%------------------------------------------------------------------------------
% File : Vampire-SAT---5.0.1
% Problem : SYO568^7 : TPTP v9.3.1. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 SAT
% Computer : n014.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 08:50:26 AM UTC 2026
% Result : Theorem 0.20s 0.28s
% Output : Refutation 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 41
% Number of leaves : 15
% Syntax : Number of formulae : 151 ( 84 unt; 0 typ; 1 def)
% Number of atoms : 1169 ( 213 equ; 0 cnn)
% Maximal formula atoms : 14 ( 7 avg)
% Number of connectives : 1690 ( 196 ~; 239 |; 13 &;1125 @)
% ( 1 <=>; 24 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 2 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 131 ( 131 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 56 usr; 8 con; 0-4 aty)
% ( 92 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 357 ( 0 sgn 81 !; 0 ?; 357 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
mu: $tType ).
thf(type_def_6,type,
sTfun: ( $tType * $tType ) > $tType ).
thf(func_def_0,type,
qmltpeq: 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,
mbox: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_6,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_7,type,
mtrue: $i > $o ).
thf(func_def_8,type,
mfalse: $i > $o ).
thf(func_def_9,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_10,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_11,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_12,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_13,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_14,type,
mdia: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_15,type,
exists_in_world: mu > $i > $o ).
thf(func_def_16,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_17,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_18,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_19,type,
mreflexive: ( $i > $i > $o ) > $o ).
thf(func_def_20,type,
msymmetric: ( $i > $i > $o ) > $o ).
thf(func_def_21,type,
mserial: ( $i > $i > $o ) > $o ).
thf(func_def_22,type,
mtransitive: ( $i > $i > $o ) > $o ).
thf(func_def_23,type,
meuclidean: ( $i > $i > $o ) > $o ).
thf(func_def_24,type,
mpartially_functional: ( $i > $i > $o ) > $o ).
thf(func_def_25,type,
mfunctional: ( $i > $i > $o ) > $o ).
thf(func_def_26,type,
mweakly_dense: ( $i > $i > $o ) > $o ).
thf(func_def_27,type,
mweakly_connected: ( $i > $i > $o ) > $o ).
thf(func_def_28,type,
mweakly_directed: ( $i > $i > $o ) > $o ).
thf(func_def_29,type,
mvalid: ( $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,
minvalid: ( $i > $o ) > $o ).
thf(func_def_33,type,
rel_s4: $i > $i > $o ).
thf(func_def_34,type,
mbox_s4: ( $i > $o ) > $i > $o ).
thf(func_def_35,type,
mdia_s4: ( $i > $o ) > $i > $o ).
thf(func_def_36,type,
f: mu > $i > $o ).
thf(func_def_37,type,
a: mu ).
thf(func_def_38,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_39,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_40,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_41,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_42,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_43,type,
vNOT: $o > $o ).
thf(func_def_44,type,
vOR: $o > $o > $o ).
thf(func_def_45,type,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_46,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_49,type,
vIMP: $o > $o > $o ).
thf(func_def_50,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_51,type,
vAND: $o > $o > $o ).
thf(func_def_52,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_53,type,
sK0: $i > mu ).
thf(f2,axiom,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mnot) ).
thf(f3,axiom,
( mor
= ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
| ( X0 @ X2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mor) ).
thf(f8,axiom,
( mand
= ( ^ [X0: $i > $o,X1: $i > $o] : ( mnot @ ( mor @ ( mnot @ X0 ) @ ( mnot @ X1 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mand) ).
thf(f9,axiom,
( mimplies
= ( ^ [X0: $i > $o,X1: $i > $o] : ( mor @ ( mnot @ X0 ) @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mimplies) ).
thf(f11,axiom,
( mequiv
= ( ^ [X0: $i > $o,X1: $i > $o] : ( mand @ ( mimplies @ X0 @ X1 ) @ ( mimplies @ X1 @ X0 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mequiv) ).
thf(f15,axiom,
( mforall_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( X0 @ X2 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mforall_ind) ).
thf(f18,axiom,
( mreflexive
= ( ^ [X0: $i > $i > $o] :
! [X1: $i] : ( X0 @ X1 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mreflexive) ).
thf(f21,axiom,
( mtransitive
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i,X3: $i] :
( ( ( X0 @ X2 @ X3 )
& ( X0 @ X1 @ X2 ) )
=> ( X0 @ X1 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mtransitive) ).
thf(f28,axiom,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL015^0.ax',mvalid) ).
thf(f32,axiom,
( mbox_s4
= ( ^ [X0: $i > $o,X1: $i] :
! [X2: $i] :
( ( X0 @ X2 )
| ~ ( rel_s4 @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^5.ax',mbox_s4) ).
thf(f34,axiom,
mreflexive @ rel_s4,
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^5.ax',a1) ).
thf(f35,axiom,
mtransitive @ rel_s4,
file('/export/starexec/sandbox2/benchmark/Axioms/LCL013^5.ax',a2) ).
thf(f37,axiom,
! [X0: $i] : ( exists_in_world @ a @ X0 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_a_ax) ).
thf(f38,conjecture,
( mvalid
@ ( mimplies
@ ( mforall_ind
@ ^ [X0: mu] : ( mor @ ( mbox_s4 @ ( f @ X0 ) ) @ ( mbox_s4 @ ( mnot @ ( f @ X0 ) ) ) ) )
@ ( mbox_s4 @ ( mequiv @ ( mbox_s4 @ ( f @ a ) ) @ ( f @ a ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',con) ).
thf(f39,negated_conjecture,
~ ( mvalid
@ ( mimplies
@ ( mforall_ind
@ ^ [X0: mu] : ( mor @ ( mbox_s4 @ ( f @ X0 ) ) @ ( mbox_s4 @ ( mnot @ ( f @ X0 ) ) ) ) )
@ ( mbox_s4 @ ( mequiv @ ( mbox_s4 @ ( f @ a ) ) @ ( f @ a ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f38]) ).
thf(f41,plain,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
inference(rectify,[],[f2]) ).
thf(f42,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(fool_elimination,[],[f41]) ).
thf(f43,plain,
( mor
= ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
| ( X0 @ X2 ) ) ) ),
inference(rectify,[],[f3]) ).
thf(f44,plain,
( mor
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
| ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f43]) ).
thf(f50,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mnot @ ( mor @ ( mnot @ Y0 ) @ ( mnot @ Y1 ) ) ) ) ),
inference(fool_elimination,[],[f8]) ).
thf(f51,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mor @ ( mnot @ Y0 ) @ Y1 ) ) ),
inference(fool_elimination,[],[f9]) ).
thf(f53,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mand @ ( mimplies @ Y0 @ Y1 ) @ ( mimplies @ Y1 @ Y0 ) ) ) ),
inference(fool_elimination,[],[f11]) ).
thf(f58,plain,
( mforall_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( X0 @ X2 @ X1 ) ) ) ),
inference(rectify,[],[f15]) ).
thf(f59,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[],[f58]) ).
thf(f64,plain,
( mreflexive
= ( ^ [X0: $i > $i > $o] :
! [X1: $i] : ( X0 @ X1 @ X1 ) ) ),
inference(rectify,[],[f18]) ).
thf(f65,plain,
( mreflexive
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f64]) ).
thf(f70,plain,
( mtransitive
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i,X3: $i] :
( ( ( X0 @ X2 @ X3 )
& ( X0 @ X1 @ X2 ) )
=> ( X0 @ X1 @ X3 ) ) ) ),
inference(rectify,[],[f21]) ).
thf(f71,plain,
( mtransitive
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( ( Y0 @ Y3 @ Y2 )
& ( Y0 @ Y2 @ Y1 ) )
=> ( Y0 @ Y3 @ Y1 ) ) ) ) ) ) ),
inference(fool_elimination,[],[f70]) ).
thf(f84,plain,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
inference(rectify,[],[f28]) ).
thf(f85,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f84]) ).
thf(f92,plain,
( mbox_s4
= ( ^ [X0: $i > $o,X1: $i] :
! [X2: $i] :
( ( X0 @ X2 )
| ~ ( rel_s4 @ X1 @ X2 ) ) ) ),
inference(rectify,[],[f32]) ).
thf(f93,plain,
( mbox_s4
= ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ) ),
inference(fool_elimination,[],[f92]) ).
thf(f95,plain,
mreflexive @ rel_s4,
inference(rectify,[],[f34]) ).
thf(f96,plain,
( ( mreflexive @ rel_s4 )
= $true ),
inference(fool_elimination,[],[f95]) ).
thf(f97,plain,
mtransitive @ rel_s4,
inference(rectify,[],[f35]) ).
thf(f98,plain,
( ( mtransitive @ rel_s4 )
= $true ),
inference(fool_elimination,[],[f97]) ).
thf(f101,plain,
! [X0: $i] : ( exists_in_world @ a @ X0 ),
inference(rectify,[],[f37]) ).
thf(f102,plain,
! [X0: $i] :
( ( exists_in_world @ a @ X0 )
= $true ),
inference(fool_elimination,[],[f101]) ).
thf(f103,plain,
~ ( mvalid
@ ( mimplies
@ ( mforall_ind
@ ^ [X0: mu] : ( mor @ ( mbox_s4 @ ( f @ X0 ) ) @ ( mbox_s4 @ ( mnot @ ( f @ X0 ) ) ) ) )
@ ( mbox_s4 @ ( mequiv @ ( mbox_s4 @ ( f @ a ) ) @ ( f @ a ) ) ) ) ),
inference(rectify,[],[f39]) ).
thf(f104,plain,
( $true
!= ( mvalid
@ ( mimplies
@ ( mforall_ind
@ ^ [Y0: mu] : ( mor @ ( mbox_s4 @ ( f @ Y0 ) ) @ ( mbox_s4 @ ( mnot @ ( f @ Y0 ) ) ) ) )
@ ( mbox_s4 @ ( mequiv @ ( mbox_s4 @ ( f @ a ) ) @ ( f @ a ) ) ) ) ) ),
inference(fool_elimination,[],[f103]) ).
thf(f105,plain,
( $true
!= ( mvalid
@ ( mimplies
@ ( mforall_ind
@ ^ [Y0: mu] : ( mor @ ( mbox_s4 @ ( f @ Y0 ) ) @ ( mbox_s4 @ ( mnot @ ( f @ Y0 ) ) ) ) )
@ ( mbox_s4 @ ( mequiv @ ( mbox_s4 @ ( f @ a ) ) @ ( f @ a ) ) ) ) ) ),
inference(flattening,[],[f104]) ).
thf(f110,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(cnf_transformation,[],[f42]) ).
thf(f111,plain,
( mor
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
| ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f44]) ).
thf(f116,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mnot @ ( mor @ ( mnot @ Y0 ) @ ( mnot @ Y1 ) ) ) ) ),
inference(cnf_transformation,[],[f50]) ).
thf(f117,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mor @ ( mnot @ Y0 ) @ Y1 ) ) ),
inference(cnf_transformation,[],[f51]) ).
thf(f119,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mand @ ( mimplies @ Y0 @ Y1 ) @ ( mimplies @ Y1 @ Y0 ) ) ) ),
inference(cnf_transformation,[],[f53]) ).
thf(f123,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[],[f59]) ).
thf(f126,plain,
( mreflexive
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f65]) ).
thf(f129,plain,
( mtransitive
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( ( Y0 @ Y3 @ Y2 )
& ( Y0 @ Y2 @ Y1 ) )
=> ( Y0 @ Y3 @ Y1 ) ) ) ) ) ) ),
inference(cnf_transformation,[],[f71]) ).
thf(f136,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f85]) ).
thf(f140,plain,
( mbox_s4
= ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[],[f93]) ).
thf(f142,plain,
( ( mreflexive @ rel_s4 )
= $true ),
inference(cnf_transformation,[],[f96]) ).
thf(f143,plain,
( ( mtransitive @ rel_s4 )
= $true ),
inference(cnf_transformation,[],[f98]) ).
thf(f145,plain,
! [X0: $i] :
( ( exists_in_world @ a @ X0 )
= $true ),
inference(cnf_transformation,[],[f102]) ).
thf(f146,plain,
( $true
!= ( mvalid
@ ( mimplies
@ ( mforall_ind
@ ^ [Y0: mu] : ( mor @ ( mbox_s4 @ ( f @ Y0 ) ) @ ( mbox_s4 @ ( mnot @ ( f @ Y0 ) ) ) ) )
@ ( mbox_s4 @ ( mequiv @ ( mbox_s4 @ ( f @ a ) ) @ ( f @ a ) ) ) ) ) ),
inference(cnf_transformation,[],[f105]) ).
thf(f150,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
| ( Y3 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y1 ) ) ) ) ),
inference(definition_unfolding,[],[f116,f110,f111,f110,f110]) ).
thf(f151,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
| ( Y3 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 ) ) ),
inference(definition_unfolding,[],[f117,f111,f110]) ).
thf(f153,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o] :
( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ ( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
| ( Y5 @ Y6 ) )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y2 )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y3 ) ) )
@ ( ^ [Y2: $i > $o,Y3: $i > $o] :
( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
| ( Y5 @ Y6 ) )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y2 )
@ Y3 )
@ Y0
@ Y1 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o] :
( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
| ( Y5 @ Y6 ) )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y2 )
@ Y3 )
@ Y1
@ Y0 ) ) ) ),
inference(definition_unfolding,[],[f119,f150,f151,f151]) ).
thf(f159,plain,
( $true
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 @ Y1 ) )
@ rel_s4 ) ),
inference(definition_unfolding,[],[f142,f126]) ).
thf(f160,plain,
( $true
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( ( Y0 @ Y3 @ Y2 )
& ( Y0 @ Y2 @ Y1 ) )
=> ( Y0 @ Y3 @ Y1 ) ) ) ) )
@ rel_s4 ) ),
inference(definition_unfolding,[],[f143,f129]) ).
thf(f161,plain,
( $true
!= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
| ( Y3 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ Y0 )
@ Y1 )
@ ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) )
@ ^ [Y0: mu] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
| ( Y2 @ Y3 ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( f @ Y0 ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( f @ Y0 ) ) ) ) )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ ( ^ [Y0: $i > $o,Y1: $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o] :
( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ ( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
| ( Y5 @ Y6 ) )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y2 )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y3 ) ) )
@ ( ^ [Y2: $i > $o,Y3: $i > $o] :
( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
| ( Y5 @ Y6 ) )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y2 )
@ Y3 )
@ Y0
@ Y1 )
@ ( ^ [Y2: $i > $o,Y3: $i > $o] :
( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
| ( Y5 @ Y6 ) )
@ ( ^ [Y4: $i > $o,Y5: $i] :
~ ( Y4 @ Y5 )
@ Y2 )
@ Y3 )
@ Y1
@ Y0 ) )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ ( f @ a ) )
@ ( f @ a ) ) ) ) ) ),
inference(definition_unfolding,[],[f146,f136,f151,f123,f111,f140,f140,f110,f140,f153,f140]) ).
thf(f162,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ Y0 )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y0 @ Y2 )
| ( f @ Y1 @ Y2 ) ) )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y0 @ Y2 )
| ~ ( f @ Y1 @ Y2 ) ) ) ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( f @ a @ Y2 ) ) )
| ( f @ a @ Y1 ) )
| ~ ( ~ ( f @ a @ Y1 )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( f @ a @ Y2 ) ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f161]) ).
thf(f163,plain,
( $false
= ( ^ [Y0: $i] :
( ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ Y0 )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y0 @ Y2 )
| ( f @ Y1 @ Y2 ) ) )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y0 @ Y2 )
| ~ ( f @ Y1 @ Y2 ) ) ) ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( f @ a @ Y2 ) ) )
| ( f @ a @ Y1 ) )
| ~ ( ~ ( f @ a @ Y1 )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( f @ a @ Y2 ) ) ) ) ) ) ) )
@ sK1 ) ),
inference(sigma_proxy_clausification,[],[f162]) ).
thf(f164,plain,
( $false
= ( ~ ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK1 )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ( f @ Y0 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ~ ( f @ Y0 @ Y1 ) ) ) ) ) )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( f @ a @ Y1 ) ) )
| ( f @ a @ Y0 ) )
| ~ ( ~ ( f @ a @ Y0 )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( f @ a @ Y1 ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f163]) ).
thf(f165,plain,
( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( f @ a @ Y1 ) ) )
| ( f @ a @ Y0 ) )
| ~ ( ~ ( f @ a @ Y0 )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( f @ a @ Y1 ) ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f164]) ).
thf(f166,plain,
( $false
= ( ~ ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK1 )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ( f @ Y0 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ~ ( f @ Y0 @ Y1 ) ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f164]) ).
thf(f167,plain,
( $true
= ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK1 )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ( f @ Y0 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ~ ( f @ Y0 @ Y1 ) ) ) ) ) ) ),
inference(not_proxy_clausification,[],[f166]) ).
thf(f168,plain,
! [X1: mu] :
( $true
= ( ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK1 )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ( f @ Y0 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK1 @ Y1 )
| ~ ( f @ Y0 @ Y1 ) ) ) ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f167]) ).
thf(f169,plain,
! [X1: mu] :
( $true
= ( ( exists_in_world @ X1 @ sK1 )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( f @ X1 @ Y0 ) ) )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( f @ X1 @ Y0 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f168]) ).
thf(f170,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $true
= ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( f @ X1 @ Y0 ) ) )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( f @ X1 @ Y0 ) ) ) ) ) ),
inference(imp_proxy_clausification,[],[f169]) ).
thf(f171,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( f @ X1 @ Y0 ) ) ) )
| ( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( f @ X1 @ Y0 ) ) ) ) ),
inference(or_proxy_clausification,[],[f170]) ).
thf(f172,plain,
! [X2: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $true
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( f @ X1 @ Y0 ) )
@ X2 ) )
| ( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( f @ X1 @ Y0 ) ) ) ) ),
inference(pi_proxy_clausification,[],[f171]) ).
thf(f173,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $true
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( f @ X1 @ Y0 ) )
@ X2 ) )
| ( $true
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( f @ X1 @ Y0 ) )
@ X3 ) ) ),
inference(pi_proxy_clausification,[],[f172]) ).
thf(f174,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X2 )
| ~ ( f @ X1 @ X2 ) ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X3 )
| ( f @ X1 @ X3 ) ) ) ),
inference(beta-eta_normalization,[],[f173]) ).
thf(f175,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $true
= ( ~ ( f @ X1 @ X2 ) ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X3 )
| ( f @ X1 @ X3 ) ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X2 ) ) ) ),
inference(or_proxy_clausification,[],[f174]) ).
thf(f176,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $false
= ( f @ X1 @ X2 ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X3 )
| ( f @ X1 @ X3 ) ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X2 ) ) ) ),
inference(not_proxy_clausification,[],[f175]) ).
thf(f177,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $false
= ( f @ X1 @ X2 ) )
| ( $true
= ( f @ X1 @ X3 ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X2 ) ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X3 ) ) ) ),
inference(or_proxy_clausification,[],[f176]) ).
thf(f178,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $false
= ( f @ X1 @ X2 ) )
| ( $true
= ( f @ X1 @ X3 ) )
| ( $false
= ( rel_s4 @ sK1 @ X2 ) )
| ( $true
= ( ~ ( rel_s4 @ sK1 @ X3 ) ) ) ),
inference(not_proxy_clausification,[],[f177]) ).
thf(f179,plain,
! [X2: $i,X3: $i,X1: mu] :
( ( $true
= ( f @ X1 @ X3 ) )
| ( $false
= ( f @ X1 @ X2 ) )
| ( $false
= ( exists_in_world @ X1 @ sK1 ) )
| ( $false
= ( rel_s4 @ sK1 @ X2 ) )
| ( $false
= ( rel_s4 @ sK1 @ X3 ) ) ),
inference(not_proxy_clausification,[],[f178]) ).
thf(f180,plain,
( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( f @ a @ Y1 ) ) )
| ( f @ a @ Y0 ) )
| ~ ( ~ ( f @ a @ Y0 )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( f @ a @ Y1 ) ) ) ) ) )
@ sK2 ) ),
inference(sigma_proxy_clausification,[],[f165]) ).
thf(f181,plain,
( $false
= ( ~ ( rel_s4 @ sK1 @ sK2 )
| ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) )
| ~ ( ~ ( f @ a @ sK2 )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f180]) ).
thf(f182,plain,
( $false
= ( ~ ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) )
| ~ ( ~ ( f @ a @ sK2 )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f181]) ).
thf(f183,plain,
( $false
= ( ~ ( rel_s4 @ sK1 @ sK2 ) ) ),
inference(or_proxy_clausification,[],[f181]) ).
thf(f184,plain,
( $true
= ( rel_s4 @ sK1 @ sK2 ) ),
inference(not_proxy_clausification,[],[f183]) ).
thf(f185,plain,
( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) )
| ~ ( ~ ( f @ a @ sK2 )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) ) ),
inference(not_proxy_clausification,[],[f182]) ).
thf(f186,plain,
( ( $true
= ( ~ ( ~ ( f @ a @ sK2 )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) ) )
| ( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ) ),
inference(or_proxy_clausification,[],[f185]) ).
thf(f187,plain,
( ( $false
= ( ~ ( f @ a @ sK2 )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) )
| ( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ) ),
inference(not_proxy_clausification,[],[f186]) ).
thf(f188,plain,
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) )
| ( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ) ),
inference(or_proxy_clausification,[],[f187]) ).
thf(f189,plain,
( ( $false
= ( ~ ( f @ a @ sK2 ) ) )
| ( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ) ),
inference(or_proxy_clausification,[],[f187]) ).
thf(f190,plain,
( ( $true
= ( f @ a @ sK2 ) )
| ( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ) ),
inference(not_proxy_clausification,[],[f189]) ).
thf(f191,plain,
( ( $true
= ( f @ a @ sK2 ) )
| ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ),
inference(not_proxy_clausification,[],[f190]) ).
thf(f193,plain,
( ( $true
= ( f @ a @ sK2 ) )
| ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f191]) ).
thf(f194,plain,
( ( $true
= ( f @ a @ sK2 ) )
| ( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) ) ) ),
inference(not_proxy_clausification,[],[f193]) ).
thf(f195,plain,
! [X1: $i] :
( ( $true
= ( f @ a @ sK2 ) )
| ( $true
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) )
@ X1 ) ) ),
inference(pi_proxy_clausification,[],[f194]) ).
thf(f196,plain,
! [X1: $i] :
( ( $true
= ( f @ a @ sK2 ) )
| ( $true
= ( ~ ( rel_s4 @ sK2 @ X1 )
| ( f @ a @ X1 ) ) ) ),
inference(beta-eta_normalization,[],[f195]) ).
thf(f197,plain,
! [X1: $i] :
( ( $true
= ( f @ a @ sK2 ) )
| ( $true
= ( f @ a @ X1 ) )
| ( $true
= ( ~ ( rel_s4 @ sK2 @ X1 ) ) ) ),
inference(or_proxy_clausification,[],[f196]) ).
thf(f198,plain,
! [X1: $i] :
( ( $true
= ( f @ a @ sK2 ) )
| ( $true
= ( f @ a @ X1 ) )
| ( $false
= ( rel_s4 @ sK2 @ X1 ) ) ),
inference(not_proxy_clausification,[],[f197]) ).
thf(f199,plain,
( ( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) )
@ sK3 ) )
| ( $true
= ( ~ ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ) ),
inference(sigma_proxy_clausification,[],[f188]) ).
thf(f200,plain,
( ( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) )
@ sK3 ) )
| ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) ) )
| ( f @ a @ sK2 ) ) ) ),
inference(not_proxy_clausification,[],[f199]) ).
thf(f201,plain,
( ( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ( f @ a @ Y0 ) )
@ sK3 ) )
| ( $false
= ( f @ a @ sK2 ) ) ),
inference(or_proxy_clausification,[],[f200]) ).
thf(f213,plain,
( ( $false
= ( ~ ( rel_s4 @ sK2 @ sK3 )
| ( f @ a @ sK3 ) ) )
| ( $false
= ( f @ a @ sK2 ) ) ),
inference(beta-eta_normalization,[],[f201]) ).
thf(f214,plain,
( ( $false
= ( f @ a @ sK3 ) )
| ( $false
= ( f @ a @ sK2 ) ) ),
inference(or_proxy_clausification,[],[f213]) ).
thf(f215,plain,
( ( $false
= ( ~ ( rel_s4 @ sK2 @ sK3 ) ) )
| ( $false
= ( f @ a @ sK2 ) ) ),
inference(or_proxy_clausification,[],[f213]) ).
thf(f216,plain,
( ( $true
= ( rel_s4 @ sK2 @ sK3 ) )
| ( $false
= ( f @ a @ sK2 ) ) ),
inference(not_proxy_clausification,[],[f215]) ).
thf(f217,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( rel_s4 @ Y2 @ Y1 )
& ( rel_s4 @ Y1 @ Y0 ) )
=> ( rel_s4 @ Y2 @ Y0 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f160]) ).
thf(f218,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( rel_s4 @ Y2 @ Y1 )
& ( rel_s4 @ Y1 @ Y0 ) )
=> ( rel_s4 @ Y2 @ Y0 ) ) ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f217]) ).
thf(f219,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( rel_s4 @ Y1 @ Y0 )
& ( rel_s4 @ Y0 @ X1 ) )
=> ( rel_s4 @ Y1 @ X1 ) ) ) ) ),
inference(beta-eta_normalization,[],[f218]) ).
thf(f220,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( rel_s4 @ Y1 @ Y0 )
& ( rel_s4 @ Y0 @ X1 ) )
=> ( rel_s4 @ Y1 @ X1 ) ) )
@ X2 ) ),
inference(pi_proxy_clausification,[],[f219]) ).
thf(f221,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( ( rel_s4 @ Y0 @ X2 )
& ( rel_s4 @ X2 @ X1 ) )
=> ( rel_s4 @ Y0 @ X1 ) ) ) ),
inference(beta-eta_normalization,[],[f220]) ).
thf(f222,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( ( rel_s4 @ Y0 @ X2 )
& ( rel_s4 @ X2 @ X1 ) )
=> ( rel_s4 @ Y0 @ X1 ) )
@ X3 ) ),
inference(pi_proxy_clausification,[],[f221]) ).
thf(f223,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ( ( rel_s4 @ X3 @ X2 )
& ( rel_s4 @ X2 @ X1 ) )
=> ( rel_s4 @ X3 @ X1 ) ) ),
inference(beta-eta_normalization,[],[f222]) ).
thf(f224,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( ( rel_s4 @ X3 @ X2 )
& ( rel_s4 @ X2 @ X1 ) ) )
| ( $true
= ( rel_s4 @ X3 @ X1 ) ) ),
inference(imp_proxy_clausification,[],[f223]) ).
thf(f225,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( rel_s4 @ X2 @ X1 ) )
| ( $true
= ( rel_s4 @ X3 @ X1 ) )
| ( $false
= ( rel_s4 @ X3 @ X2 ) ) ),
inference(and_proxy_clausification,[],[f224]) ).
thf(f226,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] : ( rel_s4 @ Y0 @ Y0 ) ) ),
inference(beta-eta_normalization,[],[f159]) ).
thf(f227,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] : ( rel_s4 @ Y0 @ Y0 )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f226]) ).
thf(f228,plain,
! [X1: $i] :
( $true
= ( rel_s4 @ X1 @ X1 ) ),
inference(beta-eta_normalization,[],[f227]) ).
thf(f230,definition,
( spl4_1
<=> ( $false
= ( f @ a @ sK2 ) ) ),
introduced(definition,[new_symbols(definition,[spl4_1])],[avatar_definition]) ).
thf(f231,plain,
( ( $false
!= ( f @ a @ sK2 ) )
| spl4_1 ),
inference(avatar_component_clause,[],[f230]) ).
thf(f232,plain,
( ( $false
= ( f @ a @ sK2 ) )
| ~ spl4_1 ),
inference(avatar_component_clause,[],[f230]) ).
thf(f238,plain,
( ! [X1: $i] :
( ( $true = $false )
| ( $true
= ( f @ a @ X1 ) )
| ( $false
= ( rel_s4 @ sK2 @ X1 ) ) )
| ~ spl4_1 ),
inference(forward_demodulation,[],[f198,f232]) ).
thf(f239,plain,
( ! [X1: $i] :
( ( $true
= ( f @ a @ X1 ) )
| ( $false
= ( rel_s4 @ sK2 @ X1 ) ) )
| ~ spl4_1 ),
inference(trivial_inequality_removal,[],[f238]) ).
thf(f246,plain,
( ( $true = $false )
| ( $false
= ( rel_s4 @ sK2 @ sK2 ) )
| ~ spl4_1 ),
inference(constrained_superposition,[],[f239,f232]) ).
thf(f249,plain,
( ( $false
= ( rel_s4 @ sK2 @ sK2 ) )
| ~ spl4_1 ),
inference(trivial_inequality_removal,[],[f246]) ).
thf(f417,plain,
( ( $true = $false )
| ~ spl4_1 ),
inference(forward_demodulation,[],[f249,f228]) ).
thf(f418,plain,
( $false
| ~ spl4_1 ),
inference(trivial_inequality_removal,[],[f417]) ).
thf(f419,plain,
~ spl4_1,
inference(avatar_contradiction_clause,[],[f418]) ).
thf(f442,plain,
( ! [X0: $i] :
( ( $false != $false )
| ( $true
= ( f @ a @ X0 ) )
| ( $false
= ( exists_in_world @ a @ sK1 ) )
| ( $false
= ( rel_s4 @ sK1 @ sK2 ) )
| ( $false
= ( rel_s4 @ sK1 @ X0 ) ) )
| spl4_1 ),
inference(constrained_superposition,[],[f231,f179]) ).
thf(f444,plain,
( ! [X0: $i] :
( ( $true
= ( f @ a @ X0 ) )
| ( $false
= ( exists_in_world @ a @ sK1 ) )
| ( $false
= ( rel_s4 @ sK1 @ sK2 ) )
| ( $false
= ( rel_s4 @ sK1 @ X0 ) ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f442]) ).
thf(f445,plain,
( ! [X0: $i] :
( ( $true = $false )
| ( $true
= ( f @ a @ X0 ) )
| ( $false
= ( rel_s4 @ sK1 @ sK2 ) )
| ( $false
= ( rel_s4 @ sK1 @ X0 ) ) )
| spl4_1 ),
inference(forward_demodulation,[],[f444,f145]) ).
thf(f446,plain,
( ! [X0: $i] :
( ( $true
= ( f @ a @ X0 ) )
| ( $false
= ( rel_s4 @ sK1 @ sK2 ) )
| ( $false
= ( rel_s4 @ sK1 @ X0 ) ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f445]) ).
thf(f447,plain,
( ! [X0: $i] :
( ( $true = $false )
| ( $true
= ( f @ a @ X0 ) )
| ( $false
= ( rel_s4 @ sK1 @ X0 ) ) )
| spl4_1 ),
inference(forward_demodulation,[],[f446,f184]) ).
thf(f448,plain,
( ! [X0: $i] :
( ( $true
= ( f @ a @ X0 ) )
| ( $false
= ( rel_s4 @ sK1 @ X0 ) ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f447]) ).
thf(f462,plain,
( ( $false
= ( f @ a @ sK3 ) )
| spl4_1 ),
inference(forward_subsumption_resolution,[],[f214,f231]) ).
thf(f473,plain,
( ( $true
= ( rel_s4 @ sK2 @ sK3 ) )
| spl4_1 ),
inference(forward_subsumption_resolution,[],[f216,f231]) ).
thf(f478,plain,
( ( $true = $false )
| ( $false
= ( rel_s4 @ sK1 @ sK3 ) )
| spl4_1 ),
inference(constrained_superposition,[],[f462,f448]) ).
thf(f479,plain,
( ( $false
= ( rel_s4 @ sK1 @ sK3 ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f478]) ).
thf(f553,plain,
! [X0: $i] :
( ( $true = $false )
| ( $true
= ( rel_s4 @ X0 @ sK2 ) )
| ( $false
= ( rel_s4 @ X0 @ sK1 ) ) ),
inference(constrained_superposition,[],[f184,f225]) ).
thf(f554,plain,
( ! [X0: $i] :
( ( $true = $false )
| ( $true
= ( rel_s4 @ X0 @ sK3 ) )
| ( $false
= ( rel_s4 @ X0 @ sK2 ) ) )
| spl4_1 ),
inference(constrained_superposition,[],[f473,f225]) ).
thf(f561,plain,
( ! [X0: $i] :
( ( $true
= ( rel_s4 @ X0 @ sK3 ) )
| ( $false
= ( rel_s4 @ X0 @ sK2 ) ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f554]) ).
thf(f562,plain,
! [X0: $i] :
( ( $true
= ( rel_s4 @ X0 @ sK2 ) )
| ( $false
= ( rel_s4 @ X0 @ sK1 ) ) ),
inference(trivial_inequality_removal,[],[f553]) ).
thf(f671,plain,
( ( $true = $false )
| ( $false
= ( rel_s4 @ sK1 @ sK2 ) )
| spl4_1 ),
inference(constrained_superposition,[],[f479,f561]) ).
thf(f672,plain,
( ( $false
= ( rel_s4 @ sK1 @ sK2 ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f671]) ).
thf(f717,plain,
( ( $true = $false )
| ( $false
= ( rel_s4 @ sK1 @ sK1 ) )
| spl4_1 ),
inference(constrained_superposition,[],[f672,f562]) ).
thf(f725,plain,
( ( $false
= ( rel_s4 @ sK1 @ sK1 ) )
| spl4_1 ),
inference(trivial_inequality_removal,[],[f717]) ).
thf(f733,plain,
( ( $true = $false )
| spl4_1 ),
inference(forward_demodulation,[],[f725,f228]) ).
thf(f734,plain,
( $false
| spl4_1 ),
inference(trivial_inequality_removal,[],[f733]) ).
thf(f735,plain,
spl4_1,
inference(avatar_contradiction_clause,[],[f734]) ).
cnf(s71,plain,
~ spl4_1,
inference(sat_conversion,[],[f419]) ).
cnf(s191,plain,
spl4_1,
inference(sat_conversion,[],[f735]) ).
cnf(s193,plain,
$false,
inference(rat,[],[s71,s191]) ).
thf(f736,plain,
$false,
inference(avatar_sat_refutation,[],[s193]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SYO568^7 : TPTP v9.3.1. Released v5.5.0.
% 0.00/0.05 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 SAT
% 0.09/0.18 % Computer : n014.cluster.edu
% 0.09/0.18 % Model : x86_64 x86_64
% 0.09/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.18 % Memory : 8046.5625MB
% 0.09/0.18 % OS : Linux 6.8.0-71-generic
% 0.09/0.18 % CPULimit : 300
% 0.09/0.18 % WCLimit : 300
% 0.09/0.18 % DateTime : Tue Sep 29 17:15:15 UTC 2026
% 0.09/0.18 % CPUTime :
% 0.09/0.18 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 SAT
% 0.09/0.21 Running first-order model finding
% 0.09/0.21 Running: /export/starexec/sandbox2/solver/bin/vampire-ho --input_syntax tptp --output_axiom_names on --mode casc --intent sat -m 16384 --cores 7 -t 300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.20/0.28 % (2951572)Will run a generic schedule for satisfiability detection.
% 0.20/0.28 % (2951592)ott+1_1_to=lpo:sil=16000:sp=reverse_arity:erd=off:random_seed=2495497594:i=131_2999 on theBenchmark for (2999ds/131Mi)
% 0.20/0.28 % (2951588)% WARNING: option uhcvi not known.
% 0.20/0.28 % (2951587)fmb+10_1_sas=cadical:bce=on:rp=on:random_seed=112923943_2999 on theBenchmark for (2999ds/0Mi)
% 0.20/0.28 % (2951591)ott+31_1_sil=16000:lcm=predicate:bce=on:newcnf=on:random_seed=1575057010:i=116_2999 on theBenchmark for (2999ds/116Mi)
% 0.20/0.28 % (2951588)dis+11_61:31_drc=ordering:lsd=5:bsr=unit_only:rp=on:newcnf=on:random_seed=3323064254:i=135531:add=off:rawr=on_2999 on theBenchmark for (2999ds/135531Mi)
% 0.20/0.28 % (2951590)dis+10_1_sil=32000:sp=arity:random_seed=727700103:i=103:fgj=on_2999 on theBenchmark for (2999ds/103Mi)
% 0.20/0.28 % (2951593)ott-3_16_to=lpo:sil=16000:sp=arity:fd=off:rp=on:random_seed=2422702662:i=159:bs=unit_only:nicw=on:fsr=off:amm=off_2999 on theBenchmark for (2999ds/159Mi)
% 0.20/0.28 % (2951589)dis+10_161_sil=256000:plsq=on:plsqr=61199697,1048576:gs=on:alpa=true:sac=on:slsq=on:cn=on:random_seed=3010994908:i=88024:add=on:rawr=on_2999 on theBenchmark for (2999ds/88024Mi)
% 0.20/0.28 % (2951591)WARNING: Not using 'newcnf' as currently not compatible with higher-order inputs.
% 0.20/0.28 % (2951588)WARNING: Not using 'newcnf' as currently not compatible with higher-order inputs.
% 0.20/0.28 % Exception at run slice level
% 0.20/0.28 User error: Finite model building is currently not compatible with polymorphism or higher-order constructs
% 0.20/0.28 % (2951588)WARNING: Look ahead literal selection is not currently compatible with higher-order. Ignoring request to use
% 0.20/0.28 % (2951589) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-2951572-2951589"...
% 0.20/0.28 % (2951606)fmb+10_1_fmbas=predicate:sil=64000:sas=cadical:random_seed=2914860143:i=714:nm=2_2999 on theBenchmark for (2999ds/714Mi)
% 0.20/0.28 % (2951592)Instruction limit reached!
% 0.20/0.28 % (2951592)------------------------------
% 0.20/0.28 % (2951592)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.20/0.28 % (2951592)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.20/0.28 % (2951592)CaDiCaL version: 2.1.3
% 0.20/0.28 % (2951592)Termination reason: Instruction limit
% 0.20/0.28 % (2951592)Termination phase: Saturation
% 0.20/0.28 % (2951592)Time elapsed: 0.032 s
% 0.20/0.28 % (2951592)Peak memory usage: 12 MB
% 0.20/0.28 % (2951592)Instructions burned: 131 (million)
% 0.20/0.28 % Exception at run slice level
% 0.20/0.28 User error: Finite model building is currently not compatible with polymorphism or higher-order constructs
% 0.20/0.28 % (2951589)...printing done.
% 0.20/0.28 % (2951591) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-2951572-2951591"...
% 0.20/0.28 % (2951589)Refutation found. Thanks to Tanya!
% 0.20/0.28 % SZS status Theorem for theBenchmark
% 0.20/0.28 % SZS output start Proof for theBenchmark
% See solution above
% 0.20/0.29 % (2951589)------------------------------
% 0.20/0.29 % (2951589)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.20/0.29 % (2951589)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.20/0.29 % (2951589)CaDiCaL version: 2.1.3
% 0.20/0.29 % (2951589)Termination reason: Refutation
% 0.20/0.29 % (2951589)Time elapsed: 0.026 s
% 0.20/0.29 % (2951589)Peak memory usage: 13 MB
% 0.20/0.29 % (2951589)Instructions burned: 48 (million)
% 0.20/0.29 % (2951572)Success in time 0.064 s
% 0.20/0.29 % Vampire exiting
%------------------------------------------------------------------------------