TSTP Solution File: SET062^7 by Lash---1.13
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%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SET062^7 : TPTP v8.1.2. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n028.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 Aug 31 15:15:38 EDT 2023
% Result : Theorem 0.18s 0.42s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 58
% Syntax : Number of formulae : 66 ( 19 unt; 8 typ; 8 def)
% Number of atoms : 163 ( 8 equ; 1 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 309 ( 35 ~; 20 |; 0 &; 186 @)
% ( 19 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 21 ( 21 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 34 usr; 33 con; 0-3 aty)
% Number of variables : 45 ( 17 ^; 28 !; 0 ?; 45 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_subset,type,
subset: mu > mu > $i > $o ).
thf(ty_empty_set,type,
empty_set: mu ).
thf(ty_member,type,
member: mu > mu > $i > $o ).
thf(ty_exists_in_world,type,
exists_in_world: mu > $i > $o ).
thf(ty_eigen__3,type,
eigen__3: mu ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__1,type,
eigen__1: mu ).
thf(h0,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__3,definition,
( eigen__3
= ( eps__0
@ ^ [X1: mu] :
~ ( ( exists_in_world @ X1 @ eigen__0 )
=> ( ( member @ X1 @ empty_set @ eigen__0 )
=> ( member @ X1 @ eigen__1 @ eigen__0 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(sP1,plain,
( sP1
<=> ( exists_in_world @ eigen__1 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__0 )
=> ( ( member @ X1 @ empty_set @ eigen__0 )
=> ( member @ X1 @ eigen__1 @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( ( subset @ empty_set @ eigen__1 @ eigen__0 )
=> sP2 )
=> ~ ( sP2
=> ( subset @ empty_set @ eigen__1 @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ~ ( member @ X2 @ empty_set @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__0 )
=> ~ ( ( ( subset @ empty_set @ X1 @ eigen__0 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__0 )
=> ( ( member @ X2 @ empty_set @ eigen__0 )
=> ( member @ X2 @ X1 @ eigen__0 ) ) ) )
=> ~ ( ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__0 )
=> ( ( member @ X2 @ empty_set @ eigen__0 )
=> ( member @ X2 @ X1 @ eigen__0 ) ) )
=> ( subset @ empty_set @ X1 @ eigen__0 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( exists_in_world @ eigen__3 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( sP2
=> ( subset @ empty_set @ eigen__1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( sP6
=> ~ ( member @ eigen__3 @ empty_set @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( subset @ empty_set @ eigen__1 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( sP6
=> ( ( member @ eigen__3 @ empty_set @ eigen__0 )
=> ( member @ eigen__3 @ eigen__1 @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__0 )
=> ~ ( member @ X1 @ empty_set @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( member @ eigen__3 @ empty_set @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( sP1
=> ~ sP3 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ! [X1: $i] : ( exists_in_world @ empty_set @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__0 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__0 )
=> ~ ( ( ( subset @ X1 @ X2 @ eigen__0 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ eigen__0 )
=> ( ( member @ X3 @ X1 @ eigen__0 )
=> ( member @ X3 @ X2 @ eigen__0 ) ) ) )
=> ~ ( ! [X3: mu] :
( ( exists_in_world @ X3 @ eigen__0 )
=> ( ( member @ X3 @ X1 @ eigen__0 )
=> ( member @ X3 @ X2 @ eigen__0 ) ) )
=> ( subset @ X1 @ X2 @ eigen__0 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( exists_in_world @ empty_set @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( sP12
=> ( member @ eigen__3 @ eigen__1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( sP16
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: $i,X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X1 )
=> ~ ( ( ( subset @ X2 @ X3 @ X1 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X1 )
=> ( ( member @ X4 @ X2 @ X1 )
=> ( member @ X4 @ X3 @ X1 ) ) ) )
=> ~ ( ! [X4: mu] :
( ( exists_in_world @ X4 @ X1 )
=> ( ( member @ X4 @ X2 @ X1 )
=> ( member @ X4 @ X3 @ X1 ) ) )
=> ( subset @ X2 @ X3 @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(def_mnot,definition,
( mnot
= ( ^ [X1: $i > $o,X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_mor,definition,
( mor
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
| ( X2 @ X3 ) ) ) ) ).
thf(def_mand,definition,
( mand
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mnot @ ( mor @ ( mnot @ X1 ) @ ( mnot @ X2 ) ) ) ) ) ).
thf(def_mimplies,definition,
( mimplies
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mor @ ( mnot @ X1 ) @ X2 ) ) ) ).
thf(def_mequiv,definition,
( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ) ).
thf(def_mforall_ind,definition,
( mforall_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( exists_in_world @ X3 @ X2 )
@ ( X1 @ X3 @ X2 ) ) ) ) ).
thf(def_mvalid,definition,
( mvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(thI15,conjecture,
! [X1: $i,X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( subset @ empty_set @ X2 @ X1 ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i,X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( subset @ empty_set @ X2 @ X1 ) ),
inference(assume_negation,[status(cth)],[thI15]) ).
thf(h2,assumption,
~ ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__0 )
=> ( subset @ empty_set @ X1 @ eigen__0 ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ( sP1
=> sP9 ),
introduced(assumption,[]) ).
thf(h4,assumption,
sP1,
introduced(assumption,[]) ).
thf(h5,assumption,
~ sP9,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP8
| ~ sP6
| ~ sP12 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP11
| sP8 ),
inference(all_rule,[status(thm)],]) ).
thf(3,plain,
( sP17
| sP12 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( sP10
| ~ sP17 ),
inference(prop_rule,[status(thm)],]) ).
thf(5,plain,
( sP10
| sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( sP2
| ~ sP10 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(7,plain,
( ~ sP7
| ~ sP2
| sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(8,plain,
( sP3
| sP7 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP13
| ~ sP1
| ~ sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP5
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP18
| ~ sP16
| sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP15
| sP18 ),
inference(all_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP14
| sP16 ),
inference(all_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP4
| sP11 ),
inference(all_rule,[status(thm)],]) ).
thf(15,plain,
( ~ sP19
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(empty_set,axiom,
sP4 ).
thf(subset,axiom,
sP19 ).
thf(existence_of_empty_set_ax,axiom,
sP14 ).
thf(16,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h4,h5,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,h4,h5,empty_set,subset,existence_of_empty_set_ax]) ).
thf(17,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h2,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,16,h4,h5]) ).
thf(18,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__1)],[h2,17,h3]) ).
thf(19,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,18,h2]) ).
thf(20,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[19,h0]) ).
thf(0,theorem,
! [X1: $i,X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( subset @ empty_set @ X2 @ X1 ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[19,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET062^7 : TPTP v8.1.2. Released v5.5.0.
% 0.00/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.16/0.33 % Computer : n028.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Sat Aug 26 13:06:37 EDT 2023
% 0.16/0.33 % CPUTime :
% 0.18/0.42 % SZS status Theorem
% 0.18/0.42 % Mode: cade22sinegrackle2x6978
% 0.18/0.42 % Steps: 413
% 0.18/0.42 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------