TSTP Solution File: SYN357^7 by Lash---1.13
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%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SYN357^7 : 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 : n017.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 03:20:38 EDT 2023
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_big_p,type,
big_p: 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(sP1,plain,
( sP1
<=> ! [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,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( exists_in_world @ eigen__2 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
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(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(h0,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(h1,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(h2,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(h3,assumption,
rel_s4 @ eigen__0 @ eigen__1,
introduced(assumption,[]) ).
thf(h4,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(h5,assumption,
~ ( sP2
=> ~ sP1 ),
introduced(assumption,[]) ).
thf(h6,assumption,
sP2,
introduced(assumption,[]) ).
thf(h7,assumption,
sP1,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP1
| ~ sP2 ),
inference(all_rule,[status(thm)],]) ).
thf(2,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0])],[1,h6,h7]) ).
thf(3,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,2,h6,h7]) ).
thf(4,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__2)],[h4,3,h5]) ).
thf(5,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,4,h3,h4]) ).
thf(6,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__1)],[h1,5,h2]) ).
thf(7,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,6,h1]) ).
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,[h0])],[7,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SYN357^7 : TPTP v8.1.2. Released v5.5.0.
% 0.06/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.33 % Computer : n017.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Sat Aug 26 16:29:56 EDT 2023
% 0.14/0.33 % CPUTime :
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % Mode: cade22grackle2xfee4
% 0.20/0.40 % Steps: 21
% 0.20/0.40 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------