TSTP Solution File: PHI005^2 by Lash---1.13
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- Process Solution
%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : PHI005^2 : TPTP v8.1.2. Released v6.1.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n028.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 12:56:23 EDT 2023
% Result : Theorem 23.70s 24.07s
% Output : Proof 23.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 84
% Syntax : Number of formulae : 94 ( 41 unt; 8 typ; 27 def)
% Number of atoms : 210 ( 28 equ; 5 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 397 ( 63 ~; 26 |; 4 &; 229 @)
% ( 23 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 110 ( 110 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 56 usr; 56 con; 0-3 aty)
% Number of variables : 149 ( 71 ^; 74 !; 4 ?; 149 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_rel,type,
rel: $i > $i > $o ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_essence,type,
essence: ( mu > $i > $o ) > mu > $i > $o ).
thf(ty_positive,type,
positive: ( mu > $i > $o ) > $i > $o ).
thf(ty_eigen__6,type,
eigen__6: mu ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
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 @ eigen__1 @ X1 )
=> ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(h1,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__6,definition,
( eigen__6
= ( eps__1
@ ^ [X1: mu] :
~ ~ ! [X2: mu > $i > $o] :
( ( positive @ X2 @ eigen__3 )
=> ( X2 @ X1 @ eigen__3 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__6])]) ).
thf(sP1,plain,
( sP1
<=> ( ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ( essence @ X3 @ X1 @ X2 )
=> ! [X4: $i] :
( ( rel @ X2 @ X4 )
=> ~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) ) )
@ eigen__3 )
=> ! [X1: mu > $i > $o] :
( ( essence @ X1 @ eigen__6 @ eigen__3 )
=> ! [X2: $i] :
( ( rel @ eigen__3 @ X2 )
=> ~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i] :
( ( rel @ eigen__3 @ X1 )
=> ~ ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( positive
@ ^ [X2: mu,X3: $i] :
! [X4: mu > $i > $o] :
( ( essence @ X4 @ X2 @ X3 )
=> ! [X5: $i] :
( ( rel @ X3 @ X5 )
=> ~ ! [X6: mu] :
~ ( X4 @ X6 @ X5 ) ) )
@ X1 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( essence
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ( positive @ X3 @ X2 )
=> ( X3 @ X1 @ X2 ) )
@ eigen__6
@ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: mu] :
( ! [X2: mu > $i > $o] :
( ( positive @ X2 @ eigen__3 )
=> ( X2 @ X1 @ eigen__3 ) )
=> ( essence
@ ^ [X2: mu,X3: $i] :
! [X4: mu > $i > $o] :
( ( positive @ X4 @ X3 )
=> ( X4 @ X2 @ X3 ) )
@ X1
@ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( rel @ eigen__3 @ eigen__1 )
=> ~ ! [X1: mu] :
~ ! [X2: mu > $i > $o] :
( ( positive @ X2 @ eigen__1 )
=> ( X2 @ X1 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i,X2: $i] :
( ( rel @ X1 @ X2 )
=> ( rel @ X2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: mu > $i > $o] :
( ( essence @ X1 @ eigen__6 @ eigen__3 )
=> ! [X2: $i] :
( ( rel @ eigen__3 @ X2 )
=> ~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( rel @ eigen__3 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( sP4
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ! [X1: mu > $i > $o] :
( ( positive @ X1 @ eigen__3 )
=> ( X1 @ eigen__6 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i] :
( ( rel @ eigen__1 @ X1 )
=> ( rel @ X1 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $i] :
( ( rel @ eigen__1 @ X1 )
=> ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ! [X1: mu] :
~ ! [X2: mu > $i > $o] :
( ( positive @ X2 @ eigen__3 )
=> ( X2 @ X1 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ! [X1: $i,X2: mu] :
( ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) )
=> ( essence
@ ^ [X3: mu,X4: $i] :
! [X5: mu > $i > $o] :
( ( positive @ X5 @ X4 )
=> ( X5 @ X3 @ X4 ) )
@ X2
@ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( rel @ eigen__1 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( sP16
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: mu] :
~ ! [X2: mu > $i > $o] :
( ( positive @ X2 @ eigen__1 )
=> ( X2 @ X1 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ( essence @ X3 @ X1 @ X2 )
=> ! [X4: $i] :
( ( rel @ X2 @ X4 )
=> ~ ! [X5: mu] :
~ ( X3 @ X5 @ X4 ) ) )
@ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( sP11
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( sP16
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ! [X1: $i] :
~ ! [X2: $i] :
( ( rel @ X1 @ X2 )
=> ! [X3: mu] :
~ ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
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(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_god,definition,
( god
= ( ^ [X1: mu] :
( mforall_indset
@ ^ [X2: mu > $i > $o] : ( mimplies @ ( positive @ X2 ) @ ( X2 @ X1 ) ) ) ) ) ).
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(thmT3,conjecture,
! [X1: $i,X2: $i] :
( ( rel @ X1 @ X2 )
=> ~ ! [X3: mu] :
~ ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) ) ) ).
thf(h2,negated_conjecture,
~ ! [X1: $i,X2: $i] :
( ( rel @ X1 @ X2 )
=> ~ ! [X3: mu] :
~ ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) ) ),
inference(assume_negation,[status(cth)],[thmT3]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( rel @ eigen__0 @ X1 )
=> ~ ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( ( rel @ eigen__0 @ eigen__1 )
=> ~ sP18 ),
introduced(assumption,[]) ).
thf(h5,assumption,
rel @ eigen__0 @ eigen__1,
introduced(assumption,[]) ).
thf(h6,assumption,
sP18,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP20
| ~ sP11
| sP4 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP5
| sP20 ),
inference(all_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP6
| ~ sP9
| ~ sP18 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP2
| sP6 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP17
| ~ sP16
| sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP10
| ~ sP4
| sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP15
| sP5 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP3
| sP19 ),
inference(all_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP12
| sP17 ),
inference(all_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP8
| sP10 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP1
| ~ sP19
| sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP11
| sP1 ),
inference(all_rule,[status(thm)],]) ).
thf(13,plain,
( sP14
| sP11 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__6]) ).
thf(14,plain,
( sP21
| ~ sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(15,plain,
( sP21
| sP16 ),
inference(prop_rule,[status(thm)],]) ).
thf(16,plain,
( sP13
| ~ sP21 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(17,plain,
( ~ sP7
| sP12 ),
inference(all_rule,[status(thm)],]) ).
thf(18,plain,
( ~ sP22
| ~ sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(sym,axiom,
sP7 ).
thf(axA5,axiom,
sP3 ).
thf(corC,axiom,
sP22 ).
thf(thmT2,axiom,
sP15 ).
thf(19,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,sym,axA5,corC,thmT2,h6]) ).
thf(20,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,19,h5,h6]) ).
thf(21,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,20,h4]) ).
thf(22,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h2,21,h3]) ).
thf(23,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[22,h1]) ).
thf(24,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[23,h0]) ).
thf(0,theorem,
! [X1: $i,X2: $i] :
( ( rel @ X1 @ X2 )
=> ~ ! [X3: mu] :
~ ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) ) ),
inference(contra,[status(thm),contra(discharge,[h2])],[22,h2]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : PHI005^2 : TPTP v8.1.2. Released v6.1.0.
% 0.00/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n028.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 : Sun Aug 27 09:14:08 EDT 2023
% 0.13/0.34 % CPUTime :
% 23.70/24.07 % SZS status Theorem
% 23.70/24.07 % Mode: cade22grackle2x798d
% 23.70/24.07 % Steps: 575
% 23.70/24.07 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------