TSTP Solution File: SYN357^7 by Satallax---3.5
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- Process Solution
%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SYN357^7 : TPTP v8.1.0. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n005.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 : 600s
% DateTime : Thu Jul 21 11:43:05 EDT 2022
% Result : Theorem 0.18s 0.35s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 62
% Syntax : Number of formulae : 72 ( 44 unt; 8 typ; 33 def)
% Number of atoms : 214 ( 37 equ; 0 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 427 ( 59 ~; 7 |; 0 &; 263 @)
% ( 7 <=>; 89 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 64 ( 64 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 46 con; 0-2 aty)
% ( 2 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 136 ( 48 ^ 88 !; 0 ?; 136 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_rel_s4,type,
rel_s4: $i > $i > $o ).
thf(ty_eigen__2,type,
eigen__2: mu ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_big_p,type,
big_p: mu > $i > $o ).
thf(ty_exists_in_world,type,
exists_in_world: mu > $i > $o ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__3,definition,
( eigen__3
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_p @ eigen__2 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_p @ eigen__2 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(sP1,plain,
( sP1
<=> ( exists_in_world @ eigen__2 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( rel_s4 @ eigen__1 @ eigen__3 )
=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( big_p @ eigen__2 @ X1 ) )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( big_p @ eigen__2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( big_p @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( sP1
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_p @ eigen__2 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_p @ eigen__2 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( big_p @ eigen__2 @ X3 ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( big_p @ X1 @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_p @ eigen__2 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_p @ eigen__2 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( sP3
=> sP3 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
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] : ~ $false ) ) ).
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] : ( mor @ ( mnot @ X1 ) ) ) ) ).
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] :
( ( 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] :
( ( 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] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) ).
thf(def_meuclidean,definition,
( meuclidean
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( 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] :
( ~ ( ( 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] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) ).
thf(def_mweakly_dense,definition,
( mweakly_dense
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ( 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] :
( ~ ( ( 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] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) ).
thf(def_mvalid,definition,
mvalid = !! ).
thf(def_msatisfiable,definition,
( msatisfiable
= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) ).
thf(def_mcountersatisfiable,definition,
( mcountersatisfiable
= ( ^ [X1: $i > $o] :
~ ( !! @ X1 ) ) ) ).
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(x2108,conjecture,
! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ~ ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ( ~ ~ ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_p @ X3 @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_p @ X4 @ X6 ) ) ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ~ ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_p @ X3 @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_p @ X4 @ X6 ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[x2108]) ).
thf(h2,assumption,
~ ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X1 )
=> ~ ! [X4: $i] :
( ( rel_s4 @ X1 @ X4 )
=> ( ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( big_p @ X2 @ X5 ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( big_p @ X3 @ X5 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ( ( rel_s4 @ eigen__0 @ eigen__1 )
=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__1 )
=> ~ ! [X3: $i] :
( ( rel_s4 @ eigen__1 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( big_p @ X1 @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( big_p @ X2 @ X4 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
rel_s4 @ eigen__0 @ eigen__1,
introduced(assumption,[]) ).
thf(h5,assumption,
~ ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__1 )
=> ~ ! [X3: $i] :
( ( rel_s4 @ eigen__1 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( big_p @ X1 @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( big_p @ X2 @ X4 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( sP1
=> ~ sP5 ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP1,
introduced(assumption,[]) ).
thf(h8,assumption,
sP5,
introduced(assumption,[]) ).
thf(1,plain,
( sP7
| ~ sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( sP7
| sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( sP2
| ~ sP7 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( sP6
| ~ sP2 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(5,plain,
( ~ sP5
| sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP4
| ~ sP1
| ~ sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(7,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h6,h4,h5,h3,h2,h1,h0])],[1,2,3,4,5,6,h7,h8]) ).
thf(8,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h6,h4,h5,h3,h2,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,7,h7,h8]) ).
thf(9,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h4,h5,h3,h2,h1,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__2)],[h5,8,h6]) ).
thf(10,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h2,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,9,h4,h5]) ).
thf(11,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__1)],[h2,10,h3]) ).
thf(12,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,11,h2]) ).
thf(13,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[12,h0]) ).
thf(0,theorem,
! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ~ ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ( ~ ~ ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_p @ X3 @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_p @ X4 @ X6 ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[12,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SYN357^7 : TPTP v8.1.0. Released v5.5.0.
% 0.03/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.11/0.33 % Computer : n005.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 600
% 0.11/0.33 % DateTime : Mon Jul 11 12:23:37 EDT 2022
% 0.11/0.33 % CPUTime :
% 0.18/0.35 % SZS status Theorem
% 0.18/0.35 % Mode: mode213
% 0.18/0.35 % Inferences: 8
% 0.18/0.35 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------