%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : PHI002^2 : TPTP v9.2.0. Released v6.1.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n006.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 02:13:31 PM UTC 2025
% Result : Theorem 0.20s 0.45s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 66
% Syntax : Number of formulae : 76 ( 36 unt; 0 typ; 24 def)
% Number of atoms : 189 ( 28 equ; 5 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 313 ( 61 ~; 23 |; 4 &; 168 @)
% ( 20 <=>; 37 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 49 usr; 49 con; 0-2 aty)
% Number of variables : 110 ( 59 ^; 47 !; 4 ?; 110 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_rel,type,
rel: $i > $i > $o ).
thf(ty_eigen__13,type,
eigen__13: mu ).
thf(ty_eigen__8,type,
eigen__8: $i ).
thf(ty_eigen__1,type,
eigen__1: mu > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_positive,type,
positive: ( mu > $i > $o ) > $i > $o ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__8,definition,
( eigen__8
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel @ eigen__0 @ X1 )
=> ! [X2: mu] :
( ( eigen__1 @ X2 @ X1 )
=> ~ ( eigen__1 @ X2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__8])]) ).
thf(h1,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__13,definition,
( eigen__13
= ( eps__1
@ ^ [X1: mu] :
~ ( ( eigen__1 @ X1 @ eigen__8 )
=> ~ ( eigen__1 @ X1 @ eigen__8 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__13])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i,X2: mu > $i > $o,X3: mu > $i > $o] :
( ~ ( ( positive @ X2 @ X1 )
=> ~ ! [X4: $i] :
( ( rel @ X1 @ X4 )
=> ! [X5: mu] :
( ( X2 @ X5 @ X4 )
=> ( X3 @ X5 @ X4 ) ) ) )
=> ( positive @ X3 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: mu] :
~ ( eigen__1 @ X1 @ eigen__8 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( eigen__1 @ eigen__13 @ eigen__8 )
=> ~ ( eigen__1 @ eigen__13 @ eigen__8 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( rel @ eigen__0 @ eigen__8 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: mu > $i > $o,X2: mu > $i > $o] :
( ~ ( ( positive @ X1 @ eigen__0 )
=> ~ ! [X3: $i] :
( ( rel @ eigen__0 @ X3 )
=> ! [X4: mu] :
( ( X1 @ X4 @ X3 )
=> ( X2 @ X4 @ X3 ) ) ) )
=> ( positive @ X2 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i] :
( ( rel @ eigen__0 @ X1 )
=> ! [X2: mu] :
( ( eigen__1 @ X2 @ X1 )
=> ~ ( eigen__1 @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: mu > $i > $o] :
( ( positive
@ ^ [X2: mu,X3: $i] :
~ ( X1 @ X2 @ X3 )
@ eigen__0 )
= ( ~ ( positive @ X1 @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( sP4
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ( positive @ eigen__1 @ eigen__0 )
= ( ~ ( positive
@ ^ [X1: mu,X2: $i] :
~ ( eigen__1 @ X1 @ X2 )
@ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: mu > $i > $o] :
( ~ ( ( positive @ eigen__1 @ eigen__0 )
=> ~ ! [X2: $i] :
( ( rel @ eigen__0 @ X2 )
=> ! [X3: mu] :
( ( eigen__1 @ X3 @ X2 )
=> ( X1 @ X3 @ X2 ) ) ) )
=> ( positive @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ! [X1: mu] :
( ( eigen__1 @ X1 @ eigen__8 )
=> ~ ( eigen__1 @ X1 @ eigen__8 ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i] :
( ( rel @ eigen__0 @ X1 )
=> ! [X2: mu] :
~ ( eigen__1 @ X2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $i,X2: mu > $i > $o] :
( ( positive
@ ^ [X3: mu,X4: $i] :
~ ( X2 @ X3 @ X4 )
@ X1 )
= ( ~ ( positive @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( sP4
=> sP11 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( eigen__1 @ eigen__13 @ eigen__8 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ( positive @ eigen__1 @ eigen__0 )
=> ~ sP6 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( positive
@ ^ [X1: mu,X2: $i] :
~ ( eigen__1 @ X1 @ X2 )
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( ~ sP16
=> sP17 ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( positive @ eigen__1 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(def_meq_ind,definition,
( meq_ind
= ( ^ [X1: mu,X2: mu,X3: $i] : ( X1 = X2 ) ) ) ).
thf(def_mtrue,definition,
( mtrue
= ( ^ [X1: $i] : $true ) ) ).
thf(def_mfalse,definition,
( mfalse
= ( ^ [X1: $i] : $false ) ) ).
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,X3: $i] :
( ( X1 @ X3 )
& ( X2 @ X3 ) ) ) ) ).
thf(def_mimplies,definition,
( mimplies
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 @ X3 ) ) ) ) ).
thf(def_mimplied,definition,
( mimplied
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( X1 @ X3 ) ) ) ) ).
thf(def_mequiv,definition,
( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
<=> ( X2 @ X3 ) ) ) ) ).
thf(def_mxor,definition,
( mxor
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
& ( (~) @ ( X2 @ X3 ) ) )
| ( ( (~) @ ( X1 @ X3 ) )
& ( X2 @ X3 ) ) ) ) ) ).
thf(def_mforall_ind,definition,
( mforall_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mforall_indset,definition,
( mforall_indset
= ( ^ [X1: ( mu > $i > $o ) > $i > $o,X2: $i] :
! [X3: mu > $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mforall_prop,definition,
( mforall_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mexists_ind,definition,
( mexists_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
? [X3: mu] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mexists_indset,definition,
( mexists_indset
= ( ^ [X1: ( mu > $i > $o ) > $i > $o,X2: $i] :
? [X3: mu > $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mexists_prop,definition,
( mexists_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
? [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mbox_generic,definition,
( mbox_generic
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( (~) @ ( X1 @ X3 @ X4 ) )
| ( X2 @ X4 ) ) ) ) ).
thf(def_mdia_generic,definition,
( mdia_generic
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
? [X4: $i] :
( ( X1 @ X3 @ X4 )
& ( X2 @ X4 ) ) ) ) ).
thf(def_mbox,definition,
( mbox
= ( mbox_generic @ rel ) ) ).
thf(def_mdia,definition,
( mdia
= ( mdia_generic @ rel ) ) ).
thf(def_mvalid,definition,
( mvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_minvalid,definition,
( minvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(thmT1,conjecture,
! [X1: $i,X2: mu > $i > $o] :
( ( positive @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( rel @ X1 @ X3 )
=> ! [X4: mu] :
~ ( X2 @ X4 @ X3 ) ) ) ).
thf(h2,negated_conjecture,
~ ! [X1: $i,X2: mu > $i > $o] :
( ( positive @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( rel @ X1 @ X3 )
=> ! [X4: mu] :
~ ( X2 @ X4 @ X3 ) ) ),
inference(assume_negation,[status(cth)],[thmT1]) ).
thf(h3,assumption,
~ ! [X1: mu > $i > $o] :
( ( positive @ X1 @ eigen__0 )
=> ~ ! [X2: $i] :
( ( rel @ eigen__0 @ X2 )
=> ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( sP19
=> ~ sP12 ),
introduced(assumption,[]) ).
thf(h5,assumption,
sP19,
introduced(assumption,[]) ).
thf(h6,assumption,
sP12,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP2
| ~ sP15 ),
inference(all_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP8
| ~ sP4
| sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP12
| sP8 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
( sP3
| sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(5,plain,
( sP11
| ~ sP3 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__13]) ).
thf(6,plain,
( sP14
| ~ sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( sP14
| sP4 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( sP6
| ~ sP14 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__8]) ).
thf(9,plain,
( ~ sP16
| ~ sP19
| ~ sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP9
| ~ sP19
| ~ sP17 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP18
| sP16
| sP17 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP7
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP10
| sP18 ),
inference(all_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP5
| sP10 ),
inference(all_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP13
| sP7 ),
inference(all_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP1
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(axA2,axiom,
sP1 ).
thf(axA1,axiom,
sP13 ).
thf(17,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,h5,h6,axA2,axA1]) ).
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__1)],[h3,18,h4]) ).
thf(20,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h2,19,h3]) ).
thf(21,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[20,h1]) ).
thf(22,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[21,h0]) ).
thf(0,theorem,
! [X1: $i,X2: mu > $i > $o] :
( ( positive @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( rel @ X1 @ X3 )
=> ! [X4: mu] :
~ ( X2 @ X4 @ X3 ) ) ),
inference(contra,[status(thm),contra(discharge,[h2])],[20,h2]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : PHI002^2 : TPTP v9.2.0. Released v6.1.0.
% 0.07/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.34 % Computer : n006.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Oct 8 23:17:23 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.45 % SZS status Theorem
% 0.20/0.45 % Mode: cade22grackle2xfee4
% 0.20/0.45 % Steps: 268
% 0.20/0.45 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------