%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU687^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n020.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 : Wed Feb 25 09:36:24 AM UTC 2026
% Result : Theorem 1.73s 1.48s
% Output : Refutation 1.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 12
% Syntax : Number of formulae : 57 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 1103 ( 291 equ; 0 cnn)
% Maximal formula atoms : 16 ( 19 avg)
% Number of connectives : 1386 ( 129 ~; 100 |; 104 &; 985 @)
% ( 5 <=>; 63 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 28 usr; 19 con; 0-4 aty)
% Number of variables : 208 ( 0 ^; 143 !; 65 ?; 208 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_4,type,
subset: $i > $i > $o ).
thf(func_def_5,type,
kpair: $i > $i > $i ).
thf(func_def_6,type,
cartprod: $i > $i > $i ).
thf(func_def_7,type,
singleton: $i > $o ).
thf(func_def_9,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_10,type,
breln: $i > $i > $i > $o ).
thf(func_def_11,type,
func: $i > $i > $i > $o ).
thf(func_def_12,type,
ap: $i > $i > $i > $i > $i ).
thf(f107,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f106,f57]) ).
thf(f57,plain,
( ( in @ sK13 @ sK9 )
= $true ),
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f44,plain,
( ( funcGraphProp1 = $true )
& ( ( func @ sK9 @ sK10 @ sK11 )
= $true )
& ! [X4: $i] :
( ( ( ap @ sK9 @ sK10 @ sK11 @ X4 )
= ( ap @ sK9 @ sK10 @ sK12 @ X4 ) )
| ( ( in @ X4 @ sK9 )
!= $true ) )
& ( ( in @ sK14 @ sK10 )
= $true )
& ( ( in @ ( kpair @ sK13 @ sK14 ) @ sK12 )
= $true )
& ( $true
!= ( in @ ( kpair @ sK13 @ sK14 ) @ sK11 ) )
& ( ( in @ sK13 @ sK9 )
= $true )
& ( ( func @ sK9 @ sK10 @ sK12 )
= $true )
& ( funcGraphProp2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11,sK12,sK13,sK14])],[f25,f43,f42,f41,f40]) ).
thf(f40,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
= $true )
& ? [X3: $i] :
( ! [X4: $i] :
( ( ( ap @ X0 @ X1 @ X3 @ X4 )
= ( ap @ X0 @ X1 @ X2 @ X4 ) )
| ( ( in @ X4 @ X0 )
!= $true ) )
& ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ X1 )
= $true )
& ( $true
= ( in @ ( kpair @ X5 @ X6 ) @ X3 ) )
& ( ( in @ ( kpair @ X5 @ X6 ) @ X2 )
!= $true ) )
& ( ( in @ X5 @ X0 )
= $true ) )
& ( ( func @ X0 @ X1 @ X3 )
= $true ) ) )
=> ( ( ( func @ sK9 @ sK10 @ sK11 )
= $true )
& ? [X3: $i] :
( ! [X4: $i] :
( ( ( ap @ sK9 @ sK10 @ sK11 @ X4 )
= ( ap @ sK9 @ sK10 @ X3 @ X4 ) )
| ( ( in @ X4 @ sK9 )
!= $true ) )
& ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ sK10 )
= $true )
& ( $true
= ( in @ ( kpair @ X5 @ X6 ) @ X3 ) )
& ( ( in @ ( kpair @ X5 @ X6 ) @ sK11 )
!= $true ) )
& ( $true
= ( in @ X5 @ sK9 ) ) )
& ( ( func @ sK9 @ sK10 @ X3 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f41,plain,
( ? [X3: $i] :
( ! [X4: $i] :
( ( ( ap @ sK9 @ sK10 @ sK11 @ X4 )
= ( ap @ sK9 @ sK10 @ X3 @ X4 ) )
| ( ( in @ X4 @ sK9 )
!= $true ) )
& ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ sK10 )
= $true )
& ( $true
= ( in @ ( kpair @ X5 @ X6 ) @ X3 ) )
& ( ( in @ ( kpair @ X5 @ X6 ) @ sK11 )
!= $true ) )
& ( $true
= ( in @ X5 @ sK9 ) ) )
& ( ( func @ sK9 @ sK10 @ X3 )
= $true ) )
=> ( ! [X4: $i] :
( ( ( ap @ sK9 @ sK10 @ sK11 @ X4 )
= ( ap @ sK9 @ sK10 @ sK12 @ X4 ) )
| ( ( in @ X4 @ sK9 )
!= $true ) )
& ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ sK10 )
= $true )
& ( ( in @ ( kpair @ X5 @ X6 ) @ sK12 )
= $true )
& ( ( in @ ( kpair @ X5 @ X6 ) @ sK11 )
!= $true ) )
& ( $true
= ( in @ X5 @ sK9 ) ) )
& ( ( func @ sK9 @ sK10 @ sK12 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f42,plain,
( ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ sK10 )
= $true )
& ( ( in @ ( kpair @ X5 @ X6 ) @ sK12 )
= $true )
& ( ( in @ ( kpair @ X5 @ X6 ) @ sK11 )
!= $true ) )
& ( $true
= ( in @ X5 @ sK9 ) ) )
=> ( ? [X6: $i] :
( ( ( in @ X6 @ sK10 )
= $true )
& ( ( in @ ( kpair @ sK13 @ X6 ) @ sK12 )
= $true )
& ( ( in @ ( kpair @ sK13 @ X6 ) @ sK11 )
!= $true ) )
& ( ( in @ sK13 @ sK9 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f43,plain,
( ? [X6: $i] :
( ( ( in @ X6 @ sK10 )
= $true )
& ( ( in @ ( kpair @ sK13 @ X6 ) @ sK12 )
= $true )
& ( ( in @ ( kpair @ sK13 @ X6 ) @ sK11 )
!= $true ) )
=> ( ( ( in @ sK14 @ sK10 )
= $true )
& ( ( in @ ( kpair @ sK13 @ sK14 ) @ sK12 )
= $true )
& ( $true
!= ( in @ ( kpair @ sK13 @ sK14 ) @ sK11 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f25,plain,
( ( funcGraphProp1 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
= $true )
& ? [X3: $i] :
( ! [X4: $i] :
( ( ( ap @ X0 @ X1 @ X3 @ X4 )
= ( ap @ X0 @ X1 @ X2 @ X4 ) )
| ( ( in @ X4 @ X0 )
!= $true ) )
& ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ X1 )
= $true )
& ( $true
= ( in @ ( kpair @ X5 @ X6 ) @ X3 ) )
& ( ( in @ ( kpair @ X5 @ X6 ) @ X2 )
!= $true ) )
& ( ( in @ X5 @ X0 )
= $true ) )
& ( ( func @ X0 @ X1 @ X3 )
= $true ) ) )
& ( funcGraphProp2 = $true ) ),
inference(flattening,[status(thm)],[f24]) ).
thf(f24,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ? [X5: $i] :
( ? [X6: $i] :
( ( ( in @ ( kpair @ X5 @ X6 ) @ X2 )
!= $true )
& ( $true
= ( in @ ( kpair @ X5 @ X6 ) @ X3 ) )
& ( ( in @ X6 @ X1 )
= $true ) )
& ( ( in @ X5 @ X0 )
= $true ) )
& ! [X4: $i] :
( ( ( ap @ X0 @ X1 @ X3 @ X4 )
= ( ap @ X0 @ X1 @ X2 @ X4 ) )
| ( ( in @ X4 @ X0 )
!= $true ) )
& ( ( func @ X0 @ X1 @ X3 )
= $true ) )
& ( ( func @ X0 @ X1 @ X2 )
= $true ) )
& ( funcGraphProp2 = $true )
& ( funcGraphProp1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
~ ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp2 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( func @ X0 @ X1 @ X3 )
= $true )
=> ( ! [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
=> ( ( ap @ X0 @ X1 @ X3 @ X4 )
= ( ap @ X0 @ X1 @ X2 @ X4 ) ) )
=> ! [X5: $i] :
( ( ( in @ X5 @ X0 )
= $true )
=> ! [X6: $i] :
( ( ( in @ X6 @ X1 )
= $true )
=> ( ( $true
= ( in @ ( kpair @ X5 @ X6 ) @ X3 ) )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X2 )
= $true ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f14]) ).
thf(f14,plain,
~ ( funcGraphProp1
=> ( funcGraphProp2
=> ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X0 @ X1 @ X2 )
=> ! [X3: $i] :
( ( func @ X0 @ X1 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( ap @ X0 @ X1 @ X3 @ X4 )
= ( ap @ X0 @ X1 @ X2 @ X4 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X1 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X2 ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,negated_conjecture,
~ ( funcGraphProp1
=> ( funcGraphProp2
=> ! [X0: $i,X3: $i,X7: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X8: $i] :
( ( func @ X0 @ X3 @ X8 )
=> ( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ( ap @ X0 @ X3 @ X7 @ X1 )
= ( ap @ X0 @ X3 @ X8 @ X1 ) ) )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X6 ) @ X8 )
=> ( in @ ( kpair @ X1 @ X6 ) @ X7 ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f7]) ).
thf(f7,conjecture,
( funcGraphProp1
=> ( funcGraphProp2
=> ! [X0: $i,X3: $i,X7: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X8: $i] :
( ( func @ X0 @ X3 @ X8 )
=> ( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ( ap @ X0 @ X3 @ X7 @ X1 )
= ( ap @ X0 @ X3 @ X8 @ X1 ) ) )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X6 ) @ X8 )
=> ( in @ ( kpair @ X1 @ X6 ) @ X7 ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2THF_3357.p',funcextLem) ).
thf(f106,plain,
( ( in @ sK13 @ sK9 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f105,f59]) ).
thf(f59,plain,
( ( in @ ( kpair @ sK13 @ sK14 ) @ sK12 )
= $true ),
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f105,plain,
( ( ( in @ ( kpair @ sK13 @ sK14 ) @ sK12 )
!= $true )
| ( ( in @ sK13 @ sK9 )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f104,f60]) ).
thf(f60,plain,
( ( in @ sK14 @ sK10 )
= $true ),
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f104,plain,
( ( ( in @ sK14 @ sK10 )
!= $true )
| ( ( in @ ( kpair @ sK13 @ sK14 ) @ sK12 )
!= $true )
| ( ( in @ sK13 @ sK9 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f103]) ).
thf(f103,plain,
( ( ( in @ ( kpair @ sK13 @ sK14 ) @ sK12 )
!= $true )
| ( $true != $true )
| ( ( in @ sK14 @ sK10 )
!= $true )
| ( ( in @ sK13 @ sK9 )
!= $true ) ),
inference(superposition,[status(thm)],[f58,f101]) ).
thf(f101,plain,
! [X0: $i,X1: $i] :
( ( ( in @ ( kpair @ X0 @ X1 ) @ sK11 )
= $true )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK12 )
!= $true )
| ( ( in @ X0 @ sK9 )
!= $true )
| ( $true
!= ( in @ X1 @ sK10 ) ) ),
inference(subsumption_resolution,[status(thm)],[f100,f62]) ).
thf(f62,plain,
( ( func @ sK9 @ sK10 @ sK11 )
= $true ),
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f100,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ sK9 )
!= $true )
| ( $true
!= ( in @ X1 @ sK10 ) )
| ( ( func @ sK9 @ sK10 @ sK11 )
!= $true )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK12 )
!= $true )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK11 )
= $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f95]) ).
thf(f95,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ sK9 )
!= $true )
| ( ( func @ sK9 @ sK10 @ sK11 )
!= $true )
| ( $true
!= ( in @ X1 @ sK10 ) )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK11 )
= $true )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK12 )
!= $true )
| ( ( in @ X0 @ sK9 )
!= $true ) ),
inference(superposition,[status(thm)],[f74,f89]) ).
thf(f89,plain,
! [X0: $i,X1: $i] :
( ( ( ap @ sK9 @ sK10 @ sK11 @ X0 )
= X1 )
| ( $true
!= ( in @ X1 @ sK10 ) )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK12 )
!= $true )
| ( ( in @ X0 @ sK9 )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f88,f56]) ).
thf(f56,plain,
( ( func @ sK9 @ sK10 @ sK12 )
= $true ),
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f88,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ sK9 )
!= $true )
| ( ( func @ sK9 @ sK10 @ sK12 )
!= $true )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK12 )
!= $true )
| ( ( ap @ sK9 @ sK10 @ sK11 @ X0 )
= X1 )
| ( $true
!= ( in @ X1 @ sK10 ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f81]) ).
thf(f81,plain,
! [X0: $i,X1: $i] :
( ( ( func @ sK9 @ sK10 @ sK12 )
!= $true )
| ( ( in @ X0 @ sK9 )
!= $true )
| ( ( in @ X0 @ sK9 )
!= $true )
| ( $true
!= ( in @ X1 @ sK10 ) )
| ( ( ap @ sK9 @ sK10 @ sK11 @ X0 )
= X1 )
| ( ( in @ ( kpair @ X0 @ X1 ) @ sK12 )
!= $true ) ),
inference(superposition,[status(thm)],[f61,f75]) ).
thf(f75,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 )
| ( ( in @ X4 @ X0 )
!= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( func @ X2 @ X0 @ X1 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f64]) ).
thf(f64,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( in @ X4 @ X0 )
!= $true )
| ( $true != $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( func @ X2 @ X0 @ X1 )
!= $true )
| ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ),
inference(definition_unfolding,[status(thm)],[f50,f55]) ).
thf(f55,plain,
funcGraphProp2 = $true,
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f50,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ X4 @ X0 )
!= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 )
| ( ( func @ X2 @ X0 @ X1 )
!= $true )
| ( funcGraphProp2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f34]) ).
thf(f34,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ) )
| ( ( func @ X2 @ X0 @ X1 )
!= $true ) )
| ( funcGraphProp2 != $true ) )
& ( ( funcGraphProp2 = $true )
| ( ( ( in @ sK3 @ sK2 )
= $true )
& ( ( in @ sK4 @ sK0 )
= $true )
& ( $true
= ( in @ ( kpair @ sK3 @ sK4 ) @ sK1 ) )
& ( sK4
!= ( ap @ sK2 @ sK0 @ sK1 @ sK3 ) )
& ( ( func @ sK2 @ sK0 @ sK1 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4])],[f30,f33,f32,f31]) ).
thf(f31,plain,
( ? [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ( ( in @ X8 @ X7 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ X5 )
= $true )
& ( ( in @ ( kpair @ X8 @ X9 ) @ X6 )
= $true )
& ( ( ap @ X7 @ X5 @ X6 @ X8 )
!= X9 ) ) )
& ( ( func @ X7 @ X5 @ X6 )
= $true ) )
=> ( ? [X8: $i] :
( ( ( in @ X8 @ sK2 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ sK0 )
= $true )
& ( ( in @ ( kpair @ X8 @ X9 ) @ sK1 )
= $true )
& ( ( ap @ sK2 @ sK0 @ sK1 @ X8 )
!= X9 ) ) )
& ( ( func @ sK2 @ sK0 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f32,plain,
( ? [X8: $i] :
( ( ( in @ X8 @ sK2 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ sK0 )
= $true )
& ( ( in @ ( kpair @ X8 @ X9 ) @ sK1 )
= $true )
& ( ( ap @ sK2 @ sK0 @ sK1 @ X8 )
!= X9 ) ) )
=> ( ( ( in @ sK3 @ sK2 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ sK0 )
= $true )
& ( ( in @ ( kpair @ sK3 @ X9 ) @ sK1 )
= $true )
& ( ( ap @ sK2 @ sK0 @ sK1 @ sK3 )
!= X9 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f33,plain,
( ? [X9: $i] :
( ( ( in @ X9 @ sK0 )
= $true )
& ( ( in @ ( kpair @ sK3 @ X9 ) @ sK1 )
= $true )
& ( ( ap @ sK2 @ sK0 @ sK1 @ sK3 )
!= X9 ) )
=> ( ( ( in @ sK4 @ sK0 )
= $true )
& ( $true
= ( in @ ( kpair @ sK3 @ sK4 ) @ sK1 ) )
& ( sK4
!= ( ap @ sK2 @ sK0 @ sK1 @ sK3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f30,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ) )
| ( ( func @ X2 @ X0 @ X1 )
!= $true ) )
| ( funcGraphProp2 != $true ) )
& ( ( funcGraphProp2 = $true )
| ? [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ( ( in @ X8 @ X7 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ X5 )
= $true )
& ( ( in @ ( kpair @ X8 @ X9 ) @ X6 )
= $true )
& ( ( ap @ X7 @ X5 @ X6 @ X8 )
!= X9 ) ) )
& ( ( func @ X7 @ X5 @ X6 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f29]) ).
thf(f29,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ) )
| ( ( func @ X2 @ X0 @ X1 )
!= $true ) )
| ( funcGraphProp2 != $true ) )
& ( ( funcGraphProp2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
= $true )
& ( ( ap @ X2 @ X0 @ X1 @ X3 )
!= X4 ) ) )
& ( ( func @ X2 @ X0 @ X1 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ) )
| ( ( func @ X2 @ X0 @ X1 )
!= $true ) )
<=> ( funcGraphProp2 = $true ) ),
inference(flattening,[status(thm)],[f27]) ).
thf(f27,plain,
( ( funcGraphProp2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ! [X4: $i] :
( ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( ( in @ X4 @ X0 )
!= $true ) )
| ( ( in @ X3 @ X2 )
!= $true ) )
| ( ( func @ X2 @ X0 @ X1 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ( funcGraphProp2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X0 @ X1 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ! [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
= $true )
=> ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
( funcGraphProp2
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X2 @ X0 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
=> ( ( ap @ X2 @ X0 @ X1 @ X3 )
= X4 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( funcGraphProp2
= ( ! [X3: $i,X7: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X6 ) @ X7 )
=> ( ( ap @ X0 @ X3 @ X7 @ X1 )
= X6 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2THF_3357.p',funcGraphProp2) ).
thf(f61,plain,
! [X4: $i] :
( ( ( ap @ sK9 @ sK10 @ sK11 @ X4 )
= ( ap @ sK9 @ sK10 @ sK12 @ X4 ) )
| ( ( in @ X4 @ sK9 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f74,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X5 @ X4 @ X6 @ X7 ) ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) )
| ( ( func @ X5 @ X4 @ X6 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f73]) ).
thf(f73,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( func @ X5 @ X4 @ X6 )
!= $true )
| ( ( in @ ( kpair @ X7 @ ( ap @ X5 @ X4 @ X6 @ X7 ) ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f51,f63]) ).
thf(f63,plain,
funcGraphProp1 = $true,
inference(cnf_transformation,[status(thm)],[f44]) ).
thf(f51,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X5 @ X4 @ X6 @ X7 ) ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) )
| ( ( func @ X5 @ X4 @ X6 )
!= $true )
| ( funcGraphProp1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f39]) ).
thf(f39,plain,
( ( ( funcGraphProp1 = $true )
| ( ( ( in @ ( kpair @ sK8 @ ( ap @ sK6 @ sK5 @ sK7 @ sK8 ) ) @ sK7 )
!= $true )
& ( ( in @ sK8 @ sK6 )
= $true )
& ( ( func @ sK6 @ sK5 @ sK7 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X5 @ X4 @ X6 @ X7 ) ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( func @ X5 @ X4 @ X6 )
!= $true ) )
| ( funcGraphProp1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7,sK8])],[f36,f38,f37]) ).
thf(f37,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( func @ X1 @ X0 @ X2 )
= $true ) )
=> ( ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ sK6 @ sK5 @ sK7 @ X3 ) ) @ sK7 )
!= $true )
& ( ( in @ X3 @ sK6 )
= $true ) )
& ( ( func @ sK6 @ sK5 @ sK7 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f38,plain,
( ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ sK6 @ sK5 @ sK7 @ X3 ) ) @ sK7 )
!= $true )
& ( ( in @ X3 @ sK6 )
= $true ) )
=> ( ( ( in @ ( kpair @ sK8 @ ( ap @ sK6 @ sK5 @ sK7 @ sK8 ) ) @ sK7 )
!= $true )
& ( ( in @ sK8 @ sK6 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f36,plain,
( ( ( funcGraphProp1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( func @ X1 @ X0 @ X2 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X5 @ X4 @ X6 @ X7 ) ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( func @ X5 @ X4 @ X6 )
!= $true ) )
| ( funcGraphProp1 != $true ) ) ),
inference(rectify,[status(thm)],[f35]) ).
thf(f35,plain,
( ( ( funcGraphProp1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( func @ X1 @ X0 @ X2 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 )
= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( func @ X1 @ X0 @ X2 )
!= $true ) )
| ( funcGraphProp1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f26]) ).
thf(f26,plain,
( ( funcGraphProp1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 )
= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( func @ X1 @ X0 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f23]) ).
thf(f23,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 )
= $true ) ) )
<=> ( funcGraphProp1 = $true ) ),
inference(fool_elimination,[status(thm)],[f22]) ).
thf(f22,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( kpair @ X3 @ ( ap @ X1 @ X0 @ X2 @ X3 ) ) @ X2 ) ) )
= funcGraphProp1 ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X3: $i,X0: $i,X7: $i] :
( ( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( kpair @ X1 @ ( ap @ X0 @ X3 @ X7 @ X1 ) ) @ X7 ) ) )
= funcGraphProp1 ),
file('/export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2THF_3357.p',funcGraphProp1) ).
thf(f58,plain,
( $true
!= ( in @ ( kpair @ sK13 @ sK14 ) @ sK11 ) ),
inference(cnf_transformation,[status(thm)],[f44]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU687^2 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n020.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:31:12 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.49 ---- Original DTF file ---
% 0.29/0.49 thf(spec,logic,$$dhol).
% 0.29/0.49 %------------------------------------------------------------------------------
% 0.29/0.49 % File : SEU687^2 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.49 % Domain : Set Theory
% 0.29/0.49 % Problem : Functions - Extensionality and Beta Reduction
% 0.29/0.49 % Version : Especial > Reduced > Especial.
% 0.29/0.49 % English : (! A:i.! B:i.! f:i.func A B f -> (! g:i.func A B g ->
% 0.29/0.49 % (! x:i.in x A -> ap A B f x = ap A B g x) -> (! x:i.in x A ->
% 0.29/0.49 % (! y:i.in y B -> in (kpair x y) g -> in (kpair x y) f))))
% 0.29/0.49
% 0.29/0.49 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.49 % Source : [Bro08]
% 0.29/0.49 % Names : ZFC189l [Bro08]
% 0.29/0.49
% 0.29/0.49 % Status : Theorem
% 0.29/0.49 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.46 v8.1.0, 0.36 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v4.0.1, 0.67 v4.0.0, 1.00 v3.7.0
% 0.29/0.49 % Syntax : Number of formulae : 21 ( 6 unt; 14 typ; 6 def)
% 0.29/0.49 % Number of atoms : 38 ( 9 equ; 0 cnn)
% 0.29/0.49 % Maximal formula atoms : 10 ( 5 avg)
% 0.29/0.49 % Number of connectives : 97 ( 0 ~; 0 |; 2 &; 79 @)
% 0.29/0.49 % ( 0 <=>; 16 =>; 0 <=; 0 <~>)
% 0.29/0.49 % Maximal formula depth : 19 ( 4 avg)
% 0.29/0.49 % Number of types : 2 ( 0 usr)
% 0.29/0.49 % Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% 0.29/0.49 % Number of symbols : 15 ( 14 usr; 3 con; 0-4 aty)
% 0.29/0.49 % Number of variables : 29 ( 11 ^; 17 !; 1 ?; 29 :)
% 0.29/0.49 % SPC : TH0_THM_EQU_NAR
% 0.29/0.49
% 0.29/0.49 % Comments : http://mathgate.info/detsetitem.php?id=414
% 0.29/0.49 %------------------------------------------------------------------------------
% 0.29/0.49 thf(in_type,type,
% 0.29/0.49 in: $i > $i > $o ).
% 0.29/0.49
% 0.29/0.49 thf(emptyset_type,type,
% 0.29/0.49 emptyset: $i ).
% 0.29/0.49
% 0.29/0.49 thf(setadjoin_type,type,
% 0.29/0.49 setadjoin: $i > $i > $i ).
% 0.29/0.49
% 0.29/0.49 thf(dsetconstr_type,type,
% 0.29/0.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.49
% 0.29/0.49 thf(subset_type,type,
% 0.29/0.49 subset: $i > $i > $o ).
% 0.29/0.49
% 0.29/0.49 thf(kpair_type,type,
% 0.29/0.49 kpair: $i > $i > $i ).
% 0.29/0.49
% 0.29/0.49 thf(cartprod_type,type,
% 0.29/0.49 cartprod: $i > $i > $i ).
% 0.29/0.49
% 0.29/0.49 thf(singleton_type,type,
% 0.29/0.49 singleton: $i > $o ).
% 0.29/0.49
% 0.29/0.49 thf(singleton,definition,
% 0.29/0.49 ( singleton
% 0.29/0.49 = ( ^ [A: $i] :
% 0.29/0.49 ? [Xx: $i] :
% 0.29/0.49 ( ( in @ Xx @ A )
% 0.29/0.49 & ( A
% 0.29/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(ex1_type,type,
% 0.29/0.49 ex1: $i > ( $i > $o ) > $o ).
% 0.29/0.49
% 0.29/0.49 thf(ex1,definition,
% 0.29/0.49 ( ex1
% 0.29/0.49 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.29/0.49 ( singleton
% 0.29/0.49 @ ( dsetconstr @ A
% 0.29/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(breln_type,type,
% 0.29/0.49 breln: $i > $i > $i > $o ).
% 0.29/0.49
% 0.29/0.49 thf(breln,definition,
% 0.29/0.49 ( breln
% 0.29/0.49 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(func_type,type,
% 0.29/0.49 func: $i > $i > $i > $o ).
% 0.29/0.49
% 0.29/0.49 thf(func,definition,
% 0.29/0.49 ( func
% 0.29/0.49 = ( ^ [A: $i,B: $i,R: $i] :
% 0.29/0.49 ( ( breln @ A @ B @ R )
% 0.29/0.49 & ! [Xx: $i] :
% 0.29/0.49 ( ( in @ Xx @ A )
% 0.29/0.49 => ( ex1 @ B
% 0.29/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(ap_type,type,
% 0.29/0.49 ap: $i > $i > $i > $i > $i ).
% 0.29/0.49
% 0.29/0.49 thf(funcGraphProp1_type,type,
% 0.29/0.49 funcGraphProp1: $o ).
% 0.29/0.49
% 0.29/0.49 thf(funcGraphProp1,definition,
% 0.29/0.49 ( funcGraphProp1
% 0.29/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.49 ( ( func @ A @ B @ Xf )
% 0.29/0.49 => ! [Xx: $i] :
% 0.29/0.49 ( ( in @ Xx @ A )
% 0.29/0.49 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(funcGraphProp2_type,type,
% 0.29/0.49 funcGraphProp2: $o ).
% 0.29/0.49
% 0.29/0.49 thf(funcGraphProp2,definition,
% 0.29/0.49 ( funcGraphProp2
% 0.29/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.49 ( ( func @ A @ B @ Xf )
% 0.29/0.49 => ! [Xx: $i] :
% 0.29/0.49 ( ( in @ Xx @ A )
% 0.29/0.49 => ! [Xy: $i] :
% 0.29/0.49 ( ( in @ Xy @ B )
% 0.29/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.29/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.49 = Xy ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(funcextLem,conjecture,
% 0.29/0.49 ( funcGraphProp1
% 0.29/0.49 => ( funcGraphProp2
% 0.29/0.49 => ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.49 ( ( func @ A @ B @ Xf )
% 0.29/0.49 => ! [Xg: $i] :
% 0.29/0.49 ( ( func @ A @ B @ Xg )
% 0.29/0.49 => ( ! [Xx: $i] :
% 0.29/0.49 ( ( in @ Xx @ A )
% 0.29/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.49 => ! [Xx: $i] :
% 0.29/0.49 ( ( in @ Xx @ A )
% 0.29/0.49 => ! [Xy: $i] :
% 0.29/0.49 ( ( in @ Xy @ B )
% 0.29/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.29/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 %------------------------------------------------------------------------------
% 0.29/0.49 ------------------------
% 1.73/1.43 ---- Embedded in THF ---
% 1.73/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.73/1.43 %%% Generated on Tue Feb 24 12:31:13 EST 2026
% 1.73/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 1.73/1.43 %%% Logic specification used:
% 1.73/1.43 %%% thf(spec, logic, $$dhol).
% 1.73/1.43
% 1.73/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2DTF3357.p
% See solution above
% 1.73/1.44 ------------------------
% 1.73/1.44 ---- Cleaned THF ---
% 1.73/1.45 thf(in_type,type,
% 1.73/1.45 in: $i > $i > $o ).
% 1.73/1.45
% 1.73/1.45 thf(emptyset_type,type,
% 1.73/1.45 emptyset: $i ).
% 1.73/1.45
% 1.73/1.45 thf(setadjoin_type,type,
% 1.73/1.45 setadjoin: $i > $i > $i ).
% 1.73/1.45
% 1.73/1.45 thf(dsetconstr_type,type,
% 1.73/1.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.73/1.45
% 1.73/1.45 thf(subset_type,type,
% 1.73/1.45 subset: $i > $i > $o ).
% 1.73/1.45
% 1.73/1.45 thf(kpair_type,type,
% 1.73/1.45 kpair: $i > $i > $i ).
% 1.73/1.45
% 1.73/1.45 thf(cartprod_type,type,
% 1.73/1.45 cartprod: $i > $i > $i ).
% 1.73/1.45
% 1.73/1.45 thf(singleton_type,type,
% 1.73/1.45 singleton: $i > $o ).
% 1.73/1.45
% 1.73/1.45 thf(singleton,definition,
% 1.73/1.45 ( singleton
% 1.73/1.45 = ( ^ [A: $i] :
% 1.73/1.45 ? [Xx: $i] :
% 1.73/1.45 ( ( in @ Xx @ A )
% 1.73/1.45 & ( A
% 1.73/1.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.73/1.45
% 1.73/1.45 thf(ex1_type,type,
% 1.73/1.45 ex1: $i > ( $i > $o ) > $o ).
% 1.73/1.45
% 1.73/1.45 thf(ex1,definition,
% 1.73/1.45 ( ex1
% 1.73/1.45 = ( ^ [A: $i,Xphi: $i > $o] :
% 1.73/1.45 ( singleton
% 1.73/1.45 @ ( dsetconstr @ A
% 1.73/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.73/1.45
% 1.73/1.45 thf(breln_type,type,
% 1.73/1.45 breln: $i > $i > $i > $o ).
% 1.73/1.45
% 1.73/1.45 thf(breln,definition,
% 1.73/1.45 ( breln
% 1.73/1.45 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 1.73/1.45
% 1.73/1.45 thf(func_type,type,
% 1.73/1.45 func: $i > $i > $i > $o ).
% 1.73/1.45
% 1.73/1.45 thf(func,definition,
% 1.73/1.45 ( func
% 1.73/1.45 = ( ^ [A: $i,B: $i,R: $i] :
% 1.73/1.45 ( ( breln @ A @ B @ R )
% 1.73/1.45 & ! [Xx: $i] :
% 1.73/1.45 ( ( in @ Xx @ A )
% 1.73/1.45 => ( ex1 @ B
% 1.73/1.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 1.73/1.45
% 1.73/1.45 thf(ap_type,type,
% 1.73/1.45 ap: $i > $i > $i > $i > $i ).
% 1.73/1.45
% 1.73/1.45 thf(funcGraphProp1_type,type,
% 1.73/1.45 funcGraphProp1: $o ).
% 1.73/1.45
% 1.73/1.45 thf(funcGraphProp1,definition,
% 1.73/1.45 ( funcGraphProp1
% 1.73/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.73/1.45 ( ( func @ A @ B @ Xf )
% 1.73/1.45 => ! [Xx: $i] :
% 1.73/1.45 ( ( in @ Xx @ A )
% 1.73/1.45 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 1.73/1.45
% 1.73/1.45 thf(funcGraphProp2_type,type,
% 1.73/1.45 funcGraphProp2: $o ).
% 1.73/1.45
% 1.73/1.45 thf(funcGraphProp2,definition,
% 1.73/1.45 ( funcGraphProp2
% 1.73/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.73/1.45 ( ( func @ A @ B @ Xf )
% 1.73/1.45 => ! [Xx: $i] :
% 1.73/1.45 ( ( in @ Xx @ A )
% 1.73/1.45 => ! [Xy: $i] :
% 1.73/1.45 ( ( in @ Xy @ B )
% 1.73/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 1.73/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 1.73/1.45 = Xy ) ) ) ) ) ) ) ).
% 1.73/1.45
% 1.73/1.45 thf(funcextLem,conjecture,
% 1.73/1.45 ( funcGraphProp1
% 1.73/1.45 => ( funcGraphProp2
% 1.73/1.45 => ! [A: $i,B: $i,Xf: $i] :
% 1.73/1.45 ( ( func @ A @ B @ Xf )
% 1.73/1.45 => ! [Xg: $i] :
% 1.73/1.45 ( ( func @ A @ B @ Xg )
% 1.73/1.45 => ( ! [Xx: $i] :
% 1.73/1.45 ( ( in @ Xx @ A )
% 1.73/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 1.73/1.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 1.73/1.45 => ! [Xx: $i] :
% 1.73/1.45 ( ( in @ Xx @ A )
% 1.73/1.45 => ! [Xy: $i] :
% 1.73/1.45 ( ( in @ Xy @ B )
% 1.73/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 1.73/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 1.73/1.45 ------------------------
% 1.73/1.47 % (3455)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/183Mi)
% 1.73/1.47 % (3457)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/27Mi)
% 1.73/1.47 % (3456)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/4Mi)
% 1.73/1.47 % (3458)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/2Mi)
% 1.73/1.47 % (3459)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/2Mi)
% 1.73/1.47 % (3460)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/275Mi)
% 1.73/1.47 % (3461)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3357 for (2999ds/18Mi)
% 1.73/1.47 % (3458)Instruction limit reached!
% 1.73/1.47 % (3458)------------------------------
% 1.73/1.47 % (3458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.73/1.47 % (3458)Termination reason: Unknown
% 1.73/1.47 % (3458)Termination phase: Unused predicate definition removal
% 1.73/1.47
% 1.73/1.47 % (3458)Memory used [KB]: 1023
% 1.73/1.47 % (3458)Time elapsed: 0.003 s
% 1.73/1.47 % (3458)Instructions burned: 3 (million)
% 1.73/1.47 % (3458)------------------------------
% 1.73/1.47 % (3458)------------------------------
% 1.73/1.47 % (3459)Instruction limit reached!
% 1.73/1.47 % (3459)------------------------------
% 1.73/1.47 % (3459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.73/1.47 % (3459)Termination reason: Unknown
% 1.73/1.47 % (3459)Termination phase: Property scanning
% 1.73/1.47
% 1.73/1.47 % (3459)Memory used [KB]: 1023
% 1.73/1.47 % (3459)Time elapsed: 0.003 s
% 1.73/1.47 % (3459)Instructions burned: 3 (million)
% 1.73/1.47 % (3459)------------------------------
% 1.73/1.47 % (3459)------------------------------
% 1.73/1.47 % (3456)Instruction limit reached!
% 1.73/1.47 % (3456)------------------------------
% 1.73/1.47 % (3456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.73/1.47 % (3456)Termination reason: Unknown
% 1.73/1.47 % (3456)Termination phase: Function definition elimination
% 1.73/1.47
% 1.73/1.47 % (3456)Memory used [KB]: 1023
% 1.73/1.47 % (3456)Time elapsed: 0.004 s
% 1.73/1.47 % (3456)Instructions burned: 5 (million)
% 1.73/1.47 % (3456)------------------------------
% 1.73/1.47 % (3456)------------------------------
% 1.73/1.47 % (3460)First to succeed.
% 1.73/1.47 % (3461)Instruction limit reached!
% 1.73/1.47 % (3461)------------------------------
% 1.73/1.47 % (3461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.73/1.47 % (3461)Termination reason: Unknown
% 1.73/1.47 % (3461)Termination phase: Saturation
% 1.73/1.47
% 1.73/1.47 % (3461)Memory used [KB]: 5756
% 1.73/1.47 % (3461)Time elapsed: 0.011 s
% 1.73/1.47 % (3461)Instructions burned: 19 (million)
% 1.73/1.47 % (3461)------------------------------
% 1.73/1.47 % (3461)------------------------------
% 1.73/1.48 % (3460)Refutation found. Thanks to Tanya!
% 1.73/1.48 % SZS status Theorem for DTF2THF_3357
% 1.73/1.48 % SZS output start Proof for DTF2THF_3357
% 1.73/1.48 thf(func_def_0, type, in: $i > $i > $o).
% 1.73/1.48 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.73/1.48 thf(func_def_3, type, dsetconstr: $i > ($i > $o) > $i).
% 1.73/1.48 thf(func_def_4, type, subset: $i > $i > $o).
% 1.73/1.48 thf(func_def_5, type, kpair: $i > $i > $i).
% 1.73/1.48 thf(func_def_6, type, cartprod: $i > $i > $i).
% 1.73/1.48 thf(func_def_7, type, singleton: $i > $o).
% 1.73/1.48 thf(func_def_9, type, ex1: $i > ($i > $o) > $o).
% 1.73/1.48 thf(func_def_10, type, breln: $i > $i > $i > $o).
% 1.73/1.48 thf(func_def_11, type, func: $i > $i > $i > $o).
% 1.73/1.48 thf(func_def_12, type, ap: $i > $i > $i > $i > $i).
% 1.73/1.48 thf(f107,plain,(
% 1.73/1.48 $false),
% 1.73/1.48 inference(subsumption_resolution,[status(thm)],[f106,f57])).
% 1.73/1.48 thf(f57,plain,(
% 1.73/1.48 ((in @ sK13 @ sK9) = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f44,plain,(
% 1.73/1.48 (funcGraphProp1 = $true) & (((func @ sK9 @ sK10 @ sK11) = $true) & (! [X4 : $i] : (((ap @ sK9 @ sK10 @ sK11 @ X4) = (ap @ sK9 @ sK10 @ sK12 @ X4)) | ((in @ X4 @ sK9) != $true)) & ((((in @ sK14 @ sK10) = $true) & ((in @ (kpair @ sK13 @ sK14) @ sK12) = $true) & ($true != (in @ (kpair @ sK13 @ sK14) @ sK11))) & ((in @ sK13 @ sK9) = $true)) & ((func @ sK9 @ sK10 @ sK12) = $true))) & (funcGraphProp2 = $true)),
% 1.73/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11,sK12,sK13,sK14])],[f25,f43,f42,f41,f40])).
% 1.73/1.48 thf(f40,plain,(
% 1.73/1.48 ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) = $true) & ? [X3 : $i] : (! [X4 : $i] : (((ap @ X0 @ X1 @ X3 @ X4) = (ap @ X0 @ X1 @ X2 @ X4)) | ((in @ X4 @ X0) != $true)) & ? [X5 : $i] : (? [X6 : $i] : (((in @ X6 @ X1) = $true) & ($true = (in @ (kpair @ X5 @ X6) @ X3)) & ((in @ (kpair @ X5 @ X6) @ X2) != $true)) & ((in @ X5 @ X0) = $true)) & ((func @ X0 @ X1 @ X3) = $true))) => (((func @ sK9 @ sK10 @ sK11) = $true) & ? [X3 : $i] : (! [X4 : $i] : (((ap @ sK9 @ sK10 @ sK11 @ X4) = (ap @ sK9 @ sK10 @ X3 @ X4)) | ((in @ X4 @ sK9) != $true)) & ? [X5 : $i] : (? [X6 : $i] : (((in @ X6 @ sK10) = $true) & ($true = (in @ (kpair @ X5 @ X6) @ X3)) & ((in @ (kpair @ X5 @ X6) @ sK11) != $true)) & ($true = (in @ X5 @ sK9))) & ((func @ sK9 @ sK10 @ X3) = $true)))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f41,plain,(
% 1.73/1.48 ? [X3 : $i] : (! [X4 : $i] : (((ap @ sK9 @ sK10 @ sK11 @ X4) = (ap @ sK9 @ sK10 @ X3 @ X4)) | ((in @ X4 @ sK9) != $true)) & ? [X5 : $i] : (? [X6 : $i] : (((in @ X6 @ sK10) = $true) & ($true = (in @ (kpair @ X5 @ X6) @ X3)) & ((in @ (kpair @ X5 @ X6) @ sK11) != $true)) & ($true = (in @ X5 @ sK9))) & ((func @ sK9 @ sK10 @ X3) = $true)) => (! [X4 : $i] : (((ap @ sK9 @ sK10 @ sK11 @ X4) = (ap @ sK9 @ sK10 @ sK12 @ X4)) | ((in @ X4 @ sK9) != $true)) & ? [X5 : $i] : (? [X6 : $i] : (((in @ X6 @ sK10) = $true) & ((in @ (kpair @ X5 @ X6) @ sK12) = $true) & ((in @ (kpair @ X5 @ X6) @ sK11) != $true)) & ($true = (in @ X5 @ sK9))) & ((func @ sK9 @ sK10 @ sK12) = $true))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f42,plain,(
% 1.73/1.48 ? [X5 : $i] : (? [X6 : $i] : (((in @ X6 @ sK10) = $true) & ((in @ (kpair @ X5 @ X6) @ sK12) = $true) & ((in @ (kpair @ X5 @ X6) @ sK11) != $true)) & ($true = (in @ X5 @ sK9))) => (? [X6 : $i] : (((in @ X6 @ sK10) = $true) & ((in @ (kpair @ sK13 @ X6) @ sK12) = $true) & ((in @ (kpair @ sK13 @ X6) @ sK11) != $true)) & ((in @ sK13 @ sK9) = $true))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f43,plain,(
% 1.73/1.48 ? [X6 : $i] : (((in @ X6 @ sK10) = $true) & ((in @ (kpair @ sK13 @ X6) @ sK12) = $true) & ((in @ (kpair @ sK13 @ X6) @ sK11) != $true)) => (((in @ sK14 @ sK10) = $true) & ((in @ (kpair @ sK13 @ sK14) @ sK12) = $true) & ($true != (in @ (kpair @ sK13 @ sK14) @ sK11)))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f25,plain,(
% 1.73/1.48 (funcGraphProp1 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) = $true) & ? [X3 : $i] : (! [X4 : $i] : (((ap @ X0 @ X1 @ X3 @ X4) = (ap @ X0 @ X1 @ X2 @ X4)) | ((in @ X4 @ X0) != $true)) & ? [X5 : $i] : (? [X6 : $i] : (((in @ X6 @ X1) = $true) & ($true = (in @ (kpair @ X5 @ X6) @ X3)) & ((in @ (kpair @ X5 @ X6) @ X2) != $true)) & ((in @ X5 @ X0) = $true)) & ((func @ X0 @ X1 @ X3) = $true))) & (funcGraphProp2 = $true)),
% 1.73/1.48 inference(flattening,[status(thm)],[f24])).
% 1.73/1.48 thf(f24,plain,(
% 1.73/1.48 (? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : ((? [X5 : $i] : (? [X6 : $i] : ((((in @ (kpair @ X5 @ X6) @ X2) != $true) & ($true = (in @ (kpair @ X5 @ X6) @ X3))) & ((in @ X6 @ X1) = $true)) & ((in @ X5 @ X0) = $true)) & ! [X4 : $i] : (((ap @ X0 @ X1 @ X3 @ X4) = (ap @ X0 @ X1 @ X2 @ X4)) | ((in @ X4 @ X0) != $true))) & ((func @ X0 @ X1 @ X3) = $true)) & ((func @ X0 @ X1 @ X2) = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp1 = $true)),
% 1.73/1.48 inference(ennf_transformation,[status(thm)],[f15])).
% 1.73/1.48 thf(f15,plain,(
% 1.73/1.48 ~((funcGraphProp1 = $true) => ((funcGraphProp2 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) = $true) => ! [X3 : $i] : (((func @ X0 @ X1 @ X3) = $true) => (! [X4 : $i] : (((in @ X4 @ X0) = $true) => ((ap @ X0 @ X1 @ X3 @ X4) = (ap @ X0 @ X1 @ X2 @ X4))) => ! [X5 : $i] : (((in @ X5 @ X0) = $true) => ! [X6 : $i] : (((in @ X6 @ X1) = $true) => (($true = (in @ (kpair @ X5 @ X6) @ X3)) => ((in @ (kpair @ X5 @ X6) @ X2) = $true)))))))))),
% 1.73/1.48 inference(fool_elimination,[status(thm)],[f14])).
% 1.73/1.48 thf(f14,plain,(
% 1.73/1.48 ~(funcGraphProp1 => (funcGraphProp2 => ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X0 @ X1 @ X2) => ! [X3 : $i] : ((func @ X0 @ X1 @ X3) => (! [X4 : $i] : ((in @ X4 @ X0) => ((ap @ X0 @ X1 @ X3 @ X4) = (ap @ X0 @ X1 @ X2 @ X4))) => ! [X5 : $i] : ((in @ X5 @ X0) => ! [X6 : $i] : ((in @ X6 @ X1) => ((in @ (kpair @ X5 @ X6) @ X3) => (in @ (kpair @ X5 @ X6) @ X2)))))))))),
% 1.73/1.48 inference(rectify,[status(thm)],[f8])).
% 1.73/1.48 thf(f8,negated_conjecture,(
% 1.73/1.48 ~(funcGraphProp1 => (funcGraphProp2 => ! [X0 : $i,X3 : $i,X7 : $i] : ((func @ X0 @ X3 @ X7) => ! [X8 : $i] : ((func @ X0 @ X3 @ X8) => (! [X1 : $i] : ((in @ X1 @ X0) => ((ap @ X0 @ X3 @ X7 @ X1) = (ap @ X0 @ X3 @ X8 @ X1))) => ! [X1 : $i] : ((in @ X1 @ X0) => ! [X6 : $i] : ((in @ X6 @ X3) => ((in @ (kpair @ X1 @ X6) @ X8) => (in @ (kpair @ X1 @ X6) @ X7)))))))))),
% 1.73/1.48 inference(negated_conjecture,[status(cth)],[f7])).
% 1.73/1.48 thf(f7,conjecture,(
% 1.73/1.48 funcGraphProp1 => (funcGraphProp2 => ! [X0 : $i,X3 : $i,X7 : $i] : ((func @ X0 @ X3 @ X7) => ! [X8 : $i] : ((func @ X0 @ X3 @ X8) => (! [X1 : $i] : ((in @ X1 @ X0) => ((ap @ X0 @ X3 @ X7 @ X1) = (ap @ X0 @ X3 @ X8 @ X1))) => ! [X1 : $i] : ((in @ X1 @ X0) => ! [X6 : $i] : ((in @ X6 @ X3) => ((in @ (kpair @ X1 @ X6) @ X8) => (in @ (kpair @ X1 @ X6) @ X7))))))))),
% 1.73/1.48 file('/export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2THF_3357.p',funcextLem)).
% 1.73/1.48 thf(f106,plain,(
% 1.73/1.48 ((in @ sK13 @ sK9) != $true)),
% 1.73/1.48 inference(subsumption_resolution,[status(thm)],[f105,f59])).
% 1.73/1.48 thf(f59,plain,(
% 1.73/1.48 ((in @ (kpair @ sK13 @ sK14) @ sK12) = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f105,plain,(
% 1.73/1.48 ((in @ (kpair @ sK13 @ sK14) @ sK12) != $true) | ((in @ sK13 @ sK9) != $true)),
% 1.73/1.48 inference(subsumption_resolution,[status(thm)],[f104,f60])).
% 1.73/1.48 thf(f60,plain,(
% 1.73/1.48 ((in @ sK14 @ sK10) = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f104,plain,(
% 1.73/1.48 ((in @ sK14 @ sK10) != $true) | ((in @ (kpair @ sK13 @ sK14) @ sK12) != $true) | ((in @ sK13 @ sK9) != $true)),
% 1.73/1.48 inference(trivial_inequality_removal,[status(thm)],[f103])).
% 1.73/1.48 thf(f103,plain,(
% 1.73/1.48 ((in @ (kpair @ sK13 @ sK14) @ sK12) != $true) | ($true != $true) | ((in @ sK14 @ sK10) != $true) | ((in @ sK13 @ sK9) != $true)),
% 1.73/1.48 inference(superposition,[status(thm)],[f58,f101])).
% 1.73/1.48 thf(f101,plain,(
% 1.73/1.48 ( ! [X0 : $i,X1 : $i] : (((in @ (kpair @ X0 @ X1) @ sK11) = $true) | ((in @ (kpair @ X0 @ X1) @ sK12) != $true) | ((in @ X0 @ sK9) != $true) | ($true != (in @ X1 @ sK10))) )),
% 1.73/1.48 inference(subsumption_resolution,[status(thm)],[f100,f62])).
% 1.73/1.48 thf(f62,plain,(
% 1.73/1.48 ((func @ sK9 @ sK10 @ sK11) = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f100,plain,(
% 1.73/1.48 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ sK9) != $true) | ($true != (in @ X1 @ sK10)) | ((func @ sK9 @ sK10 @ sK11) != $true) | ((in @ (kpair @ X0 @ X1) @ sK12) != $true) | ((in @ (kpair @ X0 @ X1) @ sK11) = $true)) )),
% 1.73/1.48 inference(duplicate_literal_removal,[status(thm)],[f95])).
% 1.73/1.48 thf(f95,plain,(
% 1.73/1.48 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ sK9) != $true) | ((func @ sK9 @ sK10 @ sK11) != $true) | ($true != (in @ X1 @ sK10)) | ((in @ (kpair @ X0 @ X1) @ sK11) = $true) | ((in @ (kpair @ X0 @ X1) @ sK12) != $true) | ((in @ X0 @ sK9) != $true)) )),
% 1.73/1.48 inference(superposition,[status(thm)],[f74,f89])).
% 1.73/1.48 thf(f89,plain,(
% 1.73/1.48 ( ! [X0 : $i,X1 : $i] : (((ap @ sK9 @ sK10 @ sK11 @ X0) = X1) | ($true != (in @ X1 @ sK10)) | ((in @ (kpair @ X0 @ X1) @ sK12) != $true) | ((in @ X0 @ sK9) != $true)) )),
% 1.73/1.48 inference(subsumption_resolution,[status(thm)],[f88,f56])).
% 1.73/1.48 thf(f56,plain,(
% 1.73/1.48 ((func @ sK9 @ sK10 @ sK12) = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f88,plain,(
% 1.73/1.48 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ sK9) != $true) | ((func @ sK9 @ sK10 @ sK12) != $true) | ((in @ (kpair @ X0 @ X1) @ sK12) != $true) | ((ap @ sK9 @ sK10 @ sK11 @ X0) = X1) | ($true != (in @ X1 @ sK10))) )),
% 1.73/1.48 inference(duplicate_literal_removal,[status(thm)],[f81])).
% 1.73/1.48 thf(f81,plain,(
% 1.73/1.48 ( ! [X0 : $i,X1 : $i] : (((func @ sK9 @ sK10 @ sK12) != $true) | ((in @ X0 @ sK9) != $true) | ((in @ X0 @ sK9) != $true) | ($true != (in @ X1 @ sK10)) | ((ap @ sK9 @ sK10 @ sK11 @ X0) = X1) | ((in @ (kpair @ X0 @ X1) @ sK12) != $true)) )),
% 1.73/1.48 inference(superposition,[status(thm)],[f61,f75])).
% 1.73/1.48 thf(f75,plain,(
% 1.73/1.48 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (((ap @ X2 @ X0 @ X1 @ X3) = X4) | ((in @ X4 @ X0) != $true) | ((in @ X3 @ X2) != $true) | ((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((func @ X2 @ X0 @ X1) != $true)) )),
% 1.73/1.48 inference(trivial_inequality_removal,[status(thm)],[f64])).
% 1.73/1.48 thf(f64,plain,(
% 1.73/1.48 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((in @ X4 @ X0) != $true) | ($true != $true) | ((in @ X3 @ X2) != $true) | ((func @ X2 @ X0 @ X1) != $true) | ((ap @ X2 @ X0 @ X1 @ X3) = X4)) )),
% 1.73/1.48 inference(definition_unfolding,[status(thm)],[f50,f55])).
% 1.73/1.48 thf(f55,plain,(
% 1.73/1.48 (funcGraphProp2 = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f50,plain,(
% 1.73/1.48 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (((in @ X3 @ X2) != $true) | ((in @ X4 @ X0) != $true) | ((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((ap @ X2 @ X0 @ X1 @ X3) = X4) | ((func @ X2 @ X0 @ X1) != $true) | (funcGraphProp2 != $true)) )),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f34])).
% 1.73/1.48 thf(f34,plain,(
% 1.73/1.48 (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ! [X4 : $i] : (((in @ X4 @ X0) != $true) | ((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((ap @ X2 @ X0 @ X1 @ X3) = X4))) | ((func @ X2 @ X0 @ X1) != $true)) | (funcGraphProp2 != $true)) & ((funcGraphProp2 = $true) | ((((in @ sK3 @ sK2) = $true) & (((in @ sK4 @ sK0) = $true) & ($true = (in @ (kpair @ sK3 @ sK4) @ sK1)) & (sK4 != (ap @ sK2 @ sK0 @ sK1 @ sK3)))) & ((func @ sK2 @ sK0 @ sK1) = $true)))),
% 1.73/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4])],[f30,f33,f32,f31])).
% 1.73/1.48 thf(f31,plain,(
% 1.73/1.48 ? [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (((in @ X8 @ X7) = $true) & ? [X9 : $i] : (((in @ X9 @ X5) = $true) & ((in @ (kpair @ X8 @ X9) @ X6) = $true) & ((ap @ X7 @ X5 @ X6 @ X8) != X9))) & ((func @ X7 @ X5 @ X6) = $true)) => (? [X8 : $i] : (((in @ X8 @ sK2) = $true) & ? [X9 : $i] : (((in @ X9 @ sK0) = $true) & ((in @ (kpair @ X8 @ X9) @ sK1) = $true) & ((ap @ sK2 @ sK0 @ sK1 @ X8) != X9))) & ((func @ sK2 @ sK0 @ sK1) = $true))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f32,plain,(
% 1.73/1.48 ? [X8 : $i] : (((in @ X8 @ sK2) = $true) & ? [X9 : $i] : (((in @ X9 @ sK0) = $true) & ((in @ (kpair @ X8 @ X9) @ sK1) = $true) & ((ap @ sK2 @ sK0 @ sK1 @ X8) != X9))) => (((in @ sK3 @ sK2) = $true) & ? [X9 : $i] : (((in @ X9 @ sK0) = $true) & ((in @ (kpair @ sK3 @ X9) @ sK1) = $true) & ((ap @ sK2 @ sK0 @ sK1 @ sK3) != X9)))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f33,plain,(
% 1.73/1.48 ? [X9 : $i] : (((in @ X9 @ sK0) = $true) & ((in @ (kpair @ sK3 @ X9) @ sK1) = $true) & ((ap @ sK2 @ sK0 @ sK1 @ sK3) != X9)) => (((in @ sK4 @ sK0) = $true) & ($true = (in @ (kpair @ sK3 @ sK4) @ sK1)) & (sK4 != (ap @ sK2 @ sK0 @ sK1 @ sK3)))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f30,plain,(
% 1.73/1.48 (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ! [X4 : $i] : (((in @ X4 @ X0) != $true) | ((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((ap @ X2 @ X0 @ X1 @ X3) = X4))) | ((func @ X2 @ X0 @ X1) != $true)) | (funcGraphProp2 != $true)) & ((funcGraphProp2 = $true) | ? [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (((in @ X8 @ X7) = $true) & ? [X9 : $i] : (((in @ X9 @ X5) = $true) & ((in @ (kpair @ X8 @ X9) @ X6) = $true) & ((ap @ X7 @ X5 @ X6 @ X8) != X9))) & ((func @ X7 @ X5 @ X6) = $true)))),
% 1.73/1.48 inference(rectify,[status(thm)],[f29])).
% 1.73/1.48 thf(f29,plain,(
% 1.73/1.48 (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ! [X4 : $i] : (((in @ X4 @ X0) != $true) | ((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((ap @ X2 @ X0 @ X1 @ X3) = X4))) | ((func @ X2 @ X0 @ X1) != $true)) | (funcGraphProp2 != $true)) & ((funcGraphProp2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ? [X4 : $i] : (((in @ X4 @ X0) = $true) & ((in @ (kpair @ X3 @ X4) @ X1) = $true) & ((ap @ X2 @ X0 @ X1 @ X3) != X4))) & ((func @ X2 @ X0 @ X1) = $true)))),
% 1.73/1.48 inference(nnf_transformation,[status(thm)],[f28])).
% 1.73/1.48 thf(f28,plain,(
% 1.73/1.48 ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ! [X4 : $i] : (((in @ X4 @ X0) != $true) | ((in @ (kpair @ X3 @ X4) @ X1) != $true) | ((ap @ X2 @ X0 @ X1 @ X3) = X4))) | ((func @ X2 @ X0 @ X1) != $true)) <=> (funcGraphProp2 = $true)),
% 1.73/1.48 inference(flattening,[status(thm)],[f27])).
% 1.73/1.48 thf(f27,plain,(
% 1.73/1.48 (funcGraphProp2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (! [X4 : $i] : ((((ap @ X2 @ X0 @ X1 @ X3) = X4) | ((in @ (kpair @ X3 @ X4) @ X1) != $true)) | ((in @ X4 @ X0) != $true)) | ((in @ X3 @ X2) != $true)) | ((func @ X2 @ X0 @ X1) != $true))),
% 1.73/1.48 inference(ennf_transformation,[status(thm)],[f13])).
% 1.73/1.48 thf(f13,plain,(
% 1.73/1.48 (funcGraphProp2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X0 @ X1) = $true) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ! [X4 : $i] : (((in @ X4 @ X0) = $true) => (((in @ (kpair @ X3 @ X4) @ X1) = $true) => ((ap @ X2 @ X0 @ X1 @ X3) = X4)))))),
% 1.73/1.48 inference(fool_elimination,[status(thm)],[f12])).
% 1.73/1.48 thf(f12,plain,(
% 1.73/1.48 (funcGraphProp2 = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X2 @ X0 @ X1) => ! [X3 : $i] : ((in @ X3 @ X2) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ X1) => ((ap @ X2 @ X0 @ X1 @ X3) = X4))))))),
% 1.73/1.48 inference(rectify,[status(thm)],[f6])).
% 1.73/1.48 thf(f6,axiom,(
% 1.73/1.48 (funcGraphProp2 = ! [X3 : $i,X7 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => ! [X6 : $i] : ((in @ X6 @ X3) => ((in @ (kpair @ X1 @ X6) @ X7) => ((ap @ X0 @ X3 @ X7 @ X1) = X6))))))),
% 1.73/1.48 file('/export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2THF_3357.p',funcGraphProp2)).
% 1.73/1.48 thf(f61,plain,(
% 1.73/1.48 ( ! [X4 : $i] : (((ap @ sK9 @ sK10 @ sK11 @ X4) = (ap @ sK9 @ sK10 @ sK12 @ X4)) | ((in @ X4 @ sK9) != $true)) )),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f74,plain,(
% 1.73/1.48 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((in @ (kpair @ X7 @ (ap @ X5 @ X4 @ X6 @ X7)) @ X6) = $true) | ($true != (in @ X7 @ X5)) | ((func @ X5 @ X4 @ X6) != $true)) )),
% 1.73/1.48 inference(trivial_inequality_removal,[status(thm)],[f73])).
% 1.73/1.48 thf(f73,plain,(
% 1.73/1.48 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((func @ X5 @ X4 @ X6) != $true) | ((in @ (kpair @ X7 @ (ap @ X5 @ X4 @ X6 @ X7)) @ X6) = $true) | ($true != (in @ X7 @ X5)) | ($true != $true)) )),
% 1.73/1.48 inference(definition_unfolding,[status(thm)],[f51,f63])).
% 1.73/1.48 thf(f63,plain,(
% 1.73/1.48 (funcGraphProp1 = $true)),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 thf(f51,plain,(
% 1.73/1.48 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((in @ (kpair @ X7 @ (ap @ X5 @ X4 @ X6 @ X7)) @ X6) = $true) | ($true != (in @ X7 @ X5)) | ((func @ X5 @ X4 @ X6) != $true) | (funcGraphProp1 != $true)) )),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f39])).
% 1.73/1.48 thf(f39,plain,(
% 1.73/1.48 ((funcGraphProp1 = $true) | ((((in @ (kpair @ sK8 @ (ap @ sK6 @ sK5 @ sK7 @ sK8)) @ sK7) != $true) & ((in @ sK8 @ sK6) = $true)) & ((func @ sK6 @ sK5 @ sK7) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (((in @ (kpair @ X7 @ (ap @ X5 @ X4 @ X6 @ X7)) @ X6) = $true) | ($true != (in @ X7 @ X5))) | ((func @ X5 @ X4 @ X6) != $true)) | (funcGraphProp1 != $true))),
% 1.73/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7,sK8])],[f36,f38,f37])).
% 1.73/1.48 thf(f37,plain,(
% 1.73/1.48 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ((func @ X1 @ X0 @ X2) = $true)) => (? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ sK6 @ sK5 @ sK7 @ X3)) @ sK7) != $true) & ((in @ X3 @ sK6) = $true)) & ((func @ sK6 @ sK5 @ sK7) = $true))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f38,plain,(
% 1.73/1.48 ? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ sK6 @ sK5 @ sK7 @ X3)) @ sK7) != $true) & ((in @ X3 @ sK6) = $true)) => (((in @ (kpair @ sK8 @ (ap @ sK6 @ sK5 @ sK7 @ sK8)) @ sK7) != $true) & ((in @ sK8 @ sK6) = $true))),
% 1.73/1.48 introduced(definition,[],[choice_axiom])).
% 1.73/1.48 thf(f36,plain,(
% 1.73/1.48 ((funcGraphProp1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ((func @ X1 @ X0 @ X2) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (((in @ (kpair @ X7 @ (ap @ X5 @ X4 @ X6 @ X7)) @ X6) = $true) | ($true != (in @ X7 @ X5))) | ((func @ X5 @ X4 @ X6) != $true)) | (funcGraphProp1 != $true))),
% 1.73/1.48 inference(rectify,[status(thm)],[f35])).
% 1.73/1.48 thf(f35,plain,(
% 1.73/1.48 ((funcGraphProp1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ((func @ X1 @ X0 @ X2) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2) = $true) | ((in @ X3 @ X1) != $true)) | ((func @ X1 @ X0 @ X2) != $true)) | (funcGraphProp1 != $true))),
% 1.73/1.48 inference(nnf_transformation,[status(thm)],[f26])).
% 1.73/1.48 thf(f26,plain,(
% 1.73/1.48 (funcGraphProp1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2) = $true) | ((in @ X3 @ X1) != $true)) | ((func @ X1 @ X0 @ X2) != $true))),
% 1.73/1.48 inference(ennf_transformation,[status(thm)],[f23])).
% 1.73/1.48 thf(f23,plain,(
% 1.73/1.48 ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2) = $true))) <=> (funcGraphProp1 = $true)),
% 1.73/1.48 inference(fool_elimination,[status(thm)],[f22])).
% 1.73/1.48 thf(f22,plain,(
% 1.73/1.48 (! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (kpair @ X3 @ (ap @ X1 @ X0 @ X2 @ X3)) @ X2))) = funcGraphProp1)),
% 1.73/1.48 inference(rectify,[status(thm)],[f5])).
% 1.73/1.48 thf(f5,axiom,(
% 1.73/1.48 (! [X3 : $i,X0 : $i,X7 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (kpair @ X1 @ (ap @ X0 @ X3 @ X7 @ X1)) @ X7))) = funcGraphProp1)),
% 1.73/1.48 file('/export/starexec/sandbox/tmp/tmp.zpMzh91CCu/DTF2THF_3357.p',funcGraphProp1)).
% 1.73/1.48 thf(f58,plain,(
% 1.73/1.48 ($true != (in @ (kpair @ sK13 @ sK14) @ sK11))),
% 1.73/1.48 inference(cnf_transformation,[status(thm)],[f44])).
% 1.73/1.48 % SZS output end Proof for DTF2THF_3357
% 1.73/1.48 % (3460)------------------------------
% 1.73/1.48 % (3460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.73/1.48 % (3460)Termination reason: Refutation
% 1.73/1.48
% 1.73/1.48 % (3460)Memory used [KB]: 5628
% 1.73/1.48 % (3460)Time elapsed: 0.014 s
% 1.73/1.48 % (3460)Instructions burned: 16 (million)
% 1.73/1.48 % (3460)------------------------------
% 1.73/1.48 % (3460)------------------------------
% 1.73/1.48 % (3454)Success in time 0.027 s
%------------------------------------------------------------------------------