%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : PHI006^4 : TPTP v9.2.0. Released v6.4.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n015.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 02:13:33 PM UTC 2025
% Result : Theorem 20.26s 20.47s
% Output : Proof 20.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 4
% Number of leaves : 67
% Syntax : Number of formulae : 74 ( 41 unt; 0 typ; 25 def)
% Number of atoms : 171 ( 26 equ; 4 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 431 ( 78 ~; 22 |; 3 &; 242 @)
% ( 21 <=>; 65 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 49 usr; 51 con; 0-2 aty)
% Number of variables : 205 ( 82 ^; 119 !; 4 ?; 205 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_eigen__20,type,
eigen__20: mu ).
thf(ty_positive,type,
positive: ( mu > $i > $o ) > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__4,type,
eigen__4: $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] :
~ ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X2 @ X1 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(h1,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__20,definition,
( eigen__20
= ( eps__1
@ ^ [X1: mu] :
~ ~ ! [X2: mu > $i > $o] :
( ! [X3: mu > $i > $o] :
( ( X3 @ X1 @ eigen__4 )
=> ! [X4: $i,X5: mu] :
( ( X2 @ X5 @ X4 )
=> ( X3 @ X5 @ X4 ) ) )
=> ! [X3: $i] :
~ ! [X4: mu] :
~ ( X2 @ X4 @ X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__20])]) ).
thf(sP1,plain,
( sP1
<=> ( positive
@ ^ [X1: mu,X2: $i] : $false
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: mu > $i > $o] :
( ( positive
@ ^ [X2: mu,X3: $i] :
~ ( X1 @ X2 @ X3 )
@ eigen__0 )
=> ~ ( positive @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ( positive @ X3 @ X2 )
=> ( X3 @ X1 @ X2 ) )
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( sP3
=> ( positive
@ ^ [X1: mu,X2: $i] : ~ $false
@ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ~ ( ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X1 @ X2 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) )
@ eigen__0 )
=> ~ ! [X1: $i,X2: mu] :
~ ! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X2 @ X1 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) ) )
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i,X2: mu > $i > $o] :
( ( positive
@ ^ [X3: mu,X4: $i] :
~ ( X2 @ X3 @ X4 )
@ X1 )
=> ~ ( positive @ X2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: mu > $i > $o] :
( ! [X2: mu > $i > $o] :
( ( X2 @ eigen__20 @ eigen__4 )
=> ! [X3: $i,X4: mu] :
( ( X1 @ X4 @ X3 )
=> ( X2 @ X4 @ X3 ) ) )
=> ! [X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( positive
@ ^ [X1: mu,X2: $i] : ~ $false
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: mu > $i > $o] :
( ~ ( ( positive
@ ^ [X2: mu,X3: $i] :
! [X4: mu > $i > $o] :
( ! [X5: mu > $i > $o] :
( ( X5 @ X2 @ X3 )
=> ! [X6: $i,X7: mu] :
( ( X4 @ X7 @ X6 )
=> ( X5 @ X7 @ X6 ) ) )
=> ! [X5: $i] :
~ ! [X6: mu] :
~ ( X4 @ X6 @ X5 ) )
@ eigen__0 )
=> ~ ! [X2: $i,X3: mu] :
( ! [X4: mu > $i > $o] :
( ! [X5: mu > $i > $o] :
( ( X5 @ X3 @ X2 )
=> ! [X6: $i,X7: mu] :
( ( X4 @ X7 @ X6 )
=> ( X5 @ X7 @ X6 ) ) )
=> ! [X5: $i] :
~ ! [X6: mu] :
~ ( X4 @ X6 @ X5 ) )
=> ( X1 @ X3 @ X2 ) ) )
=> ( positive @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: mu > $i > $o,X2: mu > $i > $o] :
( ~ ( ( positive @ X1 @ eigen__0 )
=> ~ ! [X3: $i,X4: mu] :
( ( X1 @ X4 @ X3 )
=> ( X2 @ X4 @ X3 ) ) )
=> ( positive @ X2 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X1 @ X2 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) )
@ eigen__0 )
=> ~ ! [X1: $i,X2: mu] :
~ ! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X2 @ X1 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: mu] :
~ ! [X2: mu > $i > $o] :
( ! [X3: mu > $i > $o] :
( ( X3 @ X1 @ eigen__4 )
=> ! [X4: $i,X5: mu] :
( ( X2 @ X5 @ X4 )
=> ( X3 @ X5 @ X4 ) ) )
=> ! [X3: $i] :
~ ! [X4: mu] :
~ ( X2 @ X4 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $i] :
( positive
@ ^ [X2: mu,X3: $i] :
! [X4: mu > $i > $o] :
( ( positive @ X4 @ X3 )
=> ( X4 @ X2 @ X3 ) )
@ X1 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> $false ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ! [X1: $i] :
( positive
@ ^ [X2: mu,X3: $i] :
! [X4: mu > $i > $o] :
( ! [X5: mu > $i > $o] :
( ( X5 @ X2 @ X3 )
=> ! [X6: $i,X7: mu] :
( ( X4 @ X7 @ X6 )
=> ( X5 @ X7 @ X6 ) ) )
=> ! [X5: $i] :
~ ! [X6: mu] :
~ ( X4 @ X6 @ X5 ) )
@ X1 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X1 @ X2 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) )
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( sP8
=> ~ sP1 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: $i,X2: mu] :
~ ! [X3: mu > $i > $o] :
( ! [X4: mu > $i > $o] :
( ( X4 @ X2 @ X1 )
=> ! [X5: $i,X6: mu] :
( ( X3 @ X6 @ X5 )
=> ( X4 @ X6 @ X5 ) ) )
=> ! [X4: $i] :
~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: $i,X2: mu > $i > $o,X3: mu > $i > $o] :
( ~ ( ( positive @ X2 @ X1 )
=> ~ ! [X4: $i,X5: mu] :
( ( X2 @ X5 @ X4 )
=> ( X3 @ X5 @ X4 ) ) )
=> ( positive @ X3 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ! [X1: mu > $i > $o] :
( ~ ( sP3
=> ~ ! [X2: $i,X3: mu] :
( ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) )
=> ( X1 @ X3 @ X2 ) ) )
=> ( positive @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
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] : sP14 ) ) ).
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,definition,
( mbox
= ( ^ [X1: $i > $o,X2: $i] :
! [X3: $i] : ( X1 @ X3 ) ) ) ).
thf(def_mdia,definition,
( mdia
= ( ^ [X1: $i > $o,X2: $i] :
? [X3: $i] : ( X1 @ X3 ) ) ) ).
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(def_god,definition,
( god
= ( ^ [X1: mu] :
( mforall_indset
@ ^ [X2: mu > $i > $o] : ( mimplies @ ( positive @ X2 ) @ ( X2 @ X1 ) ) ) ) ) ).
thf(def_essence,definition,
( essence
= ( ^ [X1: mu > $i > $o,X2: mu] :
( mforall_indset
@ ^ [X3: mu > $i > $o] :
( mimplies @ ( X3 @ X2 )
@ ( mbox
@ ( mforall_ind
@ ^ [X4: mu] : ( mimplies @ ( X1 @ X4 ) @ ( X3 @ X4 ) ) ) ) ) ) ) ) ).
thf(def_necessary_existence,definition,
( necessary_existence
= ( ^ [X1: mu] :
( mforall_indset
@ ^ [X2: mu > $i > $o] :
( mimplies @ ( essence @ X2 @ X1 )
@ ( mbox
@ ( mexists_ind
@ ^ [X3: mu] : ( X2 @ X3 ) ) ) ) ) ) ) ).
thf(conj,conjecture,
sP14 ).
thf(h2,negated_conjecture,
~ sP14,
inference(assume_negation,[status(cth)],[conj]) ).
thf(1,plain,
( ~ sP7
| sP14 ),
inference(all_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP4
| ~ sP3
| sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( sP12
| sP7 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__20]) ).
thf(4,plain,
( ~ sP20
| sP4 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
( sP18
| ~ sP12 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4]) ).
thf(6,plain,
~ sP14,
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP11
| ~ sP16
| ~ sP18 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP17
| ~ sP8
| ~ sP1 ),
inference(prop_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP5
| sP11
| sP1 ),
inference(prop_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP2
| sP17 ),
inference(all_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP9
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP10
| sP20 ),
inference(all_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP10
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP15
| sP16 ),
inference(all_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP13
| sP3 ),
inference(all_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP19
| sP10 ),
inference(all_rule,[status(thm)],[]) ).
thf(17,plain,
( ~ sP6
| sP2 ),
inference(all_rule,[status(thm)],[]) ).
thf(axA1a,axiom,
sP6 ).
thf(axA2,axiom,
sP19 ).
thf(axA3,axiom,
sP13 ).
thf(axA5,axiom,
sP15 ).
thf(18,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,axA1a,axA2,axA3,axA5]) ).
thf(19,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[18,h1]) ).
thf(20,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[19,h0]) ).
thf(0,theorem,
sP14,
inference(contra,[status(thm),contra(discharge,[h2])],[18,h2]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : PHI006^4 : TPTP v9.2.0. Released v6.4.0.
% 0.04/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.35 % Computer : n015.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Oct 8 23:18:08 EDT 2025
% 0.14/0.35 % CPUTime :
% 20.26/20.47 % SZS status Theorem
% 20.26/20.47 % Mode: cade22grackle2x798d
% 20.26/20.47 % Steps: 190
% 20.26/20.47 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------