%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : PHI043^2 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:20:19 AM UTC 2026
% Result : Theorem 0.09s 0.18s
% Output : Refutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 38
% Number of leaves : 14
% Syntax : Number of formulae : 123 ( 78 unt; 0 typ; 0 def)
% Number of atoms : 874 ( 231 equ; 0 cnn)
% Maximal formula atoms : 10 ( 7 avg)
% Number of connectives : 1383 ( 69 ~; 86 |; 14 &;1090 @)
% ( 2 <=>; 50 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 3 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 336 ( 336 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 48 usr; 5 con; 0-4 aty)
% ( 62 !!; 10 ??; 0 @@+; 0 @@-)
% Number of variables : 408 ( 298 ^; 108 !; 2 ?; 408 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
mu: $tType ).
thf(type_def_6,type,
sTfun: ( $tType * $tType ) > $tType ).
thf(func_def_0,type,
meq_ind: mu > mu > $i > $o ).
thf(func_def_1,type,
mtrue: $i > $o ).
thf(func_def_2,type,
mfalse: $i > $o ).
thf(func_def_3,type,
mnot: ( $i > $o ) > $i > $o ).
thf(func_def_5,type,
mor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_6,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_7,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_8,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_9,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_10,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_11,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_12,type,
mforall_indset: ( ( mu > $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_13,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_14,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_15,type,
mexists_indset: ( ( mu > $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_16,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_17,type,
mbox_generic: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_18,type,
mdia_generic: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_19,type,
rel: $i > $i > $o ).
thf(func_def_20,type,
mbox: ( $i > $o ) > $i > $o ).
thf(func_def_21,type,
mdia: ( $i > $o ) > $i > $o ).
thf(func_def_22,type,
mvalid: ( $i > $o ) > $o ).
thf(func_def_23,type,
minvalid: ( $i > $o ) > $o ).
thf(func_def_24,type,
eiw_ind: $i > mu > $o ).
thf(func_def_25,type,
mforall_eiw_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_26,type,
mexists_eiw_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_27,type,
p: ( mu > $i > $o ) > $i > $o ).
thf(func_def_28,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_29,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_30,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_31,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_32,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_33,type,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_34,type,
vOR: $o > $o > $o ).
thf(func_def_35,type,
vNOT: $o > $o ).
thf(func_def_38,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_39,type,
vIFF: $o > $o > $o ).
thf(func_def_40,type,
vIMP: $o > $o > $o ).
thf(func_def_41,type,
vAND: $o > $o > $o ).
thf(func_def_42,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_43,type,
sK0: $i > mu ).
thf(func_def_44,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_45,type,
db5:
!>[X0: $tType] : X0 ).
thf(func_def_46,type,
sK1: ( mu > $i > $o ) > $i > ( mu > $i > $o ) > $i ).
thf(func_def_47,type,
sK2: ( mu > $i > $o ) > $i > ( mu > $i > $o ) > mu ).
thf(func_def_49,type,
sK4: mu > $i > $o ).
thf(f4,axiom,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mnot) ).
thf(f7,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
=> ( X1 @ X2 ) ) )
= mimplies ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mimplies) ).
thf(f9,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
<=> ( X1 @ X2 ) ) )
= mequiv ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mequiv) ).
thf(f12,axiom,
( mforall_indset
= ( ^ [X0: ( mu > $i > $o ) > $i > $o,X1: $i] :
! [X2: mu > $i > $o] : ( X0 @ X2 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mforall_indset) ).
thf(f17,axiom,
( ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
! [X3: $i] :
( ~ ( X0 @ X2 @ X3 )
| ( X1 @ X3 ) ) )
= mbox_generic ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mbox_generic) ).
thf(f18,axiom,
( ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
? [X3: $i] :
( ( X1 @ X3 )
& ( X0 @ X2 @ X3 ) ) )
= mdia_generic ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mdia_generic) ).
thf(f19,axiom,
( mbox
= ( mbox_generic @ rel ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mbox) ).
thf(f20,axiom,
( mdia
= ( mdia_generic @ rel ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mdia) ).
thf(f21,axiom,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mvalid) ).
thf(f24,axiom,
( mforall_eiw_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] :
( ( eiw_ind @ X1 @ X2 )
=> ( X0 @ X2 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL017^1.ax',mforall_eiw_ind) ).
thf(f25,axiom,
( mexists_eiw_ind
= ( ^ [X0: mu > $i > $o] :
( mnot
@ ( mforall_eiw_ind
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL017^1.ax',mexists_eiw_ind) ).
thf(f26,axiom,
( mvalid
@ ( mnot
@ ( p
@ ^ [X0: mu,X1: $i] : ( X0 != X0 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ax16) ).
thf(f27,axiom,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mforall_indset
@ ^ [X1: mu > $i > $o] :
( mimplies
@ ( mbox
@ ( mforall_eiw_ind
@ ^ [X2: mu] : ( mequiv @ ( X0 @ X2 ) @ ( X1 @ X2 ) ) ) )
@ ( mequiv @ ( p @ X0 ) @ ( p @ X1 ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axii) ).
thf(f28,conjecture,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mimplies @ ( p @ X0 )
@ ( mdia
@ ( mexists_eiw_ind
@ ^ [X1: mu] : ( X0 @ X1 ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',possible_instantiation_of_the_positive_variable_domain) ).
thf(f29,negated_conjecture,
~ ( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mimplies @ ( p @ X0 )
@ ( mdia
@ ( mexists_eiw_ind
@ ^ [X1: mu] : ( X0 @ X1 ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f28]) ).
thf(f32,plain,
( mforall_indset
= ( ^ [X0: ( mu > $i > $o ) > $i > $o,X1: $i] :
! [X2: mu > $i > $o] : ( X0 @ X2 @ X1 ) ) ),
inference(rectify,[],[f12]) ).
thf(f33,plain,
( mforall_indset
= ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f32]) ).
thf(f38,plain,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mforall_indset
@ ^ [X1: mu > $i > $o] :
( mimplies
@ ( mbox
@ ( mforall_eiw_ind
@ ^ [X2: mu] : ( mequiv @ ( X0 @ X2 ) @ ( X1 @ X2 ) ) ) )
@ ( mequiv @ ( p @ X0 ) @ ( p @ X1 ) ) ) ) ) ),
inference(rectify,[],[f27]) ).
thf(f39,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mforall_indset
@ ^ [Y1: mu > $i > $o] :
( mimplies
@ ( mbox
@ ( mforall_eiw_ind
@ ^ [Y2: mu] : ( mequiv @ ( Y0 @ Y2 ) @ ( Y1 @ Y2 ) ) ) )
@ ( mequiv @ ( p @ Y0 ) @ ( p @ Y1 ) ) ) ) ) )
= $true ),
inference(fool_elimination,[],[f38]) ).
thf(f41,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
<=> ( X1 @ X2 ) ) )
= mequiv ),
inference(rectify,[],[f9]) ).
thf(f42,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
= ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f41]) ).
thf(f47,plain,
( ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
? [X3: $i] :
( ( X1 @ X3 )
& ( X0 @ X2 @ X3 ) ) )
= mdia_generic ),
inference(rectify,[],[f18]) ).
thf(f48,plain,
( mdia_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( Y1 @ Y3 ) ) ) ) ),
inference(fool_elimination,[],[f47]) ).
thf(f49,plain,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
inference(rectify,[],[f21]) ).
thf(f50,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f49]) ).
thf(f51,plain,
( mvalid
@ ( mnot
@ ( p
@ ^ [X0: mu,X1: $i] : ( X0 != X0 ) ) ) ),
inference(rectify,[],[f26]) ).
thf(f52,plain,
( $true
= ( mvalid
@ ( mnot
@ ( p
@ ^ [Y0: mu,Y1: $i] : ( Y0 != Y0 ) ) ) ) ),
inference(fool_elimination,[],[f51]) ).
thf(f57,plain,
( mforall_eiw_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] :
( ( eiw_ind @ X1 @ X2 )
=> ( X0 @ X2 @ X1 ) ) ) ),
inference(rectify,[],[f24]) ).
thf(f58,plain,
( mforall_eiw_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( eiw_ind @ Y1 @ Y2 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[],[f57]) ).
thf(f62,plain,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
inference(rectify,[],[f4]) ).
thf(f63,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(fool_elimination,[],[f62]) ).
thf(f64,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
=> ( X1 @ X2 ) ) )
= mimplies ),
inference(rectify,[],[f7]) ).
thf(f65,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f64]) ).
thf(f66,plain,
~ ( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mimplies @ ( p @ X0 )
@ ( mdia
@ ( mexists_eiw_ind
@ ^ [X1: mu] : ( X0 @ X1 ) ) ) ) ) ),
inference(rectify,[],[f29]) ).
thf(f67,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mimplies @ ( p @ Y0 )
@ ( mdia
@ ( mexists_eiw_ind
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(fool_elimination,[],[f66]) ).
thf(f70,plain,
( mexists_eiw_ind
= ( ^ [X0: mu > $i > $o] :
( mnot
@ ( mforall_eiw_ind
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) ) ) ) ),
inference(rectify,[],[f25]) ).
thf(f71,plain,
( mexists_eiw_ind
= ( ^ [Y0: mu > $i > $o] :
( mnot
@ ( mforall_eiw_ind
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) ) ) ) ),
inference(fool_elimination,[],[f70]) ).
thf(f77,plain,
( ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
! [X3: $i] :
( ~ ( X0 @ X2 @ X3 )
| ( X1 @ X3 ) ) )
= mbox_generic ),
inference(rectify,[],[f17]) ).
thf(f78,plain,
( mbox_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
| ~ ( Y0 @ Y2 @ Y3 ) ) ) ) ),
inference(fool_elimination,[],[f77]) ).
thf(f79,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mimplies @ ( p @ Y0 )
@ ( mdia
@ ( mexists_eiw_ind
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(flattening,[],[f67]) ).
thf(f81,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f50]) ).
thf(f82,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
= ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f42]) ).
thf(f84,plain,
( $true
= ( mvalid
@ ( mnot
@ ( p
@ ^ [Y0: mu,Y1: $i] : ( Y0 != Y0 ) ) ) ) ),
inference(cnf_transformation,[],[f52]) ).
thf(f85,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mimplies @ ( p @ Y0 )
@ ( mdia
@ ( mexists_eiw_ind
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(cnf_transformation,[],[f79]) ).
thf(f88,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(cnf_transformation,[],[f63]) ).
thf(f91,plain,
( mbox_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
| ~ ( Y0 @ Y2 @ Y3 ) ) ) ) ),
inference(cnf_transformation,[],[f78]) ).
thf(f94,plain,
( mbox
= ( mbox_generic @ rel ) ),
inference(cnf_transformation,[],[f19]) ).
thf(f95,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mforall_indset
@ ^ [Y1: mu > $i > $o] :
( mimplies
@ ( mbox
@ ( mforall_eiw_ind
@ ^ [Y2: mu] : ( mequiv @ ( Y0 @ Y2 ) @ ( Y1 @ Y2 ) ) ) )
@ ( mequiv @ ( p @ Y0 ) @ ( p @ Y1 ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[],[f39]) ).
thf(f97,plain,
( mforall_eiw_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( eiw_ind @ Y1 @ Y2 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[],[f58]) ).
thf(f99,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f65]) ).
thf(f100,plain,
( mdia
= ( mdia_generic @ rel ) ),
inference(cnf_transformation,[],[f20]) ).
thf(f101,plain,
( mexists_eiw_ind
= ( ^ [Y0: mu > $i > $o] :
( mnot
@ ( mforall_eiw_ind
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) ) ) ) ),
inference(cnf_transformation,[],[f71]) ).
thf(f104,plain,
( mdia_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( Y1 @ Y3 ) ) ) ) ),
inference(cnf_transformation,[],[f48]) ).
thf(f105,plain,
( mforall_indset
= ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f33]) ).
thf(f111,plain,
( mbox
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
| ~ ( Y0 @ Y2 @ Y3 ) ) )
@ rel ) ),
inference(definition_unfolding,[],[f94,f91]) ).
thf(f112,plain,
( mdia
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( Y1 @ Y3 ) ) )
@ rel ) ),
inference(definition_unfolding,[],[f100,f104]) ).
thf(f113,plain,
( mexists_eiw_ind
= ( ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: mu > $i > $o,Y2: $i] :
( !! @ mu
@ ^ [Y3: mu] :
( ( eiw_ind @ Y2 @ Y3 )
=> ( Y1 @ Y3 @ Y2 ) ) )
@ ^ [Y1: mu] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( Y0 @ Y1 ) ) ) ) ) ),
inference(definition_unfolding,[],[f101,f88,f97,f88]) ).
thf(f114,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 )
@ ( p
@ ^ [Y0: mu,Y1: $i] : ( Y0 != Y0 ) ) ) )
= $true ),
inference(definition_unfolding,[],[f84,f81,f88]) ).
thf(f115,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 @ Y3 ) )
@ ( p @ Y0 )
@ ( ^ [Y1: $i > $i > $o,Y2: $i > $o,Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1 @ Y3 @ Y4 )
& ( Y2 @ Y4 ) ) )
@ rel
@ ( ^ [Y1: mu > $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: mu > $i > $o,Y3: $i] :
( !! @ mu
@ ^ [Y4: mu] :
( ( eiw_ind @ Y3 @ Y4 )
=> ( Y2 @ Y4 @ Y3 ) ) )
@ ^ [Y2: mu] :
( ^ [Y3: $i > $o,Y4: $i] :
~ ( Y3 @ Y4 )
@ ( Y1 @ Y2 ) ) ) )
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(definition_unfolding,[],[f85,f81,f105,f99,f112,f113]) ).
thf(f116,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: ( mu > $i > $o ) > $i > $o,Y2: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] : ( Y1 @ Y3 @ Y2 ) )
@ ^ [Y1: mu > $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
=> ( Y3 @ Y4 ) )
@ ( ^ [Y2: $i > $i > $o,Y3: $i > $o,Y4: $i] :
( !! @ $i
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
| ~ ( Y2 @ Y4 @ Y5 ) ) )
@ rel
@ ( ^ [Y2: mu > $i > $o,Y3: $i] :
( !! @ mu
@ ^ [Y4: mu] :
( ( eiw_ind @ Y3 @ Y4 )
=> ( Y2 @ Y4 @ Y3 ) ) )
@ ^ [Y2: mu] :
( ^ [Y3: $i > $o,Y4: $i > $o,Y5: $i] :
( ( Y3 @ Y5 )
= ( Y4 @ Y5 ) )
@ ( Y0 @ Y2 )
@ ( Y1 @ Y2 ) ) ) )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
= ( Y3 @ Y4 ) )
@ ( p @ Y0 )
@ ( p @ Y1 ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f95,f81,f105,f105,f99,f111,f97,f82,f82]) ).
thf(f117,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
~ ( p
@ ^ [Y1: mu,Y2: $i] : ( Y1 != Y1 )
@ Y0 ) )
= $true ),
inference(beta-eta_normalization,[],[f114]) ).
thf(f118,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
~ ( p
@ ^ [Y1: mu,Y2: $i] : ( Y1 != Y1 )
@ Y0 )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f117]) ).
thf(f119,plain,
! [X1: $i] :
( ( ~ ( p
@ ^ [Y0: mu,Y1: $i] : ( Y0 != Y0 )
@ X1 ) )
= $true ),
inference(beta-eta_normalization,[],[f118]) ).
thf(f120,plain,
! [X1: $i] :
( $false
= ( p
@ ^ [Y0: mu,Y1: $i] : ( Y0 != Y0 )
@ X1 ) ),
inference(not_proxy_clausification,[],[f119]) ).
thf(f121,plain,
! [X1: $i] :
( $false
= ( p
@ ^ [Y0: mu,Y1: $i] : ~ $true
@ X1 ) ),
inference(boolean_simplification,[],[f120]) ).
thf(f122,plain,
! [X1: $i] :
( $false
= ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ X1 ) ),
inference(boolean_simplification,[],[f121]) ).
thf(f123,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( !! @ $i
@ ^ [Y3: $i] :
( ( !! @ mu
@ ^ [Y4: mu] :
( ( eiw_ind @ Y3 @ Y4 )
=> ( ( Y1 @ Y4 @ Y3 )
= ( Y2 @ Y4 @ Y3 ) ) ) )
| ~ ( rel @ Y0 @ Y3 ) ) )
=> ( ( p @ Y1 @ Y0 )
= ( p @ Y2 @ Y0 ) ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f116]) ).
thf(f124,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( !! @ $i
@ ^ [Y3: $i] :
( ( !! @ mu
@ ^ [Y4: mu] :
( ( eiw_ind @ Y3 @ Y4 )
=> ( ( Y1 @ Y4 @ Y3 )
= ( Y2 @ Y4 @ Y3 ) ) ) )
| ~ ( rel @ Y0 @ Y3 ) ) )
=> ( ( p @ Y1 @ Y0 )
= ( p @ Y2 @ Y0 ) ) ) ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f123]) ).
thf(f125,plain,
! [X1: $i] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( !! @ $i
@ ^ [Y2: $i] :
( ( !! @ mu
@ ^ [Y3: mu] :
( ( eiw_ind @ Y2 @ Y3 )
=> ( ( Y0 @ Y3 @ Y2 )
= ( Y1 @ Y3 @ Y2 ) ) ) )
| ~ ( rel @ X1 @ Y2 ) ) )
=> ( ( p @ Y0 @ X1 )
= ( p @ Y1 @ X1 ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f124]) ).
thf(f126,plain,
! [X2: mu > $i > $o,X1: $i] :
( $true
= ( ^ [Y0: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( !! @ $i
@ ^ [Y2: $i] :
( ( !! @ mu
@ ^ [Y3: mu] :
( ( eiw_ind @ Y2 @ Y3 )
=> ( ( Y0 @ Y3 @ Y2 )
= ( Y1 @ Y3 @ Y2 ) ) ) )
| ~ ( rel @ X1 @ Y2 ) ) )
=> ( ( p @ Y0 @ X1 )
= ( p @ Y1 @ X1 ) ) ) )
@ X2 ) ),
inference(pi_proxy_clausification,[],[f125]) ).
thf(f127,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( !! @ $i
@ ^ [Y1: $i] :
( ( !! @ mu
@ ^ [Y2: mu] :
( ( eiw_ind @ Y1 @ Y2 )
=> ( ( X2 @ Y2 @ Y1 )
= ( Y0 @ Y2 @ Y1 ) ) ) )
| ~ ( rel @ X1 @ Y1 ) ) )
=> ( ( p @ X2 @ X1 )
= ( p @ Y0 @ X1 ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f126]) ).
thf(f128,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ^ [Y0: mu > $i > $o] :
( ( !! @ $i
@ ^ [Y1: $i] :
( ( !! @ mu
@ ^ [Y2: mu] :
( ( eiw_ind @ Y1 @ Y2 )
=> ( ( X2 @ Y2 @ Y1 )
= ( Y0 @ Y2 @ Y1 ) ) ) )
| ~ ( rel @ X1 @ Y1 ) ) )
=> ( ( p @ X2 @ X1 )
= ( p @ Y0 @ X1 ) ) )
@ X3 )
= $true ),
inference(pi_proxy_clausification,[],[f127]) ).
thf(f129,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( $true
= ( ( !! @ $i
@ ^ [Y0: $i] :
( ( !! @ mu
@ ^ [Y1: mu] :
( ( eiw_ind @ Y0 @ Y1 )
=> ( ( X2 @ Y1 @ Y0 )
= ( X3 @ Y1 @ Y0 ) ) ) )
| ~ ( rel @ X1 @ Y0 ) ) )
=> ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) ) ) ),
inference(beta-eta_normalization,[],[f128]) ).
thf(f130,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ( !! @ mu
@ ^ [Y1: mu] :
( ( eiw_ind @ Y0 @ Y1 )
=> ( ( X2 @ Y1 @ Y0 )
= ( X3 @ Y1 @ Y0 ) ) ) )
| ~ ( rel @ X1 @ Y0 ) ) ) )
| ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true ) ),
inference(imp_proxy_clausification,[],[f129]) ).
thf(f131,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $false
= ( ^ [Y0: $i] :
( ( !! @ mu
@ ^ [Y1: mu] :
( ( eiw_ind @ Y0 @ Y1 )
=> ( ( X2 @ Y1 @ Y0 )
= ( X3 @ Y1 @ Y0 ) ) ) )
| ~ ( rel @ X1 @ Y0 ) )
@ ( sK1 @ X3 @ X1 @ X2 ) ) )
| ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true ) ),
inference(sigma_proxy_clausification,[],[f130]) ).
thf(f132,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true )
| ( $false
= ( ( !! @ mu
@ ^ [Y0: mu] :
( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ Y0 )
=> ( ( X2 @ Y0 @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ Y0 @ ( sK1 @ X3 @ X1 @ X2 ) ) ) ) )
| ~ ( rel @ X1 @ ( sK1 @ X3 @ X1 @ X2 ) ) ) ) ),
inference(beta-eta_normalization,[],[f131]) ).
thf(f133,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $false
= ( ~ ( rel @ X1 @ ( sK1 @ X3 @ X1 @ X2 ) ) ) )
| ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true ) ),
inference(or_proxy_clausification,[],[f132]) ).
thf(f134,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true )
| ( $false
= ( !! @ mu
@ ^ [Y0: mu] :
( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ Y0 )
=> ( ( X2 @ Y0 @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ Y0 @ ( sK1 @ X3 @ X1 @ X2 ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f132]) ).
thf(f135,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $false
= ( ^ [Y0: mu] :
( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ Y0 )
=> ( ( X2 @ Y0 @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ Y0 @ ( sK1 @ X3 @ X1 @ X2 ) ) ) )
@ ( sK2 @ X3 @ X1 @ X2 ) ) )
| ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true ) ),
inference(sigma_proxy_clausification,[],[f134]) ).
thf(f136,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true )
| ( $false
= ( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ ( sK2 @ X3 @ X1 @ X2 ) )
=> ( ( X2 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f135]) ).
thf(f137,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true )
| ( $false
= ( ( X2 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) ) ) ) ),
inference(imp_proxy_clausification,[],[f136]) ).
thf(f138,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ ( sK2 @ X3 @ X1 @ X2 ) )
= $true )
| ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true ) ),
inference(imp_proxy_clausification,[],[f136]) ).
thf(f139,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
| ( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ ( sK2 @ X3 @ X1 @ X2 ) )
= $true ) ),
inference(iff_proxy_clausification,[],[f138]) ).
thf(f141,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( eiw_ind @ ( sK1 @ X3 @ X1 @ X2 ) @ ( sK2 @ X3 @ X1 @ X2 ) )
= $true )
| ( $true
= ( p @ X3 @ X1 ) )
| ( $false
= ( p @ X2 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f139]) ).
thf(f142,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $false
= ( ( X2 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) ) ) )
| ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f137]) ).
thf(f144,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $false
= ( ( X2 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
= ( X3 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) ) ) )
| ( $false
= ( p @ X2 @ X1 ) )
| ( $true
= ( p @ X3 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f142]) ).
thf(f145,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $true
= ( p @ X3 @ X1 ) )
| ( ( X3 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
!= ( X2 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) ) )
| ( $false
= ( p @ X2 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f144]) ).
thf(f146,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( X3 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
= $true )
| ( ( X2 @ ( sK2 @ X3 @ X1 @ X2 ) @ ( sK1 @ X3 @ X1 @ X2 ) )
= $true )
| ( $false
= ( p @ X2 @ X1 ) )
| ( $true
= ( p @ X3 @ X1 ) ) ),
inference(xor_proxy_clausification,[],[f145]) ).
thf(f151,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $true
= ( rel @ X1 @ ( sK1 @ X3 @ X1 @ X2 ) ) )
| ( ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) )
= $true ) ),
inference(not_proxy_clausification,[],[f133]) ).
thf(f152,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $true
= ( rel @ X1 @ ( sK1 @ X3 @ X1 @ X2 ) ) )
| ( ( p @ X2 @ X1 )
= ( p @ X3 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f151]) ).
thf(f154,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( $true
= ( rel @ X1 @ ( sK1 @ X3 @ X1 @ X2 ) ) )
| ( $true
= ( p @ X3 @ X1 ) )
| ( $false
= ( p @ X2 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f152]) ).
thf(f155,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( p @ Y1 @ Y0 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( rel @ Y0 @ Y2 )
& ~ ( !! @ mu
@ ^ [Y3: mu] :
( ( eiw_ind @ Y2 @ Y3 )
=> ~ ( Y1 @ Y3 @ Y2 ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f115]) ).
thf(f156,plain,
( $false
= ( ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( p @ Y1 @ Y0 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( rel @ Y0 @ Y2 )
& ~ ( !! @ mu
@ ^ [Y3: mu] :
( ( eiw_ind @ Y2 @ Y3 )
=> ~ ( Y1 @ Y3 @ Y2 ) ) ) ) ) ) )
@ sK3 ) ),
inference(sigma_proxy_clausification,[],[f155]) ).
thf(f157,plain,
( $false
= ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( p @ Y0 @ sK3 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( rel @ sK3 @ Y1 )
& ~ ( !! @ mu
@ ^ [Y2: mu] :
( ( eiw_ind @ Y1 @ Y2 )
=> ~ ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f156]) ).
thf(f158,plain,
( $false
= ( ^ [Y0: mu > $i > $o] :
( ( p @ Y0 @ sK3 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( rel @ sK3 @ Y1 )
& ~ ( !! @ mu
@ ^ [Y2: mu] :
( ( eiw_ind @ Y1 @ Y2 )
=> ~ ( Y0 @ Y2 @ Y1 ) ) ) ) ) )
@ sK4 ) ),
inference(sigma_proxy_clausification,[],[f157]) ).
thf(f159,plain,
( $false
= ( ( p @ sK4 @ sK3 )
=> ( ?? @ $i
@ ^ [Y0: $i] :
( ( rel @ sK3 @ Y0 )
& ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( eiw_ind @ Y0 @ Y1 )
=> ~ ( sK4 @ Y1 @ Y0 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f158]) ).
thf(f160,plain,
( $false
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( rel @ sK3 @ Y0 )
& ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( eiw_ind @ Y0 @ Y1 )
=> ~ ( sK4 @ Y1 @ Y0 ) ) ) ) ) ),
inference(imp_proxy_clausification,[],[f159]) ).
thf(f161,plain,
( ( p @ sK4 @ sK3 )
= $true ),
inference(imp_proxy_clausification,[],[f159]) ).
thf(f162,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( rel @ sK3 @ Y0 )
& ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( eiw_ind @ Y0 @ Y1 )
=> ~ ( sK4 @ Y1 @ Y0 ) ) ) )
@ X1 )
= $false ),
inference(pi_proxy_clausification,[],[f160]) ).
thf(f163,plain,
! [X1: $i] :
( $false
= ( ( rel @ sK3 @ X1 )
& ~ ( !! @ mu
@ ^ [Y0: mu] :
( ( eiw_ind @ X1 @ Y0 )
=> ~ ( sK4 @ Y0 @ X1 ) ) ) ) ),
inference(beta-eta_normalization,[],[f162]) ).
thf(f164,plain,
! [X1: $i] :
( ( ( rel @ sK3 @ X1 )
= $false )
| ( $false
= ( ~ ( !! @ mu
@ ^ [Y0: mu] :
( ( eiw_ind @ X1 @ Y0 )
=> ~ ( sK4 @ Y0 @ X1 ) ) ) ) ) ),
inference(and_proxy_clausification,[],[f163]) ).
thf(f165,plain,
! [X1: $i] :
( ( ( rel @ sK3 @ X1 )
= $false )
| ( ( !! @ mu
@ ^ [Y0: mu] :
( ( eiw_ind @ X1 @ Y0 )
=> ~ ( sK4 @ Y0 @ X1 ) ) )
= $true ) ),
inference(not_proxy_clausification,[],[f164]) ).
thf(f166,plain,
! [X2: mu,X1: $i] :
( ( ( rel @ sK3 @ X1 )
= $false )
| ( $true
= ( ^ [Y0: mu] :
( ( eiw_ind @ X1 @ Y0 )
=> ~ ( sK4 @ Y0 @ X1 ) )
@ X2 ) ) ),
inference(pi_proxy_clausification,[],[f165]) ).
thf(f167,plain,
! [X2: mu,X1: $i] :
( ( ( rel @ sK3 @ X1 )
= $false )
| ( $true
= ( ( eiw_ind @ X1 @ X2 )
=> ~ ( sK4 @ X2 @ X1 ) ) ) ),
inference(beta-eta_normalization,[],[f166]) ).
thf(f168,plain,
! [X2: mu,X1: $i] :
( ( ( eiw_ind @ X1 @ X2 )
= $false )
| ( $true
= ( ~ ( sK4 @ X2 @ X1 ) ) )
| ( ( rel @ sK3 @ X1 )
= $false ) ),
inference(imp_proxy_clausification,[],[f167]) ).
thf(f169,plain,
! [X2: mu,X1: $i] :
( ( ( eiw_ind @ X1 @ X2 )
= $false )
| ( $false
= ( sK4 @ X2 @ X1 ) )
| ( ( rel @ sK3 @ X1 )
= $false ) ),
inference(not_proxy_clausification,[],[f168]) ).
thf(f179,plain,
! [X2: mu > $i > $o,X0: mu > $i > $o,X1: $i] :
( ( $false = $true )
| ( $false
= ( sK4 @ ( sK2 @ X0 @ X1 @ X2 ) @ ( sK1 @ X0 @ X1 @ X2 ) ) )
| ( ( p @ X0 @ X1 )
= $true )
| ( $false
= ( rel @ sK3 @ ( sK1 @ X0 @ X1 @ X2 ) ) )
| ( $false
= ( p @ X2 @ X1 ) ) ),
inference(constrained_superposition,[],[f169,f141]) ).
thf(f180,plain,
! [X2: mu > $i > $o,X0: mu > $i > $o,X1: $i] :
( ( $false
= ( sK4 @ ( sK2 @ X0 @ X1 @ X2 ) @ ( sK1 @ X0 @ X1 @ X2 ) ) )
| ( ( p @ X0 @ X1 )
= $true )
| ( $false
= ( rel @ sK3 @ ( sK1 @ X0 @ X1 @ X2 ) ) )
| ( $false
= ( p @ X2 @ X1 ) ) ),
inference(trivial_inequality_removal,[],[f179]) ).
thf(f194,plain,
! [X0: mu > $i > $o,X1: $i] :
( ( ( p @ X0 @ X1 )
= $true )
| ( $false
= ( rel @ sK3 @ ( sK1 @ X0 @ X1 @ sK4 ) ) )
| ( $false = $true )
| ( $false
= ( p @ sK4 @ X1 ) )
| ( $true
= ( X0 @ ( sK2 @ X0 @ X1 @ sK4 ) @ ( sK1 @ X0 @ X1 @ sK4 ) ) )
| ( $false
= ( p @ sK4 @ X1 ) )
| ( ( p @ X0 @ X1 )
= $true ) ),
inference(constrained_superposition,[],[f180,f146]) ).
thf(f211,plain,
! [X0: mu > $i > $o,X1: $i] :
( ( $true
= ( X0 @ ( sK2 @ X0 @ X1 @ sK4 ) @ ( sK1 @ X0 @ X1 @ sK4 ) ) )
| ( $false
= ( rel @ sK3 @ ( sK1 @ X0 @ X1 @ sK4 ) ) )
| ( $false
= ( p @ sK4 @ X1 ) )
| ( $false = $true )
| ( ( p @ X0 @ X1 )
= $true ) ),
inference(duplicate_literal_removal,[],[f194]) ).
thf(f212,plain,
! [X0: mu > $i > $o,X1: $i] :
( ( $true
= ( X0 @ ( sK2 @ X0 @ X1 @ sK4 ) @ ( sK1 @ X0 @ X1 @ sK4 ) ) )
| ( $false
= ( p @ sK4 @ X1 ) )
| ( $false
= ( rel @ sK3 @ ( sK1 @ X0 @ X1 @ sK4 ) ) )
| ( ( p @ X0 @ X1 )
= $true ) ),
inference(trivial_inequality_removal,[],[f211]) ).
thf(f235,plain,
! [X1: $i] :
( ( $false
= ( rel @ sK3
@ ( sK1
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 ) ) )
| ( $true
= ( ^ [Y0: mu,Y1: $i] : $false
@ ( sK2
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 )
@ ( sK1
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 ) ) )
| ( $false
= ( p @ sK4 @ X1 ) )
| ( $true
= ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ X1 ) ) ),
inference(primitive_instantiation,[],[f212]) ).
thf(f241,plain,
! [X1: $i] :
( ( $true
= ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ X1 ) )
| ( $false
= ( rel @ sK3
@ ( sK1
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 ) ) )
| ( $false = $true )
| ( $false
= ( p @ sK4 @ X1 ) ) ),
inference(beta-eta_normalization,[],[f235]) ).
thf(f242,plain,
! [X1: $i] :
( ( $false
= ( rel @ sK3
@ ( sK1
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 ) ) )
| ( $true
= ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ X1 ) )
| ( $false
= ( p @ sK4 @ X1 ) ) ),
inference(trivial_inequality_removal,[],[f241]) ).
thf(f250,plain,
! [X1: $i] :
( ( $false = $true )
| ( $false
= ( p @ sK4 @ X1 ) )
| ( $false
= ( rel @ sK3
@ ( sK1
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 ) ) ) ),
inference(forward_demodulation,[],[f242,f122]) ).
thf(f251,plain,
! [X1: $i] :
( ( $false
= ( rel @ sK3
@ ( sK1
@ ^ [Y0: mu,Y1: $i] : $false
@ X1
@ sK4 ) ) )
| ( $false
= ( p @ sK4 @ X1 ) ) ),
inference(trivial_inequality_removal,[],[f250]) ).
thf(f254,plain,
( ( ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ sK3 )
= $true )
| ( $false = $true )
| ( $false
= ( p @ sK4 @ sK3 ) )
| ( $false
= ( p @ sK4 @ sK3 ) ) ),
inference(constrained_superposition,[],[f154,f251]) ).
thf(f256,plain,
( ( ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ sK3 )
= $true )
| ( $false
= ( p @ sK4 @ sK3 ) )
| ( $false = $true ) ),
inference(duplicate_literal_removal,[],[f254]) ).
thf(f257,plain,
( ( ( p
@ ^ [Y0: mu,Y1: $i] : $false
@ sK3 )
= $true )
| ( $false
= ( p @ sK4 @ sK3 ) ) ),
inference(trivial_inequality_removal,[],[f256]) ).
thf(f260,plain,
( ( $false = $true )
| ( $false
= ( p @ sK4 @ sK3 ) ) ),
inference(forward_demodulation,[],[f257,f122]) ).
thf(f261,plain,
( $false
= ( p @ sK4 @ sK3 ) ),
inference(trivial_inequality_removal,[],[f260]) ).
thf(f264,plain,
$false = $true,
inference(forward_demodulation,[],[f261,f161]) ).
thf(f265,plain,
$false,
inference(trivial_inequality_removal,[],[f264]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : PHI043^2 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.03 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.00/0.11 % Computer : n012.cluster.edu
% 0.00/0.11 % Model : x86_64 x86_64
% 0.00/0.11 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.00/0.11 % Memory : 8046.5625MB
% 0.00/0.11 % OS : Linux 6.8.0-71-generic
% 0.00/0.11 % CPULimit : 300
% 0.00/0.11 % WCLimit : 300
% 0.00/0.11 % DateTime : Tue Sep 29 13:19:49 UTC 2026
% 0.00/0.11 % CPUTime :
% 0.00/0.11 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.09/0.13 Running higher-order theorem proving
% 0.09/0.14 Running: /export/starexec/sandbox2/solver/bin/vampire-ho --input_syntax tptp --output_axiom_names on --mode casc -m 16384 --cores 7 -t 300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.18 % (85025)Detected a higher-order problem, will run a greedy HOL sequence.
% 0.09/0.18 % (85038)WARNING Broken Constraint: if ho_split_queue_ratios(1,8) has been set then ho_split_queue(off) is equal to on
% 0.09/0.18 % (85038)WARNING Broken Constraint: if sine_to_age_tolerance(5) has been set then sine_to_age(off) is equal to on or sine_to_pred_levels(off) is not equal to off or sine_level_split_queue(off) is equal to on
% 0.09/0.18 % (85034)lrs+10_1_cnfonf=off:si=on:uwa=one_side_interpreted:random_seed=1429581203:i=3:rtra=on:inj=on:ntd=on_2999 on theBenchmark for (2999ds/3Mi)
% 0.09/0.18 % (85032)lrs+10_40_drc=off:e2e=on:si=on:uwa=one_side_interpreted:random_seed=3357939102:s2a=on:i=87:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/87Mi)
% 0.09/0.18 % (85033)lrs+10_16_si=on:nwc=1.5:random_seed=3847274235:i=18:kws=arity_squared:rtra=on:fe=abstraction:ntd=on_2999 on theBenchmark for (2999ds/18Mi)
% 0.09/0.18 % (85037)lrs+10_1_to=lpo:sil=128000:e2e=on:si=on:random_seed=1419618024:s2a=on:i=75:s2at=3:aac=none:bd=preordered:rtra=on:fe=abstraction_2999 on theBenchmark for (2999ds/75Mi)
% 0.09/0.18 % (85038)dis+1002_8_to=kbo:sil=128000:tgt=full:drc=off:si=on:sp=const_max:lma=off:spb=non_intro:cbe=off:uwa=interpreted_only:random_seed=3928805393:hsqr=1,8:i=157:s2at=5:add=on:nm=2:rtra=on_2999 on theBenchmark for (2999ds/157Mi)
% 0.09/0.18 % (85035)dis+1002_4:1_sfv=off:to=lpo:plsq=on:fde=none:e2e=on:si=on:spb=non_intro:acc=on:uwa=off:fd=preordered:foolp=on:s2agt=32:slsqc=1:slsq=on:random_seed=3804466584:hsq=on:hsqr=16,1:s2a=on:i=634:add=on:nm=16:nicw=on:rtra=on:gtg=position:ss=included:ixr=off:c=on:inj=on:ntd=on:rawr=on_2999 on theBenchmark for (2999ds/634Mi)
% 0.09/0.18 % (85036)dis+21_4_fde=none:e2e=on:si=on:uwa=off:foolp=on:random_seed=3013781425:i=24:av=off:rtra=on_2999 on theBenchmark for (2999ds/24Mi)
% 0.09/0.18 % (85034)Instruction limit reached!
% 0.09/0.18 % (85034)------------------------------
% 0.09/0.18 % (85034)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.09/0.18 % (85034)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.09/0.18 % (85034)CaDiCaL version: 2.1.3
% 0.09/0.18 % (85034)Termination reason: Instruction limit
% 0.09/0.18 % (85034)Termination phase: Property scanning
% 0.09/0.18 % (85034)Time elapsed: 0.001 s
% 0.09/0.18 % (85034)Peak memory usage: 10 MB
% 0.09/0.18 % (85034)Instructions burned: 3 (million)
% 0.09/0.18 % (85033)Instruction limit reached!
% 0.09/0.18 % (85033)------------------------------
% 0.09/0.18 % (85033)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.09/0.18 % (85033)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.09/0.18 % (85033)CaDiCaL version: 2.1.3
% 0.09/0.18 % (85033)Termination reason: Instruction limit
% 0.09/0.18 % (85033)Termination phase: Saturation
% 0.09/0.18 % (85033)Time elapsed: 0.005 s
% 0.09/0.18 % (85033)Peak memory usage: 12 MB
% 0.09/0.18 % (85033)Instructions burned: 18 (million)
% 0.09/0.18 % (85032) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-85025-85032"...
% 0.09/0.18 % (85038)Refutation not found, incomplete strategy
% 0.09/0.18 % (85038)------------------------------
% 0.09/0.18 % (85038)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.09/0.18 % (85038)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.09/0.18 % (85038)CaDiCaL version: 2.1.3
% 0.09/0.18 % (85038)Termination reason: Refutation not found, incomplete strategy
% 0.09/0.18 % (85038)Time elapsed: 0.007 s
% 0.09/0.18 % (85038)Peak memory usage: 12 MB
% 0.09/0.18 % (85038)Instructions burned: 24 (million)
% 0.09/0.18 % (85038)------------------------------
% 0.09/0.18 % (85038)------------------------------
% 0.09/0.18 % (85036)Instruction limit reached!
% 0.09/0.18 % (85036)------------------------------
% 0.09/0.18 % (85036)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.09/0.18 % (85036)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.09/0.18 % (85036)CaDiCaL version: 2.1.3
% 0.09/0.18 % (85036)Termination reason: Instruction limit
% 0.09/0.18 % (85036)Termination phase: Saturation
% 0.09/0.18 % (85036)Time elapsed: 0.008 s
% 0.09/0.18 % (85036)Peak memory usage: 12 MB
% 0.09/0.18 % (85036)Instructions burned: 26 (million)
% 0.09/0.18 % (85032)...printing done.
% 0.09/0.18 % (85032)Refutation found. Thanks to Tanya!
% 0.09/0.18 % SZS status Theorem for theBenchmark
% 0.09/0.18 % SZS output start Proof for theBenchmark
% See solution above
% 0.09/0.19 % (85032)------------------------------
% 0.09/0.19 % (85032)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.09/0.19 % (85032)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.09/0.19 % (85032)CaDiCaL version: 2.1.3
% 0.09/0.19 % (85032)Termination reason: Refutation
% 0.09/0.19 % (85032)Time elapsed: 0.009 s
% 0.09/0.19 % (85032)Peak memory usage: 12 MB
% 0.09/0.19 % (85032)Instructions burned: 30 (million)
% 0.09/0.19 % (85025)Success in time 0.044 s
% 0.09/0.19 % Vampire exiting
%------------------------------------------------------------------------------