TSTP Solution File: SYO572^7 by Lash---1.13
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- Process Solution
%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SYO572^7 : 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 : n021.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 : Fri Sep 1 04:47:31 EDT 2023
% Result : Theorem 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 79
% Syntax : Number of formulae : 88 ( 45 unt; 7 typ; 33 def)
% Number of atoms : 225 ( 37 equ; 5 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 385 ( 61 ~; 23 |; 8 &; 230 @)
% ( 17 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 3 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 66 ( 66 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 56 usr; 56 con; 0-2 aty)
% Number of variables : 138 ( 68 ^; 64 !; 6 ?; 138 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_f,type,
f: mu > $i > $o ).
thf(ty_rel_s4,type,
rel_s4: $i > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__2,type,
eigen__2: mu ).
thf(ty_exists_in_world,type,
exists_in_world: mu > $i > $o ).
thf(h0,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: mu] :
~ ~ ( exists_in_world @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(sP1,plain,
( sP1
<=> ( exists_in_world @ eigen__2 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ~ ( f @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( rel_s4 @ eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ( exists_in_world @ eigen__2 @ eigen__0 )
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] :
( ~ ( ( exists_in_world @ eigen__2 @ eigen__0 )
=> ~ ( rel_s4 @ eigen__0 @ X1 ) )
=> ( exists_in_world @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__0 )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__0 @ X2 )
=> ~ ( f @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i,X2: $i] :
( ~ ( ( exists_in_world @ eigen__2 @ X1 )
=> ~ ( rel_s4 @ X1 @ X2 ) )
=> ( exists_in_world @ eigen__2 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( f @ eigen__2 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ( f @ X1 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: mu] :
~ ( exists_in_world @ X1 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( exists_in_world @ eigen__2 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( sP11
=> ~ sP3 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( sP1
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ! [X1: $i] :
~ ! [X2: mu] :
~ ( exists_in_world @ X2 @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ~ sP12
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( sP3
=> ~ sP8 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ! [X1: mu,X2: $i,X3: $i] :
( ~ ( ( exists_in_world @ X1 @ X2 )
=> ~ ( rel_s4 @ X2 @ X3 ) )
=> ( exists_in_world @ X1 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
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_mbox,definition,
( mbox
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( (~) @ ( X1 @ X3 @ X4 ) )
| ( X2 @ X4 ) ) ) ) ).
thf(def_mforall_prop,definition,
( mforall_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mtrue,definition,
( mtrue
= ( ^ [X1: $i] : $true ) ) ).
thf(def_mfalse,definition,
( mfalse
= ( mnot @ mtrue ) ) ).
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_mdia,definition,
( mdia
= ( ^ [X1: $i > $i > $o,X2: $i > $o] : ( mnot @ ( mbox @ X1 @ ( mnot @ X2 ) ) ) ) ) ).
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_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_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_msatisfiable,definition,
( msatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_mcountersatisfiable,definition,
( mcountersatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_minvalid,definition,
( minvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_mbox_s4,definition,
( mbox_s4
= ( ^ [X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( (~) @ ( rel_s4 @ X2 @ X3 ) )
| ( X1 @ X3 ) ) ) ) ).
thf(def_mdia_s4,definition,
( mdia_s4
= ( ^ [X1: $i > $o] : ( mnot @ ( mbox_s4 @ ( mnot @ X1 ) ) ) ) ) ).
thf(con,conjecture,
! [X1: $i] :
( ~ ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ( f @ X3 @ X2 ) ) )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ~ ( f @ X2 @ X3 ) ) ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i] :
( ~ ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ( f @ X3 @ X2 ) ) )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ~ ( f @ X2 @ X3 ) ) ) ),
inference(assume_negation,[status(cth)],[con]) ).
thf(h2,assumption,
~ ( ~ ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( f @ X2 @ X1 ) ) )
=> ~ sP6 ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( f @ X2 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
sP6,
introduced(assumption,[]) ).
thf(h5,assumption,
~ ( sP3
=> ~ sP9 ),
introduced(assumption,[]) ).
thf(h6,assumption,
sP3,
introduced(assumption,[]) ).
thf(h7,assumption,
sP9,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP12
| ~ sP11
| ~ sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP15
| sP12
| sP1 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP5
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP7
| sP5 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP16
| ~ sP3
| ~ sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP2
| sP16 ),
inference(all_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP4
| ~ sP11
| sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP13
| ~ sP1
| sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP9
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP17
| sP7 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP6
| sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(12,plain,
( sP10
| sP11 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(13,plain,
( ~ sP14
| ~ sP10 ),
inference(all_rule,[status(thm)],]) ).
thf(cumulative_ax,axiom,
sP17 ).
thf(nonempty_ax,axiom,
sP14 ).
thf(14,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,h6,h7,h4,cumulative_ax,nonempty_ax]) ).
thf(15,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,14,h6,h7]) ).
thf(16,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__1)],[h3,15,h5]) ).
thf(17,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,16,h3,h4]) ).
thf(18,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,17,h2]) ).
thf(19,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[18,h0]) ).
thf(0,theorem,
! [X1: $i] :
( ~ ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ( f @ X3 @ X2 ) ) )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ~ ( f @ X2 @ X3 ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[18,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SYO572^7 : TPTP v8.1.2. Released v5.5.0.
% 0.00/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.16/0.34 % Computer : n021.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Sat Aug 26 05:09:56 EDT 2023
% 0.16/0.34 % CPUTime :
% 0.19/0.41 % SZS status Theorem
% 0.19/0.41 % Mode: cade22grackle2xfee4
% 0.19/0.41 % Steps: 147
% 0.19/0.41 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------