%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : SYN357^7 : TPTP v9.3.1. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% Computer : n003.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 08:41:40 AM UTC 2026
% Result : Theorem 0.22s 0.32s
% Output : Refutation 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 37
% Number of leaves : 8
% Syntax : Number of formulae : 74 ( 49 unt; 0 typ; 0 def)
% Number of atoms : 597 ( 98 equ; 0 cnn)
% Maximal formula atoms : 10 ( 8 avg)
% Number of connectives : 1078 ( 129 ~; 121 |; 0 &; 716 @)
% ( 0 <=>; 27 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 2 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 71 ( 71 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 56 usr; 6 con; 0-4 aty)
% ( 85 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 233 ( 201 ^; 32 !; 0 ?; 233 :)
% 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,
big_p: mu > $i > $o ).
thf(func_def_37,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_38,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_39,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_40,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_41,type,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_42,type,
vOR: $o > $o > $o ).
thf(func_def_43,type,
vNOT: $o > $o ).
thf(func_def_44,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_45,type,
vIMP: $o > $o > $o ).
thf(func_def_46,type,
vAND: $o > $o > $o ).
thf(func_def_47,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_48,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_51,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_52,type,
sK0: $i > mu ).
thf(func_def_55,type,
sK3: mu ).
thf(func_def_56,type,
sK4: mu > $i ).
thf(func_def_57,type,
sK5: mu > $i ).
thf(f2,axiom,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mnot) ).
thf(f3,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
| ( X0 @ X2 ) ) )
= mor ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mor) ).
thf(f9,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o] : ( mor @ ( mnot @ X0 ) @ X1 ) )
= mimplies ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mimplies) ).
thf(f15,axiom,
( ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( X0 @ X2 @ X1 ) ) )
= mforall_ind ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mforall_ind) ).
thf(f16,axiom,
( ( ^ [X0: mu > $i > $o] :
( mnot
@ ( mforall_ind
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) ) ) )
= mexists_ind ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mexists_ind) ).
thf(f28,axiom,
( ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) )
= mvalid ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mvalid) ).
thf(f32,axiom,
( ( ^ [X0: $i > $o,X1: $i] :
! [X2: $i] :
( ( X0 @ X2 )
| ~ ( rel_s4 @ X1 @ X2 ) ) )
= mbox_s4 ),
file('/export/starexec/sandbox/benchmark/Axioms/LCL013^5.ax',mbox_s4) ).
thf(f37,conjecture,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X0: mu] :
( mexists_ind
@ ^ [X1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( big_p @ X0 ) ) @ ( mbox_s4 @ ( big_p @ X1 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',x2108) ).
thf(f38,negated_conjecture,
~ ( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X0: mu] :
( mexists_ind
@ ^ [X1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( big_p @ X0 ) ) @ ( mbox_s4 @ ( big_p @ X1 ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f37]) ).
thf(f41,plain,
( ( ^ [X0: $i > $o,X1: $i] :
! [X2: $i] :
( ( X0 @ X2 )
| ~ ( rel_s4 @ X1 @ X2 ) ) )
= mbox_s4 ),
inference(rectify,[],[f32]) ).
thf(f42,plain,
( mbox_s4
= ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ) ),
inference(fool_elimination,[],[f41]) ).
thf(f49,plain,
( ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) )
= mvalid ),
inference(rectify,[],[f28]) ).
thf(f50,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f49]) ).
thf(f60,plain,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
inference(rectify,[],[f2]) ).
thf(f61,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(fool_elimination,[],[f60]) ).
thf(f64,plain,
( ( ^ [X0: mu > $i > $o] :
( mnot
@ ( mforall_ind
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) ) ) )
= mexists_ind ),
inference(rectify,[],[f16]) ).
thf(f65,plain,
( mexists_ind
= ( ^ [Y0: mu > $i > $o] :
( mnot
@ ( mforall_ind
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) ) ) ) ),
inference(fool_elimination,[],[f64]) ).
thf(f76,plain,
( ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ( X0 @ X2 @ X1 ) ) )
= mforall_ind ),
inference(rectify,[],[f15]) ).
thf(f77,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[],[f76]) ).
thf(f79,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
| ( X0 @ X2 ) ) )
= mor ),
inference(rectify,[],[f3]) ).
thf(f80,plain,
( mor
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
| ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f79]) ).
thf(f95,plain,
~ ( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X0: mu] :
( mexists_ind
@ ^ [X1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( big_p @ X0 ) ) @ ( mbox_s4 @ ( big_p @ X1 ) ) ) ) ) ) ) ),
inference(rectify,[],[f38]) ).
thf(f96,plain,
( ( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [Y0: mu] :
( mexists_ind
@ ^ [Y1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( big_p @ Y0 ) ) @ ( mbox_s4 @ ( big_p @ Y1 ) ) ) ) ) ) ) )
!= $true ),
inference(fool_elimination,[],[f95]) ).
thf(f101,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mor @ ( mnot @ Y0 ) @ Y1 ) ) ),
inference(fool_elimination,[],[f9]) ).
thf(f102,plain,
( ( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [Y0: mu] :
( mexists_ind
@ ^ [Y1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( big_p @ Y0 ) ) @ ( mbox_s4 @ ( big_p @ Y1 ) ) ) ) ) ) ) )
!= $true ),
inference(flattening,[],[f96]) ).
thf(f119,plain,
( ( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [Y0: mu] :
( mexists_ind
@ ^ [Y1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( big_p @ Y0 ) ) @ ( mbox_s4 @ ( big_p @ Y1 ) ) ) ) ) ) ) )
!= $true ),
inference(cnf_transformation,[],[f102]) ).
thf(f123,plain,
( mor
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
| ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f80]) ).
thf(f125,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[],[f77]) ).
thf(f129,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o] : ( mor @ ( mnot @ Y0 ) @ Y1 ) ) ),
inference(cnf_transformation,[],[f101]) ).
thf(f130,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f50]) ).
thf(f133,plain,
( mexists_ind
= ( ^ [Y0: mu > $i > $o] :
( mnot
@ ( mforall_ind
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) ) ) ) ),
inference(cnf_transformation,[],[f65]) ).
thf(f135,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(cnf_transformation,[],[f61]) ).
thf(f143,plain,
( mbox_s4
= ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[],[f42]) ).
thf(f148,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,[],[f129,f123,f135]) ).
thf(f153,plain,
( mexists_ind
= ( ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( ^ [Y1: mu > $i > $o,Y2: $i] :
( !! @ mu
@ ^ [Y3: mu] :
( ( exists_in_world @ Y3 @ Y2 )
=> ( Y1 @ Y3 @ Y2 ) ) )
@ ^ [Y1: mu] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( Y0 @ Y1 ) ) ) ) ) ),
inference(definition_unfolding,[],[f133,f135,f125,f135]) ).
thf(f157,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: $i > $o,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( Y0 @ Y2 ) ) )
@ ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) )
@ ^ [Y0: mu] :
( ^ [Y1: mu > $i > $o] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( ^ [Y2: mu > $i > $o,Y3: $i] :
( !! @ mu
@ ^ [Y4: mu] :
( ( exists_in_world @ Y4 @ Y3 )
=> ( Y2 @ Y4 @ Y3 ) ) )
@ ^ [Y2: mu] :
( ^ [Y3: $i > $o,Y4: $i] :
~ ( Y3 @ Y4 )
@ ( Y1 @ Y2 ) ) ) )
@ ^ [Y1: mu] :
( ^ [Y2: $i > $o,Y3: $i] :
( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( Y2 @ Y4 ) ) )
@ ( ^ [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 )
@ ( ^ [Y2: $i > $o,Y3: $i] :
( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( Y2 @ Y4 ) ) )
@ ( big_p @ Y0 ) )
@ ( ^ [Y2: $i > $o,Y3: $i] :
( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( Y2 @ Y4 ) ) )
@ ( big_p @ Y1 ) ) ) ) ) ) ) )
!= $true ),
inference(definition_unfolding,[],[f119,f130,f143,f125,f153,f143,f148,f143,f143]) ).
thf(f171,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ~ ( !! @ mu
@ ^ [Y3: mu] :
( ( exists_in_world @ Y3 @ Y1 )
=> ~ ( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y1 @ Y4 )
| ~ ( !! @ $i
@ ^ [Y5: $i] :
( ~ ( rel_s4 @ Y4 @ Y5 )
| ( big_p @ Y2 @ Y5 ) ) )
| ( !! @ $i
@ ^ [Y5: $i] :
( ~ ( rel_s4 @ Y4 @ Y5 )
| ( big_p @ Y3 @ Y5 ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f157]) ).
thf(f172,plain,
( $false
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y1 )
=> ~ ( !! @ mu
@ ^ [Y3: mu] :
( ( exists_in_world @ Y3 @ Y1 )
=> ~ ( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y1 @ Y4 )
| ~ ( !! @ $i
@ ^ [Y5: $i] :
( ~ ( rel_s4 @ Y4 @ Y5 )
| ( big_p @ Y2 @ Y5 ) ) )
| ( !! @ $i
@ ^ [Y5: $i] :
( ~ ( rel_s4 @ Y4 @ Y5 )
| ( big_p @ Y3 @ Y5 ) ) ) ) ) ) ) ) ) ) )
@ sK1 ) ),
inference(sigma_proxy_clausification,[],[f171]) ).
thf(f173,plain,
( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ Y0 )
=> ~ ( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y0 )
=> ~ ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y0 @ Y3 )
| ~ ( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( big_p @ Y1 @ Y4 ) ) )
| ( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( big_p @ Y2 @ Y4 ) ) ) ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f172]) ).
thf(f174,plain,
( ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK1 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ Y0 )
=> ~ ( !! @ mu
@ ^ [Y2: mu] :
( ( exists_in_world @ Y2 @ Y0 )
=> ~ ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y0 @ Y3 )
| ~ ( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( big_p @ Y1 @ Y4 ) ) )
| ( !! @ $i
@ ^ [Y4: $i] :
( ~ ( rel_s4 @ Y3 @ Y4 )
| ( big_p @ Y2 @ Y4 ) ) ) ) ) ) ) ) ) )
@ sK2 )
= $false ),
inference(sigma_proxy_clausification,[],[f173]) ).
thf(f175,plain,
( $false
= ( ~ ( rel_s4 @ sK1 @ sK2 )
| ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ sK2 @ Y2 )
| ~ ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( big_p @ Y0 @ Y3 ) ) )
| ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( big_p @ Y1 @ Y3 ) ) ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f174]) ).
thf(f176,plain,
( $false
= ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ sK2 @ Y2 )
| ~ ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( big_p @ Y0 @ Y3 ) ) )
| ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( big_p @ Y1 @ Y3 ) ) ) ) ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f175]) ).
thf(f179,plain,
( $false
= ( ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ mu
@ ^ [Y1: mu] :
( ( exists_in_world @ Y1 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ sK2 @ Y2 )
| ~ ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( big_p @ Y0 @ Y3 ) ) )
| ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel_s4 @ Y2 @ Y3 )
| ( big_p @ Y1 @ Y3 ) ) ) ) ) ) ) )
@ sK3 ) ),
inference(sigma_proxy_clausification,[],[f176]) ).
thf(f180,plain,
( $false
= ( ( exists_in_world @ sK3 @ sK2 )
=> ~ ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK2 @ Y1 )
| ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ sK3 @ Y2 ) ) )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ Y0 @ Y2 ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f179]) ).
thf(f181,plain,
( $false
= ( ~ ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK2 @ Y1 )
| ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ sK3 @ Y2 ) ) )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ Y0 @ Y2 ) ) ) ) ) ) ) ) ),
inference(imp_proxy_clausification,[],[f180]) ).
thf(f182,plain,
( ( exists_in_world @ sK3 @ sK2 )
= $true ),
inference(imp_proxy_clausification,[],[f180]) ).
thf(f183,plain,
( ( !! @ mu
@ ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK2 @ Y1 )
| ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ sK3 @ Y2 ) ) )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ Y0 @ Y2 ) ) ) ) ) ) )
= $true ),
inference(not_proxy_clausification,[],[f181]) ).
thf(f184,plain,
! [X1: mu] :
( ( ^ [Y0: mu] :
( ( exists_in_world @ Y0 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ sK2 @ Y1 )
| ~ ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ sK3 @ Y2 ) ) )
| ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel_s4 @ Y1 @ Y2 )
| ( big_p @ Y0 @ Y2 ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f183]) ).
thf(f185,plain,
! [X1: mu] :
( ( ( exists_in_world @ X1 @ sK2 )
=> ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ sK3 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ X1 @ Y1 ) ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f184]) ).
thf(f186,plain,
! [X1: mu] :
( ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ sK3 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ X1 @ Y1 ) ) ) ) ) )
= $true )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(imp_proxy_clausification,[],[f185]) ).
thf(f187,plain,
! [X1: mu] :
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ sK3 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ X1 @ Y1 ) ) ) ) ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(not_proxy_clausification,[],[f186]) ).
thf(f188,plain,
! [X1: mu] :
( ( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ sK2 @ Y0 )
| ~ ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ sK3 @ Y1 ) ) )
| ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel_s4 @ Y0 @ Y1 )
| ( big_p @ X1 @ Y1 ) ) ) )
@ ( sK4 @ X1 ) ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(sigma_proxy_clausification,[],[f187]) ).
thf(f189,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( $false
= ( ~ ( rel_s4 @ sK2 @ ( sK4 @ X1 ) )
| ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ sK3 @ Y0 ) ) )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ X1 @ Y0 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f188]) ).
thf(f190,plain,
! [X1: mu] :
( ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ sK3 @ Y0 ) ) )
| ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ X1 @ Y0 ) ) ) ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(or_proxy_clausification,[],[f189]) ).
thf(f193,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ X1 @ Y0 ) ) ) ) ),
inference(or_proxy_clausification,[],[f190]) ).
thf(f194,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( $false
= ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ sK3 @ Y0 ) ) ) ) ) ),
inference(or_proxy_clausification,[],[f190]) ).
thf(f195,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ sK3 @ Y0 ) ) )
= $true ) ),
inference(not_proxy_clausification,[],[f194]) ).
thf(f196,plain,
! [X2: $i,X1: mu] :
( ( ( ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ sK3 @ Y0 ) )
@ X2 )
= $true )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(pi_proxy_clausification,[],[f195]) ).
thf(f197,plain,
! [X2: $i,X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( ( ~ ( rel_s4 @ ( sK4 @ X1 ) @ X2 )
| ( big_p @ sK3 @ X2 ) )
= $true ) ),
inference(beta-eta_normalization,[],[f196]) ).
thf(f198,plain,
! [X2: $i,X1: mu] :
( ( $true
= ( big_p @ sK3 @ X2 ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( ( ~ ( rel_s4 @ ( sK4 @ X1 ) @ X2 ) )
= $true ) ),
inference(or_proxy_clausification,[],[f197]) ).
thf(f199,plain,
! [X2: $i,X1: mu] :
( ( $false
= ( rel_s4 @ ( sK4 @ X1 ) @ X2 ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( $true
= ( big_p @ sK3 @ X2 ) ) ),
inference(not_proxy_clausification,[],[f198]) ).
thf(f200,plain,
! [X1: mu] :
( ( $false
= ( ^ [Y0: $i] :
( ~ ( rel_s4 @ ( sK4 @ X1 ) @ Y0 )
| ( big_p @ X1 @ Y0 ) )
@ ( sK5 @ X1 ) ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(sigma_proxy_clausification,[],[f193]) ).
thf(f201,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( $false
= ( ~ ( rel_s4 @ ( sK4 @ X1 ) @ ( sK5 @ X1 ) )
| ( big_p @ X1 @ ( sK5 @ X1 ) ) ) ) ),
inference(beta-eta_normalization,[],[f200]) ).
thf(f202,plain,
! [X1: mu] :
( ( $false
= ( big_p @ X1 @ ( sK5 @ X1 ) ) )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(or_proxy_clausification,[],[f201]) ).
thf(f203,plain,
! [X1: mu] :
( ( $false
= ( exists_in_world @ X1 @ sK2 ) )
| ( $false
= ( ~ ( rel_s4 @ ( sK4 @ X1 ) @ ( sK5 @ X1 ) ) ) ) ),
inference(or_proxy_clausification,[],[f201]) ).
thf(f204,plain,
! [X1: mu] :
( ( ( rel_s4 @ ( sK4 @ X1 ) @ ( sK5 @ X1 ) )
= $true )
| ( $false
= ( exists_in_world @ X1 @ sK2 ) ) ),
inference(not_proxy_clausification,[],[f203]) ).
thf(f206,plain,
! [X0: mu] :
( ( ( big_p @ sK3 @ ( sK5 @ X0 ) )
= $true )
| ( $false
= ( exists_in_world @ X0 @ sK2 ) )
| ( $false
= ( exists_in_world @ X0 @ sK2 ) )
| ( $false = $true ) ),
inference(constrained_superposition,[],[f204,f199]) ).
thf(f207,plain,
! [X0: mu] :
( ( $false
= ( exists_in_world @ X0 @ sK2 ) )
| ( $false = $true )
| ( ( big_p @ sK3 @ ( sK5 @ X0 ) )
= $true ) ),
inference(duplicate_literal_removal,[],[f206]) ).
thf(f208,plain,
! [X0: mu] :
( ( ( big_p @ sK3 @ ( sK5 @ X0 ) )
= $true )
| ( $false
= ( exists_in_world @ X0 @ sK2 ) ) ),
inference(trivial_inequality_removal,[],[f207]) ).
thf(f212,plain,
( ( $false
= ( exists_in_world @ sK3 @ sK2 ) )
| ( $false = $true )
| ( $false
= ( exists_in_world @ sK3 @ sK2 ) ) ),
inference(constrained_superposition,[],[f202,f208]) ).
thf(f213,plain,
( ( $false
= ( exists_in_world @ sK3 @ sK2 ) )
| ( $false = $true ) ),
inference(duplicate_literal_removal,[],[f212]) ).
thf(f214,plain,
( $false
= ( exists_in_world @ sK3 @ sK2 ) ),
inference(trivial_inequality_removal,[],[f213]) ).
thf(f217,plain,
$false = $true,
inference(forward_demodulation,[],[f214,f182]) ).
thf(f218,plain,
$false,
inference(trivial_inequality_removal,[],[f217]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SYN357^7 : TPTP v9.3.1. Released v5.5.0.
% 0.00/0.06 % Command : run_vampire /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% 0.09/0.20 % Computer : n003.cluster.edu
% 0.09/0.20 % Model : x86_64 x86_64
% 0.09/0.20 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.20 % Memory : 8046.5625MB
% 0.09/0.20 % OS : Linux 6.8.0-71-generic
% 0.09/0.20 % CPULimit : 300
% 0.09/0.20 % WCLimit : 300
% 0.09/0.20 % DateTime : Tue Sep 29 16:16:59 UTC 2026
% 0.09/0.20 % CPUTime :
% 0.09/0.20 Running run_vampire /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% 0.09/0.24 Running higher-order theorem proving
% 0.09/0.26 Running: /export/starexec/sandbox/solver/bin/vampire-ho --input_syntax tptp --output_axiom_names on --mode casc -m 16384 --cores 7 -t 300 /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.22/0.32 % (2690041)Detected a higher-order problem, will run a greedy HOL sequence.
% 0.22/0.32 % (2690051)lrs+10_1_to=lpo:sil=128000:e2e=on:si=on:random_seed=2339767285:s2a=on:i=75:s2at=3:aac=none:bd=preordered:rtra=on:fe=abstraction_2999 on theBenchmark for (2999ds/75Mi)
% 0.22/0.32 % (2690048)lrs+10_1_cnfonf=off:si=on:uwa=one_side_interpreted:random_seed=4275312996:i=3:rtra=on:inj=on:ntd=on_2999 on theBenchmark for (2999ds/3Mi)
% 0.22/0.32 % (2690046)lrs+10_40_drc=off:e2e=on:si=on:uwa=one_side_interpreted:random_seed=3852457482:s2a=on:i=87:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/87Mi)
% 0.22/0.32 % (2690047)lrs+10_16_si=on:nwc=1.5:random_seed=2528762654:i=18:kws=arity_squared:rtra=on:fe=abstraction:ntd=on_2999 on theBenchmark for (2999ds/18Mi)
% 0.22/0.32 % (2690049)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=3090364204: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)
% 0.22/0.32 % (2690048)Instruction limit reached!
% 0.22/0.32 % (2690048)------------------------------
% 0.22/0.32 % (2690048)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.32 % (2690048)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.32 % (2690048)CaDiCaL version: 2.1.3
% 0.22/0.32 % (2690048)Termination reason: Instruction limit
% 0.22/0.32 % (2690048)Termination phase: Property scanning
% 0.22/0.32 % (2690048)Time elapsed: 0.003 s
% 0.22/0.32 % (2690048)Peak memory usage: 10 MB
% 0.22/0.32 % (2690048)Instructions burned: 5 (million)
% 0.22/0.32 % (2690046) found proof, printing to "/export/starexec/sandbox/tmp/vampire-proof-2690041-2690046"...
% 0.22/0.32 % (2690047) found proof, printing to "/export/starexec/sandbox/tmp/vampire-proof-2690041-2690047"...
% 0.22/0.32 % (2690046)...printing done.
% 0.22/0.32 % (2690047)...printing done.
% 0.22/0.32 % (2690052)WARNING Broken Constraint: if ho_split_queue_ratios(1,8) has been set then ho_split_queue(off) is equal to on
% 0.22/0.32 % (2690052)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
% 0.22/0.32 % (2690046)Refutation found. Thanks to Tanya!
% 0.22/0.32 % SZS status Theorem for theBenchmark
% 0.22/0.32 % SZS output start Proof for theBenchmark
% See solution above
% 0.22/0.32 % (2690046)------------------------------
% 0.22/0.32 % (2690046)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.32 % (2690046)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.32 % (2690046)CaDiCaL version: 2.1.3
% 0.22/0.32 % (2690046)Termination reason: Refutation
% 0.22/0.32 % (2690046)Time elapsed: 0.010 s
% 0.22/0.32 % (2690046)Peak memory usage: 12 MB
% 0.22/0.32 % (2690046)Instructions burned: 19 (million)
% 0.22/0.32 % (2690041)Success in time 0.052 s
% 0.22/0.32 % Vampire exiting
%------------------------------------------------------------------------------