%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SET027^3 : TPTP v9.2.0. Released v8.1.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n016.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 : Thu Oct 9 02:23:29 PM UTC 2025
% Result : Theorem 0.22s 0.44s
% Output : Proof 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 53
% Syntax : Number of formulae : 63 ( 23 unt; 0 typ; 11 def)
% Number of atoms : 153 ( 17 equ; 1 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 268 ( 40 ~; 21 |; 2 &; 158 @)
% ( 19 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 17 ( 17 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 37 usr; 37 con; 0-3 aty)
% Number of variables : 58 ( 28 ^; 28 !; 2 ?; 58 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mworld,type,
mworld: $tType ).
thf(ty_eigen__4,type,
eigen__4: $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_mactual,type,
mactual: mworld ).
thf(ty_subset,type,
subset: $i > $i > mworld > $o ).
thf(ty_member,type,
member: $i > $i > mworld > $o ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__4,definition,
( eigen__4
= ( eps__0
@ ^ [X1: $i] :
~ ( ( member @ X1 @ eigen__0 @ mactual )
=> ( member @ X1 @ eigen__2 @ mactual ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 @ mactual )
= ( ! [X3: $i] :
( ( member @ X3 @ X1 @ mactual )
=> ( member @ X3 @ X2 @ mactual ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( subset @ eigen__1 @ eigen__2 @ mactual ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( sP2
= ( ! [X1: $i] :
( ( member @ X1 @ eigen__1 @ mactual )
=> ( member @ X1 @ eigen__2 @ mactual ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i] :
( ( member @ X1 @ eigen__0 @ mactual )
=> ( member @ X1 @ eigen__1 @ mactual ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( member @ eigen__4 @ eigen__1 @ mactual ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i] :
( ( member @ X1 @ eigen__1 @ mactual )
=> ( member @ X1 @ eigen__2 @ mactual ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( member @ eigen__4 @ eigen__2 @ mactual ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( subset @ eigen__0 @ eigen__1 @ mactual ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ( member @ eigen__4 @ eigen__0 @ mactual )
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i] :
( ( subset @ eigen__0 @ X1 @ mactual )
= ( ! [X2: $i] :
( ( member @ X2 @ eigen__0 @ mactual )
=> ( member @ X2 @ X1 @ mactual ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( member @ eigen__4 @ eigen__0 @ mactual ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( sP11
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ( subset @ eigen__0 @ eigen__2 @ mactual )
= ( ! [X1: $i] :
( ( member @ X1 @ eigen__0 @ mactual )
=> ( member @ X1 @ eigen__2 @ mactual ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( subset @ eigen__0 @ eigen__2 @ mactual ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( sP8 = sP4 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( sP5
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ! [X1: $i] :
( ( subset @ eigen__1 @ X1 @ mactual )
= ( ! [X2: $i] :
( ( member @ X2 @ eigen__1 @ mactual )
=> ( member @ X2 @ X1 @ mactual ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: $i] :
( ( member @ X1 @ eigen__0 @ mactual )
=> ( member @ X1 @ eigen__2 @ mactual ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(def_mlocal,definition,
( mlocal
= ( ^ [X1: mworld > $o] : ( X1 @ mactual ) ) ) ).
thf(def_mnot,definition,
( mnot
= ( ^ [X1: mworld > $o,X2: mworld] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_mand,definition,
( mand
= ( ^ [X1: mworld > $o,X2: mworld > $o,X3: mworld] :
( ( X1 @ X3 )
& ( X2 @ X3 ) ) ) ) ).
thf(def_mor,definition,
( mor
= ( ^ [X1: mworld > $o,X2: mworld > $o,X3: mworld] :
( ( X1 @ X3 )
| ( X2 @ X3 ) ) ) ) ).
thf(def_mimplies,definition,
( mimplies
= ( ^ [X1: mworld > $o,X2: mworld > $o,X3: mworld] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 @ X3 ) ) ) ) ).
thf(def_mequiv,definition,
( mequiv
= ( ^ [X1: mworld > $o,X2: mworld > $o,X3: mworld] :
( ( X1 @ X3 )
<=> ( X2 @ X3 ) ) ) ) ).
thf(def_mbox,definition,
( mbox
= ( ^ [X1: mworld > $o,X2: mworld] :
! [X3: mworld] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( mrel @ X2 @ X3 )
@ ( X1 @ X3 ) ) ) ) ).
thf(def_mdia,definition,
( mdia
= ( ^ [X1: mworld > $o,X2: mworld] :
? [X3: mworld] :
( ( mrel @ X2 @ X3 )
& ( X1 @ X3 ) ) ) ) ).
thf(def_mforall_di,definition,
( mforall_di
= ( ^ [X1: $i > mworld > $o,X2: mworld] :
! [X3: $i] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mexists_di,definition,
( mexists_di
= ( ^ [X1: $i > mworld > $o,X2: mworld] :
? [X3: $i] : ( X1 @ X3 @ X2 ) ) ) ).
thf(prove_transitivity_of_subset,conjecture,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( subset @ X1 @ X2 @ mactual )
=> ~ ( subset @ X2 @ X3 @ mactual ) )
=> ( subset @ X1 @ X3 @ mactual ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( subset @ X1 @ X2 @ mactual )
=> ~ ( subset @ X2 @ X3 @ mactual ) )
=> ( subset @ X1 @ X3 @ mactual ) ),
inference(assume_negation,[status(cth)],[prove_transitivity_of_subset]) ).
thf(h2,assumption,
~ ! [X1: $i,X2: $i] :
( ~ ( ( subset @ eigen__0 @ X1 @ mactual )
=> ~ ( subset @ X1 @ X2 @ mactual ) )
=> ( subset @ eigen__0 @ X2 @ mactual ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ~ ( sP8
=> ~ ( subset @ eigen__1 @ X1 @ mactual ) )
=> ( subset @ eigen__0 @ X1 @ mactual ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( ~ ( sP8
=> ~ sP2 )
=> sP14 ),
introduced(assumption,[]) ).
thf(h5,assumption,
~ ( sP8
=> ~ sP2 ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ sP14,
introduced(assumption,[]) ).
thf(h7,assumption,
sP8,
introduced(assumption,[]) ).
thf(h8,assumption,
sP2,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP9
| ~ sP11
| sP5 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP16
| ~ sP5
| sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP4
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP6
| sP16 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
( ~ sP3
| ~ sP2
| sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP17
| sP3 ),
inference(all_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP15
| ~ sP8
| sP4 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP1
| sP17 ),
inference(all_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP10
| sP15 ),
inference(all_rule,[status(thm)],[]) ).
thf(10,plain,
( sP12
| ~ sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( sP12
| sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( sP18
| ~ sP12 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4]) ).
thf(13,plain,
( ~ sP13
| sP14
| ~ sP18 ),
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP10
| sP13 ),
inference(all_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP1
| sP10 ),
inference(all_rule,[status(thm)],[]) ).
thf(subset_defn,axiom,
sP1 ).
thf(16,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,h7,h8,h6,subset_defn]) ).
thf(17,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h7,h8])],[h5,16,h7,h8]) ).
thf(18,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,17,h5,h6]) ).
thf(19,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__2)],[h3,18,h4]) ).
thf(20,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__1)],[h2,19,h3]) ).
thf(21,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,20,h2]) ).
thf(22,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[21,h0]) ).
thf(0,theorem,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( subset @ X1 @ X2 @ mactual )
=> ~ ( subset @ X2 @ X3 @ mactual ) )
=> ( subset @ X1 @ X3 @ mactual ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[21,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET027^3 : TPTP v9.2.0. Released v8.1.0.
% 0.07/0.14 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.36 % Computer : n016.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Wed Oct 8 14:33:38 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.22/0.44 % SZS status Theorem
% 0.22/0.44 % Mode: cade22grackle2xfee4
% 0.22/0.44 % Steps: 228
% 0.22/0.44 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------