TSTP Solution File: SET669^3 by Lash---1.13
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- Process Solution
%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SET669^3 : TPTP v8.1.2. Released v3.6.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n008.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 15:18:10 EDT 2023
% Result : Theorem 0.23s 0.46s
% Output : Proof 0.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 5
% Number of leaves : 68
% Syntax : Number of formulae : 76 ( 45 unt; 3 typ; 37 def)
% Number of atoms : 183 ( 59 equ; 8 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 304 ( 70 ~; 18 |; 18 &; 160 @)
% ( 14 <=>; 24 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 2 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 76 ( 76 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 53 con; 0-2 aty)
% Number of variables : 172 ( 110 ^; 54 !; 8 ?; 172 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_eigen__0,type,
eigen__0: $i > $i > $o ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ~ ! [X2: $i] :
~ ( eigen__0 @ X1 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
( ~ $false
!= ( ~ ! [X2: $i] :
~ ( eigen__0 @ X2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( eigen__0 @ X1 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ~ $false
= ( ~ ! [X1: $i] :
~ ( eigen__0 @ X1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( eigen__0 @ eigen__2 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( eigen__0 @ eigen__1 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] :
~ ( eigen__0 @ eigen__1 @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i] :
( ( eigen__2 = X1 )
=> ( eigen__0 @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i] :
( ( eigen__1 = X1 )
=> ( eigen__0 @ eigen__1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: $i] :
( ~ $false
= ( ~ ! [X2: $i] :
~ ( eigen__0 @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: $i] :
~ ( eigen__0 @ X1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i] :
~ ! [X2: $i] :
~ ( eigen__0 @ X1 @ X2 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( sP10
=> ( ( ^ [X1: $i] : ~ $false )
!= ( ^ [X1: $i] :
~ ! [X2: $i] :
~ ( eigen__0 @ X2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> $false ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ( ^ [X1: $i] : ~ sP12 )
= ( ^ [X1: $i] :
~ ! [X2: $i] :
~ ( eigen__0 @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(def_in,definition,
( in
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_is_a,definition,
( is_a
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_emptyset,definition,
( emptyset
= ( ^ [X1: $i] : sP12 ) ) ).
thf(def_unord_pair,definition,
( unord_pair
= ( ^ [X1: $i,X2: $i,X3: $i] :
( ( X3 = X1 )
| ( X3 = X2 ) ) ) ) ).
thf(def_singleton,definition,
( singleton
= ( ^ [X1: $i,X2: $i] : ( X2 = X1 ) ) ) ).
thf(def_union,definition,
( union
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
| ( X2 @ X3 ) ) ) ) ).
thf(def_excl_union,definition,
( excl_union
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
& ( (~) @ ( X2 @ X3 ) ) )
| ( ( (~) @ ( X1 @ X3 ) )
& ( X2 @ X3 ) ) ) ) ) ).
thf(def_intersection,definition,
( intersection
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
& ( X2 @ X3 ) ) ) ) ).
thf(def_setminus,definition,
( setminus
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
& ( (~) @ ( X2 @ X3 ) ) ) ) ) ).
thf(def_complement,definition,
( complement
= ( ^ [X1: $i > $o,X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_disjoint,definition,
( disjoint
= ( ^ [X1: $i > $o,X2: $i > $o] :
( ( intersection @ X1 @ X2 )
= emptyset ) ) ) ).
thf(def_subset,definition,
( subset
= ( ^ [X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 @ X3 ) ) ) ) ).
thf(def_meets,definition,
( meets
= ( ^ [X1: $i > $o,X2: $i > $o] :
? [X3: $i] :
( ( X1 @ X3 )
& ( X2 @ X3 ) ) ) ) ).
thf(def_misses,definition,
( misses
= ( ^ [X1: $i > $o,X2: $i > $o] :
( (~)
@ ? [X3: $i] :
( ( X1 @ X3 )
& ( X2 @ X3 ) ) ) ) ) ).
thf(def_cartesian_product,definition,
( cartesian_product
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i,X4: $i] :
( ( X1 @ X3 )
& ( X2 @ X4 ) ) ) ) ).
thf(def_pair_rel,definition,
( pair_rel
= ( ^ [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( X3 = X1 )
| ( X4 = X2 ) ) ) ) ).
thf(def_id_rel,definition,
( id_rel
= ( ^ [X1: $i > $o,X2: $i,X3: $i] :
( ( X1 @ X2 )
& ( X2 = X3 ) ) ) ) ).
thf(def_sub_rel,definition,
( sub_rel
= ( ^ [X1: $i > $i > $o,X2: $i > $i > $o] :
! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X1 @ X3 @ X4 )
@ ( X2 @ X3 @ X4 ) ) ) ) ).
thf(def_is_rel_on,definition,
( is_rel_on
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i > $o] :
! [X4: $i,X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( X1 @ X4 @ X5 )
@ ( ( X2 @ X4 )
& ( X3 @ X5 ) ) ) ) ) ).
thf(def_restrict_rel_domain,definition,
( restrict_rel_domain
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i,X4: $i] :
( ( X2 @ X3 )
& ( X1 @ X3 @ X4 ) ) ) ) ).
thf(def_rel_diagonal,definition,
( rel_diagonal
= ( ^ [X1: $i,X2: $i] : ( X1 = X2 ) ) ) ).
thf(def_rel_composition,definition,
( rel_composition
= ( ^ [X1: $i > $i > $o,X2: $i > $i > $o,X3: $i,X4: $i] :
? [X5: $i] :
( ( X1 @ X3 @ X5 )
& ( X2 @ X5 @ X4 ) ) ) ) ).
thf(def_reflexive,definition,
( reflexive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) ).
thf(def_irreflexive,definition,
( irreflexive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( (~) @ ( X1 @ X2 @ X2 ) ) ) ) ).
thf(def_symmetric,definition,
( symmetric
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 @ X3 )
@ ( X1 @ X3 @ X2 ) ) ) ) ).
thf(def_transitive,definition,
( transitive
= ( ^ [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_equiv_rel,definition,
( equiv_rel
= ( ^ [X1: $i > $i > $o] :
( ( reflexive @ X1 )
& ( symmetric @ X1 )
& ( transitive @ X1 ) ) ) ) ).
thf(def_rel_codomain,definition,
( rel_codomain
= ( ^ [X1: $i > $i > $o,X2: $i] :
? [X3: $i] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_rel_domain,definition,
( rel_domain
= ( ^ [X1: $i > $i > $o,X2: $i] :
? [X3: $i] : ( X1 @ X2 @ X3 ) ) ) ).
thf(def_rel_inverse,definition,
( rel_inverse
= ( ^ [X1: $i > $i > $o,X2: $i,X3: $i] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_equiv_classes,definition,
( equiv_classes
= ( ^ [X1: $i > $i > $o,X2: $i > $o] :
? [X3: $i] :
( ( X2 @ X3 )
& ! [X4: $i] :
( ( X2 @ X4 )
<=> ( X1 @ X3 @ X4 ) ) ) ) ) ).
thf(def_restrict_rel_codomain,definition,
( restrict_rel_codomain
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i,X4: $i] :
( ( X2 @ X4 )
& ( X1 @ X3 @ X4 ) ) ) ) ).
thf(def_rel_field,definition,
( rel_field
= ( ^ [X1: $i > $i > $o,X2: $i] :
( ( rel_domain @ X1 @ X2 )
| ( rel_codomain @ X1 @ X2 ) ) ) ) ).
thf(def_well_founded,definition,
( well_founded
= ( ^ [X1: $i > $i > $o] :
! [X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ? [X4: $i] :
( ( X2 @ X4 )
& ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( X1 @ X4 @ X5 )
@ ( (~) @ ( X2 @ X5 ) ) ) ) ) ) ) ).
thf(def_upwards_well_founded,definition,
( upwards_well_founded
= ( ^ [X1: $i > $i > $o] :
! [X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ? [X4: $i] :
( ( X2 @ X4 )
& ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( X1 @ X4 @ X4 )
@ ( (~) @ ( X2 @ X5 ) ) ) ) ) ) ) ).
thf(thm,conjecture,
! [X1: $i > $i > $o] :
( ! [X2: $i,X3: $i] :
( ( X2 = X3 )
=> ( X1 @ X2 @ X3 ) )
=> ~ ( ! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 )
=> ( ( ^ [X2: $i] : ~ sP12 )
!= ( ^ [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X3 @ X2 ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i > $i > $o] :
( ! [X2: $i,X3: $i] :
( ( X2 = X3 )
=> ( X1 @ X2 @ X3 ) )
=> ~ ( ! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 )
=> ( ( ^ [X2: $i] : ~ sP12 )
!= ( ^ [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X3 @ X2 ) ) ) ) ),
inference(assume_negation,[status(cth)],[thm]) ).
thf(h2,assumption,
~ ( sP1
=> ~ sP11 ),
introduced(assumption,[]) ).
thf(h3,assumption,
sP1,
introduced(assumption,[]) ).
thf(h4,assumption,
sP11,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP6
| sP3 ),
inference(all_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP1
| sP6 ),
inference(all_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP9
| ~ sP3 ),
inference(all_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP7
| sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP1
| sP7 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP5
| ~ sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(7,plain,
~ sP12,
inference(prop_rule,[status(thm)],]) ).
thf(8,plain,
( sP2
| sP12
| sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( sP8
| ~ sP2 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(10,plain,
( sP13
| ~ sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(11,plain,
( sP10
| sP5 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(12,plain,
( ~ sP11
| ~ sP10
| ~ sP13 ),
inference(prop_rule,[status(thm)],]) ).
thf(13,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,h3,h4]) ).
thf(14,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,13,h3,h4]) ).
thf(15,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,14,h2]) ).
thf(16,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[15,h0]) ).
thf(0,theorem,
! [X1: $i > $i > $o] :
( ! [X2: $i,X3: $i] :
( ( X2 = X3 )
=> ( X1 @ X2 @ X3 ) )
=> ~ ( ! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 )
=> ( ( ^ [X2: $i] : ~ sP12 )
!= ( ^ [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X3 @ X2 ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[15,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.15 % Problem : SET669^3 : TPTP v8.1.2. Released v3.6.0.
% 0.00/0.16 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.16/0.38 % Computer : n008.cluster.edu
% 0.16/0.38 % Model : x86_64 x86_64
% 0.16/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.38 % Memory : 8042.1875MB
% 0.16/0.38 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.38 % CPULimit : 300
% 0.16/0.38 % WCLimit : 300
% 0.16/0.38 % DateTime : Sat Aug 26 08:32:33 EDT 2023
% 0.16/0.39 % CPUTime :
% 0.23/0.46 % SZS status Theorem
% 0.23/0.46 % Mode: cade22grackle2xfee4
% 0.23/0.46 % Steps: 72
% 0.23/0.46 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------