%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : SYO433^1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% Computer : n011.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 08:47:41 AM UTC 2026
% Result : Theorem 0.19s 0.27s
% Output : Refutation 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 29
% Number of leaves : 17
% Syntax : Number of formulae : 141 ( 58 unt; 0 typ; 7 def)
% Number of atoms : 1064 ( 211 equ; 0 cnn)
% Maximal formula atoms : 12 ( 7 avg)
% Number of connectives : 1583 ( 281 ~; 265 |; 13 &; 926 @)
% ( 6 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 181 ( 181 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 62 usr; 12 con; 0-4 aty)
% ( 80 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 416 ( 0 sgn 59 !; 0 ?; 416 :)
% 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,
meq_prop: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_3,type,
mnot: ( $i > $o ) > $i > $o ).
thf(func_def_4,type,
mor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_5,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_6,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_7,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_8,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_9,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_10,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_11,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_12,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_13,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_14,type,
mtrue: $i > $o ).
thf(func_def_15,type,
mfalse: $i > $o ).
thf(func_def_16,type,
mbox: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_17,type,
mdia: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_18,type,
mreflexive: ( $i > $i > $o ) > $o ).
thf(func_def_19,type,
msymmetric: ( $i > $i > $o ) > $o ).
thf(func_def_20,type,
mserial: ( $i > $i > $o ) > $o ).
thf(func_def_21,type,
mtransitive: ( $i > $i > $o ) > $o ).
thf(func_def_22,type,
meuclidean: ( $i > $i > $o ) > $o ).
thf(func_def_23,type,
mpartially_functional: ( $i > $i > $o ) > $o ).
thf(func_def_24,type,
mfunctional: ( $i > $i > $o ) > $o ).
thf(func_def_25,type,
mweakly_dense: ( $i > $i > $o ) > $o ).
thf(func_def_26,type,
mweakly_connected: ( $i > $i > $o ) > $o ).
thf(func_def_27,type,
mweakly_directed: ( $i > $i > $o ) > $o ).
thf(func_def_28,type,
mvalid: ( $i > $o ) > $o ).
thf(func_def_29,type,
minvalid: ( $i > $o ) > $o ).
thf(func_def_30,type,
msatisfiable: ( $i > $o ) > $o ).
thf(func_def_31,type,
mcountersatisfiable: ( $i > $o ) > $o ).
thf(func_def_32,type,
rel_s4: $i > $i > $o ).
thf(func_def_33,type,
mbox_s4: ( $i > $o ) > $i > $o ).
thf(func_def_34,type,
mdia_s4: ( $i > $o ) > $i > $o ).
thf(func_def_35,type,
p: $i > $o ).
thf(func_def_36,type,
q: $i > $o ).
thf(func_def_37,type,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_38,type,
vOR: $o > $o > $o ).
thf(func_def_39,type,
vNOT: $o > $o ).
thf(func_def_40,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_41,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_42,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_43,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_44,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_45,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_46,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_47,type,
vIMP: $o > $o > $o ).
thf(func_def_48,type,
vAND: $o > $o > $o ).
thf(func_def_51,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_52,type,
sF0: $o ).
thf(func_def_54,type,
sK2: $i > $i ).
thf(func_def_56,type,
sK4: $i > $i ).
thf(func_def_57,type,
sK5: $i > $i ).
thf(f3,axiom,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^0.ax',mnot) ).
thf(f4,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
| ( X1 @ X2 ) ) )
= mor ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^0.ax',mor) ).
thf(f5,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o] : ( mnot @ ( mor @ ( mnot @ X0 ) @ ( mnot @ X1 ) ) ) )
= mand ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^0.ax',mand) ).
thf(f6,axiom,
( mimplies
= ( ^ [X0: $i > $o,X1: $i > $o] : ( mor @ ( mnot @ X0 ) @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^0.ax',mimplies) ).
thf(f21,axiom,
( ( ^ [X0: $i > $i > $o] :
! [X2: $i,X1: $i,X3: $i] :
( ( ( X0 @ X1 @ X2 )
& ( X0 @ X2 @ X3 ) )
=> ( X0 @ X1 @ X3 ) ) )
= mtransitive ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^0.ax',mtransitive) ).
thf(f28,axiom,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^0.ax',mvalid) ).
thf(f32,axiom,
( ( ^ [X0: $i > $o,X1: $i] :
! [X2: $i] :
( ( X0 @ X2 )
| ~ ( rel_s4 @ X1 @ X2 ) ) )
= mbox_s4 ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^5.ax',mbox_s4) ).
thf(f33,axiom,
( ( ^ [X0: $i > $o] : ( mnot @ ( mbox_s4 @ ( mnot @ X0 ) ) ) )
= mdia_s4 ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^5.ax',mdia_s4) ).
thf(f35,axiom,
mtransitive @ rel_s4,
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^5.ax',a2) ).
thf(f36,conjecture,
mvalid @ ( mimplies @ ( mand @ ( mdia_s4 @ p ) @ ( mbox_s4 @ q ) ) @ ( mdia_s4 @ ( mand @ p @ ( mbox_s4 @ q ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove) ).
thf(f37,negated_conjecture,
~ ( mvalid @ ( mimplies @ ( mand @ ( mdia_s4 @ p ) @ ( mbox_s4 @ q ) ) @ ( mdia_s4 @ ( mand @ p @ ( mbox_s4 @ q ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f36]) ).
thf(f49,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mor @ ( mnot @ Y0 ) @ Y1 ) ) ),
inference(fool_elimination,[],[f6]) ).
thf(f50,plain,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
inference(rectify,[],[f3]) ).
thf(f51,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(fool_elimination,[],[f50]) ).
thf(f59,plain,
mtransitive @ rel_s4,
inference(rectify,[],[f35]) ).
thf(f60,plain,
( ( mtransitive @ rel_s4 )
= $true ),
inference(fool_elimination,[],[f59]) ).
thf(f65,plain,
( mdia_s4
= ( ^ [Y0: $i > $o] : ( mnot @ ( mbox_s4 @ ( mnot @ Y0 ) ) ) ) ),
inference(fool_elimination,[],[f33]) ).
thf(f69,plain,
~ ( mvalid @ ( mimplies @ ( mand @ ( mdia_s4 @ p ) @ ( mbox_s4 @ q ) ) @ ( mdia_s4 @ ( mand @ p @ ( mbox_s4 @ q ) ) ) ) ),
inference(rectify,[],[f37]) ).
thf(f70,plain,
( ( mvalid @ ( mimplies @ ( mand @ ( mdia_s4 @ p ) @ ( mbox_s4 @ q ) ) @ ( mdia_s4 @ ( mand @ p @ ( mbox_s4 @ q ) ) ) ) )
!= $true ),
inference(fool_elimination,[],[f69]) ).
thf(f71,plain,
( ( ^ [X0: $i > $o,X1: $i] :
! [X2: $i] :
( ( X0 @ X2 )
| ~ ( rel_s4 @ X1 @ X2 ) ) )
= mbox_s4 ),
inference(rectify,[],[f32]) ).
thf(f72,plain,
( mbox_s4
= ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ) ),
inference(fool_elimination,[],[f71]) ).
thf(f76,plain,
( ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i,X3: $i] :
( ( ( X0 @ X2 @ X1 )
& ( X0 @ X1 @ X3 ) )
=> ( X0 @ X2 @ X3 ) ) )
= mtransitive ),
inference(rectify,[],[f21]) ).
thf(f77,plain,
( mtransitive
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( ( Y0 @ Y3 @ Y1 )
& ( Y0 @ Y2 @ Y3 ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ),
inference(fool_elimination,[],[f76]) ).
thf(f80,plain,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
inference(rectify,[],[f28]) ).
thf(f81,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f80]) ).
thf(f84,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
| ( X1 @ X2 ) ) )
= mor ),
inference(rectify,[],[f4]) ).
thf(f85,plain,
( mor
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f84]) ).
thf(f95,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mnot @ ( mor @ ( mnot @ Y0 ) @ ( mnot @ Y1 ) ) ) ) ),
inference(fool_elimination,[],[f5]) ).
thf(f98,plain,
( ( mvalid @ ( mimplies @ ( mand @ ( mdia_s4 @ p ) @ ( mbox_s4 @ q ) ) @ ( mdia_s4 @ ( mand @ p @ ( mbox_s4 @ q ) ) ) ) )
!= $true ),
inference(flattening,[],[f70]) ).
thf(f112,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mnot @ ( mor @ ( mnot @ Y0 ) @ ( mnot @ Y1 ) ) ) ) ),
inference(cnf_transformation,[],[f95]) ).
thf(f116,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f81]) ).
thf(f117,plain,
( mtransitive
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( ( Y0 @ Y3 @ Y1 )
& ( Y0 @ Y2 @ Y3 ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ),
inference(cnf_transformation,[],[f77]) ).
thf(f122,plain,
( mor
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f85]) ).
thf(f124,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(cnf_transformation,[],[f51]) ).
thf(f126,plain,
( ( mtransitive @ rel_s4 )
= $true ),
inference(cnf_transformation,[],[f60]) ).
thf(f127,plain,
( ( mvalid @ ( mimplies @ ( mand @ ( mdia_s4 @ p ) @ ( mbox_s4 @ q ) ) @ ( mdia_s4 @ ( mand @ p @ ( mbox_s4 @ q ) ) ) ) )
!= $true ),
inference(cnf_transformation,[],[f98]) ).
thf(f129,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mor @ ( mnot @ Y0 ) @ Y1 ) ) ),
inference(cnf_transformation,[],[f49]) ).
thf(f130,plain,
( mdia_s4
= ( ^ [Y0: $i > $o] : ( mnot @ ( mbox_s4 @ ( mnot @ Y0 ) ) ) ) ),
inference(cnf_transformation,[],[f65]) ).
thf(f131,plain,
( mbox_s4
= ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[],[f72]) ).
thf(f137,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) ) ) ),
inference(definition_unfolding,[],[f112,f124,f122,f124,f124]) ).
thf(f138,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 ) ) ),
inference(definition_unfolding,[],[f129,f122,f124]) ).
thf(f146,plain,
( mdia_s4
= ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) ) ) ),
inference(definition_unfolding,[],[f130,f124,f131,f124]) ).
thf(f147,plain,
( ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( ( Y0 @ Y3 @ Y1 )
& ( Y0 @ Y2 @ Y3 ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) )
@ rel_s4 )
= $true ),
inference(definition_unfolding,[],[f126,f117]) ).
thf(f148,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ p )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ p
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) ) ) ) )
!= $true ),
inference(definition_unfolding,[],[f127,f116,f138,f137,f146,f131,f146,f137,f131]) ).
thf(f150,definition,
( sF0
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ p )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ p
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sF0])],[function_definition]) ).
thf(f151,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ p )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ p
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) ) ) ) )
= sF0 ),
inference(reorient_equations,[],[f150]) ).
thf(f152,plain,
sF0 != $true,
inference(definition_folding,[],[f148,f151]) ).
thf(f157,plain,
( ( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ p )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) )
@ ( ^ [Y0: $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ Y0 ) ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
| ( Y2 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) )
@ p
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ q ) ) ) ) )
= $false )
| ( sF0 = $true ) ),
inference(iff_proxy_clausification,[],[f151]) ).
thf(f158,plain,
( ( sF0 = $true )
| ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( q @ Y2 ) ) )
| ~ ( p @ Y1 ) ) ) )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( q @ Y1 ) ) )
| ~ ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ~ ( p @ Y1 ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f157]) ).
thf(f159,plain,
( ( sF0 = $true )
| ( $false
= ( ^ [Y0: $i] :
( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( q @ Y2 ) ) )
| ~ ( p @ Y1 ) ) ) )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( q @ Y1 ) ) )
| ~ ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ~ ( p @ Y1 ) ) ) ) )
@ sK1 ) ) ),
inference(sigma_proxy_clausification,[],[f158]) ).
thf(f160,plain,
( ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( q @ Y1 ) ) )
| ~ ( p @ Y0 ) ) ) )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) ) ) ) )
| ( sF0 = $true ) ),
inference(beta-eta_normalization,[],[f159]) ).
thf(f161,plain,
( ( sF0 = $true )
| ( $false
= ( ~ ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f160]) ).
thf(f162,plain,
( ( sF0 = $true )
| ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( q @ Y1 ) ) )
| ~ ( p @ Y0 ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f160]) ).
thf(f163,plain,
( ( sF0 = $true )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( q @ Y1 ) ) )
| ~ ( p @ Y0 ) ) ) )
= $true ) ),
inference(not_proxy_clausification,[],[f162]) ).
thf(f164,plain,
! [X1: $i] :
( ( ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( q @ Y1 ) ) )
| ~ ( p @ Y0 ) ) )
@ X1 )
= $true )
| ( sF0 = $true ) ),
inference(pi_proxy_clausification,[],[f163]) ).
thf(f165,plain,
! [X1: $i] :
( ( ( ~ ( rel_s4 @ sK1 @ X1 )
| ~ ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ( p @ X1 ) ) )
= $true )
| ( sF0 = $true ) ),
inference(beta-eta_normalization,[],[f164]) ).
thf(f166,plain,
! [X1: $i] :
( ( sF0 = $true )
| ( ( ~ ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ( p @ X1 ) ) )
= $true )
| ( ( ~ ( rel_s4 @ sK1 @ X1 ) )
= $true ) ),
inference(or_proxy_clausification,[],[f165]) ).
thf(f167,plain,
! [X1: $i] :
( ( ( ~ ( rel_s4 @ sK1 @ X1 ) )
= $true )
| ( sF0 = $true )
| ( $false
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ( p @ X1 ) ) ) ) ),
inference(not_proxy_clausification,[],[f166]) ).
thf(f168,plain,
! [X1: $i] :
( ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( sF0 = $true )
| ( $false
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ( p @ X1 ) ) ) ) ),
inference(not_proxy_clausification,[],[f167]) ).
thf(f169,plain,
! [X1: $i] :
( ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ( p @ X1 ) )
= $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( sF0 = $true ) ),
inference(not_proxy_clausification,[],[f168]) ).
thf(f170,plain,
! [X1: $i] :
( ( ( ~ ( p @ X1 ) )
= $true )
| ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) ) )
= $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( sF0 = $true ) ),
inference(or_proxy_clausification,[],[f169]) ).
thf(f171,plain,
! [X1: $i] :
( ( sF0 = $true )
| ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) ) )
= $true )
| ( $false
= ( p @ X1 ) )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) ) ),
inference(not_proxy_clausification,[],[f170]) ).
thf(f172,plain,
! [X1: $i] :
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) ) ) )
| ( sF0 = $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( p @ X1 ) ) ),
inference(not_proxy_clausification,[],[f171]) ).
thf(f173,plain,
! [X1: $i] :
( ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( p @ X1 ) )
| ( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ X1 @ Y0 )
| ( q @ Y0 ) )
@ ( sK2 @ X1 ) ) )
| ( sF0 = $true ) ),
inference(sigma_proxy_clausification,[],[f172]) ).
thf(f174,plain,
! [X1: $i] :
( ( $false
= ( ~ ( rel_s4 @ X1 @ ( sK2 @ X1 ) )
| ( q @ ( sK2 @ X1 ) ) ) )
| ( sF0 = $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( p @ X1 ) ) ),
inference(beta-eta_normalization,[],[f173]) ).
thf(f175,plain,
! [X1: $i] :
( ( $false
= ( p @ X1 ) )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( sF0 = $true )
| ( $false
= ( q @ ( sK2 @ X1 ) ) ) ),
inference(or_proxy_clausification,[],[f174]) ).
thf(f176,plain,
! [X1: $i] :
( ( sF0 = $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( p @ X1 ) )
| ( $false
= ( ~ ( rel_s4 @ X1 @ ( sK2 @ X1 ) ) ) ) ),
inference(or_proxy_clausification,[],[f174]) ).
thf(f177,plain,
! [X1: $i] :
( ( sF0 = $true )
| ( ( rel_s4 @ X1 @ ( sK2 @ X1 ) )
= $true )
| ( $false
= ( p @ X1 ) )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) ) ),
inference(not_proxy_clausification,[],[f176]) ).
thf(f178,plain,
( ( sF0 = $true )
| ( ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) ) ) )
= $true ) ),
inference(not_proxy_clausification,[],[f161]) ).
thf(f179,plain,
( ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) ) )
| ~ ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) ) ) )
| ( sF0 = $true ) ),
inference(not_proxy_clausification,[],[f178]) ).
thf(f180,plain,
( ( ( ~ ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) ) )
= $false )
| ( sF0 = $true ) ),
inference(or_proxy_clausification,[],[f179]) ).
thf(f181,plain,
( ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) ) ) ) )
| ( sF0 = $true ) ),
inference(or_proxy_clausification,[],[f179]) ).
thf(f182,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) ) )
= $true )
| ( sF0 = $true ) ),
inference(not_proxy_clausification,[],[f181]) ).
thf(f183,plain,
! [X1: $i] :
( ( $true
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( q @ Y0 ) )
@ X1 ) )
| ( sF0 = $true ) ),
inference(pi_proxy_clausification,[],[f182]) ).
thf(f184,plain,
! [X1: $i] :
( ( ( ~ ( rel_s4 @ sK1 @ X1 )
| ( q @ X1 ) )
= $true )
| ( sF0 = $true ) ),
inference(beta-eta_normalization,[],[f183]) ).
thf(f185,plain,
! [X1: $i] :
( ( sF0 = $true )
| ( ( ~ ( rel_s4 @ sK1 @ X1 ) )
= $true )
| ( ( q @ X1 )
= $true ) ),
inference(or_proxy_clausification,[],[f184]) ).
thf(f186,plain,
! [X1: $i] :
( ( ( q @ X1 )
= $true )
| ( sF0 = $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) ) ),
inference(not_proxy_clausification,[],[f185]) ).
thf(f187,plain,
( ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) ) )
= $true )
| ( sF0 = $true ) ),
inference(not_proxy_clausification,[],[f180]) ).
thf(f188,plain,
( ( sF0 = $true )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) ) )
= $false ) ),
inference(not_proxy_clausification,[],[f187]) ).
thf(f189,plain,
( ( sF0 = $true )
| ( ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( p @ Y0 ) )
@ sK3 )
= $false ) ),
inference(sigma_proxy_clausification,[],[f188]) ).
thf(f190,plain,
( ( $false
= ( ~ ( rel_s4 @ sK1 @ sK3 )
| ~ ( p @ sK3 ) ) )
| ( sF0 = $true ) ),
inference(beta-eta_normalization,[],[f189]) ).
thf(f191,plain,
( ( ( ~ ( p @ sK3 ) )
= $false )
| ( sF0 = $true ) ),
inference(or_proxy_clausification,[],[f190]) ).
thf(f192,plain,
( ( sF0 = $true )
| ( ( ~ ( rel_s4 @ sK1 @ sK3 ) )
= $false ) ),
inference(or_proxy_clausification,[],[f190]) ).
thf(f193,plain,
( ( sF0 = $true )
| ( ( rel_s4 @ sK1 @ sK3 )
= $true ) ),
inference(not_proxy_clausification,[],[f192]) ).
thf(f194,plain,
( ( sF0 = $true )
| ( $true
= ( p @ sK3 ) ) ),
inference(not_proxy_clausification,[],[f191]) ).
thf(f252,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( rel_s4 @ Y2 @ Y0 )
& ( rel_s4 @ Y1 @ Y2 ) )
=> ( rel_s4 @ Y1 @ Y0 ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f147]) ).
thf(f253,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( rel_s4 @ Y2 @ Y0 )
& ( rel_s4 @ Y1 @ Y2 ) )
=> ( rel_s4 @ Y1 @ Y0 ) ) ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f252]) ).
thf(f254,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( rel_s4 @ Y1 @ X1 )
& ( rel_s4 @ Y0 @ Y1 ) )
=> ( rel_s4 @ Y0 @ X1 ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f253]) ).
thf(f255,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( rel_s4 @ Y1 @ X1 )
& ( rel_s4 @ Y0 @ Y1 ) )
=> ( rel_s4 @ Y0 @ X1 ) ) )
@ X2 )
= $true ),
inference(pi_proxy_clausification,[],[f254]) ).
thf(f256,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( ( rel_s4 @ Y0 @ X1 )
& ( rel_s4 @ X2 @ Y0 ) )
=> ( rel_s4 @ X2 @ X1 ) ) )
= $true ),
inference(beta-eta_normalization,[],[f255]) ).
thf(f257,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( ( rel_s4 @ Y0 @ X1 )
& ( rel_s4 @ X2 @ Y0 ) )
=> ( rel_s4 @ X2 @ X1 ) )
@ X3 ) ),
inference(pi_proxy_clausification,[],[f256]) ).
thf(f258,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( ( rel_s4 @ X3 @ X1 )
& ( rel_s4 @ X2 @ X3 ) )
=> ( rel_s4 @ X2 @ X1 ) )
= $true ),
inference(beta-eta_normalization,[],[f257]) ).
thf(f259,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( rel_s4 @ X2 @ X1 )
= $true )
| ( ( ( rel_s4 @ X3 @ X1 )
& ( rel_s4 @ X2 @ X3 ) )
= $false ) ),
inference(imp_proxy_clausification,[],[f258]) ).
thf(f260,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( rel_s4 @ X3 @ X1 ) )
| ( ( rel_s4 @ X2 @ X1 )
= $true )
| ( $false
= ( rel_s4 @ X2 @ X3 ) ) ),
inference(and_proxy_clausification,[],[f259]) ).
thf(f262,definition,
( spl6_1
<=> ( sF0 = $true ) ),
introduced(definition,[new_symbols(definition,[spl6_1])],[avatar_definition]) ).
thf(f266,definition,
( spl6_2
<=> ! [X1: $i] :
( ( ( rel_s4 @ X1 @ ( sK2 @ X1 ) )
= $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[spl6_2])],[avatar_definition]) ).
thf(f267,plain,
( ! [X1: $i] :
( ( ( rel_s4 @ X1 @ ( sK2 @ X1 ) )
= $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( p @ X1 ) ) )
| ~ spl6_2 ),
inference(avatar_component_clause,[],[f266]) ).
thf(f268,plain,
( spl6_1
| spl6_2 ),
inference(avatar_split_clause,[],[f177,f266,f262]) ).
thf(f270,definition,
( spl6_3
<=> ! [X1: $i] :
( ( $false
= ( p @ X1 ) )
| ( $false
= ( q @ ( sK2 @ X1 ) ) )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[spl6_3])],[avatar_definition]) ).
thf(f271,plain,
( ! [X1: $i] :
( ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( $false
= ( q @ ( sK2 @ X1 ) ) )
| ( $false
= ( p @ X1 ) ) )
| ~ spl6_3 ),
inference(avatar_component_clause,[],[f270]) ).
thf(f272,plain,
( spl6_3
| spl6_1 ),
inference(avatar_split_clause,[],[f175,f262,f270]) ).
thf(f274,definition,
( spl6_4
<=> ! [X1: $i] :
( ( ( q @ X1 )
= $true )
| ( $false
= ( rel_s4 @ sK1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[spl6_4])],[avatar_definition]) ).
thf(f275,plain,
( ! [X1: $i] :
( ( $false
= ( rel_s4 @ sK1 @ X1 ) )
| ( ( q @ X1 )
= $true ) )
| ~ spl6_4 ),
inference(avatar_component_clause,[],[f274]) ).
thf(f276,plain,
( spl6_4
| spl6_1 ),
inference(avatar_split_clause,[],[f186,f262,f274]) ).
thf(f278,definition,
( spl6_5
<=> ( ( rel_s4 @ sK1 @ sK3 )
= $true ) ),
introduced(definition,[new_symbols(definition,[spl6_5])],[avatar_definition]) ).
thf(f280,plain,
( ( ( rel_s4 @ sK1 @ sK3 )
= $true )
| ~ spl6_5 ),
inference(avatar_component_clause,[],[f278]) ).
thf(f281,plain,
( spl6_1
| spl6_5 ),
inference(avatar_split_clause,[],[f193,f278,f262]) ).
thf(f283,definition,
( spl6_6
<=> ( $true
= ( p @ sK3 ) ) ),
introduced(definition,[new_symbols(definition,[spl6_6])],[avatar_definition]) ).
thf(f285,plain,
( ( $true
= ( p @ sK3 ) )
| ~ spl6_6 ),
inference(avatar_component_clause,[],[f283]) ).
thf(f286,plain,
( spl6_6
| spl6_1 ),
inference(avatar_split_clause,[],[f194,f262,f283]) ).
thf(f315,plain,
~ spl6_1,
inference(avatar_split_clause,[],[f152,f262]) ).
thf(f361,plain,
( ( $false = $true )
| ( $false
= ( p @ sK3 ) )
| ( $false
= ( q @ ( sK2 @ sK3 ) ) )
| ~ spl6_3
| ~ spl6_5 ),
inference(constrained_superposition,[],[f280,f271]) ).
thf(f363,plain,
( ( $false
= ( q @ ( sK2 @ sK3 ) ) )
| ( $false
= ( p @ sK3 ) )
| ~ spl6_3
| ~ spl6_5 ),
inference(trivial_inequality_removal,[],[f361]) ).
thf(f376,plain,
( ( $false = $true )
| ( $false
= ( q @ ( sK2 @ sK3 ) ) )
| ~ spl6_3
| ~ spl6_5
| ~ spl6_6 ),
inference(forward_demodulation,[],[f363,f285]) ).
thf(f377,plain,
( ( $false
= ( q @ ( sK2 @ sK3 ) ) )
| ~ spl6_3
| ~ spl6_5
| ~ spl6_6 ),
inference(trivial_inequality_removal,[],[f376]) ).
thf(f387,plain,
( ! [X0: $i,X1: $i] :
( ( ( p @ X0 )
= $false )
| ( ( rel_s4 @ sK1 @ X0 )
= $false )
| ( $false = $true )
| ( ( rel_s4 @ X1 @ ( sK2 @ X0 ) )
= $true )
| ( ( rel_s4 @ X1 @ X0 )
= $false ) )
| ~ spl6_2 ),
inference(constrained_superposition,[],[f267,f260]) ).
thf(f394,plain,
( ! [X0: $i,X1: $i] :
( ( ( rel_s4 @ X1 @ ( sK2 @ X0 ) )
= $true )
| ( ( rel_s4 @ X1 @ X0 )
= $false )
| ( ( rel_s4 @ sK1 @ X0 )
= $false )
| ( ( p @ X0 )
= $false ) )
| ~ spl6_2 ),
inference(trivial_inequality_removal,[],[f387]) ).
thf(f426,plain,
( ! [X0: $i] :
( ( ( rel_s4 @ sK1 @ X0 )
= $false )
| ( ( rel_s4 @ sK1 @ X0 )
= $false )
| ( ( p @ X0 )
= $false )
| ( $false = $true )
| ( $true
= ( q @ ( sK2 @ X0 ) ) ) )
| ~ spl6_2
| ~ spl6_4 ),
inference(constrained_superposition,[],[f275,f394]) ).
thf(f428,plain,
( ! [X0: $i] :
( ( ( p @ X0 )
= $false )
| ( ( rel_s4 @ sK1 @ X0 )
= $false )
| ( $true
= ( q @ ( sK2 @ X0 ) ) )
| ( $false = $true ) )
| ~ spl6_2
| ~ spl6_4 ),
inference(duplicate_literal_removal,[],[f426]) ).
thf(f429,plain,
( ! [X0: $i] :
( ( ( rel_s4 @ sK1 @ X0 )
= $false )
| ( ( p @ X0 )
= $false )
| ( $true
= ( q @ ( sK2 @ X0 ) ) ) )
| ~ spl6_2
| ~ spl6_4 ),
inference(trivial_inequality_removal,[],[f428]) ).
thf(f446,plain,
( ( ( q @ ( sK2 @ sK3 ) )
= $true )
| ( $false = $true )
| ( $false
= ( p @ sK3 ) )
| ~ spl6_2
| ~ spl6_4
| ~ spl6_5 ),
inference(constrained_superposition,[],[f280,f429]) ).
thf(f456,plain,
( ( $false
= ( p @ sK3 ) )
| ( ( q @ ( sK2 @ sK3 ) )
= $true )
| ~ spl6_2
| ~ spl6_4
| ~ spl6_5 ),
inference(trivial_inequality_removal,[],[f446]) ).
thf(f460,plain,
( ( $false = $true )
| ( ( q @ ( sK2 @ sK3 ) )
= $true )
| ~ spl6_2
| ~ spl6_4
| ~ spl6_5
| ~ spl6_6 ),
inference(forward_demodulation,[],[f456,f285]) ).
thf(f461,plain,
( ( ( q @ ( sK2 @ sK3 ) )
= $true )
| ~ spl6_2
| ~ spl6_4
| ~ spl6_5
| ~ spl6_6 ),
inference(trivial_inequality_removal,[],[f460]) ).
thf(f465,plain,
( ( $false = $true )
| ~ spl6_2
| ~ spl6_3
| ~ spl6_4
| ~ spl6_5
| ~ spl6_6 ),
inference(forward_demodulation,[],[f461,f377]) ).
thf(f466,plain,
( $false
| ~ spl6_2
| ~ spl6_3
| ~ spl6_4
| ~ spl6_5
| ~ spl6_6 ),
inference(trivial_inequality_removal,[],[f465]) ).
thf(f467,plain,
( ~ spl6_2
| ~ spl6_3
| ~ spl6_4
| ~ spl6_5
| ~ spl6_6 ),
inference(avatar_contradiction_clause,[],[f466]) ).
cnf(s1,plain,
( spl6_1
| spl6_2 ),
inference(sat_conversion,[],[f268]) ).
cnf(s2,plain,
( spl6_1
| spl6_3 ),
inference(sat_conversion,[],[f272]) ).
cnf(s3,plain,
( spl6_1
| spl6_4 ),
inference(sat_conversion,[],[f276]) ).
cnf(s4,plain,
( spl6_1
| spl6_5 ),
inference(sat_conversion,[],[f281]) ).
cnf(s5,plain,
( spl6_1
| spl6_6 ),
inference(sat_conversion,[],[f286]) ).
cnf(s12,plain,
~ spl6_1,
inference(sat_conversion,[],[f315]) ).
cnf(s20,plain,
( ~ spl6_2
| ~ spl6_3
| ~ spl6_4
| ~ spl6_5
| ~ spl6_6 ),
inference(sat_conversion,[],[f467]) ).
cnf(s21,plain,
spl6_6,
inference(rat,[],[s5,s12]) ).
cnf(s22,plain,
spl6_5,
inference(rat,[],[s4,s12]) ).
cnf(s23,plain,
spl6_4,
inference(rat,[],[s3,s12]) ).
cnf(s24,plain,
spl6_3,
inference(rat,[],[s2,s12]) ).
cnf(s25,plain,
~ spl6_2,
inference(rat,[],[s20,s21,s22,s23,s24]) ).
cnf(s26,plain,
$false,
inference(rat,[],[s1,s25,s12]) ).
thf(f468,plain,
$false,
inference(avatar_sat_refutation,[],[s26]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SYO433^1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.05 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.10/0.18 % Computer : n011.cluster.edu
% 0.10/0.18 % Model : x86_64 x86_64
% 0.10/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.18 % Memory : 8046.5625MB
% 0.10/0.18 % OS : Linux 6.8.0-71-generic
% 0.10/0.18 % CPULimit : 300
% 0.10/0.18 % WCLimit : 300
% 0.10/0.18 % DateTime : Tue Sep 29 16:51:46 UTC 2026
% 0.10/0.18 % CPUTime :
% 0.10/0.18 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.10/0.22 Running higher-order theorem proving
% 0.10/0.23 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
% 0.19/0.27 % (343070)Detected a higher-order problem, will run a greedy HOL sequence.
% 0.19/0.27 % (343093)WARNING Broken Constraint: if ho_split_queue_ratios(1,8) has been set then ho_split_queue(off) is equal to on
% 0.19/0.27 % (343093)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
% 0.19/0.27 % (343093)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=1206935091:hsqr=1,8:i=157:s2at=5:add=on:nm=2:rtra=on_2999 on theBenchmark for (2999ds/157Mi)
% 0.19/0.27 % (343089)lrs+10_1_cnfonf=off:si=on:uwa=one_side_interpreted:random_seed=2148279884:i=3:rtra=on:inj=on:ntd=on_2999 on theBenchmark for (2999ds/3Mi)
% 0.19/0.27 % (343093) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-343070-343093"...
% 0.19/0.27 % (343087)lrs+10_40_drc=off:e2e=on:si=on:uwa=one_side_interpreted:random_seed=1600860548:s2a=on:i=87:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/87Mi)
% 0.19/0.27 % (343092)lrs+10_1_to=lpo:sil=128000:e2e=on:si=on:random_seed=1675584304:s2a=on:i=75:s2at=3:aac=none:bd=preordered:rtra=on:fe=abstraction_2999 on theBenchmark for (2999ds/75Mi)
% 0.19/0.27 % (343088)lrs+10_16_si=on:nwc=1.5:random_seed=1502696229:i=18:kws=arity_squared:rtra=on:fe=abstraction:ntd=on_2999 on theBenchmark for (2999ds/18Mi)
% 0.19/0.27 % (343091)dis+21_4_fde=none:e2e=on:si=on:uwa=off:foolp=on:random_seed=1185676129:i=24:av=off:rtra=on_2999 on theBenchmark for (2999ds/24Mi)
% 0.19/0.27 % (343090)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=736715469: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)
% 0.19/0.27 % (343089)Instruction limit reached!
% 0.19/0.27 % (343089)------------------------------
% 0.19/0.27 % (343089)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.19/0.27 % (343089)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.19/0.27 % (343089)CaDiCaL version: 2.1.3
% 0.19/0.27 % (343089)Termination reason: Instruction limit
% 0.19/0.27 % (343089)Termination phase: Property scanning
% 0.19/0.27 % (343089)Time elapsed: 0.002 s
% 0.19/0.27 % (343089)Peak memory usage: 10 MB
% 0.19/0.27 % (343089)Instructions burned: 4 (million)
% 0.19/0.27 % (343093)...printing done.
% 0.19/0.27 % (343093)Refutation found. Thanks to Tanya!
% 0.19/0.27 % SZS status Theorem for theBenchmark
% 0.19/0.27 % SZS output start Proof for theBenchmark
% See solution above
% 0.19/0.28 % (343093)------------------------------
% 0.19/0.28 % (343093)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.19/0.28 % (343093)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.19/0.28 % (343093)CaDiCaL version: 2.1.3
% 0.19/0.28 % (343093)Termination reason: Refutation
% 0.19/0.28 % (343093)Time elapsed: 0.009 s
% 0.19/0.28 % (343093)Peak memory usage: 13 MB
% 0.19/0.28 % (343093)Instructions burned: 30 (million)
% 0.19/0.28 % (343070)Success in time 0.041 s
% 0.19/0.28 % Vampire exiting
%------------------------------------------------------------------------------