TSTP Solution File: PHI003^1 by Lash---1.13
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%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : PHI003^1 : 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 : n002.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 12:56:22 EDT 2023
% Result : Theorem 20.28s 20.50s
% Output : Proof 20.28s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_rel,type,
rel: $i > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_positive,type,
positive: ( mu > $i > $o ) > $i > $o ).
thf(sP1,plain,
( sP1
<=> ( positive
@ ^ [X1: mu,X2: $i] :
! [X3: mu > $i > $o] :
( ( positive @ X3 @ X2 )
=> ( X3 @ X1 @ X2 ) )
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( positive
@ ^ [X1: mu,X2: $i] : $false
@ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: mu > $i > $o] :
( sP2
=> ( positive @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: mu > $i > $o,X3: mu > $i > $o] :
( ~ ( ( positive @ X2 @ X1 )
=> ~ ! [X4: $i] :
( ( rel @ X1 @ X4 )
=> ! [X5: mu] :
( ( X2 @ X5 @ X4 )
=> ( X3 @ X5 @ X4 ) ) ) )
=> ( positive @ X3 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( positive
@ ^ [X1: mu,X2: $i] : ~ $false
@ eigen__0 ) ),
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
<=> ( sP1
=> ~ ! [X1: $i] :
( ( rel @ eigen__0 @ X1 )
=> ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( sP5 = ~ sP2 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: mu > $i > $o,X2: mu > $i > $o] :
( ~ ( ( positive @ X1 @ eigen__0 )
=> ~ ! [X3: $i] :
( ( rel @ eigen__0 @ X3 )
=> ! [X4: mu] :
( ( X1 @ X4 @ X3 )
=> ( X2 @ X4 @ X3 ) ) ) )
=> ( positive @ X2 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i] :
( ( rel @ eigen__0 @ X1 )
=> ! [X2: mu] :
~ ! [X3: mu > $i > $o] :
( ( positive @ X3 @ X1 )
=> ( X3 @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( sP2
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: mu > $i > $o] :
( ( positive
@ ^ [X2: mu,X3: $i] :
~ ( X1 @ X2 @ X3 )
@ eigen__0 )
= ( ~ ( positive @ X1 @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ~ sP7
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ! [X1: $i] :
( positive
@ ^ [X2: mu,X3: $i] :
! [X4: mu > $i > $o] :
( ( positive @ X4 @ X3 )
=> ( X4 @ X2 @ X3 ) )
@ X1 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ! [X1: mu > $i > $o] :
( ~ ( sP1
=> ~ ! [X2: $i] :
( ( rel @ eigen__0 @ X2 )
=> ! [X3: mu] :
( ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) )
=> ( X1 @ X3 @ X2 ) ) ) )
=> ( positive @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
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_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] :
( mand @ ( X1 @ X2 )
@ ( 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(corC,conjecture,
! [X1: $i] :
~ ! [X2: $i] :
( ( rel @ X1 @ X2 )
=> ! [X3: mu] :
~ ! [X4: mu > $i > $o] :
( ( positive @ X4 @ X2 )
=> ( X4 @ X3 @ X2 ) ) ) ).
thf(h0,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)],[corC]) ).
thf(h1,assumption,
sP10,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP11
| ~ sP2
| sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP3
| sP11 ),
inference(all_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP7
| ~ sP1
| ~ sP10 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP8
| ~ sP5
| ~ sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP13
| sP7
| sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP9
| sP3 ),
inference(all_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP12
| sP8 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP15
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP9
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP6
| sP12 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP4
| sP9 ),
inference(all_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP14
| sP1 ),
inference(all_rule,[status(thm)],]) ).
thf(axA1,axiom,
sP6 ).
thf(axA2,axiom,
sP4 ).
thf(axA3,axiom,
sP14 ).
thf(13,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,axA1,axA2,axA3,h1]) ).
thf(14,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,13,h1]) ).
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,[h0])],[14,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : PHI003^1 : TPTP v8.1.2. Released v6.1.0.
% 0.13/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.35 % Computer : n002.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sun Aug 27 09:14:03 EDT 2023
% 0.13/0.35 % CPUTime :
% 20.28/20.50 % SZS status Theorem
% 20.28/20.50 % Mode: cade22grackle2x798d
% 20.28/20.50 % Steps: 168
% 20.28/20.50 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------