def_76,type,
c_tuple_p: $i > $i > $o ).
thf(func_def_77,type,
c_Pi: $i > ( $i > $i ) > $i ).
thf(func_def_78,type,
c_setexp: $i > $i > $i ).
thf(func_def_79,type,
c_Sep2: $i > ( $i > $i ) > ( $i > $i > $o ) > $i ).
thf(func_def_80,type,
c_set_of_pairs: $i > $o ).
thf(func_def_81,type,
c_lam2: $i > ( $i > $i ) > ( $i > $i > $i ) > $i ).
thf(func_def_82,type,
c_PNoEq_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
thf(func_def_83,type,
c_PNoLt_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
thf(func_def_84,type,
c_PNoLt: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
thf(func_def_85,type,
c_PNoLe: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
thf(func_def_86,type,
c_PNo_downc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
thf(func_def_87,type,
c_PNo_upc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
thf(func_def_88,type,
c_SNoElts_: $i > $i ).
thf(func_def_89,type,
c_SNo_: $i > $i > $o ).
thf(func_def_90,type,
c_PSNo: $i > ( $i > $o ) > $i ).
thf(func_def_91,type,
c_SNo: $i > $o ).
thf(func_def_92,type,
c_SNoLev: $i > $i ).
thf(func_def_93,type,
c_SNoEq_: $i > $i > $i > $o ).
thf(func_def_94,type,
c_SNoLt: $i > $i > $o ).
thf(func_def_95,type,
c_SNoLe: $i > $i > $o ).
thf(func_def_96,type,
c_binop_on: $i > ( $i > $i > $i ) > $o ).
thf(func_def_97,type,
c_Loop: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > $o ).
thf(func_def_98,type,
c_Loop_with_defs: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
thf(func_def_99,type,
c_Loop_with_defs_cex1: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
thf(func_def_100,type,
c_Loop_with_defs_cex2: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
thf(func_def_101,type,
c_combinator: $i > $o ).
thf(func_def_102,type,
c_combinator_equiv: $i > $i > $o ).
thf(func_def_103,type,
c_equip_mod: $i > $i > $i > $o ).
thf(func_def_134,type,
sK3: $i > $i ).
thf(func_def_135,type,
sK4: ( $i > $o ) > $i ).
thf(func_def_137,type,
sK6: $i > $i > $o ).
thf(func_def_139,type,
ph7:
!>[X0: $tType] : X0 ).
thf(f421,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f418]) ).
thf(f418,plain,
$false = $true,
inference(superposition,[status(thm)],[f370,f371]) ).
thf(f371,plain,
! [X3: $i] :
( $false
= ( sK6 @ X3 @ X3 ) ),
inference(binary_proxy_clausification,[status(thm)],[f369]) ).
thf(f369,plain,
! [X3: $i] :
( $false
= ( ( !! @ $i
@ ^ [Y0: $i] :
( ( sK6 @ X3 @ Y0 )
=> ( sK6 @ X3 @ ( c_ordsucc @ Y0 ) ) ) )
=> ( sK6 @ X3 @ X3 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f368]) ).
thf(f368,plain,
! [X3: $i] :
( $false
= ( ( sK6 @ X3 @ c_Empty )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( sK6 @ X3 @ Y0 )
=> ( sK6 @ X3 @ ( c_ordsucc @ Y0 ) ) ) )
=> ( sK6 @ X3 @ X3 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f367]) ).
thf(f367,plain,
! [X3: $i] :
( $false
= ( ^ [Y0: $i > $o] :
( ( Y0 @ c_Empty )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( c_ordsucc @ Y1 ) ) ) )
=> ( Y0 @ X3 ) ) )
@ ( sK6 @ X3 ) ) ),
inference(sigma_clausification,[status(thm)],[f366]) ).
thf(f366,plain,
! [X3: $i] :
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( Y0 @ c_Empty )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( c_ordsucc @ Y1 ) ) ) )
=> ( Y0 @ X3 ) ) ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f354]) ).
thf(f354,plain,
! [X3: $i] :
( ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( ( Y1 @ c_Empty )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( c_ordsucc @ Y2 ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
@ X3 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f308,f325]) ).
thf(f325,plain,
( c_nat_p
= ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( ( Y1 @ c_Empty )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( c_ordsucc @ Y2 ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f225]) ).
thf(f225,plain,
( c_nat_p
= ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( ( Y1 @ c_Empty )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( c_ordsucc @ Y2 ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f224]) ).
thf(f224,plain,
( c_nat_p
= ( ^ [X0: $i] :
! [X1: $i > $o] :
( ( X1 @ c_Empty )
=> ( ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( c_ordsucc @ X2 ) ) )
=> ( X1 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f63]) ).
thf(f63,axiom,
( c_nat_p
= ( ^ [X0: $i] :
! [X1: $i > $o] :
( ( X1 @ c_Empty )
=> ( ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( c_ordsucc @ X2 ) ) )
=> ( X1 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.meWfZCKs7P/DTF2THF_9274.p',ax63) ).
thf(f308,plain,
! [X3: $i] :
( ( c_nat_p @ X3 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f296]) ).
thf(f296,plain,
( ( ( c_In @ sK0 @ c_Empty )
= $true )
& ! [X3: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ ( sK3 @ X3 ) ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ sK2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ sK2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ ( sK3 @ X3 ) ) ) ) )
= $true )
| ( ( ( ( sK3 @ X3 )
= X3 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( ( c_In @ sK2 @ sK1 )
= $true )
& ( ( c_In @ sK1 @ sK0 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f291,f295,f294,f293,f292]) ).
thf(f292,plain,
( ? [X0: $i] :
( ( ( c_In @ X0 @ c_Empty )
= $true )
& ? [X1: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( ( c_In @ X2 @ X1 )
= $true ) )
& ( ( c_In @ X1 @ X0 )
= $true ) ) )
=> ( ( ( c_In @ sK0 @ c_Empty )
= $true )
& ? [X1: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( ( c_In @ X2 @ X1 )
= $true ) )
& ( ( c_In @ X1 @ sK0 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f293,plain,
( ? [X1: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( ( c_In @ X2 @ X1 )
= $true ) )
& ( ( c_In @ X1 @ sK0 )
= $true ) )
=> ( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( $true
= ( c_In @ X2 @ sK1 ) ) )
& ( ( c_In @ sK1 @ sK0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f294,plain,
( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( $true
= ( c_In @ X2 @ sK1 ) ) )
=> ( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( $true
= ( c_and @ ( c_atleast5 @ sK2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ sK2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( ( c_In @ sK2 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f295,plain,
! [X3: $i] :
( ? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( $true
= ( c_and @ ( c_atleast5 @ sK2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ sK2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
=> ( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ ( sK3 @ X3 ) ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ sK2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ sK2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ ( sK3 @ X3 ) ) ) ) )
= $true )
| ( ( ( ( sK3 @ X3 )
= X3 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f291,plain,
? [X0: $i] :
( ( ( c_In @ X0 @ c_Empty )
= $true )
& ? [X1: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
& ( ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) )
| ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) ) )
& ( ( c_nat_p @ X3 )
!= $true ) )
& ( ( c_In @ X2 @ X1 )
= $true ) )
& ( ( c_In @ X1 @ X0 )
= $true ) ) ),
inference(flattening,[status(thm)],[f290]) ).
thf(f290,plain,
? [X0: $i] :
( ? [X1: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
? [X4: $i] :
( ( ( c_nat_p @ X3 )
!= $true )
& ( ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true )
| ( ( ( X3 = X4 )
| ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
!= $true ) )
& ( ( c_atleast3 @ X3 )
!= $true ) )
| ( ( c_atleast2 @ X3 )
!= $true )
| ( ( ( c_ordinal @ X3 )
= $true )
& ( ( c_exactly2 @ X3 )
!= $true ) ) )
& ( ( ( c_ordinal @ X3 )
!= $true )
| ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) ) )
& ( ( c_In @ X2 @ X1 )
= $true ) )
& ( ( c_In @ X1 @ X0 )
= $true ) )
& ( ( c_In @ X0 @ c_Empty )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f262]) ).
thf(f262,plain,
~ ! [X0: $i] :
( ( ( c_In @ X0 @ c_Empty )
= $true )
=> ! [X1: $i] :
( ( ( c_In @ X1 @ X0 )
= $true )
=> ! [X2: $i] :
( ( ( c_In @ X2 @ X1 )
= $true )
=> ? [X3: $i] :
! [X4: $i] :
( ( ( ( c_ordinal @ X3 )
= $true )
=> ( ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) )
= $true ) )
=> ( ( ( ( ( c_ordinal @ X3 )
= $true )
=> ( ( c_exactly2 @ X3 )
= $true ) )
=> ( ( ( c_atleast2 @ X3 )
= $true )
=> ( ( ( ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
= $true )
=> ( X3 = X4 ) )
=> ( ( c_atleast3 @ X3 )
= $true ) )
=> ( ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) )
= $true ) ) ) )
=> ( ( c_nat_p @ X3 )
= $true ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f261]) ).
thf(f261,plain,
~ ! [X0: $i] :
( ( c_In @ X0 @ c_Empty )
=> ! [X1: $i] :
( ( c_In @ X1 @ X0 )
=> ! [X2: $i] :
( ( c_In @ X2 @ X1 )
=> ? [X3: $i] :
! [X4: $i] :
( ( ( c_ordinal @ X3 )
=> ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) ) )
=> ( ( ( ( c_ordinal @ X3 )
=> ( c_exactly2 @ X3 ) )
=> ( ( c_atleast2 @ X3 )
=> ( ( ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
=> ( X3 = X4 ) )
=> ( c_atleast3 @ X3 ) )
=> ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) ) ) )
=> ( c_nat_p @ X3 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f110]) ).
thf(f110,negated_conjecture,
~ ! [X0: $i] :
( ( c_In @ X0 @ c_Empty )
=> ! [X1: $i] :
( ( c_In @ X1 @ X0 )
=> ! [X2: $i] :
( ( c_In @ X2 @ X1 )
=> ? [X3: $i] :
! [X4: $i] :
( ( ( c_ordinal @ X3 )
=> ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) ) )
=> ( ( ( ( c_ordinal @ X3 )
=> ( c_exactly2 @ X3 ) )
=> ( ( c_atleast2 @ X3 )
=> ( ( ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
=> ( X3 = X4 ) )
=> ( c_atleast3 @ X3 ) )
=> ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) ) ) )
=> ( c_nat_p @ X3 ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f109]) ).
thf(f109,conjecture,
! [X0: $i] :
( ( c_In @ X0 @ c_Empty )
=> ! [X1: $i] :
( ( c_In @ X1 @ X0 )
=> ! [X2: $i] :
( ( c_In @ X2 @ X1 )
=> ? [X3: $i] :
! [X4: $i] :
( ( ( c_ordinal @ X3 )
=> ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) ) )
=> ( ( ( ( c_ordinal @ X3 )
=> ( c_exactly2 @ X3 ) )
=> ( ( c_atleast2 @ X3 )
=> ( ( ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
=> ( X3 = X4 ) )
=> ( c_atleast3 @ X3 ) )
=> ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) ) ) )
=> ( c_nat_p @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.meWfZCKs7P/DTF2THF_9274.p',conj) ).
thf(f370,plain,
! [X3: $i] :
( ( sK6 @ X3 @ c_Empty )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f368]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SYO868^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n003.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 14:49:10 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.45 ---- Original DTF file ---
% 0.28/0.45 thf(spec,logic,$$dhol).
% 0.28/0.45 %------------------------------------------------------------------------------
% 0.28/0.45 % File : SYO868^1 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.45 % Domain : Syntactic
% 0.28/0.45 % Problem : ProofGold problem Random1_197_pos_th0
% 0.28/0.45 % Version : Especial.
% 0.28/0.45 % English :
% 0.28/0.45
% 0.28/0.45 % Refs : [Urb20] Urban (2020) Email to Geoff Sutcliffe
% 0.28/0.45 % Source : [Urb20]
% 0.28/0.45 % Names : Random1_197_pos_th0.p [Urb20]
% 0.28/0.45
% 0.28/0.45 % Status : Theorem
% 0.28/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% 0.28/0.45 % Syntax : Number of formulae : 213 ( 99 unt; 104 typ; 0 def)
% 0.28/0.45 % Number of atoms : 501 ( 136 equ; 0 cnn)
% 0.28/0.45 % Maximal formula atoms : 22 ( 4 avg)
% 0.28/0.45 % Number of connectives : 1173 ( 0 ~; 0 |; 0 &;1072 @)
% 0.28/0.45 % ( 0 <=>; 101 =>; 0 <=; 0 <~>)
% 0.28/0.45 % Maximal formula depth : 25 ( 2 avg)
% 0.28/0.45 % Number of types : 2 ( 0 usr)
% 0.28/0.45 % Number of type conns : 557 ( 557 >; 0 *; 0 +; 0 <<)
% 0.28/0.45 % Number of symbols : 105 ( 104 usr; 3 con; 0-14 aty)
% 0.28/0.45 % Number of variables : 410 ( 275 ^; 97 !; 38 ?; 410 :)
% 0.28/0.45 % SPC : TH0_THM_EQU_NAR
% 0.28/0.45
% 0.28/0.45 % Comments : See https://proofgold.org
% 0.28/0.45 %------------------------------------------------------------------------------
% 0.28/0.45 thf(c_Eps_i_tp,type,
% 0.28/0.45 c_Eps_i: ( $i > $o ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_False_tp,type,
% 0.28/0.45 c_False: $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_True_tp,type,
% 0.28/0.45 c_True: $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_not_tp,type,
% 0.28/0.45 c_not: $o > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_and_tp,type,
% 0.28/0.45 c_and: $o > $o > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_or_tp,type,
% 0.28/0.45 c_or: $o > $o > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_iff_tp,type,
% 0.28/0.45 c_iff: $o > $o > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_In_tp,type,
% 0.28/0.45 c_In: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Subq_tp,type,
% 0.28/0.45 c_Subq: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Empty_tp,type,
% 0.28/0.45 c_Empty: $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Union_tp,type,
% 0.28/0.45 c_Union: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Power_tp,type,
% 0.28/0.45 c_Power: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Repl_tp,type,
% 0.28/0.45 c_Repl: $i > ( $i > $i ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_TransSet_tp,type,
% 0.28/0.45 c_TransSet: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_atleast2_tp,type,
% 0.28/0.45 c_atleast2: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_atleast3_tp,type,
% 0.28/0.45 c_atleast3: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_atleast4_tp,type,
% 0.28/0.45 c_atleast4: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_atleast5_tp,type,
% 0.28/0.45 c_atleast5: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_atleast6_tp,type,
% 0.28/0.45 c_atleast6: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_exactly2_tp,type,
% 0.28/0.45 c_exactly2: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_exactly3_tp,type,
% 0.28/0.45 c_exactly3: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_exactly4_tp,type,
% 0.28/0.45 c_exactly4: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_exactly5_tp,type,
% 0.28/0.45 c_exactly5: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_exu_i_tp,type,
% 0.28/0.45 c_exu_i: ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_reflexive_i_tp,type,
% 0.28/0.45 c_reflexive_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_irreflexive_i_tp,type,
% 0.28/0.45 c_irreflexive_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_symmetric_i_tp,type,
% 0.28/0.45 c_symmetric_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_antisymmetric_i_tp,type,
% 0.28/0.45 c_antisymmetric_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_transitive_i_tp,type,
% 0.28/0.45 c_transitive_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_eqreln_i_tp,type,
% 0.28/0.45 c_eqreln_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_per_i_tp,type,
% 0.28/0.45 c_per_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_linear_i_tp,type,
% 0.28/0.45 c_linear_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_trichotomous_or_i_tp,type,
% 0.28/0.45 c_trichotomous_or_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_partialorder_i_tp,type,
% 0.28/0.45 c_partialorder_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_totalorder_i_tp,type,
% 0.28/0.45 c_totalorder_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_strictpartialorder_i_tp,type,
% 0.28/0.45 c_strictpartialorder_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_stricttotalorder_i_tp,type,
% 0.28/0.45 c_stricttotalorder_i: ( $i > $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_If_i_tp,type,
% 0.28/0.45 c_If_i: $o > $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_exactly1of2_tp,type,
% 0.28/0.45 c_exactly1of2: $o > $o > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_exactly1of3_tp,type,
% 0.28/0.45 c_exactly1of3: $o > $o > $o > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_nIn_tp,type,
% 0.28/0.45 c_nIn: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_nSubq_tp,type,
% 0.28/0.45 c_nSubq: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_UPair_tp,type,
% 0.28/0.45 c_UPair: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Sing_tp,type,
% 0.28/0.45 c_Sing: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_binunion_tp,type,
% 0.28/0.45 c_binunion: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_SetAdjoin_tp,type,
% 0.28/0.45 c_SetAdjoin: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_famunion_tp,type,
% 0.28/0.45 c_famunion: $i > ( $i > $i ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Sep_tp,type,
% 0.28/0.45 c_Sep: $i > ( $i > $o ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_ReplSep_tp,type,
% 0.28/0.45 c_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_binintersect_tp,type,
% 0.28/0.45 c_binintersect: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_setminus_tp,type,
% 0.28/0.45 c_setminus: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_inj_tp,type,
% 0.28/0.45 c_inj: $i > $i > ( $i > $i ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_bij_tp,type,
% 0.28/0.45 c_bij: $i > $i > ( $i > $i ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_atleastp_tp,type,
% 0.28/0.45 c_atleastp: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_equip_tp,type,
% 0.28/0.45 c_equip: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_In_rec_poly_G_i_tp,type,
% 0.28/0.45 c_In_rec_poly_G_i: ( $i > ( $i > $i ) > $i ) > $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_In_rec_poly_i_tp,type,
% 0.28/0.45 c_In_rec_poly_i: ( $i > ( $i > $i ) > $i ) > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_ordsucc_tp,type,
% 0.28/0.45 c_ordsucc: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_nat_p_tp,type,
% 0.28/0.45 c_nat_p: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_nat_primrec_tp,type,
% 0.28/0.45 c_nat_primrec: $i > ( $i > $i > $i ) > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_add_nat_tp,type,
% 0.28/0.45 c_add_nat: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_mul_nat_tp,type,
% 0.28/0.45 c_mul_nat: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_ordinal_tp,type,
% 0.28/0.45 c_ordinal: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_V__tp,type,
% 0.28/0.45 c_V_: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Inj1_tp,type,
% 0.28/0.45 c_Inj1: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Inj0_tp,type,
% 0.28/0.45 c_Inj0: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Unj_tp,type,
% 0.28/0.45 c_Unj: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_combine_funcs_tp,type,
% 0.28/0.45 c_combine_funcs: $i > $i > ( $i > $i ) > ( $i > $i ) > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_setsum_tp,type,
% 0.28/0.45 c_setsum: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_proj0_tp,type,
% 0.28/0.45 c_proj0: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_proj1_tp,type,
% 0.28/0.45 c_proj1: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_binrep_tp,type,
% 0.28/0.45 c_binrep: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_lam_tp,type,
% 0.28/0.45 c_lam: $i > ( $i > $i ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_setprod_tp,type,
% 0.28/0.45 c_setprod: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_ap_tp,type,
% 0.28/0.45 c_ap: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_setsum_p_tp,type,
% 0.28/0.45 c_setsum_p: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_tuple_p_tp,type,
% 0.28/0.45 c_tuple_p: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Pi_tp,type,
% 0.28/0.45 c_Pi: $i > ( $i > $i ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_setexp_tp,type,
% 0.28/0.45 c_setexp: $i > $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_Sep2_tp,type,
% 0.28/0.45 c_Sep2: $i > ( $i > $i ) > ( $i > $i > $o ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_set_of_pairs_tp,type,
% 0.28/0.45 c_set_of_pairs: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_lam2_tp,type,
% 0.28/0.45 c_lam2: $i > ( $i > $i ) > ( $i > $i > $i ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_PNoEq__tp,type,
% 0.28/0.45 c_PNoEq_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_PNoLt__tp,type,
% 0.28/0.45 c_PNoLt_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_PNoLt_tp,type,
% 0.28/0.45 c_PNoLt: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_PNoLe_tp,type,
% 0.28/0.45 c_PNoLe: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_PNo_downc_tp,type,
% 0.28/0.45 c_PNo_downc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_PNo_upc_tp,type,
% 0.28/0.45 c_PNo_upc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNoElts__tp,type,
% 0.28/0.45 c_SNoElts_: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNo__tp,type,
% 0.28/0.45 c_SNo_: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_PSNo_tp,type,
% 0.28/0.45 c_PSNo: $i > ( $i > $o ) > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNo_tp,type,
% 0.28/0.45 c_SNo: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNoLev_tp,type,
% 0.28/0.45 c_SNoLev: $i > $i ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNoEq__tp,type,
% 0.28/0.45 c_SNoEq_: $i > $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNoLt_tp,type,
% 0.28/0.45 c_SNoLt: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_SNoLe_tp,type,
% 0.28/0.45 c_SNoLe: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_binop_on_tp,type,
% 0.28/0.45 c_binop_on: $i > ( $i > $i > $i ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Loop_tp,type,
% 0.28/0.45 c_Loop: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Loop_with_defs_tp,type,
% 0.28/0.45 c_Loop_with_defs: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Loop_with_defs_cex1_tp,type,
% 0.28/0.45 c_Loop_with_defs_cex1: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_Loop_with_defs_cex2_tp,type,
% 0.28/0.45 c_Loop_with_defs_cex2: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_combinator_tp,type,
% 0.28/0.45 c_combinator: $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_combinator_equiv_tp,type,
% 0.28/0.45 c_combinator_equiv: $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(c_equip_mod_tp,type,
% 0.28/0.45 c_equip_mod: $i > $i > $i > $o ).
% 0.28/0.45
% 0.28/0.45 thf(ax1,axiom,
% 0.28/0.45 ! [X0: $i > $o,X1: $i] :
% 0.28/0.45 ( ( X0 @ X1 )
% 0.28/0.45 => ( X0 @ ( c_Eps_i @ X0 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax2,axiom,
% 0.28/0.45 ! [X0: $o] :
% 0.28/0.45 ( ( c_not @ ( c_not @ X0 ) )
% 0.28/0.45 => X0 ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax3,axiom,
% 0.28/0.45 ! [X0: $o,X1: $o] :
% 0.28/0.45 ( ( c_iff @ X0 @ X1 )
% 0.28/0.45 => ( X0 = X1 ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax4,axiom,
% 0.28/0.45 ! [X0: $i,X1: $i] :
% 0.28/0.45 ( ( c_Subq @ X0 @ X1 )
% 0.28/0.45 => ( ( c_Subq @ X1 @ X0 )
% 0.28/0.45 => ( X0 = X1 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax5,axiom,
% 0.28/0.45 ( c_not
% 0.28/0.45 @ ? [X0: $i] : ( c_In @ X0 @ c_Empty ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax6,axiom,
% 0.28/0.45 ! [X0: $i,X1: $i] :
% 0.28/0.45 ( c_iff @ ( c_In @ X1 @ ( c_Union @ X0 ) )
% 0.28/0.45 @ ? [X2: $i] : ( c_and @ ( c_In @ X1 @ X2 ) @ ( c_In @ X2 @ X0 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax7,axiom,
% 0.28/0.45 ! [X0: $i,X1: $i] : ( c_iff @ ( c_In @ X1 @ ( c_Power @ X0 ) ) @ ( c_Subq @ X1 @ X0 ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax8,axiom,
% 0.28/0.45 ! [X0: $i,X1: $i > $i,X2: $i] :
% 0.28/0.45 ( c_iff
% 0.28/0.45 @ ( c_In @ X2
% 0.28/0.45 @ ( c_Repl @ X0
% 0.28/0.45 @ ^ [X3: $i] : ( X1 @ X3 ) ) )
% 0.28/0.45 @ ? [X3: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X3 @ X0 )
% 0.28/0.45 @ ( X2
% 0.28/0.45 = ( X1 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax9,axiom,
% 0.28/0.45 ! [X0: $i > $o] :
% 0.28/0.45 ( ! [X1: $i] :
% 0.28/0.45 ( ( X0 @ X1 )
% 0.28/0.45 => ! [X2: $i] :
% 0.28/0.45 ( ( c_In @ X2 @ X1 )
% 0.28/0.45 => ( X0 @ X2 ) ) )
% 0.28/0.45 => ( ( X0 @ c_Empty )
% 0.28/0.45 => ( ! [X1: $i] :
% 0.28/0.45 ( ( X0 @ X1 )
% 0.28/0.45 => ( X0 @ ( c_Union @ X1 ) ) )
% 0.28/0.45 => ( ! [X1: $i] :
% 0.28/0.45 ( ( X0 @ X1 )
% 0.28/0.45 => ( X0 @ ( c_Power @ X1 ) ) )
% 0.28/0.45 => ( ! [X1: $i] :
% 0.28/0.45 ( ( X0 @ X1 )
% 0.28/0.45 => ! [X2: $i > $i] :
% 0.28/0.45 ( ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X1 )
% 0.28/0.45 => ( X0 @ ( X2 @ X3 ) ) )
% 0.28/0.45 => ( X0
% 0.28/0.45 @ ( c_Repl @ X1
% 0.28/0.45 @ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) )
% 0.28/0.45 => ! [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax10,axiom,
% 0.28/0.45 ! [X0: $i > $o] :
% 0.28/0.45 ( ! [X1: $i] :
% 0.28/0.45 ( ! [X2: $i] :
% 0.28/0.45 ( ( c_In @ X2 @ X1 )
% 0.28/0.45 => ( X0 @ X2 ) )
% 0.28/0.45 => ( X0 @ X1 ) )
% 0.28/0.45 => ! [X1: $i] : ( X0 @ X1 ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax11,axiom,
% 0.28/0.45 ( c_False
% 0.28/0.45 = ( ! [X0: $o] : X0 ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax12,axiom,
% 0.28/0.45 ( c_True
% 0.28/0.45 = ( ! [X0: $o] :
% 0.28/0.45 ( X0
% 0.28/0.45 => X0 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax13,axiom,
% 0.28/0.45 ( c_not
% 0.28/0.45 = ( ^ [X0: $o] :
% 0.28/0.45 ( X0
% 0.28/0.45 => c_False ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax14,axiom,
% 0.28/0.45 ( c_and
% 0.28/0.45 = ( ^ [X0: $o,X1: $o] :
% 0.28/0.45 ! [X2: $o] :
% 0.28/0.45 ( ( X0
% 0.28/0.45 => ( X1
% 0.28/0.45 => X2 ) )
% 0.28/0.45 => X2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax15,axiom,
% 0.28/0.45 ( c_or
% 0.28/0.45 = ( ^ [X0: $o,X1: $o] :
% 0.28/0.45 ! [X2: $o] :
% 0.28/0.45 ( ( X0
% 0.28/0.45 => X2 )
% 0.28/0.45 => ( ( X1
% 0.28/0.45 => X2 )
% 0.28/0.45 => X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax16,axiom,
% 0.28/0.45 ( c_iff
% 0.28/0.45 = ( ^ [X0: $o,X1: $o] :
% 0.28/0.45 ( c_and
% 0.28/0.45 @ ( X0
% 0.28/0.45 => X1 )
% 0.28/0.45 @ ( X1
% 0.28/0.45 => X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax17,axiom,
% 0.28/0.45 ( c_Subq
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ! [X2: $i] :
% 0.28/0.45 ( ( c_In @ X2 @ X0 )
% 0.28/0.45 => ( c_In @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax18,axiom,
% 0.28/0.45 ( c_TransSet
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ! [X1: $i] :
% 0.28/0.45 ( ( c_In @ X1 @ X0 )
% 0.28/0.45 => ( c_Subq @ X1 @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax19,axiom,
% 0.28/0.45 ( c_atleast2
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ? [X1: $i] : ( c_and @ ( c_In @ X1 @ X0 ) @ ( c_not @ ( c_Subq @ X0 @ ( c_Power @ X1 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax20,axiom,
% 0.28/0.45 ( c_atleast3
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast2 @ X1 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax21,axiom,
% 0.28/0.45 ( c_atleast4
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast3 @ X1 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax22,axiom,
% 0.28/0.45 ( c_atleast5
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast4 @ X1 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax23,axiom,
% 0.28/0.45 ( c_atleast6
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast5 @ X1 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax24,axiom,
% 0.28/0.45 ( c_exactly2
% 0.28/0.45 = ( ^ [X0: $i] : ( c_and @ ( c_atleast2 @ X0 ) @ ( c_not @ ( c_atleast3 @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax25,axiom,
% 0.28/0.45 ( c_exactly3
% 0.28/0.45 = ( ^ [X0: $i] : ( c_and @ ( c_atleast3 @ X0 ) @ ( c_not @ ( c_atleast4 @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax26,axiom,
% 0.28/0.45 ( c_exactly4
% 0.28/0.45 = ( ^ [X0: $i] : ( c_and @ ( c_atleast4 @ X0 ) @ ( c_not @ ( c_atleast5 @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax27,axiom,
% 0.28/0.45 ( c_exactly5
% 0.28/0.45 = ( ^ [X0: $i] : ( c_and @ ( c_atleast5 @ X0 ) @ ( c_not @ ( c_atleast6 @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax28,axiom,
% 0.28/0.45 ( c_exu_i
% 0.28/0.45 = ( ^ [X0: $i > $o] :
% 0.28/0.45 ( c_and
% 0.28/0.45 @ ? [X1: $i] : ( X0 @ X1 )
% 0.28/0.45 @ ! [X1: $i,X2: $i] :
% 0.28/0.45 ( ( X0 @ X1 )
% 0.28/0.45 => ( ( X0 @ X2 )
% 0.28/0.45 => ( X1 = X2 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax29,axiom,
% 0.28/0.45 ( c_reflexive_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i] : ( X0 @ X1 @ X1 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax30,axiom,
% 0.28/0.45 ( c_irreflexive_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i] : ( c_not @ ( X0 @ X1 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax31,axiom,
% 0.28/0.45 ( c_symmetric_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i,X2: $i] :
% 0.28/0.45 ( ( X0 @ X1 @ X2 )
% 0.28/0.45 => ( X0 @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax32,axiom,
% 0.28/0.45 ( c_antisymmetric_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i,X2: $i] :
% 0.28/0.45 ( ( X0 @ X1 @ X2 )
% 0.28/0.45 => ( ( X0 @ X2 @ X1 )
% 0.28/0.45 => ( X1 = X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax33,axiom,
% 0.28/0.45 ( c_transitive_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i,X2: $i,X3: $i] :
% 0.28/0.45 ( ( X0 @ X1 @ X2 )
% 0.28/0.45 => ( ( X0 @ X2 @ X3 )
% 0.28/0.45 => ( X0 @ X1 @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax34,axiom,
% 0.28/0.45 ( c_eqreln_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_symmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax35,axiom,
% 0.28/0.45 ( c_per_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_symmetric_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax36,axiom,
% 0.28/0.45 ( c_linear_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i,X2: $i] : ( c_or @ ( X0 @ X1 @ X2 ) @ ( X0 @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax37,axiom,
% 0.28/0.45 ( c_trichotomous_or_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] :
% 0.28/0.45 ! [X1: $i,X2: $i] : ( c_or @ ( c_or @ ( X0 @ X1 @ X2 ) @ ( X1 = X2 ) ) @ ( X0 @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax38,axiom,
% 0.28/0.45 ( c_partialorder_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_antisymmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax39,axiom,
% 0.28/0.45 ( c_totalorder_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_partialorder_i @ X0 ) @ ( c_linear_i @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax40,axiom,
% 0.28/0.45 ( c_strictpartialorder_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_irreflexive_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax41,axiom,
% 0.28/0.45 ( c_stricttotalorder_i
% 0.28/0.45 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_strictpartialorder_i @ X0 ) @ ( c_trichotomous_or_i @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax42,axiom,
% 0.28/0.45 ( c_If_i
% 0.28/0.45 = ( ^ [X0: $o,X1: $i,X2: $i] :
% 0.28/0.45 ( c_Eps_i
% 0.28/0.45 @ ^ [X3: $i] : ( c_or @ ( c_and @ X0 @ ( X3 = X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ ( X3 = X2 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax43,axiom,
% 0.28/0.45 ( c_exactly1of2
% 0.28/0.45 = ( ^ [X0: $o,X1: $o] : ( c_or @ ( c_and @ X0 @ ( c_not @ X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax44,axiom,
% 0.28/0.45 ( c_exactly1of3
% 0.28/0.45 = ( ^ [X0: $o,X1: $o,X2: $o] : ( c_or @ ( c_and @ ( c_exactly1of2 @ X0 @ X1 ) @ ( c_not @ X2 ) ) @ ( c_and @ ( c_and @ ( c_not @ X0 ) @ ( c_not @ X1 ) ) @ X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax45,axiom,
% 0.28/0.45 ( c_nIn
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] : ( c_not @ ( c_In @ X0 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax46,axiom,
% 0.28/0.45 ( c_nSubq
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] : ( c_not @ ( c_Subq @ X0 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax47,axiom,
% 0.28/0.45 ( c_UPair
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_Repl @ ( c_Power @ ( c_Power @ c_Empty ) )
% 0.28/0.45 @ ^ [X2: $i] : ( c_If_i @ ( c_In @ c_Empty @ X2 ) @ X0 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax48,axiom,
% 0.28/0.45 ( c_Sing
% 0.28/0.45 = ( ^ [X0: $i] : ( c_UPair @ X0 @ X0 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax49,axiom,
% 0.28/0.45 ( c_binunion
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] : ( c_Union @ ( c_UPair @ X0 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax50,axiom,
% 0.28/0.45 ( c_SetAdjoin
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] : ( c_binunion @ X0 @ ( c_Sing @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax51,axiom,
% 0.28/0.45 ( c_famunion
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i] :
% 0.28/0.45 ( c_Union
% 0.28/0.45 @ ( c_Repl @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax52,axiom,
% 0.28/0.45 ( c_Sep
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o] :
% 0.28/0.45 ( c_If_i
% 0.28/0.45 @ ? [X2: $i] : ( c_and @ ( c_In @ X2 @ X0 ) @ ( X1 @ X2 ) )
% 0.28/0.45 @ ( c_Repl @ X0
% 0.28/0.45 @ ^ [X2: $i] :
% 0.28/0.45 ( ^ [X3: $i] :
% 0.28/0.45 ( c_If_i @ ( X1 @ X3 ) @ X3
% 0.28/0.45 @ ( c_Eps_i
% 0.28/0.45 @ ^ [X4: $i] : ( c_and @ ( c_In @ X4 @ X0 ) @ ( X1 @ X4 ) ) ) )
% 0.28/0.45 @ X2 ) )
% 0.28/0.45 @ c_Empty ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax53,axiom,
% 0.28/0.45 ( c_ReplSep
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o,X2: $i > $i] :
% 0.28/0.45 ( c_Repl
% 0.28/0.45 @ ( c_Sep @ X0
% 0.28/0.45 @ ^ [X3: $i] : ( X1 @ X3 ) )
% 0.28/0.45 @ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax54,axiom,
% 0.28/0.45 ( c_binintersect
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_Sep @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax55,axiom,
% 0.28/0.45 ( c_setminus
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_Sep @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( c_nIn @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax56,axiom,
% 0.28/0.45 ( c_inj
% 0.28/0.45 = ( ^ [X0: $i,X1: $i,X2: $i > $i] :
% 0.28/0.45 ( c_and
% 0.28/0.45 @ ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X0 )
% 0.28/0.45 => ( c_In @ ( X2 @ X3 ) @ X1 ) )
% 0.28/0.45 @ ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X0 )
% 0.28/0.45 => ! [X4: $i] :
% 0.28/0.45 ( ( c_In @ X4 @ X0 )
% 0.28/0.45 => ( ( ( X2 @ X3 )
% 0.28/0.45 = ( X2 @ X4 ) )
% 0.28/0.45 => ( X3 = X4 ) ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax57,axiom,
% 0.28/0.45 ( c_bij
% 0.28/0.45 = ( ^ [X0: $i,X1: $i,X2: $i > $i] :
% 0.28/0.45 ( c_and @ ( c_inj @ X0 @ X1 @ X2 )
% 0.28/0.45 @ ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X1 )
% 0.28/0.45 => ? [X4: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X4 @ X0 )
% 0.28/0.45 @ ( ( X2 @ X4 )
% 0.28/0.45 = X3 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax58,axiom,
% 0.28/0.45 ( c_atleastp
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ? [X2: $i > $i] : ( c_inj @ X0 @ X1 @ X2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax59,axiom,
% 0.28/0.45 ( c_equip
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ? [X2: $i > $i] : ( c_bij @ X0 @ X1 @ X2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax60,axiom,
% 0.28/0.45 ( c_In_rec_poly_G_i
% 0.28/0.45 = ( ^ [X0: $i > ( $i > $i ) > $i,X1: $i,X2: $i] :
% 0.28/0.45 ! [X3: $i > $i > $o] :
% 0.28/0.45 ( ! [X4: $i,X5: $i > $i] :
% 0.28/0.45 ( ! [X6: $i] :
% 0.28/0.45 ( ( c_In @ X6 @ X4 )
% 0.28/0.45 => ( X3 @ X6 @ ( X5 @ X6 ) ) )
% 0.28/0.45 => ( X3 @ X4 @ ( X0 @ X4 @ X5 ) ) )
% 0.28/0.45 => ( X3 @ X1 @ X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax61,axiom,
% 0.28/0.45 ( c_In_rec_poly_i
% 0.28/0.45 = ( ^ [X0: $i > ( $i > $i ) > $i,X1: $i] :
% 0.28/0.45 ( c_Eps_i
% 0.28/0.45 @ ^ [X2: $i] : ( c_In_rec_poly_G_i @ X0 @ X1 @ X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax62,axiom,
% 0.28/0.45 ( c_ordsucc
% 0.28/0.45 = ( ^ [X0: $i] : ( c_binunion @ X0 @ ( c_Sing @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax63,axiom,
% 0.28/0.45 ( c_nat_p
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ! [X1: $i > $o] :
% 0.28/0.45 ( ( X1 @ c_Empty )
% 0.28/0.45 => ( ! [X2: $i] :
% 0.28/0.45 ( ( X1 @ X2 )
% 0.28/0.45 => ( X1 @ ( c_ordsucc @ X2 ) ) )
% 0.28/0.45 => ( X1 @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax64,axiom,
% 0.28/0.45 ( c_nat_primrec
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i > $i] :
% 0.28/0.45 ( c_In_rec_poly_i
% 0.28/0.45 @ ^ [X2: $i,X3: $i > $i] : ( c_If_i @ ( c_In @ ( c_Union @ X2 ) @ X2 ) @ ( X1 @ ( c_Union @ X2 ) @ ( X3 @ ( c_Union @ X2 ) ) ) @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax65,axiom,
% 0.28/0.45 ( c_add_nat
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_nat_primrec @ X0
% 0.28/0.45 @ ^ [X2: $i,X3: $i] : ( c_ordsucc @ X3 )
% 0.28/0.45 @ X1 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax66,axiom,
% 0.28/0.45 ( c_mul_nat
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_nat_primrec @ c_Empty
% 0.28/0.45 @ ^ [X2: $i,X3: $i] : ( c_add_nat @ X0 @ X3 )
% 0.28/0.45 @ X1 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax67,axiom,
% 0.28/0.45 ( c_ordinal
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( c_and @ ( c_TransSet @ X0 )
% 0.28/0.45 @ ! [X1: $i] :
% 0.28/0.45 ( ( c_In @ X1 @ X0 )
% 0.28/0.45 => ( c_TransSet @ X1 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax68,axiom,
% 0.28/0.45 ( c_V_
% 0.28/0.45 = ( c_In_rec_poly_i
% 0.28/0.45 @ ^ [X0: $i,X1: $i > $i] :
% 0.28/0.45 ( c_famunion @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( c_Power @ ( X1 @ X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax69,axiom,
% 0.28/0.45 ( c_Inj1
% 0.28/0.45 = ( c_In_rec_poly_i
% 0.28/0.45 @ ^ [X0: $i,X1: $i > $i] :
% 0.28/0.45 ( c_binunion @ ( c_Sing @ c_Empty )
% 0.28/0.45 @ ( c_Repl @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax70,axiom,
% 0.28/0.45 ( c_Inj0
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( c_Repl @ X0
% 0.28/0.45 @ ^ [X1: $i] : ( c_Inj1 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax71,axiom,
% 0.28/0.45 ( c_Unj
% 0.28/0.45 = ( c_In_rec_poly_i
% 0.28/0.45 @ ^ [X0: $i,X1: $i > $i] :
% 0.28/0.45 ( c_Repl @ ( c_setminus @ X0 @ ( c_Sing @ c_Empty ) )
% 0.28/0.45 @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax72,axiom,
% 0.28/0.45 ( c_combine_funcs
% 0.28/0.45 = ( ^ [X0: $i,X1: $i,X2: $i > $i,X3: $i > $i,X4: $i] :
% 0.28/0.45 ( c_If_i
% 0.28/0.45 @ ( X4
% 0.28/0.45 = ( c_Inj0 @ ( c_Unj @ X4 ) ) )
% 0.28/0.45 @ ( X2 @ ( c_Unj @ X4 ) )
% 0.28/0.45 @ ( X3 @ ( c_Unj @ X4 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax73,axiom,
% 0.28/0.45 ( c_setsum
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_binunion
% 0.28/0.45 @ ( c_Repl @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( c_Inj0 @ X2 ) )
% 0.28/0.45 @ ( c_Repl @ X1
% 0.28/0.45 @ ^ [X2: $i] : ( c_Inj1 @ X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax74,axiom,
% 0.28/0.45 ( c_proj0
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( c_ReplSep @ X0
% 0.28/0.45 @ ^ [X1: $i] :
% 0.28/0.45 ? [X2: $i] :
% 0.28/0.45 ( ( c_Inj0 @ X2 )
% 0.28/0.45 = X1 )
% 0.28/0.45 @ ^ [X1: $i] : ( c_Unj @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax75,axiom,
% 0.28/0.45 ( c_proj1
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( c_ReplSep @ X0
% 0.28/0.45 @ ^ [X1: $i] :
% 0.28/0.45 ? [X2: $i] :
% 0.28/0.45 ( ( c_Inj1 @ X2 )
% 0.28/0.45 = X1 )
% 0.28/0.45 @ ^ [X1: $i] : ( c_Unj @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax76,axiom,
% 0.28/0.45 ( c_binrep
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] : ( c_setsum @ X0 @ ( c_Power @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax77,axiom,
% 0.28/0.45 ( c_lam
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i] :
% 0.28/0.45 ( c_famunion @ X0
% 0.28/0.45 @ ^ [X2: $i] :
% 0.28/0.45 ( c_Repl @ ( X1 @ X2 )
% 0.28/0.45 @ ^ [X3: $i] : ( c_setsum @ X2 @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax78,axiom,
% 0.28/0.45 ( c_setprod
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_lam @ X0
% 0.28/0.45 @ ^ [X2: $i] : X1 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax79,axiom,
% 0.28/0.45 ( c_ap
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_ReplSep @ X0
% 0.28/0.45 @ ^ [X2: $i] :
% 0.28/0.45 ? [X3: $i] :
% 0.28/0.45 ( X2
% 0.28/0.45 = ( c_setsum @ X1 @ X3 ) )
% 0.28/0.45 @ ^ [X2: $i] : ( c_proj1 @ X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax80,axiom,
% 0.28/0.45 ( c_setsum_p
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( ( c_setsum @ ( c_ap @ X0 @ c_Empty ) @ ( c_ap @ X0 @ ( c_ordsucc @ c_Empty ) ) )
% 0.28/0.45 = X0 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax81,axiom,
% 0.28/0.45 ( c_tuple_p
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ! [X2: $i] :
% 0.28/0.45 ( ( c_In @ X2 @ X1 )
% 0.28/0.45 => ? [X3: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X3 @ X0 )
% 0.28/0.45 @ ? [X4: $i] :
% 0.28/0.45 ( X2
% 0.28/0.45 = ( c_setsum @ X3 @ X4 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax82,axiom,
% 0.28/0.45 ( c_Pi
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i] :
% 0.28/0.45 ( c_Sep
% 0.28/0.45 @ ( c_Power
% 0.28/0.45 @ ( c_lam @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( c_Union @ ( X1 @ X2 ) ) ) )
% 0.28/0.45 @ ^ [X2: $i] :
% 0.28/0.45 ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X0 )
% 0.28/0.45 => ( c_In @ ( c_ap @ X2 @ X3 ) @ ( X1 @ X3 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax83,axiom,
% 0.28/0.45 ( c_setexp
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_Pi @ X1
% 0.28/0.45 @ ^ [X2: $i] : X0 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax84,axiom,
% 0.28/0.45 ( c_Sep2
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i,X2: $i > $i > $o] :
% 0.28/0.45 ( c_Sep
% 0.28/0.45 @ ( c_lam @ X0
% 0.28/0.45 @ ^ [X3: $i] : ( X1 @ X3 ) )
% 0.28/0.45 @ ^ [X3: $i] : ( X2 @ ( c_ap @ X3 @ c_Empty ) @ ( c_ap @ X3 @ ( c_ordsucc @ c_Empty ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax85,axiom,
% 0.28/0.45 ( c_set_of_pairs
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ! [X1: $i] :
% 0.28/0.45 ( ( c_In @ X1 @ X0 )
% 0.28/0.45 => ? [X2: $i,X3: $i] :
% 0.28/0.45 ( X1
% 0.28/0.45 = ( c_lam @ ( c_ordsucc @ ( c_ordsucc @ c_Empty ) )
% 0.28/0.45 @ ^ [X4: $i] : ( c_If_i @ ( X4 = c_Empty ) @ X2 @ X3 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax86,axiom,
% 0.28/0.45 ( c_lam2
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i,X2: $i > $i > $i] :
% 0.28/0.45 ( c_lam @ X0
% 0.28/0.45 @ ^ [X3: $i] :
% 0.28/0.45 ( c_lam @ ( X1 @ X3 )
% 0.28/0.45 @ ^ [X4: $i] : ( X2 @ X3 @ X4 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax87,axiom,
% 0.28/0.45 ( c_PNoEq_
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o,X2: $i > $o] :
% 0.28/0.45 ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X0 )
% 0.28/0.45 => ( c_iff @ ( X1 @ X3 ) @ ( X2 @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax88,axiom,
% 0.28/0.45 ( c_PNoLt_
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o,X2: $i > $o] :
% 0.28/0.45 ? [X3: $i] : ( c_and @ ( c_In @ X3 @ X0 ) @ ( c_and @ ( c_and @ ( c_PNoEq_ @ X3 @ X1 @ X2 ) @ ( c_not @ ( X1 @ X3 ) ) ) @ ( X2 @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax89,axiom,
% 0.28/0.45 ( c_PNoLt
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o,X2: $i,X3: $i > $o] : ( c_or @ ( c_or @ ( c_PNoLt_ @ ( c_binintersect @ X0 @ X2 ) @ X1 @ X3 ) @ ( c_and @ ( c_and @ ( c_In @ X0 @ X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) @ ( X3 @ X0 ) ) ) @ ( c_and @ ( c_and @ ( c_In @ X2 @ X0 ) @ ( c_PNoEq_ @ X2 @ X1 @ X3 ) ) @ ( c_not @ ( X1 @ X2 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax90,axiom,
% 0.28/0.45 ( c_PNoLe
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o,X2: $i,X3: $i > $o] : ( c_or @ ( c_PNoLt @ X0 @ X1 @ X2 @ X3 ) @ ( c_and @ ( X0 = X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax91,axiom,
% 0.28/0.45 ( c_PNo_downc
% 0.28/0.45 = ( ^ [X0: $i > ( $i > $o ) > $o,X1: $i,X2: $i > $o] :
% 0.28/0.45 ? [X3: $i] :
% 0.28/0.45 ( c_and @ ( c_ordinal @ X3 )
% 0.28/0.45 @ ? [X4: $i > $o] : ( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax92,axiom,
% 0.28/0.45 ( c_PNo_upc
% 0.28/0.45 = ( ^ [X0: $i > ( $i > $o ) > $o,X1: $i,X2: $i > $o] :
% 0.28/0.45 ? [X3: $i] :
% 0.28/0.45 ( c_and @ ( c_ordinal @ X3 )
% 0.28/0.45 @ ? [X4: $i > $o] : ( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X3 @ X4 @ X1 @ X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax93,axiom,
% 0.28/0.45 ( c_SNoElts_
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( c_binunion @ X0
% 0.28/0.45 @ ( c_Repl @ X0
% 0.28/0.45 @ ^ [X1: $i] :
% 0.28/0.45 ( ^ [X2: $i] : ( c_SetAdjoin @ X2 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
% 0.28/0.45 @ X1 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax94,axiom,
% 0.28/0.45 ( c_SNo_
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_and @ ( c_Subq @ X1 @ ( c_SNoElts_ @ X0 ) )
% 0.28/0.45 @ ! [X2: $i] :
% 0.28/0.45 ( ( c_In @ X2 @ X0 )
% 0.28/0.45 => ( c_exactly1of2
% 0.28/0.45 @ ( c_In
% 0.28/0.45 @ ( ^ [X3: $i] : ( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
% 0.28/0.45 @ X2 )
% 0.28/0.45 @ X1 )
% 0.28/0.45 @ ( c_In @ X2 @ X1 ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax95,axiom,
% 0.28/0.45 ( c_PSNo
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $o] :
% 0.28/0.45 ( c_binunion
% 0.28/0.45 @ ( c_Sep @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( X1 @ X2 ) )
% 0.28/0.45 @ ( c_ReplSep @ X0
% 0.28/0.45 @ ^ [X2: $i] : ( c_not @ ( X1 @ X2 ) )
% 0.28/0.45 @ ^ [X2: $i] :
% 0.28/0.45 ( ^ [X3: $i] : ( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
% 0.28/0.45 @ X2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax96,axiom,
% 0.28/0.45 ( c_SNo
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ? [X1: $i] : ( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax97,axiom,
% 0.28/0.45 ( c_SNoLev
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ( c_Eps_i
% 0.28/0.45 @ ^ [X1: $i] : ( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax98,axiom,
% 0.28/0.45 ( c_SNoEq_
% 0.28/0.45 = ( ^ [X0: $i,X1: $i,X2: $i] :
% 0.28/0.45 ( c_PNoEq_ @ X0
% 0.28/0.45 @ ^ [X3: $i] : ( c_In @ X3 @ X1 )
% 0.28/0.45 @ ^ [X3: $i] : ( c_In @ X3 @ X2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax99,axiom,
% 0.28/0.45 ( c_SNoLt
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_PNoLt @ ( c_SNoLev @ X0 )
% 0.28/0.45 @ ^ [X2: $i] : ( c_In @ X2 @ X0 )
% 0.28/0.45 @ ( c_SNoLev @ X1 )
% 0.28/0.45 @ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax100,axiom,
% 0.28/0.45 ( c_SNoLe
% 0.28/0.45 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.45 ( c_PNoLe @ ( c_SNoLev @ X0 )
% 0.28/0.45 @ ^ [X2: $i] : ( c_In @ X2 @ X0 )
% 0.28/0.45 @ ( c_SNoLev @ X1 )
% 0.28/0.45 @ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax101,axiom,
% 0.28/0.45 ( c_binop_on
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i > $i] :
% 0.28/0.45 ! [X2: $i] :
% 0.28/0.45 ( ( c_In @ X2 @ X0 )
% 0.28/0.45 => ! [X3: $i] :
% 0.28/0.45 ( ( c_In @ X3 @ X0 )
% 0.28/0.45 => ( c_In @ ( X1 @ X2 @ X3 ) @ X0 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax102,axiom,
% 0.28/0.45 ( c_Loop
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i] :
% 0.28/0.45 ( c_and
% 0.28/0.45 @ ( c_and @ ( c_and @ ( c_and @ ( c_binop_on @ X0 @ X1 ) @ ( c_binop_on @ X0 @ X2 ) ) @ ( c_binop_on @ X0 @ X3 ) )
% 0.28/0.45 @ ! [X5: $i] :
% 0.28/0.45 ( ( c_In @ X5 @ X0 )
% 0.28/0.45 => ( c_and
% 0.28/0.45 @ ( ( X1 @ X4 @ X5 )
% 0.28/0.45 = X5 )
% 0.28/0.45 @ ( ( X1 @ X5 @ X4 )
% 0.28/0.45 = X5 ) ) ) )
% 0.28/0.45 @ ! [X5: $i] :
% 0.28/0.45 ( ( c_In @ X5 @ X0 )
% 0.28/0.45 => ! [X6: $i] :
% 0.28/0.45 ( ( c_In @ X6 @ X0 )
% 0.28/0.45 => ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( ( X2 @ X5 @ ( X1 @ X5 @ X6 ) )
% 0.28/0.45 = X6 )
% 0.28/0.45 @ ( ( X1 @ X5 @ ( X2 @ X5 @ X6 ) )
% 0.28/0.45 = X6 ) )
% 0.28/0.45 @ ( ( X3 @ ( X1 @ X5 @ X6 ) @ X6 )
% 0.28/0.45 = X5 ) )
% 0.28/0.45 @ ( ( X1 @ ( X3 @ X5 @ X6 ) @ X6 )
% 0.28/0.45 = X5 ) ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax103,axiom,
% 0.28/0.45 ( c_Loop_with_defs
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
% 0.28/0.45 ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( c_and @ ( c_Loop @ X0 @ X1 @ X2 @ X3 @ X4 )
% 0.28/0.45 @ ! [X14: $i] :
% 0.28/0.45 ( ( c_In @ X14 @ X0 )
% 0.28/0.45 => ! [X15: $i] :
% 0.28/0.45 ( ( c_In @ X15 @ X0 )
% 0.28/0.45 => ( ( X5 @ X14 @ X15 )
% 0.28/0.45 = ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) )
% 0.28/0.45 @ ! [X14: $i] :
% 0.28/0.45 ( ( c_In @ X14 @ X0 )
% 0.28/0.45 => ! [X15: $i] :
% 0.28/0.45 ( ( c_In @ X15 @ X0 )
% 0.28/0.45 => ! [X16: $i] :
% 0.28/0.45 ( ( c_In @ X16 @ X0 )
% 0.28/0.45 => ( ( X6 @ X14 @ X15 @ X16 )
% 0.28/0.45 = ( X2 @ ( X1 @ X14 @ ( X1 @ X15 @ X16 ) ) @ ( X1 @ ( X1 @ X14 @ X15 ) @ X16 ) ) ) ) ) ) )
% 0.28/0.45 @ ! [X14: $i] :
% 0.28/0.45 ( ( c_In @ X14 @ X0 )
% 0.28/0.45 => ! [X15: $i] :
% 0.28/0.45 ( ( c_In @ X15 @ X0 )
% 0.28/0.45 => ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( c_and
% 0.28/0.45 @ ( ( X7 @ X14 @ X15 )
% 0.28/0.45 = ( X2 @ X14 @ ( X1 @ X15 @ X14 ) ) )
% 0.28/0.45 @ ( ( X10 @ X14 @ X15 )
% 0.28/0.45 = ( X1 @ X14 @ ( X1 @ X15 @ ( X2 @ X14 @ X4 ) ) ) ) )
% 0.28/0.45 @ ( ( X11 @ X14 @ X15 )
% 0.28/0.45 = ( X1 @ ( X1 @ ( X3 @ X4 @ X14 ) @ X15 ) @ X14 ) ) )
% 0.28/0.45 @ ( ( X12 @ X14 @ X15 )
% 0.28/0.45 = ( X1 @ ( X2 @ X14 @ X15 ) @ ( X2 @ ( X2 @ X14 @ X4 ) @ X4 ) ) ) )
% 0.28/0.45 @ ( ( X13 @ X14 @ X15 )
% 0.28/0.45 = ( X1 @ ( X3 @ X4 @ ( X3 @ X4 @ X14 ) ) @ ( X3 @ X15 @ X14 ) ) ) ) ) ) )
% 0.28/0.45 @ ! [X14: $i] :
% 0.28/0.45 ( ( c_In @ X14 @ X0 )
% 0.28/0.45 => ! [X15: $i] :
% 0.28/0.45 ( ( c_In @ X15 @ X0 )
% 0.28/0.45 => ! [X16: $i] :
% 0.28/0.45 ( ( c_In @ X16 @ X0 )
% 0.28/0.45 => ( c_and
% 0.28/0.45 @ ( ( X8 @ X14 @ X15 @ X16 )
% 0.28/0.45 = ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X15 @ ( X1 @ X14 @ X16 ) ) ) )
% 0.28/0.45 @ ( ( X9 @ X14 @ X15 @ X16 )
% 0.28/0.45 = ( X3 @ ( X1 @ ( X1 @ X16 @ X14 ) @ X15 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax104,axiom,
% 0.28/0.45 ( c_Loop_with_defs_cex1
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
% 0.28/0.45 ( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
% 0.28/0.45 @ ? [X14: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X14 @ X0 )
% 0.28/0.45 @ ? [X15: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X15 @ X0 )
% 0.28/0.45 @ ? [X16: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X16 @ X0 )
% 0.28/0.45 @ ? [X17: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X17 @ X0 )
% 0.28/0.45 @ ( c_not
% 0.28/0.45 @ ( ( X5 @ ( X1 @ ( X2 @ ( X8 @ X15 @ X16 @ X14 ) @ X4 ) @ X14 ) @ X17 )
% 0.28/0.45 = X4 ) ) ) ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax105,axiom,
% 0.28/0.45 ( c_Loop_with_defs_cex2
% 0.28/0.45 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
% 0.28/0.45 ( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
% 0.28/0.45 @ ? [X14: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X14 @ X0 )
% 0.28/0.45 @ ? [X15: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X15 @ X0 )
% 0.28/0.45 @ ? [X16: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X16 @ X0 )
% 0.28/0.45 @ ? [X17: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X17 @ X0 )
% 0.28/0.45 @ ? [X18: $i] :
% 0.28/0.45 ( c_and @ ( c_In @ X18 @ X0 )
% 0.28/0.45 @ ( c_not
% 0.28/0.45 @ ( ( X6 @ X18 @ ( X1 @ ( X3 @ X4 @ X14 ) @ ( X9 @ X15 @ X16 @ X14 ) ) @ X17 )
% 0.28/0.45 = X4 ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(ax106,axiom,
% 0.28/0.45 ( c_combinator
% 0.28/0.45 = ( ^ [X0: $i] :
% 0.28/0.45 ! [X1: $i > $o] :
% 0.28/0.45 ( ( X1 @ ( c_Inj0 @ c_Empty ) )
% 0.28/0.46 => ( ( X1 @ ( c_Inj0 @ ( c_Power @ c_Empty ) ) )
% 0.28/0.46 => ( ! [X2: $i,X3: $i] :
% 0.28/0.46 ( ( X1 @ X2 )
% 0.28/0.46 => ( ( X1 @ X3 )
% 0.28/0.46 => ( X1 @ ( c_Inj1 @ ( c_setsum @ X2 @ X3 ) ) ) ) )
% 0.28/0.46 => ( X1 @ X0 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(ax107,axiom,
% 0.28/0.46 ( c_combinator_equiv
% 0.28/0.46 = ( ^ [X0: $i,X1: $i] :
% 0.28/0.46 ! [X2: $i > $i > $o] :
% 0.28/0.46 ( ^ [X3: $i,X4: $i,X5: $i > $i > $i] :
% 0.28/0.46 ( ( c_per_i @ X2 )
% 0.28/0.46 => ( ! [X6: $i] :
% 0.28/0.46 ( ( c_combinator @ X6 )
% 0.28/0.46 => ( X2 @ X6 @ X6 ) )
% 0.28/0.46 => ( ! [X6: $i,X7: $i,X8: $i,X9: $i] :
% 0.28/0.46 ( ( c_combinator @ X6 )
% 0.28/0.46 => ( ( c_combinator @ X7 )
% 0.28/0.46 => ( ( c_combinator @ X8 )
% 0.28/0.46 => ( ( c_combinator @ X9 )
% 0.28/0.46 => ( ( X2 @ X6 @ X8 )
% 0.28/0.46 => ( ( X2 @ X7 @ X9 )
% 0.28/0.46 => ( X2 @ ( X5 @ X6 @ X7 ) @ ( X5 @ X8 @ X9 ) ) ) ) ) ) ) )
% 0.28/0.46 => ( ! [X6: $i,X7: $i] : ( X2 @ ( X5 @ ( X5 @ X3 @ X6 ) @ X7 ) @ X6 )
% 0.28/0.46 => ( ! [X6: $i,X7: $i,X8: $i] : ( X2 @ ( X5 @ ( X5 @ ( X5 @ X4 @ X6 ) @ X7 ) @ X8 ) @ ( X5 @ ( X5 @ X6 @ X8 ) @ ( X5 @ X7 @ X8 ) ) )
% 0.28/0.46 => ( X2 @ X0 @ X1 ) ) ) ) ) )
% 0.28/0.46 @ ( c_Inj0 @ c_Empty )
% 0.28/0.46 @ ( c_Inj0 @ ( c_Power @ c_Empty ) )
% 0.28/0.46 @ ^ [X3: $i,X4: $i] : ( c_Inj1 @ ( c_setsum @ X3 @ X4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(ax108,axiom,
% 0.28/0.46 ( c_equip_mod
% 0.28/0.46 = ( ^ [X0: $i,X1: $i,X2: $i] :
% 0.28/0.46 ? [X3: $i,X4: $i] : ( c_or @ ( c_and @ ( c_equip @ ( c_setsum @ X0 @ X3 ) @ X1 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) @ ( c_and @ ( c_equip @ ( c_setsum @ X1 @ X3 ) @ X0 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(conj,conjecture,
% 0.28/0.46 ! [X0: $i] :
% 0.28/0.46 ( ( c_In @ X0 @ c_Empty )
% 0.28/0.46 => ! [X1: $i] :
% 0.28/0.46 ( ( c_In @ X1 @ X0 )
% 0.28/0.46 => ! [X2: $i] :
% 0.28/0.46 ( ( c_In @ X2 @ X1 )
% 0.28/0.46 => ? [X3: $i] :
% 0.28/0.46 ! [X4: $i] :
% 0.28/0.46 ( ( ( c_ordinal @ X3 )
% 0.28/0.46 => ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) ) )
% 0.28/0.46 => ( ( ( ( c_ordinal @ X3 )
% 0.28/0.46 => ( c_exactly2 @ X3 ) )
% 0.28/0.46 => ( ( c_atleast2 @ X3 )
% 0.28/0.46 => ( ( ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
% 0.28/0.46 => ( X3 = X4 ) )
% 0.28/0.46 => ( c_atleast3 @ X3 ) )
% 0.28/0.46 => ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) ) ) )
% 0.28/0.46 => ( c_nat_p @ X3 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 2.38/1.43 ---- Embedded in THF ---
% 2.38/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.38/1.43 %%% Generated on Tue Feb 24 14:49:11 EST 2026
% 2.38/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.38/1.43 %%% Logic specification used:
% 2.38/1.43 %%% thf(spec, logic, $$dhol).
% 2.38/1.43
% 2.38/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.meWfZCKs7P/DTF2DTF9274.p
% See solution above
% 2.38/1.43 ------------------------
% 2.38/1.46 ---- Cleaned THF ---
% 2.38/1.46 thf(c_Eps_i_tp,type,
% 2.38/1.46 c_Eps_i: ( $i > $o ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_False_tp,type,
% 2.38/1.46 c_False: $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_True_tp,type,
% 2.38/1.46 c_True: $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_not_tp,type,
% 2.38/1.46 c_not: $o > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_and_tp,type,
% 2.38/1.46 c_and: $o > $o > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_or_tp,type,
% 2.38/1.46 c_or: $o > $o > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_iff_tp,type,
% 2.38/1.46 c_iff: $o > $o > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_In_tp,type,
% 2.38/1.46 c_In: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Subq_tp,type,
% 2.38/1.46 c_Subq: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Empty_tp,type,
% 2.38/1.46 c_Empty: $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Union_tp,type,
% 2.38/1.46 c_Union: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Power_tp,type,
% 2.38/1.46 c_Power: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Repl_tp,type,
% 2.38/1.46 c_Repl: $i > ( $i > $i ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_TransSet_tp,type,
% 2.38/1.46 c_TransSet: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_atleast2_tp,type,
% 2.38/1.46 c_atleast2: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_atleast3_tp,type,
% 2.38/1.46 c_atleast3: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_atleast4_tp,type,
% 2.38/1.46 c_atleast4: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_atleast5_tp,type,
% 2.38/1.46 c_atleast5: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_atleast6_tp,type,
% 2.38/1.46 c_atleast6: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_exactly2_tp,type,
% 2.38/1.46 c_exactly2: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_exactly3_tp,type,
% 2.38/1.46 c_exactly3: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_exactly4_tp,type,
% 2.38/1.46 c_exactly4: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_exactly5_tp,type,
% 2.38/1.46 c_exactly5: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_exu_i_tp,type,
% 2.38/1.46 c_exu_i: ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_reflexive_i_tp,type,
% 2.38/1.46 c_reflexive_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_irreflexive_i_tp,type,
% 2.38/1.46 c_irreflexive_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_symmetric_i_tp,type,
% 2.38/1.46 c_symmetric_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_antisymmetric_i_tp,type,
% 2.38/1.46 c_antisymmetric_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_transitive_i_tp,type,
% 2.38/1.46 c_transitive_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_eqreln_i_tp,type,
% 2.38/1.46 c_eqreln_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_per_i_tp,type,
% 2.38/1.46 c_per_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_linear_i_tp,type,
% 2.38/1.46 c_linear_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_trichotomous_or_i_tp,type,
% 2.38/1.46 c_trichotomous_or_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_partialorder_i_tp,type,
% 2.38/1.46 c_partialorder_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_totalorder_i_tp,type,
% 2.38/1.46 c_totalorder_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_strictpartialorder_i_tp,type,
% 2.38/1.46 c_strictpartialorder_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_stricttotalorder_i_tp,type,
% 2.38/1.46 c_stricttotalorder_i: ( $i > $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_If_i_tp,type,
% 2.38/1.46 c_If_i: $o > $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_exactly1of2_tp,type,
% 2.38/1.46 c_exactly1of2: $o > $o > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_exactly1of3_tp,type,
% 2.38/1.46 c_exactly1of3: $o > $o > $o > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_nIn_tp,type,
% 2.38/1.46 c_nIn: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_nSubq_tp,type,
% 2.38/1.46 c_nSubq: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_UPair_tp,type,
% 2.38/1.46 c_UPair: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Sing_tp,type,
% 2.38/1.46 c_Sing: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_binunion_tp,type,
% 2.38/1.46 c_binunion: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_SetAdjoin_tp,type,
% 2.38/1.46 c_SetAdjoin: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_famunion_tp,type,
% 2.38/1.46 c_famunion: $i > ( $i > $i ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Sep_tp,type,
% 2.38/1.46 c_Sep: $i > ( $i > $o ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_ReplSep_tp,type,
% 2.38/1.46 c_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_binintersect_tp,type,
% 2.38/1.46 c_binintersect: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_setminus_tp,type,
% 2.38/1.46 c_setminus: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_inj_tp,type,
% 2.38/1.46 c_inj: $i > $i > ( $i > $i ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_bij_tp,type,
% 2.38/1.46 c_bij: $i > $i > ( $i > $i ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_atleastp_tp,type,
% 2.38/1.46 c_atleastp: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_equip_tp,type,
% 2.38/1.46 c_equip: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_In_rec_poly_G_i_tp,type,
% 2.38/1.46 c_In_rec_poly_G_i: ( $i > ( $i > $i ) > $i ) > $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_In_rec_poly_i_tp,type,
% 2.38/1.46 c_In_rec_poly_i: ( $i > ( $i > $i ) > $i ) > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_ordsucc_tp,type,
% 2.38/1.46 c_ordsucc: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_nat_p_tp,type,
% 2.38/1.46 c_nat_p: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_nat_primrec_tp,type,
% 2.38/1.46 c_nat_primrec: $i > ( $i > $i > $i ) > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_add_nat_tp,type,
% 2.38/1.46 c_add_nat: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_mul_nat_tp,type,
% 2.38/1.46 c_mul_nat: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_ordinal_tp,type,
% 2.38/1.46 c_ordinal: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_V__tp,type,
% 2.38/1.46 c_V_: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Inj1_tp,type,
% 2.38/1.46 c_Inj1: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Inj0_tp,type,
% 2.38/1.46 c_Inj0: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Unj_tp,type,
% 2.38/1.46 c_Unj: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_combine_funcs_tp,type,
% 2.38/1.46 c_combine_funcs: $i > $i > ( $i > $i ) > ( $i > $i ) > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_setsum_tp,type,
% 2.38/1.46 c_setsum: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_proj0_tp,type,
% 2.38/1.46 c_proj0: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_proj1_tp,type,
% 2.38/1.46 c_proj1: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_binrep_tp,type,
% 2.38/1.46 c_binrep: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_lam_tp,type,
% 2.38/1.46 c_lam: $i > ( $i > $i ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_setprod_tp,type,
% 2.38/1.46 c_setprod: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_ap_tp,type,
% 2.38/1.46 c_ap: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_setsum_p_tp,type,
% 2.38/1.46 c_setsum_p: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_tuple_p_tp,type,
% 2.38/1.46 c_tuple_p: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Pi_tp,type,
% 2.38/1.46 c_Pi: $i > ( $i > $i ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_setexp_tp,type,
% 2.38/1.46 c_setexp: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_Sep2_tp,type,
% 2.38/1.46 c_Sep2: $i > ( $i > $i ) > ( $i > $i > $o ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_set_of_pairs_tp,type,
% 2.38/1.46 c_set_of_pairs: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_lam2_tp,type,
% 2.38/1.46 c_lam2: $i > ( $i > $i ) > ( $i > $i > $i ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_PNoEq__tp,type,
% 2.38/1.46 c_PNoEq_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_PNoLt__tp,type,
% 2.38/1.46 c_PNoLt_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_PNoLt_tp,type,
% 2.38/1.46 c_PNoLt: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_PNoLe_tp,type,
% 2.38/1.46 c_PNoLe: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_PNo_downc_tp,type,
% 2.38/1.46 c_PNo_downc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_PNo_upc_tp,type,
% 2.38/1.46 c_PNo_upc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNoElts__tp,type,
% 2.38/1.46 c_SNoElts_: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNo__tp,type,
% 2.38/1.46 c_SNo_: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_PSNo_tp,type,
% 2.38/1.46 c_PSNo: $i > ( $i > $o ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNo_tp,type,
% 2.38/1.46 c_SNo: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNoLev_tp,type,
% 2.38/1.46 c_SNoLev: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNoEq__tp,type,
% 2.38/1.46 c_SNoEq_: $i > $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNoLt_tp,type,
% 2.38/1.46 c_SNoLt: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_SNoLe_tp,type,
% 2.38/1.46 c_SNoLe: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_binop_on_tp,type,
% 2.38/1.46 c_binop_on: $i > ( $i > $i > $i ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Loop_tp,type,
% 2.38/1.46 c_Loop: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Loop_with_defs_tp,type,
% 2.38/1.46 c_Loop_with_defs: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Loop_with_defs_cex1_tp,type,
% 2.38/1.46 c_Loop_with_defs_cex1: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_Loop_with_defs_cex2_tp,type,
% 2.38/1.46 c_Loop_with_defs_cex2: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_combinator_tp,type,
% 2.38/1.46 c_combinator: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_combinator_equiv_tp,type,
% 2.38/1.46 c_combinator_equiv: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(c_equip_mod_tp,type,
% 2.38/1.46 c_equip_mod: $i > $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(ax1,axiom,
% 2.38/1.46 ! [X0: $i > $o,X1: $i] :
% 2.38/1.46 ( ( X0 @ X1 )
% 2.38/1.46 => ( X0 @ ( c_Eps_i @ X0 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax2,axiom,
% 2.38/1.46 ! [X0: $o] :
% 2.38/1.46 ( ( c_not @ ( c_not @ X0 ) )
% 2.38/1.46 => X0 ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax3,axiom,
% 2.38/1.46 ! [X0: $o,X1: $o] :
% 2.38/1.46 ( ( c_iff @ X0 @ X1 )
% 2.38/1.46 => ( X0 = X1 ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax4,axiom,
% 2.38/1.46 ! [X0: $i,X1: $i] :
% 2.38/1.46 ( ( c_Subq @ X0 @ X1 )
% 2.38/1.46 => ( ( c_Subq @ X1 @ X0 )
% 2.38/1.46 => ( X0 = X1 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax5,axiom,
% 2.38/1.46 ( c_not
% 2.38/1.46 @ ? [X0: $i] : ( c_In @ X0 @ c_Empty ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax6,axiom,
% 2.38/1.46 ! [X0: $i,X1: $i] :
% 2.38/1.46 ( c_iff @ ( c_In @ X1 @ ( c_Union @ X0 ) )
% 2.38/1.46 @ ? [X2: $i] : ( c_and @ ( c_In @ X1 @ X2 ) @ ( c_In @ X2 @ X0 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax7,axiom,
% 2.38/1.46 ! [X0: $i,X1: $i] : ( c_iff @ ( c_In @ X1 @ ( c_Power @ X0 ) ) @ ( c_Subq @ X1 @ X0 ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax8,axiom,
% 2.38/1.46 ! [X0: $i,X1: $i > $i,X2: $i] :
% 2.38/1.46 ( c_iff
% 2.38/1.46 @ ( c_In @ X2
% 2.38/1.46 @ ( c_Repl @ X0
% 2.38/1.46 @ ^ [X3: $i] : ( X1 @ X3 ) ) )
% 2.38/1.46 @ ? [X3: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X3 @ X0 )
% 2.38/1.46 @ ( X2
% 2.38/1.46 = ( X1 @ X3 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax9,axiom,
% 2.38/1.46 ! [X0: $i > $o] :
% 2.38/1.46 ( ! [X1: $i] :
% 2.38/1.46 ( ( X0 @ X1 )
% 2.38/1.46 => ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X1 )
% 2.38/1.46 => ( X0 @ X2 ) ) )
% 2.38/1.46 => ( ( X0 @ c_Empty )
% 2.38/1.46 => ( ! [X1: $i] :
% 2.38/1.46 ( ( X0 @ X1 )
% 2.38/1.46 => ( X0 @ ( c_Union @ X1 ) ) )
% 2.38/1.46 => ( ! [X1: $i] :
% 2.38/1.46 ( ( X0 @ X1 )
% 2.38/1.46 => ( X0 @ ( c_Power @ X1 ) ) )
% 2.38/1.46 => ( ! [X1: $i] :
% 2.38/1.46 ( ( X0 @ X1 )
% 2.38/1.46 => ! [X2: $i > $i] :
% 2.38/1.46 ( ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X1 )
% 2.38/1.46 => ( X0 @ ( X2 @ X3 ) ) )
% 2.38/1.46 => ( X0
% 2.38/1.46 @ ( c_Repl @ X1
% 2.38/1.46 @ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) )
% 2.38/1.46 => ! [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax10,axiom,
% 2.38/1.46 ! [X0: $i > $o] :
% 2.38/1.46 ( ! [X1: $i] :
% 2.38/1.46 ( ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X1 )
% 2.38/1.46 => ( X0 @ X2 ) )
% 2.38/1.46 => ( X0 @ X1 ) )
% 2.38/1.46 => ! [X1: $i] : ( X0 @ X1 ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax11,axiom,
% 2.38/1.46 ( c_False
% 2.38/1.46 = ( ! [X0: $o] : X0 ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax12,axiom,
% 2.38/1.46 ( c_True
% 2.38/1.46 = ( ! [X0: $o] :
% 2.38/1.46 ( X0
% 2.38/1.46 => X0 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax13,axiom,
% 2.38/1.46 ( c_not
% 2.38/1.46 = ( ^ [X0: $o] :
% 2.38/1.46 ( X0
% 2.38/1.46 => c_False ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax14,axiom,
% 2.38/1.46 ( c_and
% 2.38/1.46 = ( ^ [X0: $o,X1: $o] :
% 2.38/1.46 ! [X2: $o] :
% 2.38/1.46 ( ( X0
% 2.38/1.46 => ( X1
% 2.38/1.46 => X2 ) )
% 2.38/1.46 => X2 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax15,axiom,
% 2.38/1.46 ( c_or
% 2.38/1.46 = ( ^ [X0: $o,X1: $o] :
% 2.38/1.46 ! [X2: $o] :
% 2.38/1.46 ( ( X0
% 2.38/1.46 => X2 )
% 2.38/1.46 => ( ( X1
% 2.38/1.46 => X2 )
% 2.38/1.46 => X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax16,axiom,
% 2.38/1.46 ( c_iff
% 2.38/1.46 = ( ^ [X0: $o,X1: $o] :
% 2.38/1.46 ( c_and
% 2.38/1.46 @ ( X0
% 2.38/1.46 => X1 )
% 2.38/1.46 @ ( X1
% 2.38/1.46 => X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax17,axiom,
% 2.38/1.46 ( c_Subq
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X0 )
% 2.38/1.46 => ( c_In @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax18,axiom,
% 2.38/1.46 ( c_TransSet
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ! [X1: $i] :
% 2.38/1.46 ( ( c_In @ X1 @ X0 )
% 2.38/1.46 => ( c_Subq @ X1 @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax19,axiom,
% 2.38/1.46 ( c_atleast2
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ? [X1: $i] : ( c_and @ ( c_In @ X1 @ X0 ) @ ( c_not @ ( c_Subq @ X0 @ ( c_Power @ X1 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax20,axiom,
% 2.38/1.46 ( c_atleast3
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast2 @ X1 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax21,axiom,
% 2.38/1.46 ( c_atleast4
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast3 @ X1 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax22,axiom,
% 2.38/1.46 ( c_atleast5
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast4 @ X1 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax23,axiom,
% 2.38/1.46 ( c_atleast6
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast5 @ X1 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax24,axiom,
% 2.38/1.46 ( c_exactly2
% 2.38/1.46 = ( ^ [X0: $i] : ( c_and @ ( c_atleast2 @ X0 ) @ ( c_not @ ( c_atleast3 @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax25,axiom,
% 2.38/1.46 ( c_exactly3
% 2.38/1.46 = ( ^ [X0: $i] : ( c_and @ ( c_atleast3 @ X0 ) @ ( c_not @ ( c_atleast4 @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax26,axiom,
% 2.38/1.46 ( c_exactly4
% 2.38/1.46 = ( ^ [X0: $i] : ( c_and @ ( c_atleast4 @ X0 ) @ ( c_not @ ( c_atleast5 @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax27,axiom,
% 2.38/1.46 ( c_exactly5
% 2.38/1.46 = ( ^ [X0: $i] : ( c_and @ ( c_atleast5 @ X0 ) @ ( c_not @ ( c_atleast6 @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax28,axiom,
% 2.38/1.46 ( c_exu_i
% 2.38/1.46 = ( ^ [X0: $i > $o] :
% 2.38/1.46 ( c_and
% 2.38/1.46 @ ? [X1: $i] : ( X0 @ X1 )
% 2.38/1.46 @ ! [X1: $i,X2: $i] :
% 2.38/1.46 ( ( X0 @ X1 )
% 2.38/1.46 => ( ( X0 @ X2 )
% 2.38/1.46 => ( X1 = X2 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax29,axiom,
% 2.38/1.46 ( c_reflexive_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i] : ( X0 @ X1 @ X1 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax30,axiom,
% 2.38/1.46 ( c_irreflexive_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i] : ( c_not @ ( X0 @ X1 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax31,axiom,
% 2.38/1.46 ( c_symmetric_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i,X2: $i] :
% 2.38/1.46 ( ( X0 @ X1 @ X2 )
% 2.38/1.46 => ( X0 @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax32,axiom,
% 2.38/1.46 ( c_antisymmetric_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i,X2: $i] :
% 2.38/1.46 ( ( X0 @ X1 @ X2 )
% 2.38/1.46 => ( ( X0 @ X2 @ X1 )
% 2.38/1.46 => ( X1 = X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax33,axiom,
% 2.38/1.46 ( c_transitive_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i,X2: $i,X3: $i] :
% 2.38/1.46 ( ( X0 @ X1 @ X2 )
% 2.38/1.46 => ( ( X0 @ X2 @ X3 )
% 2.38/1.46 => ( X0 @ X1 @ X3 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax34,axiom,
% 2.38/1.46 ( c_eqreln_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_symmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax35,axiom,
% 2.38/1.46 ( c_per_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_symmetric_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax36,axiom,
% 2.38/1.46 ( c_linear_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i,X2: $i] : ( c_or @ ( X0 @ X1 @ X2 ) @ ( X0 @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax37,axiom,
% 2.38/1.46 ( c_trichotomous_or_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] :
% 2.38/1.46 ! [X1: $i,X2: $i] : ( c_or @ ( c_or @ ( X0 @ X1 @ X2 ) @ ( X1 = X2 ) ) @ ( X0 @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax38,axiom,
% 2.38/1.46 ( c_partialorder_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_antisymmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax39,axiom,
% 2.38/1.46 ( c_totalorder_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_partialorder_i @ X0 ) @ ( c_linear_i @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax40,axiom,
% 2.38/1.46 ( c_strictpartialorder_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_irreflexive_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax41,axiom,
% 2.38/1.46 ( c_stricttotalorder_i
% 2.38/1.46 = ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_strictpartialorder_i @ X0 ) @ ( c_trichotomous_or_i @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax42,axiom,
% 2.38/1.46 ( c_If_i
% 2.38/1.46 = ( ^ [X0: $o,X1: $i,X2: $i] :
% 2.38/1.46 ( c_Eps_i
% 2.38/1.46 @ ^ [X3: $i] : ( c_or @ ( c_and @ X0 @ ( X3 = X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ ( X3 = X2 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax43,axiom,
% 2.38/1.46 ( c_exactly1of2
% 2.38/1.46 = ( ^ [X0: $o,X1: $o] : ( c_or @ ( c_and @ X0 @ ( c_not @ X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax44,axiom,
% 2.38/1.46 ( c_exactly1of3
% 2.38/1.46 = ( ^ [X0: $o,X1: $o,X2: $o] : ( c_or @ ( c_and @ ( c_exactly1of2 @ X0 @ X1 ) @ ( c_not @ X2 ) ) @ ( c_and @ ( c_and @ ( c_not @ X0 ) @ ( c_not @ X1 ) ) @ X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax45,axiom,
% 2.38/1.46 ( c_nIn
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] : ( c_not @ ( c_In @ X0 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax46,axiom,
% 2.38/1.46 ( c_nSubq
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] : ( c_not @ ( c_Subq @ X0 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax47,axiom,
% 2.38/1.46 ( c_UPair
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_Repl @ ( c_Power @ ( c_Power @ c_Empty ) )
% 2.38/1.46 @ ^ [X2: $i] : ( c_If_i @ ( c_In @ c_Empty @ X2 ) @ X0 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax48,axiom,
% 2.38/1.46 ( c_Sing
% 2.38/1.46 = ( ^ [X0: $i] : ( c_UPair @ X0 @ X0 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax49,axiom,
% 2.38/1.46 ( c_binunion
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] : ( c_Union @ ( c_UPair @ X0 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax50,axiom,
% 2.38/1.46 ( c_SetAdjoin
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] : ( c_binunion @ X0 @ ( c_Sing @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax51,axiom,
% 2.38/1.46 ( c_famunion
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i] :
% 2.38/1.46 ( c_Union
% 2.38/1.46 @ ( c_Repl @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax52,axiom,
% 2.38/1.46 ( c_Sep
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o] :
% 2.38/1.46 ( c_If_i
% 2.38/1.46 @ ? [X2: $i] : ( c_and @ ( c_In @ X2 @ X0 ) @ ( X1 @ X2 ) )
% 2.38/1.46 @ ( c_Repl @ X0
% 2.38/1.46 @ ^ [X2: $i] :
% 2.38/1.46 ( ^ [X3: $i] :
% 2.38/1.46 ( c_If_i @ ( X1 @ X3 ) @ X3
% 2.38/1.46 @ ( c_Eps_i
% 2.38/1.46 @ ^ [X4: $i] : ( c_and @ ( c_In @ X4 @ X0 ) @ ( X1 @ X4 ) ) ) )
% 2.38/1.46 @ X2 ) )
% 2.38/1.46 @ c_Empty ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax53,axiom,
% 2.38/1.46 ( c_ReplSep
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o,X2: $i > $i] :
% 2.38/1.46 ( c_Repl
% 2.38/1.46 @ ( c_Sep @ X0
% 2.38/1.46 @ ^ [X3: $i] : ( X1 @ X3 ) )
% 2.38/1.46 @ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax54,axiom,
% 2.38/1.46 ( c_binintersect
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_Sep @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax55,axiom,
% 2.38/1.46 ( c_setminus
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_Sep @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( c_nIn @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax56,axiom,
% 2.38/1.46 ( c_inj
% 2.38/1.46 = ( ^ [X0: $i,X1: $i,X2: $i > $i] :
% 2.38/1.46 ( c_and
% 2.38/1.46 @ ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X0 )
% 2.38/1.46 => ( c_In @ ( X2 @ X3 ) @ X1 ) )
% 2.38/1.46 @ ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X0 )
% 2.38/1.46 => ! [X4: $i] :
% 2.38/1.46 ( ( c_In @ X4 @ X0 )
% 2.38/1.46 => ( ( ( X2 @ X3 )
% 2.38/1.46 = ( X2 @ X4 ) )
% 2.38/1.46 => ( X3 = X4 ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax57,axiom,
% 2.38/1.46 ( c_bij
% 2.38/1.46 = ( ^ [X0: $i,X1: $i,X2: $i > $i] :
% 2.38/1.46 ( c_and @ ( c_inj @ X0 @ X1 @ X2 )
% 2.38/1.46 @ ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X1 )
% 2.38/1.46 => ? [X4: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X4 @ X0 )
% 2.38/1.46 @ ( ( X2 @ X4 )
% 2.38/1.46 = X3 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax58,axiom,
% 2.38/1.46 ( c_atleastp
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ? [X2: $i > $i] : ( c_inj @ X0 @ X1 @ X2 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax59,axiom,
% 2.38/1.46 ( c_equip
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ? [X2: $i > $i] : ( c_bij @ X0 @ X1 @ X2 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax60,axiom,
% 2.38/1.46 ( c_In_rec_poly_G_i
% 2.38/1.46 = ( ^ [X0: $i > ( $i > $i ) > $i,X1: $i,X2: $i] :
% 2.38/1.46 ! [X3: $i > $i > $o] :
% 2.38/1.46 ( ! [X4: $i,X5: $i > $i] :
% 2.38/1.46 ( ! [X6: $i] :
% 2.38/1.46 ( ( c_In @ X6 @ X4 )
% 2.38/1.46 => ( X3 @ X6 @ ( X5 @ X6 ) ) )
% 2.38/1.46 => ( X3 @ X4 @ ( X0 @ X4 @ X5 ) ) )
% 2.38/1.46 => ( X3 @ X1 @ X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax61,axiom,
% 2.38/1.46 ( c_In_rec_poly_i
% 2.38/1.46 = ( ^ [X0: $i > ( $i > $i ) > $i,X1: $i] :
% 2.38/1.46 ( c_Eps_i
% 2.38/1.46 @ ^ [X2: $i] : ( c_In_rec_poly_G_i @ X0 @ X1 @ X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax62,axiom,
% 2.38/1.46 ( c_ordsucc
% 2.38/1.46 = ( ^ [X0: $i] : ( c_binunion @ X0 @ ( c_Sing @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax63,axiom,
% 2.38/1.46 ( c_nat_p
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ! [X1: $i > $o] :
% 2.38/1.46 ( ( X1 @ c_Empty )
% 2.38/1.46 => ( ! [X2: $i] :
% 2.38/1.46 ( ( X1 @ X2 )
% 2.38/1.46 => ( X1 @ ( c_ordsucc @ X2 ) ) )
% 2.38/1.46 => ( X1 @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax64,axiom,
% 2.38/1.46 ( c_nat_primrec
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i > $i] :
% 2.38/1.46 ( c_In_rec_poly_i
% 2.38/1.46 @ ^ [X2: $i,X3: $i > $i] : ( c_If_i @ ( c_In @ ( c_Union @ X2 ) @ X2 ) @ ( X1 @ ( c_Union @ X2 ) @ ( X3 @ ( c_Union @ X2 ) ) ) @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax65,axiom,
% 2.38/1.46 ( c_add_nat
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_nat_primrec @ X0
% 2.38/1.46 @ ^ [X2: $i,X3: $i] : ( c_ordsucc @ X3 )
% 2.38/1.46 @ X1 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax66,axiom,
% 2.38/1.46 ( c_mul_nat
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_nat_primrec @ c_Empty
% 2.38/1.46 @ ^ [X2: $i,X3: $i] : ( c_add_nat @ X0 @ X3 )
% 2.38/1.46 @ X1 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax67,axiom,
% 2.38/1.46 ( c_ordinal
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( c_and @ ( c_TransSet @ X0 )
% 2.38/1.46 @ ! [X1: $i] :
% 2.38/1.46 ( ( c_In @ X1 @ X0 )
% 2.38/1.46 => ( c_TransSet @ X1 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax68,axiom,
% 2.38/1.46 ( c_V_
% 2.38/1.46 = ( c_In_rec_poly_i
% 2.38/1.46 @ ^ [X0: $i,X1: $i > $i] :
% 2.38/1.46 ( c_famunion @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( c_Power @ ( X1 @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax69,axiom,
% 2.38/1.46 ( c_Inj1
% 2.38/1.46 = ( c_In_rec_poly_i
% 2.38/1.46 @ ^ [X0: $i,X1: $i > $i] :
% 2.38/1.46 ( c_binunion @ ( c_Sing @ c_Empty )
% 2.38/1.46 @ ( c_Repl @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax70,axiom,
% 2.38/1.46 ( c_Inj0
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( c_Repl @ X0
% 2.38/1.46 @ ^ [X1: $i] : ( c_Inj1 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax71,axiom,
% 2.38/1.46 ( c_Unj
% 2.38/1.46 = ( c_In_rec_poly_i
% 2.38/1.46 @ ^ [X0: $i,X1: $i > $i] :
% 2.38/1.46 ( c_Repl @ ( c_setminus @ X0 @ ( c_Sing @ c_Empty ) )
% 2.38/1.46 @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax72,axiom,
% 2.38/1.46 ( c_combine_funcs
% 2.38/1.46 = ( ^ [X0: $i,X1: $i,X2: $i > $i,X3: $i > $i,X4: $i] :
% 2.38/1.46 ( c_If_i
% 2.38/1.46 @ ( X4
% 2.38/1.46 = ( c_Inj0 @ ( c_Unj @ X4 ) ) )
% 2.38/1.46 @ ( X2 @ ( c_Unj @ X4 ) )
% 2.38/1.46 @ ( X3 @ ( c_Unj @ X4 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax73,axiom,
% 2.38/1.46 ( c_setsum
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_binunion
% 2.38/1.46 @ ( c_Repl @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( c_Inj0 @ X2 ) )
% 2.38/1.46 @ ( c_Repl @ X1
% 2.38/1.46 @ ^ [X2: $i] : ( c_Inj1 @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax74,axiom,
% 2.38/1.46 ( c_proj0
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( c_ReplSep @ X0
% 2.38/1.46 @ ^ [X1: $i] :
% 2.38/1.46 ? [X2: $i] :
% 2.38/1.46 ( ( c_Inj0 @ X2 )
% 2.38/1.46 = X1 )
% 2.38/1.46 @ ^ [X1: $i] : ( c_Unj @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax75,axiom,
% 2.38/1.46 ( c_proj1
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( c_ReplSep @ X0
% 2.38/1.46 @ ^ [X1: $i] :
% 2.38/1.46 ? [X2: $i] :
% 2.38/1.46 ( ( c_Inj1 @ X2 )
% 2.38/1.46 = X1 )
% 2.38/1.46 @ ^ [X1: $i] : ( c_Unj @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax76,axiom,
% 2.38/1.46 ( c_binrep
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] : ( c_setsum @ X0 @ ( c_Power @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax77,axiom,
% 2.38/1.46 ( c_lam
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i] :
% 2.38/1.46 ( c_famunion @ X0
% 2.38/1.46 @ ^ [X2: $i] :
% 2.38/1.46 ( c_Repl @ ( X1 @ X2 )
% 2.38/1.46 @ ^ [X3: $i] : ( c_setsum @ X2 @ X3 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax78,axiom,
% 2.38/1.46 ( c_setprod
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_lam @ X0
% 2.38/1.46 @ ^ [X2: $i] : X1 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax79,axiom,
% 2.38/1.46 ( c_ap
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_ReplSep @ X0
% 2.38/1.46 @ ^ [X2: $i] :
% 2.38/1.46 ? [X3: $i] :
% 2.38/1.46 ( X2
% 2.38/1.46 = ( c_setsum @ X1 @ X3 ) )
% 2.38/1.46 @ ^ [X2: $i] : ( c_proj1 @ X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax80,axiom,
% 2.38/1.46 ( c_setsum_p
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( ( c_setsum @ ( c_ap @ X0 @ c_Empty ) @ ( c_ap @ X0 @ ( c_ordsucc @ c_Empty ) ) )
% 2.38/1.46 = X0 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax81,axiom,
% 2.38/1.46 ( c_tuple_p
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X1 )
% 2.38/1.46 => ? [X3: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X3 @ X0 )
% 2.38/1.46 @ ? [X4: $i] :
% 2.38/1.46 ( X2
% 2.38/1.46 = ( c_setsum @ X3 @ X4 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax82,axiom,
% 2.38/1.46 ( c_Pi
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i] :
% 2.38/1.46 ( c_Sep
% 2.38/1.46 @ ( c_Power
% 2.38/1.46 @ ( c_lam @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( c_Union @ ( X1 @ X2 ) ) ) )
% 2.38/1.46 @ ^ [X2: $i] :
% 2.38/1.46 ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X0 )
% 2.38/1.46 => ( c_In @ ( c_ap @ X2 @ X3 ) @ ( X1 @ X3 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax83,axiom,
% 2.38/1.46 ( c_setexp
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_Pi @ X1
% 2.38/1.46 @ ^ [X2: $i] : X0 ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax84,axiom,
% 2.38/1.46 ( c_Sep2
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i,X2: $i > $i > $o] :
% 2.38/1.46 ( c_Sep
% 2.38/1.46 @ ( c_lam @ X0
% 2.38/1.46 @ ^ [X3: $i] : ( X1 @ X3 ) )
% 2.38/1.46 @ ^ [X3: $i] : ( X2 @ ( c_ap @ X3 @ c_Empty ) @ ( c_ap @ X3 @ ( c_ordsucc @ c_Empty ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax85,axiom,
% 2.38/1.46 ( c_set_of_pairs
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ! [X1: $i] :
% 2.38/1.46 ( ( c_In @ X1 @ X0 )
% 2.38/1.46 => ? [X2: $i,X3: $i] :
% 2.38/1.46 ( X1
% 2.38/1.46 = ( c_lam @ ( c_ordsucc @ ( c_ordsucc @ c_Empty ) )
% 2.38/1.46 @ ^ [X4: $i] : ( c_If_i @ ( X4 = c_Empty ) @ X2 @ X3 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax86,axiom,
% 2.38/1.46 ( c_lam2
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i,X2: $i > $i > $i] :
% 2.38/1.46 ( c_lam @ X0
% 2.38/1.46 @ ^ [X3: $i] :
% 2.38/1.46 ( c_lam @ ( X1 @ X3 )
% 2.38/1.46 @ ^ [X4: $i] : ( X2 @ X3 @ X4 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax87,axiom,
% 2.38/1.46 ( c_PNoEq_
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o,X2: $i > $o] :
% 2.38/1.46 ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X0 )
% 2.38/1.46 => ( c_iff @ ( X1 @ X3 ) @ ( X2 @ X3 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax88,axiom,
% 2.38/1.46 ( c_PNoLt_
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o,X2: $i > $o] :
% 2.38/1.46 ? [X3: $i] : ( c_and @ ( c_In @ X3 @ X0 ) @ ( c_and @ ( c_and @ ( c_PNoEq_ @ X3 @ X1 @ X2 ) @ ( c_not @ ( X1 @ X3 ) ) ) @ ( X2 @ X3 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax89,axiom,
% 2.38/1.46 ( c_PNoLt
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o,X2: $i,X3: $i > $o] : ( c_or @ ( c_or @ ( c_PNoLt_ @ ( c_binintersect @ X0 @ X2 ) @ X1 @ X3 ) @ ( c_and @ ( c_and @ ( c_In @ X0 @ X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) @ ( X3 @ X0 ) ) ) @ ( c_and @ ( c_and @ ( c_In @ X2 @ X0 ) @ ( c_PNoEq_ @ X2 @ X1 @ X3 ) ) @ ( c_not @ ( X1 @ X2 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax90,axiom,
% 2.38/1.46 ( c_PNoLe
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o,X2: $i,X3: $i > $o] : ( c_or @ ( c_PNoLt @ X0 @ X1 @ X2 @ X3 ) @ ( c_and @ ( X0 = X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax91,axiom,
% 2.38/1.46 ( c_PNo_downc
% 2.38/1.46 = ( ^ [X0: $i > ( $i > $o ) > $o,X1: $i,X2: $i > $o] :
% 2.38/1.46 ? [X3: $i] :
% 2.38/1.46 ( c_and @ ( c_ordinal @ X3 )
% 2.38/1.46 @ ? [X4: $i > $o] : ( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax92,axiom,
% 2.38/1.46 ( c_PNo_upc
% 2.38/1.46 = ( ^ [X0: $i > ( $i > $o ) > $o,X1: $i,X2: $i > $o] :
% 2.38/1.46 ? [X3: $i] :
% 2.38/1.46 ( c_and @ ( c_ordinal @ X3 )
% 2.38/1.46 @ ? [X4: $i > $o] : ( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X3 @ X4 @ X1 @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax93,axiom,
% 2.38/1.46 ( c_SNoElts_
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( c_binunion @ X0
% 2.38/1.46 @ ( c_Repl @ X0
% 2.38/1.46 @ ^ [X1: $i] :
% 2.38/1.46 ( ^ [X2: $i] : ( c_SetAdjoin @ X2 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
% 2.38/1.46 @ X1 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax94,axiom,
% 2.38/1.46 ( c_SNo_
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_and @ ( c_Subq @ X1 @ ( c_SNoElts_ @ X0 ) )
% 2.38/1.46 @ ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X0 )
% 2.38/1.46 => ( c_exactly1of2
% 2.38/1.46 @ ( c_In
% 2.38/1.46 @ ( ^ [X3: $i] : ( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
% 2.38/1.46 @ X2 )
% 2.38/1.46 @ X1 )
% 2.38/1.46 @ ( c_In @ X2 @ X1 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax95,axiom,
% 2.38/1.46 ( c_PSNo
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $o] :
% 2.38/1.46 ( c_binunion
% 2.38/1.46 @ ( c_Sep @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( X1 @ X2 ) )
% 2.38/1.46 @ ( c_ReplSep @ X0
% 2.38/1.46 @ ^ [X2: $i] : ( c_not @ ( X1 @ X2 ) )
% 2.38/1.46 @ ^ [X2: $i] :
% 2.38/1.46 ( ^ [X3: $i] : ( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
% 2.38/1.46 @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax96,axiom,
% 2.38/1.46 ( c_SNo
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ? [X1: $i] : ( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax97,axiom,
% 2.38/1.46 ( c_SNoLev
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ( c_Eps_i
% 2.38/1.46 @ ^ [X1: $i] : ( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax98,axiom,
% 2.38/1.46 ( c_SNoEq_
% 2.38/1.46 = ( ^ [X0: $i,X1: $i,X2: $i] :
% 2.38/1.46 ( c_PNoEq_ @ X0
% 2.38/1.46 @ ^ [X3: $i] : ( c_In @ X3 @ X1 )
% 2.38/1.46 @ ^ [X3: $i] : ( c_In @ X3 @ X2 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax99,axiom,
% 2.38/1.46 ( c_SNoLt
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_PNoLt @ ( c_SNoLev @ X0 )
% 2.38/1.46 @ ^ [X2: $i] : ( c_In @ X2 @ X0 )
% 2.38/1.46 @ ( c_SNoLev @ X1 )
% 2.38/1.46 @ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax100,axiom,
% 2.38/1.46 ( c_SNoLe
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ( c_PNoLe @ ( c_SNoLev @ X0 )
% 2.38/1.46 @ ^ [X2: $i] : ( c_In @ X2 @ X0 )
% 2.38/1.46 @ ( c_SNoLev @ X1 )
% 2.38/1.46 @ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax101,axiom,
% 2.38/1.46 ( c_binop_on
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i > $i] :
% 2.38/1.46 ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X0 )
% 2.38/1.46 => ! [X3: $i] :
% 2.38/1.46 ( ( c_In @ X3 @ X0 )
% 2.38/1.46 => ( c_In @ ( X1 @ X2 @ X3 ) @ X0 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax102,axiom,
% 2.38/1.46 ( c_Loop
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i] :
% 2.38/1.46 ( c_and
% 2.38/1.46 @ ( c_and @ ( c_and @ ( c_and @ ( c_binop_on @ X0 @ X1 ) @ ( c_binop_on @ X0 @ X2 ) ) @ ( c_binop_on @ X0 @ X3 ) )
% 2.38/1.46 @ ! [X5: $i] :
% 2.38/1.46 ( ( c_In @ X5 @ X0 )
% 2.38/1.46 => ( c_and
% 2.38/1.46 @ ( ( X1 @ X4 @ X5 )
% 2.38/1.46 = X5 )
% 2.38/1.46 @ ( ( X1 @ X5 @ X4 )
% 2.38/1.46 = X5 ) ) ) )
% 2.38/1.46 @ ! [X5: $i] :
% 2.38/1.46 ( ( c_In @ X5 @ X0 )
% 2.38/1.46 => ! [X6: $i] :
% 2.38/1.46 ( ( c_In @ X6 @ X0 )
% 2.38/1.46 => ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( ( X2 @ X5 @ ( X1 @ X5 @ X6 ) )
% 2.38/1.46 = X6 )
% 2.38/1.46 @ ( ( X1 @ X5 @ ( X2 @ X5 @ X6 ) )
% 2.38/1.46 = X6 ) )
% 2.38/1.46 @ ( ( X3 @ ( X1 @ X5 @ X6 ) @ X6 )
% 2.38/1.46 = X5 ) )
% 2.38/1.46 @ ( ( X1 @ ( X3 @ X5 @ X6 ) @ X6 )
% 2.38/1.46 = X5 ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax103,axiom,
% 2.38/1.46 ( c_Loop_with_defs
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
% 2.38/1.46 ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( c_and @ ( c_Loop @ X0 @ X1 @ X2 @ X3 @ X4 )
% 2.38/1.46 @ ! [X14: $i] :
% 2.38/1.46 ( ( c_In @ X14 @ X0 )
% 2.38/1.46 => ! [X15: $i] :
% 2.38/1.46 ( ( c_In @ X15 @ X0 )
% 2.38/1.46 => ( ( X5 @ X14 @ X15 )
% 2.38/1.46 = ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) )
% 2.38/1.46 @ ! [X14: $i] :
% 2.38/1.46 ( ( c_In @ X14 @ X0 )
% 2.38/1.46 => ! [X15: $i] :
% 2.38/1.46 ( ( c_In @ X15 @ X0 )
% 2.38/1.46 => ! [X16: $i] :
% 2.38/1.46 ( ( c_In @ X16 @ X0 )
% 2.38/1.46 => ( ( X6 @ X14 @ X15 @ X16 )
% 2.38/1.46 = ( X2 @ ( X1 @ X14 @ ( X1 @ X15 @ X16 ) ) @ ( X1 @ ( X1 @ X14 @ X15 ) @ X16 ) ) ) ) ) ) )
% 2.38/1.46 @ ! [X14: $i] :
% 2.38/1.46 ( ( c_In @ X14 @ X0 )
% 2.38/1.46 => ! [X15: $i] :
% 2.38/1.46 ( ( c_In @ X15 @ X0 )
% 2.38/1.46 => ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( c_and
% 2.38/1.46 @ ( ( X7 @ X14 @ X15 )
% 2.38/1.46 = ( X2 @ X14 @ ( X1 @ X15 @ X14 ) ) )
% 2.38/1.46 @ ( ( X10 @ X14 @ X15 )
% 2.38/1.46 = ( X1 @ X14 @ ( X1 @ X15 @ ( X2 @ X14 @ X4 ) ) ) ) )
% 2.38/1.46 @ ( ( X11 @ X14 @ X15 )
% 2.38/1.46 = ( X1 @ ( X1 @ ( X3 @ X4 @ X14 ) @ X15 ) @ X14 ) ) )
% 2.38/1.46 @ ( ( X12 @ X14 @ X15 )
% 2.38/1.46 = ( X1 @ ( X2 @ X14 @ X15 ) @ ( X2 @ ( X2 @ X14 @ X4 ) @ X4 ) ) ) )
% 2.38/1.46 @ ( ( X13 @ X14 @ X15 )
% 2.38/1.46 = ( X1 @ ( X3 @ X4 @ ( X3 @ X4 @ X14 ) ) @ ( X3 @ X15 @ X14 ) ) ) ) ) ) )
% 2.38/1.46 @ ! [X14: $i] :
% 2.38/1.46 ( ( c_In @ X14 @ X0 )
% 2.38/1.46 => ! [X15: $i] :
% 2.38/1.46 ( ( c_In @ X15 @ X0 )
% 2.38/1.46 => ! [X16: $i] :
% 2.38/1.46 ( ( c_In @ X16 @ X0 )
% 2.38/1.46 => ( c_and
% 2.38/1.46 @ ( ( X8 @ X14 @ X15 @ X16 )
% 2.38/1.46 = ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X15 @ ( X1 @ X14 @ X16 ) ) ) )
% 2.38/1.46 @ ( ( X9 @ X14 @ X15 @ X16 )
% 2.38/1.46 = ( X3 @ ( X1 @ ( X1 @ X16 @ X14 ) @ X15 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax104,axiom,
% 2.38/1.46 ( c_Loop_with_defs_cex1
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
% 2.38/1.46 ( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
% 2.38/1.46 @ ? [X14: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X14 @ X0 )
% 2.38/1.46 @ ? [X15: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X15 @ X0 )
% 2.38/1.46 @ ? [X16: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X16 @ X0 )
% 2.38/1.46 @ ? [X17: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X17 @ X0 )
% 2.38/1.46 @ ( c_not
% 2.38/1.46 @ ( ( X5 @ ( X1 @ ( X2 @ ( X8 @ X15 @ X16 @ X14 ) @ X4 ) @ X14 ) @ X17 )
% 2.38/1.46 = X4 ) ) ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax105,axiom,
% 2.38/1.46 ( c_Loop_with_defs_cex2
% 2.38/1.46 = ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
% 2.38/1.46 ( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
% 2.38/1.46 @ ? [X14: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X14 @ X0 )
% 2.38/1.46 @ ? [X15: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X15 @ X0 )
% 2.38/1.46 @ ? [X16: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X16 @ X0 )
% 2.38/1.46 @ ? [X17: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X17 @ X0 )
% 2.38/1.46 @ ? [X18: $i] :
% 2.38/1.46 ( c_and @ ( c_In @ X18 @ X0 )
% 2.38/1.46 @ ( c_not
% 2.38/1.46 @ ( ( X6 @ X18 @ ( X1 @ ( X3 @ X4 @ X14 ) @ ( X9 @ X15 @ X16 @ X14 ) ) @ X17 )
% 2.38/1.46 = X4 ) ) ) ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax106,axiom,
% 2.38/1.46 ( c_combinator
% 2.38/1.46 = ( ^ [X0: $i] :
% 2.38/1.46 ! [X1: $i > $o] :
% 2.38/1.46 ( ( X1 @ ( c_Inj0 @ c_Empty ) )
% 2.38/1.46 => ( ( X1 @ ( c_Inj0 @ ( c_Power @ c_Empty ) ) )
% 2.38/1.46 => ( ! [X2: $i,X3: $i] :
% 2.38/1.46 ( ( X1 @ X2 )
% 2.38/1.46 => ( ( X1 @ X3 )
% 2.38/1.46 => ( X1 @ ( c_Inj1 @ ( c_setsum @ X2 @ X3 ) ) ) ) )
% 2.38/1.46 => ( X1 @ X0 ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax107,axiom,
% 2.38/1.46 ( c_combinator_equiv
% 2.38/1.46 = ( ^ [X0: $i,X1: $i] :
% 2.38/1.46 ! [X2: $i > $i > $o] :
% 2.38/1.46 ( ^ [X3: $i,X4: $i,X5: $i > $i > $i] :
% 2.38/1.46 ( ( c_per_i @ X2 )
% 2.38/1.46 => ( ! [X6: $i] :
% 2.38/1.46 ( ( c_combinator @ X6 )
% 2.38/1.46 => ( X2 @ X6 @ X6 ) )
% 2.38/1.46 => ( ! [X6: $i,X7: $i,X8: $i,X9: $i] :
% 2.38/1.46 ( ( c_combinator @ X6 )
% 2.38/1.46 => ( ( c_combinator @ X7 )
% 2.38/1.46 => ( ( c_combinator @ X8 )
% 2.38/1.46 => ( ( c_combinator @ X9 )
% 2.38/1.46 => ( ( X2 @ X6 @ X8 )
% 2.38/1.46 => ( ( X2 @ X7 @ X9 )
% 2.38/1.46 => ( X2 @ ( X5 @ X6 @ X7 ) @ ( X5 @ X8 @ X9 ) ) ) ) ) ) ) )
% 2.38/1.46 => ( ! [X6: $i,X7: $i] : ( X2 @ ( X5 @ ( X5 @ X3 @ X6 ) @ X7 ) @ X6 )
% 2.38/1.46 => ( ! [X6: $i,X7: $i,X8: $i] : ( X2 @ ( X5 @ ( X5 @ ( X5 @ X4 @ X6 ) @ X7 ) @ X8 ) @ ( X5 @ ( X5 @ X6 @ X8 ) @ ( X5 @ X7 @ X8 ) ) )
% 2.38/1.46 => ( X2 @ X0 @ X1 ) ) ) ) ) )
% 2.38/1.46 @ ( c_Inj0 @ c_Empty )
% 2.38/1.46 @ ( c_Inj0 @ ( c_Power @ c_Empty ) )
% 2.38/1.46 @ ^ [X3: $i,X4: $i] : ( c_Inj1 @ ( c_setsum @ X3 @ X4 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(ax108,axiom,
% 2.38/1.46 ( c_equip_mod
% 2.38/1.46 = ( ^ [X0: $i,X1: $i,X2: $i] :
% 2.38/1.46 ? [X3: $i,X4: $i] : ( c_or @ ( c_and @ ( c_equip @ ( c_setsum @ X0 @ X3 ) @ X1 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) @ ( c_and @ ( c_equip @ ( c_setsum @ X1 @ X3 ) @ X0 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(conj,conjecture,
% 2.38/1.46 ! [X0: $i] :
% 2.38/1.46 ( ( c_In @ X0 @ c_Empty )
% 2.38/1.46 => ! [X1: $i] :
% 2.38/1.46 ( ( c_In @ X1 @ X0 )
% 2.38/1.46 => ! [X2: $i] :
% 2.38/1.46 ( ( c_In @ X2 @ X1 )
% 2.38/1.46 => ? [X3: $i] :
% 2.38/1.46 ! [X4: $i] :
% 2.38/1.46 ( ( ( c_ordinal @ X3 )
% 2.38/1.46 => ( c_and @ ( c_exactly2 @ X3 ) @ ( c_not @ ( c_exactly5 @ X4 ) ) ) )
% 2.38/1.46 => ( ( ( ( c_ordinal @ X3 )
% 2.38/1.46 => ( c_exactly2 @ X3 ) )
% 2.38/1.46 => ( ( c_atleast2 @ X3 )
% 2.38/1.46 => ( ( ( ( c_not @ ( c_atleast2 @ ( c_Power @ ( c_binrep @ ( c_Power @ ( c_Power @ c_Empty ) ) @ c_Empty ) ) ) )
% 2.38/1.46 => ( X3 = X4 ) )
% 2.38/1.46 => ( c_atleast3 @ X3 ) )
% 2.38/1.46 => ( c_and @ ( c_atleast5 @ X2 ) @ ( c_not @ ( c_exactly1of2 @ ( c_SNoLt @ X2 @ ( c_binrep @ ( c_Power @ ( c_Power @ ( c_Power @ ( c_Power @ c_Empty ) ) ) ) @ c_Empty ) ) @ ( c_TransSet @ X4 ) ) ) ) ) ) )
% 2.38/1.46 => ( c_nat_p @ X3 ) ) ) ) ) ) ).
% 2.38/1.46 ------------------------
% 2.38/1.48 % (9391)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/183Mi)
% 2.38/1.48 % (9394)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/2Mi)
% 2.38/1.48 % (9397)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/18Mi)
% 2.38/1.48 % (9394)Instruction limit reached!
% 2.38/1.48 % (9394)------------------------------
% 2.38/1.48 % (9394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.48 % (9394)Termination reason: Unknown
% 2.38/1.48 % (9394)Termination phase: shuffling
% 2.38/1.48
% 2.38/1.48 % (9394)Memory used [KB]: 1279
% 2.38/1.48 % (9394)Time elapsed: 0.002 s
% 2.38/1.48 % (9394)Instructions burned: 3 (million)
% 2.38/1.48 % (9394)------------------------------
% 2.38/1.48 % (9394)------------------------------
% 2.38/1.48 % (9397)Instruction limit reached!
% 2.38/1.48 % (9397)------------------------------
% 2.38/1.48 % (9397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.48 % (9397)Termination reason: Unknown
% 2.38/1.48 % (9397)Termination phase: Property scanning
% 2.38/1.48
% 2.38/1.48 % (9397)Memory used [KB]: 1535
% 2.38/1.48 % (9397)Time elapsed: 0.008 s
% 2.38/1.48 % (9397)Instructions burned: 21 (million)
% 2.38/1.48 % (9397)------------------------------
% 2.38/1.48 % (9397)------------------------------
% 2.38/1.48 % (9392)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_9274 for (2999ds/4Mi)
% 2.38/1.48 % (9393)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_9274 for (2999ds/27Mi)
% 2.38/1.48 % (9395)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_9274 for (2999ds/2Mi)
% 2.38/1.49 % (9396)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_9274 for (2999ds/275Mi)
% 2.38/1.49 % (9395)Instruction limit reached!
% 2.38/1.49 % (9395)------------------------------
% 2.38/1.49 % (9395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (9395)Termination reason: Unknown
% 2.38/1.49 % (9395)Termination phase: shuffling
% 2.38/1.49
% 2.38/1.49 % (9395)Memory used [KB]: 1151
% 2.38/1.49 % (9395)Time elapsed: 0.003 s
% 2.38/1.49 % (9395)Instructions burned: 2 (million)
% 2.38/1.49 % (9395)------------------------------
% 2.38/1.49 % (9395)------------------------------
% 2.38/1.49 % (9392)Instruction limit reached!
% 2.38/1.49 % (9392)------------------------------
% 2.38/1.49 % (9392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (9392)Termination reason: Unknown
% 2.38/1.49 % (9392)Termination phase: shuffling
% 2.38/1.49
% 2.38/1.49 % (9392)Memory used [KB]: 1279
% 2.38/1.49 % (9392)Time elapsed: 0.004 s
% 2.38/1.49 % (9392)Instructions burned: 4 (million)
% 2.38/1.49 % (9392)------------------------------
% 2.38/1.49 % (9392)------------------------------
% 2.38/1.49 % (9398)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/3Mi)
% 2.38/1.49 % (9398)Instruction limit reached!
% 2.38/1.49 % (9398)------------------------------
% 2.38/1.49 % (9398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (9398)Termination reason: Unknown
% 2.38/1.49 % (9398)Termination phase: shuffling
% 2.38/1.49
% 2.38/1.49 % (9398)Memory used [KB]: 1279
% 2.38/1.49 % (9398)Time elapsed: 0.003 s
% 2.38/1.49 % (9398)Instructions burned: 4 (million)
% 2.38/1.49 % (9398)------------------------------
% 2.38/1.49 % (9398)------------------------------
% 2.38/1.49 % (9399)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_9274 for (2999ds/37Mi)
% 2.45/1.50 % (9393)Instruction limit reached!
% 2.45/1.50 % (9393)------------------------------
% 2.45/1.50 % (9393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (9393)Termination reason: Unknown
% 2.45/1.50 % (9393)Termination phase: Preprocessing 3
% 2.45/1.50
% 2.45/1.50 % (9393)Memory used [KB]: 1535
% 2.45/1.50 % (9393)Time elapsed: 0.016 s
% 2.45/1.50 % (9393)Instructions burned: 28 (million)
% 2.45/1.50 % (9393)------------------------------
% 2.45/1.50 % (9393)------------------------------
% 2.45/1.50 % (9402)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_9274 for (2999ds/1041Mi)
% 2.45/1.50 % (9400)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_9274 for (2999ds/15Mi)
% 2.45/1.50 % (9399)Instruction limit reached!
% 2.45/1.50 % (9399)------------------------------
% 2.45/1.50 % (9399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (9399)Termination reason: Unknown
% 2.45/1.50 % (9399)Termination phase: Function definition elimination
% 2.45/1.50
% 2.45/1.50 % (9399)Memory used [KB]: 1535
% 2.45/1.50 % (9399)Time elapsed: 0.012 s
% 2.45/1.50 % (9399)Instructions burned: 38 (million)
% 2.45/1.50 % (9399)------------------------------
% 2.45/1.50 % (9399)------------------------------
% 2.45/1.50 % (9401)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_9274 for (2999ds/3Mi)
% 2.45/1.51 % (9401)Instruction limit reached!
% 2.45/1.51 % (9401)------------------------------
% 2.45/1.51 % (9401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.51 % (9401)Termination reason: Unknown
% 2.45/1.51 % (9401)Termination phase: shuffling
% 2.45/1.51
% 2.45/1.51 % (9401)Memory used [KB]: 1279
% 2.45/1.51 % (9401)Time elapsed: 0.004 s
% 2.45/1.51 % (9401)Instructions burned: 4 (million)
% 2.45/1.51 % (9401)------------------------------
% 2.45/1.51 % (9401)------------------------------
% 2.45/1.51 % (9400)Instruction limit reached!
% 2.45/1.51 % (9400)------------------------------
% 2.45/1.51 % (9400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.51 % (9400)Termination reason: Unknown
% 2.45/1.51 % (9400)Termination phase: shuffling
% 2.45/1.51
% 2.45/1.51 % (9400)Memory used [KB]: 1407
% 2.45/1.51 % (9400)Time elapsed: 0.010 s
% 2.45/1.51 % (9400)Instructions burned: 16 (million)
% 2.45/1.51 % (9400)------------------------------
% 2.45/1.51 % (9400)------------------------------
% 2.45/1.51 % (9404)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/16Mi)
% 2.45/1.52 % (9403)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/7Mi)
% 2.45/1.52 % (9404)Instruction limit reached!
% 2.45/1.52 % (9404)------------------------------
% 2.45/1.52 % (9404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (9404)Termination reason: Unknown
% 2.45/1.52 % (9404)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52 % (9404)Memory used [KB]: 1407
% 2.45/1.52 % (9404)Time elapsed: 0.005 s
% 2.45/1.52 % (9404)Instructions burned: 16 (million)
% 2.45/1.52 % (9404)------------------------------
% 2.45/1.52 % (9404)------------------------------
% 2.45/1.52 % (9391)Instruction limit reached!
% 2.45/1.52 % (9391)------------------------------
% 2.45/1.52 % (9391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (9391)Termination reason: Unknown
% 2.45/1.52 % (9391)Termination phase: Saturation
% 2.45/1.52
% 2.45/1.52 % (9391)Memory used [KB]: 6268
% 2.45/1.52 % (9391)Time elapsed: 0.043 s
% 2.45/1.52 % (9391)Instructions burned: 184 (million)
% 2.45/1.52 % (9391)------------------------------
% 2.45/1.52 % (9391)------------------------------
% 2.45/1.52 % (9403)Instruction limit reached!
% 2.45/1.52 % (9403)------------------------------
% 2.45/1.52 % (9403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (9403)Termination reason: Unknown
% 2.45/1.52 % (9403)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52 % (9403)Memory used [KB]: 1279
% 2.45/1.52 % (9403)Time elapsed: 0.006 s
% 2.45/1.52 % (9403)Instructions burned: 8 (million)
% 2.45/1.52 % (9403)------------------------------
% 2.45/1.52 % (9403)------------------------------
% 2.45/1.52 % (9405)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_9274 for (2999ds/3Mi)
% 2.45/1.53 % (9405)Instruction limit reached!
% 2.45/1.53 % (9405)------------------------------
% 2.45/1.53 % (9405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.53 % (9405)Termination reason: Unknown
% 2.45/1.53 % (9405)Termination phase: shuffling
% 2.45/1.53
% 2.45/1.53 % (9405)Memory used [KB]: 1279
% 2.45/1.53 % (9405)Time elapsed: 0.004 s
% 2.45/1.53 % (9405)Instructions burned: 4 (million)
% 2.45/1.53 % (9405)------------------------------
% 2.45/1.53 % (9405)------------------------------
% 2.45/1.53 % (9407)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/7Mi)
% 2.45/1.53 % (9408)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/3Mi)
% 2.45/1.53 % (9408)Instruction limit reached!
% 2.45/1.53 % (9408)------------------------------
% 2.45/1.53 % (9408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.53 % (9408)Termination reason: Unknown
% 2.45/1.53 % (9408)Termination phase: shuffling
% 2.45/1.53
% 2.45/1.53 % (9408)Memory used [KB]: 1279
% 2.45/1.53 % (9408)Time elapsed: 0.002 s
% 2.45/1.53 % (9407)Instruction limit reached!
% 2.45/1.53 % (9407)------------------------------
% 2.45/1.53 % (9407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.53 % (9408)Instructions burned: 4 (million)
% 2.45/1.53 % (9408)------------------------------
% 2.45/1.53 % (9408)------------------------------
% 2.45/1.53 % (9407)Termination reason: Unknown
% 2.45/1.53 % (9407)Termination phase: shuffling
% 2.45/1.53
% 2.45/1.53 % (9407)Memory used [KB]: 1279
% 2.45/1.53 % (9407)Time elapsed: 0.003 s
% 2.45/1.53 % (9407)Instructions burned: 8 (million)
% 2.45/1.53 % (9406)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_9274 for (2999ds/3Mi)
% 2.45/1.53 % (9407)------------------------------
% 2.45/1.53 % (9407)------------------------------
% 2.45/1.53 % (9406)Instruction limit reached!
% 2.45/1.53 % (9406)------------------------------
% 2.45/1.53 % (9406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.53 % (9406)Termination reason: Unknown
% 2.45/1.53 % (9406)Termination phase: shuffling
% 2.45/1.53
% 2.45/1.53 % (9406)Memory used [KB]: 1279
% 2.45/1.53 % (9406)Time elapsed: 0.004 s
% 2.45/1.53 % (9406)Instructions burned: 4 (million)
% 2.45/1.53 % (9406)------------------------------
% 2.45/1.53 % (9406)------------------------------
% 2.45/1.54 % (9409)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/4Mi)
% 2.45/1.54 % (9409)Instruction limit reached!
% 2.45/1.54 % (9409)------------------------------
% 2.45/1.54 % (9409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.54 % (9409)Termination reason: Unknown
% 2.45/1.54 % (9409)Termination phase: shuffling
% 2.45/1.54
% 2.45/1.54 % (9409)Memory used [KB]: 1279
% 2.45/1.54 % (9409)Time elapsed: 0.003 s
% 2.45/1.54 % (9409)Instructions burned: 5 (million)
% 2.45/1.54 % (9409)------------------------------
% 2.45/1.54 % (9409)------------------------------
% 2.45/1.54 % (9411)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/710Mi)
% 2.45/1.54 % (9412)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/6Mi)
% 2.45/1.54 % (9412)Instruction limit reached!
% 2.45/1.54 % (9412)------------------------------
% 2.45/1.54 % (9412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.54 % (9412)Termination reason: Unknown
% 2.45/1.54 % (9412)Termination phase: shuffling
% 2.45/1.54
% 2.45/1.54 % (9412)Memory used [KB]: 1279
% 2.45/1.54 % (9412)Time elapsed: 0.003 s
% 2.45/1.54 % (9412)Instructions burned: 8 (million)
% 2.45/1.54 % (9412)------------------------------
% 2.45/1.54 % (9412)------------------------------
% 2.45/1.54 % (9410)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/18Mi)
% 2.45/1.55 % (9414)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_9274 for (2999ds/21Mi)
% 2.45/1.55 % (9413)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_9274 for (2999ds/902Mi)
% 2.45/1.55 % (9415)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9274 for (2999ds/5Mi)
% 2.45/1.55 % (9415)Instruction limit reached!
% 2.45/1.55 % (9415)------------------------------
% 2.45/1.55 % (9415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.55 % (9415)Termination reason: Unknown
% 2.45/1.55 % (9415)Termination phase: shuffling
% 2.45/1.55
% 2.45/1.55 % (9415)Memory used [KB]: 1279
% 2.45/1.55 % (9415)Time elapsed: 0.003 s
% 2.45/1.55 % (9415)Instructions burned: 8 (million)
% 2.45/1.55 % (9415)------------------------------
% 2.45/1.55 % (9415)------------------------------
% 2.45/1.55 % (9410)Instruction limit reached!
% 2.45/1.55 % (9410)------------------------------
% 2.45/1.55 % (9410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.55 % (9410)Termination reason: Unknown
% 2.45/1.55 % (9410)Termination phase: shuffling
% 2.45/1.55
% 2.45/1.55 % (9410)Memory used [KB]: 1535
% 2.45/1.55 % (9410)Time elapsed: 0.011 s
% 2.45/1.55 % (9410)Instructions burned: 18 (million)
% 2.45/1.55 % (9410)------------------------------
% 2.45/1.55 % (9410)------------------------------
% 2.45/1.55 % (9414)Instruction limit reached!
% 2.45/1.55 % (9414)------------------------------
% 2.45/1.55 % (9414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.55 % (9414)Termination reason: Unknown
% 2.45/1.55 % (9414)Termination phase: Property scanning
% 2.45/1.55
% 2.45/1.55 % (9414)Memory used [KB]: 1535
% 2.45/1.55 % (9414)Time elapsed: 0.008 s
% 2.45/1.55 % (9414)Instructions burned: 22 (million)
% 2.45/1.55 % (9414)------------------------------
% 2.45/1.55 % (9414)------------------------------
% 2.45/1.56 % (9411)First to succeed.
% 2.45/1.56 % (9416)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_9274 for (2999ds/6Mi)
% 2.45/1.56 % (9411)Refutation found. Thanks to Tanya!
% 2.45/1.56 % SZS status Theorem for DTF2THF_9274
% 2.45/1.56 % SZS output start Proof for DTF2THF_9274
% 2.45/1.56 thf(func_def_0, type, c_Eps_i: ($i > $o) > $i).
% 2.45/1.56 thf(func_def_3, type, c_not: $o > $o).
% 2.45/1.56 thf(func_def_4, type, c_and: $o > $o > $o).
% 2.45/1.56 thf(func_def_5, type, c_or: $o > $o > $o).
% 2.45/1.56 thf(func_def_6, type, c_iff: $o > $o > $o).
% 2.45/1.56 thf(func_def_7, type, c_In: $i > $i > $o).
% 2.45/1.56 thf(func_def_8, type, c_Subq: $i > $i > $o).
% 2.45/1.56 thf(func_def_10, type, c_Union: $i > $i).
% 2.45/1.56 thf(func_def_11, type, c_Power: $i > $i).
% 2.45/1.56 thf(func_def_12, type, c_Repl: $i > ($i > $i) > $i).
% 2.45/1.56 thf(func_def_13, type, c_TransSet: $i > $o).
% 2.45/1.56 thf(func_def_14, type, c_atleast2: $i > $o).
% 2.45/1.56 thf(func_def_15, type, c_atleast3: $i > $o).
% 2.45/1.56 thf(func_def_16, type, c_atleast4: $i > $o).
% 2.45/1.56 thf(func_def_17, type, c_atleast5: $i > $o).
% 2.45/1.56 thf(func_def_18, type, c_atleast6: $i > $o).
% 2.45/1.56 thf(func_def_19, type, c_exactly2: $i > $o).
% 2.45/1.56 thf(func_def_20, type, c_exactly3: $i > $o).
% 2.45/1.56 thf(func_def_21, type, c_exactly4: $i > $o).
% 2.45/1.56 thf(func_def_22, type, c_exactly5: $i > $o).
% 2.45/1.56 thf(func_def_23, type, c_exu_i: ($i > $o) > $o).
% 2.45/1.56 thf(func_def_24, type, c_reflexive_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_25, type, c_irreflexive_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_26, type, c_symmetric_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_27, type, c_antisymmetric_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_28, type, c_transitive_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_29, type, c_eqreln_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_30, type, c_per_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_31, type, c_linear_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_32, type, c_trichotomous_or_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_33, type, c_partialorder_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_34, type, c_totalorder_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_35, type, c_strictpartialorder_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_36, type, c_stricttotalorder_i: ($i > $i > $o) > $o).
% 2.45/1.56 thf(func_def_37, type, c_If_i: $o > $i > $i > $i).
% 2.45/1.56 thf(func_def_38, type, c_exactly1of2: $o > $o > $o).
% 2.45/1.56 thf(func_def_39, type, c_exactly1of3: $o > $o > $o > $o).
% 2.45/1.56 thf(func_def_40, type, c_nIn: $i > $i > $o).
% 2.45/1.56 thf(func_def_41, type, c_nSubq: $i > $i > $o).
% 2.45/1.56 thf(func_def_42, type, c_UPair: $i > $i > $i).
% 2.45/1.56 thf(func_def_43, type, c_Sing: $i > $i).
% 2.45/1.56 thf(func_def_44, type, c_binunion: $i > $i > $i).
% 2.45/1.56 thf(func_def_45, type, c_SetAdjoin: $i > $i > $i).
% 2.45/1.56 thf(func_def_46, type, c_famunion: $i > ($i > $i) > $i).
% 2.45/1.56 thf(func_def_47, type, c_Sep: $i > ($i > $o) > $i).
% 2.45/1.56 thf(func_def_48, type, c_ReplSep: $i > ($i > $o) > ($i > $i) > $i).
% 2.45/1.56 thf(func_def_49, type, c_binintersect: $i > $i > $i).
% 2.45/1.56 thf(func_def_50, type, c_setminus: $i > $i > $i).
% 2.45/1.56 thf(func_def_51, type, c_inj: $i > $i > ($i > $i) > $o).
% 2.45/1.56 thf(func_def_52, type, c_bij: $i > $i > ($i > $i) > $o).
% 2.45/1.56 thf(func_def_53, type, c_atleastp: $i > $i > $o).
% 2.45/1.56 thf(func_def_54, type, c_equip: $i > $i > $o).
% 2.45/1.56 thf(func_def_55, type, c_In_rec_poly_G_i: ($i > ($i > $i) > $i) > $i > $i > $o).
% 2.45/1.56 thf(func_def_56, type, c_In_rec_poly_i: ($i > ($i > $i) > $i) > $i > $i).
% 2.45/1.56 thf(func_def_57, type, c_ordsucc: $i > $i).
% 2.45/1.56 thf(func_def_58, type, c_nat_p: $i > $o).
% 2.45/1.56 thf(func_def_59, type, c_nat_primrec: $i > ($i > $i > $i) > $i > $i).
% 2.45/1.56 thf(func_def_60, type, c_add_nat: $i > $i > $i).
% 2.45/1.56 thf(func_def_61, type, c_mul_nat: $i > $i > $i).
% 2.45/1.56 thf(func_def_62, type, c_ordinal: $i > $o).
% 2.45/1.56 thf(func_def_63, type, c_V_: $i > $i).
% 2.45/1.56 thf(func_def_64, type, c_Inj1: $i > $i).
% 2.45/1.56 thf(func_def_65, type, c_Inj0: $i > $i).
% 2.45/1.56 thf(func_def_66, type, c_Unj: $i > $i).
% 2.45/1.56 thf(func_def_67, type, c_combine_funcs: $i > $i > ($i > $i) > ($i > $i) > $i > $i).
% 2.45/1.56 thf(func_def_68, type, c_setsum: $i > $i > $i).
% 2.45/1.56 thf(func_def_69, type, c_proj0: $i > $i).
% 2.45/1.56 thf(func_def_70, type, c_proj1: $i > $i).
% 2.45/1.56 thf(func_def_71, type, c_binrep: $i > $i > $i).
% 2.45/1.56 thf(func_def_72, type, c_lam: $i > ($i > $i) > $i).
% 2.45/1.56 thf(func_def_73, type, c_setprod: $i > $i > $i).
% 2.45/1.56 thf(func_def_74, type, c_ap: $i > $i > $i).
% 2.45/1.56 thf(func_def_75, type, c_setsum_p: $i > $o).
% 2.45/1.56 thf(func_def_76, type, c_tuple_p: $i > $i > $o).
% 2.45/1.56 thf(func_def_77, type, c_Pi: $i > ($i > $i) > $i).
% 2.45/1.56 thf(func_def_78, type, c_setexp: $i > $i > $i).
% 2.45/1.56 thf(func_def_79, type, c_Sep2: $i > ($i > $i) > ($i > $i > $o) > $i).
% 2.45/1.56 thf(func_def_80, type, c_set_of_pairs: $i > $o).
% 2.45/1.56 thf(func_def_81, type, c_lam2: $i > ($i > $i) > ($i > $i > $i) > $i).
% 2.45/1.56 thf(func_def_82, type, c_PNoEq_: $i > ($i > $o) > ($i > $o) > $o).
% 2.45/1.56 thf(func_def_83, type, c_PNoLt_: $i > ($i > $o) > ($i > $o) > $o).
% 2.45/1.56 thf(func_def_84, type, c_PNoLt: $i > ($i > $o) > $i > ($i > $o) > $o).
% 2.45/1.56 thf(func_def_85, type, c_PNoLe: $i > ($i > $o) > $i > ($i > $o) > $o).
% 2.45/1.56 thf(func_def_86, type, c_PNo_downc: ($i > ($i > $o) > $o) > $i > ($i > $o) > $o).
% 2.45/1.56 thf(func_def_87, type, c_PNo_upc: ($i > ($i > $o) > $o) > $i > ($i > $o) > $o).
% 2.45/1.56 thf(func_def_88, type, c_SNoElts_: $i > $i).
% 2.45/1.56 thf(func_def_89, type, c_SNo_: $i > $i > $o).
% 2.45/1.56 thf(func_def_90, type, c_PSNo: $i > ($i > $o) > $i).
% 2.45/1.56 thf(func_def_91, type, c_SNo: $i > $o).
% 2.45/1.56 thf(func_def_92, type, c_SNoLev: $i > $i).
% 2.45/1.56 thf(func_def_93, type, c_SNoEq_: $i > $i > $i > $o).
% 2.45/1.56 thf(func_def_94, type, c_SNoLt: $i > $i > $o).
% 2.45/1.56 thf(func_def_95, type, c_SNoLe: $i > $i > $o).
% 2.45/1.56 thf(func_def_96, type, c_binop_on: $i > ($i > $i > $i) > $o).
% 2.45/1.56 thf(func_def_97, type, c_Loop: $i > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $i > $o).
% 2.45/1.56 thf(func_def_98, type, c_Loop_with_defs: $i > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $i > ($i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $o).
% 2.45/1.56 thf(func_def_99, type, c_Loop_with_defs_cex1: $i > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $i > ($i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $o).
% 2.45/1.56 thf(func_def_100, type, c_Loop_with_defs_cex2: $i > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $i > ($i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > ($i > $i > $i) > $o).
% 2.45/1.56 thf(func_def_101, type, c_combinator: $i > $o).
% 2.45/1.56 thf(func_def_102, type, c_combinator_equiv: $i > $i > $o).
% 2.45/1.56 thf(func_def_103, type, c_equip_mod: $i > $i > $i > $o).
% 2.45/1.56 thf(func_def_134, type, sK3: $i > $i).
% 2.45/1.56 thf(func_def_135, type, sK4: ($i > $o) > $i).
% 2.45/1.56 thf(func_def_137, type, sK6: $i > $i > $o).
% 2.45/1.56 thf(func_def_139, type, ph7: !>[X0: $tType]:(X0)).
% 2.45/1.56 thf(f421,plain,(
% 2.45/1.56 $false),
% 2.45/1.56 inference(trivial_inequality_removal,[status(thm)],[f418])).
% 2.45/1.56 thf(f418,plain,(
% 2.45/1.56 ($false = $true)),
% 2.45/1.56 inference(superposition,[status(thm)],[f370,f371])).
% 2.45/1.56 thf(f371,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (($false = (sK6 @ X3 @ X3))) )),
% 2.45/1.56 inference(binary_proxy_clausification,[status(thm)],[f369])).
% 2.45/1.56 thf(f369,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (($false = ((!! @ $i @ (^[Y0 : $i]: ((sK6 @ X3 @ Y0) => (sK6 @ X3 @ (c_ordsucc @ Y0))))) => (sK6 @ X3 @ X3)))) )),
% 2.45/1.56 inference(binary_proxy_clausification,[status(thm)],[f368])).
% 2.45/1.56 thf(f368,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (($false = ((sK6 @ X3 @ c_Empty) => ((!! @ $i @ (^[Y0 : $i]: ((sK6 @ X3 @ Y0) => (sK6 @ X3 @ (c_ordsucc @ Y0))))) => (sK6 @ X3 @ X3))))) )),
% 2.45/1.56 inference(beta_eta_normalization,[status(thm)],[f367])).
% 2.45/1.56 thf(f367,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (($false = ((^[Y0 : $i > $o]: ((Y0 @ c_Empty) => ((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (c_ordsucc @ Y1))))) => (Y0 @ X3)))) @ (sK6 @ X3)))) )),
% 2.45/1.56 inference(sigma_clausification,[status(thm)],[f366])).
% 2.45/1.56 thf(f366,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (((!! @ ($i > $o) @ (^[Y0 : $i > $o]: ((Y0 @ c_Empty) => ((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (c_ordsucc @ Y1))))) => (Y0 @ X3))))) != $true)) )),
% 2.45/1.56 inference(beta_eta_normalization,[status(thm)],[f354])).
% 2.45/1.56 thf(f354,plain,(
% 2.45/1.56 ( ! [X3 : $i] : ((((^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((Y1 @ c_Empty) => ((!! @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) => (Y1 @ (c_ordsucc @ Y2))))) => (Y1 @ Y0)))))) @ X3) != $true)) )),
% 2.45/1.56 inference(definition_unfolding,[status(thm)],[f308,f325])).
% 2.45/1.56 thf(f325,plain,(
% 2.45/1.56 (c_nat_p = (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((Y1 @ c_Empty) => ((!! @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) => (Y1 @ (c_ordsucc @ Y2))))) => (Y1 @ Y0)))))))),
% 2.45/1.56 inference(cnf_transformation,[status(thm)],[f225])).
% 2.45/1.56 thf(f225,plain,(
% 2.45/1.56 (c_nat_p = (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((Y1 @ c_Empty) => ((!! @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) => (Y1 @ (c_ordsucc @ Y2))))) => (Y1 @ Y0)))))))),
% 2.45/1.56 inference(fool_elimination,[status(thm)],[f224])).
% 2.45/1.56 thf(f224,plain,(
% 2.45/1.56 (c_nat_p = (^[X0 : $i] : (! [X1 : $i > $o] : ((X1 @ c_Empty) => (! [X2 : $i] : ((X1 @ X2) => (X1 @ (c_ordsucc @ X2))) => (X1 @ X0))))))),
% 2.45/1.56 inference(rectify,[status(thm)],[f63])).
% 2.45/1.56 thf(f63,axiom,(
% 2.45/1.56 (c_nat_p = (^[X0 : $i] : (! [X1 : $i > $o] : ((X1 @ c_Empty) => (! [X2 : $i] : ((X1 @ X2) => (X1 @ (c_ordsucc @ X2))) => (X1 @ X0))))))),
% 2.45/1.56 file('/export/starexec/sandbox/tmp/tmp.meWfZCKs7P/DTF2THF_9274.p',ax63)).
% 2.45/1.56 thf(f308,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (((c_nat_p @ X3) != $true)) )),
% 2.45/1.56 inference(cnf_transformation,[status(thm)],[f296])).
% 2.45/1.56 thf(f296,plain,(
% 2.45/1.56 ((c_In @ sK0 @ c_Empty) = $true) & ((! [X3 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ (sK3 @ X3)))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ sK2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ sK2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ (sK3 @ X3))))) = $true) | ((((sK3 @ X3) = X3) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ((c_In @ sK2 @ sK1) = $true)) & ((c_In @ sK1 @ sK0) = $true))),
% 2.45/1.56 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f291,f295,f294,f293,f292])).
% 2.45/1.56 thf(f292,plain,(
% 2.45/1.56 ? [X0 : $i] : (((c_In @ X0 @ c_Empty) = $true) & ? [X1 : $i] : (? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ((c_In @ X2 @ X1) = $true)) & ((c_In @ X1 @ X0) = $true))) => (((c_In @ sK0 @ c_Empty) = $true) & ? [X1 : $i] : (? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ((c_In @ X2 @ X1) = $true)) & ((c_In @ X1 @ sK0) = $true)))),
% 2.45/1.56 introduced(definition,[],[choice_axiom])).
% 2.45/1.56 thf(f293,plain,(
% 2.45/1.56 ? [X1 : $i] : (? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ((c_In @ X2 @ X1) = $true)) & ((c_In @ X1 @ sK0) = $true)) => (? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ($true = (c_In @ X2 @ sK1))) & ((c_In @ sK1 @ sK0) = $true))),
% 2.45/1.56 introduced(definition,[],[choice_axiom])).
% 2.45/1.56 thf(f294,plain,(
% 2.45/1.56 ? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ($true = (c_In @ X2 @ sK1))) => (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ($true = (c_and @ (c_atleast5 @ sK2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ sK2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4))))) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ((c_In @ sK2 @ sK1) = $true))),
% 2.45/1.56 introduced(definition,[],[choice_axiom])).
% 2.45/1.56 thf(f295,plain,(
% 2.45/1.56 ! [X3 : $i] : (? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ($true = (c_and @ (c_atleast5 @ sK2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ sK2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4))))) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) => ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ (sK3 @ X3)))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ sK2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ sK2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ (sK3 @ X3))))) = $true) | ((((sK3 @ X3) = X3) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)))),
% 2.45/1.56 introduced(definition,[],[choice_axiom])).
% 2.45/1.56 thf(f291,plain,(
% 2.45/1.56 ? [X0 : $i] : (((c_In @ X0 @ c_Empty) = $true) & ? [X1 : $i] : (? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) & (((c_atleast2 @ X3) != $true) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)) | ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) & ((c_nat_p @ X3) != $true)) & ((c_In @ X2 @ X1) = $true)) & ((c_In @ X1 @ X0) = $true)))),
% 2.45/1.56 inference(flattening,[status(thm)],[f290])).
% 2.45/1.56 thf(f290,plain,(
% 2.45/1.56 ? [X0 : $i] : (? [X1 : $i] : (? [X2 : $i] : (! [X3 : $i] : ? [X4 : $i] : ((((c_nat_p @ X3) != $true) & (((((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true) | (((X3 = X4) | ((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) != $true)) & ((c_atleast3 @ X3) != $true))) | ((c_atleast2 @ X3) != $true)) | (((c_ordinal @ X3) = $true) & ((c_exactly2 @ X3) != $true)))) & (((c_ordinal @ X3) != $true) | ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true))) & ((c_In @ X2 @ X1) = $true)) & ((c_In @ X1 @ X0) = $true)) & ((c_In @ X0 @ c_Empty) = $true))),
% 2.45/1.56 inference(ennf_transformation,[status(thm)],[f262])).
% 2.45/1.56 thf(f262,plain,(
% 2.45/1.56 ~! [X0 : $i] : (((c_In @ X0 @ c_Empty) = $true) => ! [X1 : $i] : (((c_In @ X1 @ X0) = $true) => ! [X2 : $i] : (((c_In @ X2 @ X1) = $true) => ? [X3 : $i] : ! [X4 : $i] : ((((c_ordinal @ X3) = $true) => ((c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4))) = $true)) => (((((c_ordinal @ X3) = $true) => ((c_exactly2 @ X3) = $true)) => (((c_atleast2 @ X3) = $true) => (((((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) = $true) => (X3 = X4)) => ((c_atleast3 @ X3) = $true)) => ((c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4)))) = $true)))) => ((c_nat_p @ X3) = $true))))))),
% 2.45/1.56 inference(fool_elimination,[status(thm)],[f261])).
% 2.45/1.56 thf(f261,plain,(
% 2.45/1.56 ~! [X0 : $i] : ((c_In @ X0 @ c_Empty) => ! [X1 : $i] : ((c_In @ X1 @ X0) => ! [X2 : $i] : ((c_In @ X2 @ X1) => ? [X3 : $i] : ! [X4 : $i] : (((c_ordinal @ X3) => (c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4)))) => ((((c_ordinal @ X3) => (c_exactly2 @ X3)) => ((c_atleast2 @ X3) => ((((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) => (X3 = X4)) => (c_atleast3 @ X3)) => (c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4))))))) => (c_nat_p @ X3))))))),
% 2.45/1.56 inference(rectify,[status(thm)],[f110])).
% 2.45/1.56 thf(f110,negated_conjecture,(
% 2.45/1.56 ~! [X0 : $i] : ((c_In @ X0 @ c_Empty) => ! [X1 : $i] : ((c_In @ X1 @ X0) => ! [X2 : $i] : ((c_In @ X2 @ X1) => ? [X3 : $i] : ! [X4 : $i] : (((c_ordinal @ X3) => (c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4)))) => ((((c_ordinal @ X3) => (c_exactly2 @ X3)) => ((c_atleast2 @ X3) => ((((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) => (X3 = X4)) => (c_atleast3 @ X3)) => (c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4))))))) => (c_nat_p @ X3))))))),
% 2.45/1.56 inference(negated_conjecture,[status(cth)],[f109])).
% 2.45/1.56 thf(f109,conjecture,(
% 2.45/1.56 ! [X0 : $i] : ((c_In @ X0 @ c_Empty) => ! [X1 : $i] : ((c_In @ X1 @ X0) => ! [X2 : $i] : ((c_In @ X2 @ X1) => ? [X3 : $i] : ! [X4 : $i] : (((c_ordinal @ X3) => (c_and @ (c_exactly2 @ X3) @ (c_not @ (c_exactly5 @ X4)))) => ((((c_ordinal @ X3) => (c_exactly2 @ X3)) => ((c_atleast2 @ X3) => ((((c_not @ (c_atleast2 @ (c_Power @ (c_binrep @ (c_Power @ (c_Power @ c_Empty)) @ c_Empty)))) => (X3 = X4)) => (c_atleast3 @ X3)) => (c_and @ (c_atleast5 @ X2) @ (c_not @ (c_exactly1of2 @ (c_SNoLt @ X2 @ (c_binrep @ (c_Power @ (c_Power @ (c_Power @ (c_Power @ c_Empty)))) @ c_Empty)) @ (c_TransSet @ X4))))))) => (c_nat_p @ X3))))))),
% 2.45/1.56 file('/export/starexec/sandbox/tmp/tmp.meWfZCKs7P/DTF2THF_9274.p',conj)).
% 2.45/1.56 thf(f370,plain,(
% 2.45/1.56 ( ! [X3 : $i] : (((sK6 @ X3 @ c_Empty) = $true)) )),
% 2.45/1.56 inference(binary_proxy_clausification,[status(thm)],[f368])).
% 2.45/1.56 % SZS output end Proof for DTF2THF_9274
% 2.45/1.56 % (9411)------------------------------
% 2.45/1.56 % (9411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.56 % (9411)Termination reason: Refutation
% 2.45/1.56
% 2.45/1.56 % (9411)Memory used [KB]: 6268
% 2.45/1.56 % (9411)Time elapsed: 0.023 s
% 2.45/1.56 % (9411)Instructions burned: 80 (million)
% 2.45/1.56 % (9411)------------------------------
% 2.45/1.56 % (9411)------------------------------
% 2.45/1.56 % (9390)Success in time 0.079 s
%------------------------------------------------------------------------------