TSTP Solution File: LCL874^1 by Lash---1.13
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- Process Solution
%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : LCL874^1 : TPTP v8.1.2. Bugfixed v5.0.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n009.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 08:05:24 EDT 2023
% Result : Theorem 20.24s 20.54s
% Output : Proof 20.24s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 94
% Syntax : Number of formulae : 104 ( 47 unt; 6 typ; 32 def)
% Number of atoms : 247 ( 37 equ; 4 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 410 ( 48 ~; 29 |; 8 &; 243 @)
% ( 25 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 71 ( 71 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 64 usr; 64 con; 0-2 aty)
% Number of variables : 151 ( 66 ^; 79 !; 6 ?; 151 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__4,type,
eigen__4: $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_rk,type,
rk: $i > $i > $o ).
thf(ty_rb,type,
rb: $i > $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i > $o ).
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] :
~ ~ ( rb @ eigen__0 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(sP1,plain,
( sP1
<=> ( rk @ eigen__0 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i] :
( ( rb @ eigen__0 @ X1 )
=> ( rk @ eigen__0 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( rk @ eigen__0 @ eigen__4 )
=> ~ sP1 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( rk @ X1 @ X2 )
=> ~ ( rk @ X1 @ X3 ) )
=> ( rk @ X2 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] :
~ ! [X2: $i] :
~ ( rb @ X1 @ X2 ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i] :
( ~ ( ( rk @ eigen__0 @ eigen__4 )
=> ~ ( rk @ eigen__0 @ X1 ) )
=> ( rk @ eigen__4 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( rk @ X1 @ X3 )
=> ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( rb @ X1 @ X3 )
=> ( X2 @ X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( rb @ eigen__0 @ eigen__4 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( sP8
=> ! [X1: $i] :
( ( rk @ eigen__4 @ X1 )
=> ( rb @ eigen__0 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i] :
( ( rb @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rk @ X1 @ X2 )
=> ( rb @ eigen__0 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ! [X1: $i] :
~ ( rb @ eigen__0 @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i > $o] :
( ! [X2: $i] :
( ( rk @ eigen__0 @ X2 )
=> ( X1 @ X2 ) )
=> ! [X2: $i] :
( ( rb @ eigen__0 @ X2 )
=> ( X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $i,X2: $i] :
( ~ ( ( rk @ eigen__0 @ X1 )
=> ~ ( rk @ eigen__0 @ X2 ) )
=> ( rk @ X1 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ! [X1: $i] :
( ( rb @ eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( rk @ eigen__4 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( sP15
=> ( rb @ eigen__0 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( rk @ eigen__0 @ eigen__4 ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ( rb @ eigen__0 @ eigen__2 )
=> sP17 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( ~ sP3
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i > $o] :
( ! [X2: $i] :
( ( rb @ eigen__0 @ X2 )
=> ( X1 @ X2 ) )
=> ! [X2: $i] :
( ( rb @ eigen__0 @ X2 )
=> ! [X3: $i] :
( ( rk @ X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( sP8
=> sP18 ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( rb @ eigen__0 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ! [X1: $i] :
( ( rk @ eigen__4 @ X1 )
=> ( rb @ eigen__0 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( rb @ X1 @ X3 )
=> ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( rb @ X1 @ X3 )
=> ! [X4: $i] :
( ( rk @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(def_meq_ind,definition,
( meq_ind
= ( ^ [X1: mu,X2: mu,X3: $i] : ( X1 = X2 ) ) ) ).
thf(def_meq_prop,definition,
( meq_prop
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) ).
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_mimplied,definition,
( mimplied
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mor @ ( mnot @ X2 ) @ X1 ) ) ) ).
thf(def_mequiv,definition,
( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ) ).
thf(def_mxor,definition,
( mxor
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mnot @ ( mequiv @ X1 @ X2 ) ) ) ) ).
thf(def_mforall_ind,definition,
( mforall_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( 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] :
( mnot
@ ( mforall_ind
@ ^ [X2: mu] : ( mnot @ ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_mexists_prop,definition,
( mexists_prop
= ( ^ [X1: ( $i > $o ) > $i > $o] :
( mnot
@ ( mforall_prop
@ ^ [X2: $i > $o] : ( mnot @ ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_mtrue,definition,
( mtrue
= ( ^ [X1: $i] : $true ) ) ).
thf(def_mfalse,definition,
( mfalse
= ( mnot @ mtrue ) ) ).
thf(def_mbox,definition,
( mbox
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( (~) @ ( X1 @ X3 @ X4 ) )
| ( X2 @ X4 ) ) ) ) ).
thf(def_mdia,definition,
( mdia
= ( ^ [X1: $i > $i > $o,X2: $i > $o] : ( mnot @ ( mbox @ X1 @ ( mnot @ X2 ) ) ) ) ) ).
thf(def_mreflexive,definition,
( mreflexive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) ).
thf(def_msymmetric,definition,
( msymmetric
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 @ X3 )
@ ( X1 @ X3 @ X2 ) ) ) ) ).
thf(def_mserial,definition,
( mserial
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
? [X3: $i] : ( X1 @ X2 @ X3 ) ) ) ).
thf(def_mtransitive,definition,
( mtransitive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X3 @ X4 ) )
@ ( X1 @ X2 @ X4 ) ) ) ) ).
thf(def_meuclidean,definition,
( meuclidean
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ( X1 @ X3 @ X4 ) ) ) ) ).
thf(def_mpartially_functional,definition,
( mpartially_functional
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ( X3 = X4 ) ) ) ) ).
thf(def_mfunctional,definition,
( mfunctional
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
? [X3: $i] :
( ( X1 @ X2 @ X3 )
& ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X1 @ X2 @ X4 )
@ ( X3 = X4 ) ) ) ) ) ).
thf(def_mweakly_dense,definition,
( mweakly_dense
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X1 @ X2 @ X3 )
@ ? [X5: $i] :
( ( X1 @ X2 @ X5 )
& ( X1 @ X5 @ X3 ) ) ) ) ) ).
thf(def_mweakly_connected,definition,
( mweakly_connected
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ( ( X1 @ X3 @ X4 )
| ( X3 = X4 )
| ( X1 @ X4 @ X3 ) ) ) ) ) ).
thf(def_mweakly_directed,definition,
( mweakly_directed
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ? [X5: $i] :
( ( X1 @ X3 @ X5 )
& ( X1 @ X4 @ X5 ) ) ) ) ) ).
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_msatisfiable,definition,
( msatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_mcountersatisfiable,definition,
( mcountersatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(conj,conjecture,
! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( rb @ X1 @ X3 )
=> ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( rk @ X1 @ X3 )
=> ( X2 @ X3 ) ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( rb @ X1 @ X3 )
=> ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( rk @ X1 @ X3 )
=> ( X2 @ X3 ) ) ),
inference(assume_negation,[status(cth)],[conj]) ).
thf(h2,assumption,
~ ! [X1: $i > $o] :
( ! [X2: $i] :
( ( rb @ eigen__0 @ X2 )
=> ( X1 @ X2 ) )
=> ! [X2: $i] :
( ( rk @ eigen__0 @ X2 )
=> ( X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ( sP14
=> ! [X1: $i] :
( ( rk @ eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
sP14,
introduced(assumption,[]) ).
thf(h5,assumption,
~ ! [X1: $i] :
( ( rk @ eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( sP1
=> sP17 ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP1,
introduced(assumption,[]) ).
thf(h8,assumption,
~ sP17,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP3
| ~ sP18
| ~ sP1 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP16
| ~ sP15
| sP23 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP20
| sP3
| sP15 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP24
| sP16 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP6
| sP20 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP9
| ~ sP8
| sP24 ),
inference(prop_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP22
| ~ sP8
| sP18 ),
inference(prop_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP13
| sP6 ),
inference(all_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP10
| sP9 ),
inference(all_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP2
| sP22 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( sP11
| sP8 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4]) ).
thf(12,plain,
( ~ sP12
| sP2 ),
inference(all_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP21
| sP10 ),
inference(all_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP19
| ~ sP23
| sP17 ),
inference(prop_rule,[status(thm)],]) ).
thf(15,plain,
( ~ sP5
| ~ sP11 ),
inference(all_rule,[status(thm)],]) ).
thf(16,plain,
( ~ sP4
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(17,plain,
( ~ sP7
| sP12 ),
inference(all_rule,[status(thm)],]) ).
thf(18,plain,
( ~ sP25
| sP21 ),
inference(all_rule,[status(thm)],]) ).
thf(19,plain,
( ~ sP14
| sP19 ),
inference(all_rule,[status(thm)],]) ).
thf(ax2,axiom,
sP5 ).
thf(ax5,axiom,
sP4 ).
thf(ax7,axiom,
sP7 ).
thf(ax8,axiom,
sP25 ).
thf(20,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h6,h4,h5,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,ax2,ax5,ax7,ax8,h4,h7,h8]) ).
thf(21,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h6,h4,h5,h3,h2,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,20,h7,h8]) ).
thf(22,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h4,h5,h3,h2,h1,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__2)],[h5,21,h6]) ).
thf(23,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h2,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,22,h4,h5]) ).
thf(24,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__1)],[h2,23,h3]) ).
thf(25,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,24,h2]) ).
thf(26,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[25,h0]) ).
thf(0,theorem,
! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( rb @ X1 @ X3 )
=> ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( rk @ X1 @ X3 )
=> ( X2 @ X3 ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[25,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL874^1 : TPTP v8.1.2. Bugfixed v5.0.0.
% 0.07/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n009.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 00:41:06 EDT 2023
% 0.20/0.34 % CPUTime :
% 20.24/20.54 % SZS status Theorem
% 20.24/20.54 % Mode: cade22grackle2x798d
% 20.24/20.54 % Steps: 1083
% 20.24/20.54 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------