%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM782^4 : TPTP v9.2.0. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n022.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 Oct 9 02:12:08 PM UTC 2025
% Result : Theorem 23.12s 23.09s
% Output : Proof 23.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 40
% Syntax : Number of formulae : 47 ( 18 unt; 0 typ; 11 def)
% Number of atoms : 173 ( 46 equ; 1 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 442 ( 19 ~; 17 |; 0 &; 352 @)
% ( 15 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 37 usr; 30 con; 0-3 aty)
% Number of variables : 63 ( 46 ^; 17 !; 0 ?; 63 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_nat,type,
nat: $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_power,type,
power: $i > $i ).
thf(ty_num,type,
num: $i > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_d_Sep,type,
d_Sep: $i > ( $i > $o ) > $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_den,type,
den: $i > $i ).
thf(ty_n_ts,type,
n_ts: $i > $i > $i ).
thf(ty_pair1type,type,
pair1type: $i > $i ).
thf(ty_n_is,type,
n_is: $i > $i > $o ).
thf(ty_anec,type,
anec: $i > ( $i > $i > $o ) > $i > $o ).
thf(ty_in,type,
in: $i > $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] :
~ ( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ( ( eigen__1 = eigen__2 )
=> ( ( eigen__2 = X1 )
=> ( eigen__1 = X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X3: $i,X4: $i] : ( n_is @ ( n_ts @ ( num @ X3 ) @ ( den @ X4 ) ) @ ( n_ts @ ( num @ X4 ) @ ( den @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X4: $i,X5: $i] : ( n_is @ ( n_ts @ ( num @ X4 ) @ ( den @ X5 ) ) @ ( n_ts @ ( num @ X5 ) @ ( den @ X4 ) ) ) ) ) )
=> ( ( X1 = X2 )
=> ( ( X2 = X3 )
=> ( X1 = X3 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X3: $i,X4: $i] : ( n_is @ ( n_ts @ ( num @ X3 ) @ ( den @ X4 ) ) @ ( n_ts @ ( num @ X4 ) @ ( den @ X3 ) ) ) ) ) )
=> ( ( eigen__1 = X1 )
=> ( ( X1 = X2 )
=> ( eigen__1 = X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(sP1,plain,
( sP1
<=> ( ( eigen__1 = eigen__2 )
=> ( ( eigen__2 = eigen__3 )
=> ( eigen__1 = eigen__3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( eigen__2 = eigen__3 )
=> ( eigen__1 = eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ( ( eigen__1 = eigen__2 )
=> ( ( eigen__2 = X1 )
=> ( eigen__1 = X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] :
( ( eigen__1 = X1 )
=> ( X1 = eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( eigen__2 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( ( in @ eigen__3
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X1: $i,X2: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X2 ) ) @ ( n_ts @ ( num @ X2 ) @ ( den @ X1 ) ) ) ) ) )
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( eigen__1 = eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( eigen__1 = eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i] :
( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X3: $i,X4: $i] : ( n_is @ ( n_ts @ ( num @ X3 ) @ ( den @ X4 ) ) @ ( n_ts @ ( num @ X4 ) @ ( den @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X4: $i,X5: $i] : ( n_is @ ( n_ts @ ( num @ X4 ) @ ( den @ X5 ) ) @ ( n_ts @ ( num @ X5 ) @ ( den @ X4 ) ) ) ) ) )
=> ( ( X1 = X2 )
=> ( ( X2 = X3 )
=> ( X1 = X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ( in @ eigen__2
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X1: $i,X2: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X2 ) ) @ ( n_ts @ ( num @ X2 ) @ ( den @ X1 ) ) ) ) ) )
=> sP3 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( ( in @ eigen__1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X1: $i,X2: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X2 ) ) @ ( n_ts @ ( num @ X2 ) @ ( den @ X1 ) ) ) ) ) )
=> ! [X1: $i] :
( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X3: $i,X4: $i] : ( n_is @ ( n_ts @ ( num @ X3 ) @ ( den @ X4 ) ) @ ( n_ts @ ( num @ X4 ) @ ( den @ X3 ) ) ) ) ) )
=> ( ( eigen__1 = X1 )
=> ( ( X1 = X2 )
=> ( eigen__1 = X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( sP8
=> sP6 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( eigen__2 = eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ! [X1: $i] :
( ( in @ X1
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X2: $i,X3: $i] : ( n_is @ ( n_ts @ ( num @ X2 ) @ ( den @ X3 ) ) @ ( n_ts @ ( num @ X3 ) @ ( den @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( d_Sep @ ( power @ ( pair1type @ nat ) )
@ ( anec @ ( pair1type @ nat )
@ ^ [X3: $i,X4: $i] : ( n_is @ ( n_ts @ ( num @ X3 ) @ ( den @ X4 ) ) @ ( n_ts @ ( num @ X4 ) @ ( den @ X3 ) ) ) ) ) )
=> ( ( eigen__1 = X1 )
=> ( ( X1 = X2 )
=> ( eigen__1 = X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(def_is_of,definition,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_all_of,definition,
( all_of
= ( ^ [X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ( is_of @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i] : (=) ) ) ).
thf(def_ect,definition,
( ect
= ( ^ [X1: $i,X2: $i > $i > $o] : ( d_Sep @ ( power @ X1 ) @ ( anec @ X1 @ X2 ) ) ) ) ).
thf(def_frac,definition,
( frac
= ( pair1type @ nat ) ) ).
thf(def_n_eq,definition,
( n_eq
= ( ^ [X1: $i,X2: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X2 ) ) @ ( n_ts @ ( num @ X2 ) @ ( den @ X1 ) ) ) ) ) ).
thf(def_rat,definition,
( rat
= ( ect @ frac @ n_eq ) ) ).
thf(def_rt_is,definition,
( rt_is
= ( e_is @ rat ) ) ).
thf(satz80,conjecture,
sP10 ).
thf(h1,negated_conjecture,
~ sP10,
inference(assume_negation,[status(cth)],[satz80]) ).
thf(1,plain,
( ~ sP13
| ~ sP8
| sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP5
| sP13 ),
inference(all_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP14
| sP9
| ~ sP6
| ~ sP14 ),
inference(confrontation_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP4
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
sP4,
inference(eq_sym,[status(thm)],[]) ).
thf(6,plain,
( sP2
| ~ sP9 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( sP2
| sP14 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( sP1
| ~ sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(9,plain,
( sP1
| sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(10,plain,
( sP7
| ~ sP1 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( sP3
| ~ sP7 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(12,plain,
( sP11
| ~ sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
( sP15
| ~ sP11 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(14,plain,
( sP12
| ~ sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( sP10
| ~ sP12 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(16,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,h1]) ).
thf(17,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[16,h0]) ).
thf(0,theorem,
sP10,
inference(contra,[status(thm),contra(discharge,[h1])],[16,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.14 % Problem : NUM782^4 : TPTP v9.2.0. Released v7.1.0.
% 0.04/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.14/0.36 % Computer : n022.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Wed Oct 8 22:37:23 EDT 2025
% 0.14/0.36 % CPUTime :
% 23.12/23.09 % SZS status Theorem
% 23.12/23.09 % Mode: mode84:USE_SINE=true:SINE_TOLERANCE=1.2:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
% 23.12/23.09 % Inferences: 77
% 23.12/23.09 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------