%------------------------------------------------------------------------------
% File : E---3.5.1
% Problem : NUM750^4 : TPTP v9.3.1. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_E /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:17:01 AM UTC 2026
% Result : Theorem 0.84s 0.66s
% Output : CNFRefutation 0.84s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 16
% Syntax : Number of formulae : 55 ( 30 unt; 0 typ; 0 def)
% Number of atoms : 550 ( 446 equ; 0 cnn)
% Maximal formula atoms : 54 ( 10 avg)
% Number of connectives : 2993 ( 444 ~; 21 |; 4 &;2506 @)
% ( 0 <=>; 18 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 32 ( 30 usr; 9 con; 0-3 aty)
% Number of variables : 529 ( 503 ^; 26 !; 0 ?; 529 :)
% Comments :
%------------------------------------------------------------------------------
thf(decl_23,type,
is_of: $i > ( $i > $o ) > $o ).
thf(decl_24,type,
all_of: ( $i > $o ) > ( $i > $o ) > $o ).
thf(decl_26,type,
in: $i > $i > $o ).
thf(decl_28,type,
emptyset: $i ).
thf(decl_43,type,
d_Sep: $i > ( $i > $o ) > $i ).
thf(decl_50,type,
omega: $i ).
thf(decl_59,type,
ap: $i > $i > $i ).
thf(decl_61,type,
d_Pi: $i > ( $i > $i ) > $i ).
thf(decl_78,type,
e_is: $i > $i > $i > $o ).
thf(decl_124,type,
nat: $i ).
thf(decl_125,type,
n_is: $i > $i > $o ).
thf(decl_142,type,
plus: $i > $i ).
thf(decl_143,type,
n_pl: $i > $i > $i ).
thf(decl_162,type,
times: $i > $i ).
thf(decl_163,type,
n_ts: $i > $i > $i ).
thf(decl_168,type,
d_1to: $i > $i ).
thf(decl_173,type,
n_2: $i ).
thf(decl_177,type,
pair1type: $i > $i ).
thf(decl_178,type,
pair1: $i > $i > $i > $i ).
thf(decl_179,type,
first1: $i > $i > $i ).
thf(decl_180,type,
second1: $i > $i > $i ).
thf(decl_190,type,
frac: $i ).
thf(decl_191,type,
n_fr: $i > $i > $i ).
thf(decl_192,type,
num: $i > $i ).
thf(decl_193,type,
den: $i > $i ).
thf(decl_194,type,
n_eq: $i > $i > $o ).
thf(decl_199,type,
n_pf: $i > $i > $i ).
thf(decl_200,type,
esk1_0: $i ).
thf(decl_201,type,
esk2_0: $i ).
thf(decl_202,type,
esk3_0: $i ).
thf(def_nat,axiom,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( X1 != emptyset ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_nat) ).
thf(def_pair1type,axiom,
( pair1type
= ( ^ [X1: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [X425: $i] : X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_pair1type) ).
thf(def_n_pf,axiom,
( n_pf
= ( ^ [X1: $i,X507: $i] : ( n_fr @ ( n_pl @ ( n_ts @ ( num @ X1 ) @ ( den @ X507 ) ) @ ( n_ts @ ( num @ X507 ) @ ( den @ X1 ) ) ) @ ( n_ts @ ( den @ X1 ) @ ( den @ X507 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_n_pf) ).
thf(def_n_pl,axiom,
( n_pl
= ( ^ [X1: $i] : ( ap @ ( plus @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_n_pl) ).
thf(def_n_fr,axiom,
( n_fr
= ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_n_fr) ).
thf(def_num,axiom,
( num
= ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_num) ).
thf(def_den,axiom,
( den
= ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_den) ).
thf(def_n_ts,axiom,
( n_ts
= ( ^ [X1: $i] : ( ap @ ( times @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_n_ts) ).
thf(def_n_eq,axiom,
( n_eq
= ( ^ [X1: $i,X436: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X436 ) ) @ ( n_ts @ ( num @ X436 ) @ ( den @ X1 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_n_eq) ).
thf(def_n_is,axiom,
( n_is
= ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_n_is) ).
thf(def_all_of,axiom,
( all_of
= ( ^ [X3: $i > $o,X2: $i > $o] :
! [X4: $i] :
( ( is_of @ X4 @ X3 )
=> ( X2 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_all_of) ).
thf(def_is_of,axiom,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_is_of) ).
thf(satz62b,conjecture,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X524: $i] : ( in @ X524 @ frac )
@ ^ [X525: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( ( n_eq @ X1 @ X525 )
=> ( n_eq @ ( n_pf @ X1 @ X4 ) @ ( n_pf @ X525 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',satz62b) ).
thf(def_frac,axiom,
( frac
= ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',def_frac) ).
thf(satz56,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X508: $i] : ( in @ X508 @ frac )
@ ^ [X509: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( all_of
@ ^ [X13: $i] : ( in @ X13 @ frac )
@ ^ [X13: $i] :
( ( n_eq @ X1 @ X509 )
=> ( ( n_eq @ X4 @ X13 )
=> ( n_eq @ ( n_pf @ X1 @ X4 ) @ ( n_pf @ X509 @ X13 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',satz56) ).
thf(satz37,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( n_eq @ X1 @ X1 ) ),
file('/export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p',satz37) ).
thf(c_0_16,plain,
( nat
= ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ),
inference(fof_simplification,[status(thm)],[def_nat]) ).
thf(c_0_17,plain,
( pair1type
= ( ^ [Z0: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z1: $i] : Z0 ) ) ),
inference(fof_simplification,[status(thm)],[def_pair1type]) ).
thf(c_0_18,plain,
( n_pf
= ( ^ [Z0: $i,Z1: $i] :
( pair1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[def_n_pf]) ).
thf(c_0_19,plain,
( n_pl
= ( ^ [Z0: $i] : ( ap @ ( plus @ Z0 ) ) ) ),
inference(fof_simplification,[status(thm)],[def_n_pl]) ).
thf(c_0_20,axiom,
( n_fr
= ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[def_n_fr,c_0_16]) ).
thf(c_0_21,axiom,
( num
= ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[def_num,c_0_16]) ).
thf(c_0_22,axiom,
( den
= ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[def_den,c_0_16]) ).
thf(c_0_23,plain,
( n_ts
= ( ^ [Z0: $i] : ( ap @ ( times @ Z0 ) ) ) ),
inference(fof_simplification,[status(thm)],[def_n_ts]) ).
thf(c_0_24,plain,
( n_eq
= ( ^ [Z0: $i,Z1: $i] :
( e_is
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[def_n_eq]) ).
thf(c_0_25,axiom,
( n_is
= ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[def_n_is,c_0_16]) ).
thf(c_0_26,plain,
( all_of
= ( ^ [Z0: $i > $o,Z1: $i > $o] :
! [X4: $i] :
( ( Z0 @ X4 )
=> ( Z1 @ X4 ) ) ) ),
inference(fof_simplification,[status(thm)],[def_all_of]) ).
thf(c_0_27,plain,
( is_of
= ( ^ [Z0: $i,Z1: $i > $o] : ( Z1 @ Z0 ) ) ),
inference(fof_simplification,[status(thm)],[def_is_of]) ).
thf(c_0_28,negated_conjecture,
~ ( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X524: $i] : ( in @ X524 @ frac )
@ ^ [X525: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( ( n_eq @ X1 @ X525 )
=> ( n_eq @ ( n_pf @ X1 @ X4 ) @ ( n_pf @ X525 @ X4 ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[satz62b]) ).
thf(c_0_29,axiom,
( frac
= ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[def_frac,c_0_16]),c_0_17]) ).
thf(c_0_30,plain,
( n_pf
= ( ^ [Z0: $i,Z1: $i] :
( pair1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_18,c_0_19]),c_0_20]),c_0_21]),c_0_22]),c_0_23]) ).
thf(c_0_31,plain,
( n_eq
= ( ^ [Z0: $i,Z1: $i] :
( e_is
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_24,c_0_21]),c_0_22]),c_0_23]),c_0_25]) ).
thf(c_0_32,plain,
( all_of
= ( ^ [Z0: $i > $o,Z1: $i > $o] :
! [X4: $i] :
( ( Z0 @ X4 )
=> ( Z1 @ X4 ) ) ) ),
inference(apply_def,[status(thm)],[c_0_26,c_0_27]) ).
thf(c_0_33,negated_conjecture,
~ ! [X534: $i] :
( ( in @ X534
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X533: $i] :
( ( in @ X533
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X532: $i] :
( ( in @ X532
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) ) )
=> ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X533 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X534 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X532 ) ) ) ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[c_0_28]),c_0_29]),c_0_30]),c_0_31]),c_0_32]) ).
thf(c_0_34,plain,
! [X544: $i] :
( ( in @ X544
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X543: $i] :
( ( in @ X543
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X542: $i] :
( ( in @ X542
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X541: $i] :
( ( in @ X541
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) ) )
=> ( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) ) )
=> ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X543 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X541 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X544 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X542 ) ) ) ) ) ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz56]),c_0_29]),c_0_30]),c_0_31]),c_0_32]) ).
thf(c_0_35,negated_conjecture,
( ( in @ esk1_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
& ( in @ esk2_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
& ( in @ esk3_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
& ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
& ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_33])])])]) ).
thf(c_0_36,plain,
! [X643: $i,X644: $i,X645: $i,X646: $i] :
( ~ ( in @ X643
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X644
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X645
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X646
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) ) )
| ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) ) )
| ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X644 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X646 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X643 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X645 ) ) ) ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_34])])])]) ).
thf(c_0_37,plain,
! [X535: $i] :
( ( in @ X535
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X535 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X535 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X535 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X535 ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz37]),c_0_29]),c_0_31]),c_0_32]) ).
thf(c_0_38,negated_conjecture,
~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
thf(c_0_39,plain,
! [X1: $i,X4: $i,X6: $i,X5: $i] :
( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X6
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_36]) ).
thf(c_0_40,negated_conjecture,
( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) ),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
thf(c_0_41,negated_conjecture,
( in @ esk3_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
thf(c_0_42,negated_conjecture,
( in @ esk2_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
thf(c_0_43,negated_conjecture,
( in @ esk1_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
thf(c_0_44,plain,
! [X637: $i] :
( ~ ( in @ X637
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X637 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X637 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X637 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X637 ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_37])])]) ).
thf(c_0_45,negated_conjecture,
~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) ) ),
inference(cn,[status(thm)],[c_0_38]) ).
thf(c_0_46,plain,
! [X6: $i,X5: $i,X4: $i,X1: $i] :
( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) ) ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( pair1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( plus
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
@ ( ap
@ ( times
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ) )
| ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X6
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_39]) ).
thf(c_0_47,negated_conjecture,
( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) ),
inference(cn,[status(thm)],[c_0_40]) ).
thf(c_0_48,negated_conjecture,
( in @ esk3_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(cn,[status(thm)],[c_0_41]) ).
thf(c_0_49,negated_conjecture,
( in @ esk2_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(cn,[status(thm)],[c_0_42]) ).
thf(c_0_50,negated_conjecture,
( in @ esk1_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(cn,[status(thm)],[c_0_43]) ).
thf(c_0_51,plain,
! [X1: $i] :
( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_44]) ).
thf(c_0_52,negated_conjecture,
~ ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_45,c_0_46]),c_0_47]),c_0_48]),c_0_49]),c_0_50])]) ).
thf(c_0_53,plain,
! [X1: $i] :
( ( e_is
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_51]) ).
thf(c_0_54,negated_conjecture,
$false,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_52,c_0_53]),c_0_48])]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : NUM750^4 : TPTP v9.3.1. Released v7.1.0.
% 0.00/0.06 % Command : run_E /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% 0.11/0.24 % Computer : n020.cluster.edu
% 0.11/0.24 % Model : x86_64 x86_64
% 0.11/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.24 % Memory : 8046.5625MB
% 0.11/0.24 % OS : Linux 6.8.0-71-generic
% 0.11/0.24 % CPULimit : 300
% 0.11/0.24 % WCLimit : 300
% 0.11/0.24 % DateTime : Tue Sep 29 12:51:04 UTC 2026
% 0.11/0.24 % CPUTime :
% 0.11/0.24 Running run_E /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% 0.24/0.29 Running higher-order theorem proving
% 0.24/0.36 Running: /export/starexec/sandbox/solver/bin/eprover-ho --delete-bad-limit=2000000000 --definitional-cnf=24 -s --print-statistics -R --print-version --proof-object --auto-schedule=8 --cpu-limit=300 /export/starexec/sandbox/tmp/tmp.S9dcS69u4X/E---3.1_1003476.p
% 0.84/0.66 % Version: 3.5.1-ho
% 0.84/0.66 % Preprocessing class: HSLMSMSMLLLCHSA.
% 0.84/0.66 % Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 0.84/0.66 % Starting pre_casc_4 with 1200s (4) cores
% 0.84/0.66 % Starting full_lambda_6 with 300s (1) cores
% 0.84/0.66 % Starting sh10 with 300s (1) cores
% 0.84/0.66 % Starting post_as_ho9 with 300s (1) cores
% 0.84/0.66 % Starting post_as_ho8 with 300s (1) cores
% 0.84/0.66 % post_as_ho9 with pid 1003492 completed with status 0
% 0.84/0.66 % Result found by post_as_ho9
% 0.84/0.66 % Preprocessing class: HSLMSMSMLLLCHSA.
% 0.84/0.66 % Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 0.84/0.66 % Starting pre_casc_4 with 1200s (4) cores
% 0.84/0.66 % Starting full_lambda_6 with 300s (1) cores
% 0.84/0.66 % Starting sh10 with 300s (1) cores
% 0.84/0.66 % Starting post_as_ho9 with 300s (1) cores
% 0.84/0.66 % (lift_lambdas = 0, lambda_to_forall = 1,unroll_only_formulas = 0, sine = GSinE(CountFormulas,,true,1,0,2,20000,1.0,true))
% 0.84/0.66 % SinE strategy is GSinE(CountFormulas,,true,1,0,2,20000,1.0,true)
% 0.84/0.66 % Search class: HGUNF-FFLS31-DHSMMFBN
% 0.84/0.66 % partial match(5): HGUNF-FFSS32-SHSSMMBN
% 0.84/0.66 % Scheduled 7 strats onto 1 cores with 300 seconds (300 total)
% 0.84/0.66 % Starting SubtermCWHack with 28s (1) cores
% 0.84/0.66 % SubtermCWHack with pid 1003498 completed with status 0
% 0.84/0.66 % Result found by SubtermCWHack
% 0.84/0.66 % Preprocessing class: HSLMSMSMLLLCHSA.
% 0.84/0.66 % Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 0.84/0.66 % Starting pre_casc_4 with 1200s (4) cores
% 0.84/0.66 % Starting full_lambda_6 with 300s (1) cores
% 0.84/0.66 % Starting sh10 with 300s (1) cores
% 0.84/0.66 % Starting post_as_ho9 with 300s (1) cores
% 0.84/0.66 % (lift_lambdas = 0, lambda_to_forall = 1,unroll_only_formulas = 0, sine = GSinE(CountFormulas,,true,1,0,2,20000,1.0,true))
% 0.84/0.66 % SinE strategy is GSinE(CountFormulas,,true,1,0,2,20000,1.0,true)
% 0.84/0.66 % Search class: HGUNF-FFLS31-DHSMMFBN
% 0.84/0.66 % partial match(5): HGUNF-FFSS32-SHSSMMBN
% 0.84/0.66 % Scheduled 7 strats onto 1 cores with 300 seconds (300 total)
% 0.84/0.66 % Starting SubtermCWHack with 28s (1) cores
% 0.84/0.66 % Preprocessing time : 0.008 s
% 0.84/0.66
% 0.84/0.66 % Proof found!
% 0.84/0.66 % SZS status Theorem
% 0.84/0.66 % SZS output start CNFRefutation
% See solution above
% 0.84/0.66 % Parsed axioms : 583
% 0.84/0.66 % Removed by relevancy pruning/SinE : 526
% 0.84/0.66 % Initial clauses : 47
% 0.84/0.66 % Removed in clause preprocessing : 0
% 0.84/0.66 % Initial clauses in saturation : 47
% 0.84/0.66 % Processed clauses : 144
% 0.84/0.66 % ...of these trivial : 0
% 0.84/0.66 % ...subsumed : 53
% 0.84/0.66 % ...remaining for further processing : 91
% 0.84/0.66 % Other redundant clauses eliminated : 0
% 0.84/0.66 % Clauses deleted for lack of memory : 0
% 0.84/0.66 % Backward-subsumed : 1
% 0.84/0.66 % Backward-rewritten : 0
% 0.84/0.66 % Generated clauses : 239
% 0.84/0.66 % ...of the previous two non-redundant : 188
% 0.84/0.66 % ...aggressively subsumed : 0
% 0.84/0.66 % Contextual simplify-reflections : 0
% 0.84/0.66 % Paramodulations : 239
% 0.84/0.66 % Factorizations : 0
% 0.84/0.66 % NegExts : 0
% 0.84/0.66 % Equation resolutions : 0
% 0.84/0.66 % Disequality decompositions : 0
% 0.84/0.66 % Total rewrite steps : 152
% 0.84/0.66 % ...of those cached : 145
% 0.84/0.66 % Propositional unsat checks : 0
% 0.84/0.66 % Propositional check models : 0
% 0.84/0.66 % Propositional check unsatisfiable : 0
% 0.84/0.66 % Propositional clauses : 0
% 0.84/0.66 % Propositional clauses after purity: 0
% 0.84/0.66 % Propositional unsat core size : 0
% 0.84/0.66 % Propositional preprocessing time : 0.000
% 0.84/0.66 % Propositional encoding time : 0.000
% 0.84/0.66 % Propositional solver time : 0.000
% 0.84/0.66 % Success case prop preproc time : 0.000
% 0.84/0.66 % Success case prop encoding time : 0.000
% 0.84/0.66 % Success case prop solver time : 0.000
% 0.84/0.66 % Current number of processed clauses : 90
% 0.84/0.66 % Positive orientable unit clauses : 9
% 0.84/0.66 % Positive unorientable unit clauses: 0
% 0.84/0.66 % Negative unit clauses : 2
% 0.84/0.66 % Non-unit-clauses : 79
% 0.84/0.66 % Current number of unprocessed clauses: 88
% 0.84/0.66 % ...number of literals in the above : 519
% 0.84/0.66 % Current number of archived formulas : 0
% 0.84/0.66 % Current number of archived clauses : 1
% 0.84/0.66 % Clause-clause subsumption calls (NU) : 5271
% 0.84/0.66 % Rec. Clause-clause subsumption calls : 2062
% 0.84/0.66 % Non-unit clause-clause subsumptions : 54
% 0.84/0.66 % Unit Clause-clause subsumption calls : 43
% 0.84/0.66 % Rewrite failures with RHS unbound : 0
% 0.84/0.66 % BW rewrite match attempts : 33
% 0.84/0.66 % BW rewrite match successes : 0
% 0.84/0.66 % Condensation attempts : 0
% 0.84/0.66 % Condensation successes : 0
% 0.84/0.66 % Termbank termtop insertions : 84326
% 0.84/0.66 % Search garbage collected termcells : 11081
% 0.84/0.66
% 0.84/0.66 % -------------------------------------------------
% 0.84/0.66 % User time : 0.115 s
% 0.84/0.66 % System time : 0.015 s
% 0.84/0.66 % Total time : 0.131 s
% 0.84/0.66 % Maximum resident set size: 5992 pages
% 0.84/0.66
% 0.84/0.66 % -------------------------------------------------
% 0.84/0.66 % User time : 0.146 s
% 0.84/0.66 % System time : 0.023 s
% 0.84/0.66 % Total time : 0.170 s
% 0.84/0.66 % Maximum resident set size: 5888 pages
% 0.84/0.66 % E exiting
%------------------------------------------------------------------------------