%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : PHI002^1 : TPTP v9.3.1. Released v6.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:20:16 AM UTC 2026
% Result : Theorem 0.22s 0.31s
% Output : Refutation 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 36
% Number of leaves : 19
% Syntax : Number of formulae : 131 ( 77 unt; 0 typ; 4 def)
% Number of atoms : 764 ( 172 equ; 0 cnn)
% Maximal formula atoms : 10 ( 5 avg)
% Number of connectives : 1207 ( 72 ~; 95 |; 29 &; 885 @)
% ( 5 <=>; 35 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 3 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 387 ( 387 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 53 usr; 9 con; 0-4 aty)
% ( 60 !!; 26 ??; 0 @@+; 0 @@-)
% Number of variables : 408 ( 0 sgn 84 !; 4 ?; 408 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
mu: $tType ).
thf(type_def_6,type,
sTfun: ( $tType * $tType ) > $tType ).
thf(func_def_0,type,
meq_ind: mu > mu > $i > $o ).
thf(func_def_1,type,
mtrue: $i > $o ).
thf(func_def_2,type,
mfalse: $i > $o ).
thf(func_def_3,type,
mnot: ( $i > $o ) > $i > $o ).
thf(func_def_5,type,
mor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_6,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_7,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_8,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_9,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_10,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_11,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_12,type,
mforall_indset: ( ( mu > $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_13,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_14,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(func_def_15,type,
mexists_indset: ( ( mu > $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_16,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(func_def_17,type,
mbox_generic: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_18,type,
mdia_generic: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_19,type,
rel: $i > $i > $o ).
thf(func_def_20,type,
mbox: ( $i > $o ) > $i > $o ).
thf(func_def_21,type,
mdia: ( $i > $o ) > $i > $o ).
thf(func_def_22,type,
mvalid: ( $i > $o ) > $o ).
thf(func_def_23,type,
minvalid: ( $i > $o ) > $o ).
thf(func_def_24,type,
positive: ( mu > $i > $o ) > $i > $o ).
thf(func_def_25,type,
god: mu > $i > $o ).
thf(func_def_26,type,
essence: ( mu > $i > $o ) > mu > $i > $o ).
thf(func_def_27,type,
necessary_existence: mu > $i > $o ).
thf(func_def_28,type,
vOR: $o > $o > $o ).
thf(func_def_29,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_30,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_31,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_32,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_33,type,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_34,type,
vNOT: $o > $o ).
thf(func_def_35,type,
vIFF: $o > $o > $o ).
thf(func_def_36,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_39,type,
vAND: $o > $o > $o ).
thf(func_def_40,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_41,type,
vIMP: $o > $o > $o ).
thf(func_def_42,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_43,type,
sF0: $o ).
thf(func_def_44,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_45,type,
db5:
!>[X0: $tType] : X0 ).
thf(func_def_46,type,
sK1: $i > ( mu > $i > $o ) > $i ).
thf(func_def_47,type,
sK2: $i > ( mu > $i > $o ) > mu ).
thf(func_def_49,type,
sK4: mu > $i > $o ).
thf(func_def_50,type,
sK5: ( mu > $i > $o ) > $i > ( mu > $i > $o ) > $i ).
thf(func_def_51,type,
sK6: ( mu > $i > $o ) > $i > ( mu > $i > $o ) > mu ).
thf(f4,axiom,
( ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) )
= mnot ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mnot) ).
thf(f6,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
& ( X0 @ X2 ) ) )
= mand ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mand) ).
thf(f7,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
=> ( X1 @ X2 ) ) )
= mimplies ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mimplies) ).
thf(f9,axiom,
( mequiv
= ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
<=> ( X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mequiv) ).
thf(f11,axiom,
( mforall_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] : ( X0 @ X2 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mforall_ind) ).
thf(f12,axiom,
( ( ^ [X0: ( mu > $i > $o ) > $i > $o,X1: $i] :
! [X2: mu > $i > $o] : ( X0 @ X2 @ X1 ) )
= mforall_indset ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mforall_indset) ).
thf(f14,axiom,
( ( ^ [X0: mu > $i > $o,X1: $i] :
? [X2: mu] : ( X0 @ X2 @ X1 ) )
= mexists_ind ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mexists_ind) ).
thf(f17,axiom,
( mbox_generic
= ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( X1 @ X3 )
| ~ ( X0 @ X2 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mbox_generic) ).
thf(f18,axiom,
( mdia_generic
= ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
? [X3: $i] :
( ( X1 @ X3 )
& ( X0 @ X2 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mdia_generic) ).
thf(f19,axiom,
( mbox
= ( mbox_generic @ rel ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mbox) ).
thf(f20,axiom,
( mdia
= ( mdia_generic @ rel ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mdia) ).
thf(f21,axiom,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mvalid) ).
thf(f23,axiom,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) )
@ ( mnot @ ( positive @ X0 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/PHI001^0.ax',axA1) ).
thf(f24,axiom,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mforall_indset
@ ^ [X1: mu > $i > $o] :
( mimplies
@ ( mand @ ( positive @ X0 )
@ ( mbox
@ ( mforall_ind
@ ^ [X2: mu] : ( mimplies @ ( X0 @ X2 ) @ ( X1 @ X2 ) ) ) ) )
@ ( positive @ X1 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/PHI001^0.ax',axA2) ).
thf(f31,conjecture,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mimplies @ ( positive @ X0 )
@ ( mdia
@ ( mexists_ind
@ ^ [X1: mu] : ( X0 @ X1 ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',thmT1) ).
thf(f32,negated_conjecture,
~ ( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mimplies @ ( positive @ X0 )
@ ( mdia
@ ( mexists_ind
@ ^ [X1: mu] : ( X0 @ X1 ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f31]) ).
thf(f37,plain,
( mequiv
= ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
<=> ( X1 @ X2 ) ) ) ),
inference(rectify,[],[f9]) ).
thf(f38,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
= ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f37]) ).
thf(f41,plain,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mforall_indset
@ ^ [X1: mu > $i > $o] :
( mimplies
@ ( mand @ ( positive @ X0 )
@ ( mbox
@ ( mforall_ind
@ ^ [X2: mu] : ( mimplies @ ( X0 @ X2 ) @ ( X1 @ X2 ) ) ) ) )
@ ( positive @ X1 ) ) ) ) ),
inference(rectify,[],[f24]) ).
thf(f42,plain,
( $true
= ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mforall_indset
@ ^ [Y1: mu > $i > $o] :
( mimplies
@ ( mand @ ( positive @ Y0 )
@ ( mbox
@ ( mforall_ind
@ ^ [Y2: mu] : ( mimplies @ ( Y0 @ Y2 ) @ ( Y1 @ Y2 ) ) ) ) )
@ ( positive @ Y1 ) ) ) ) ) ),
inference(fool_elimination,[],[f41]) ).
thf(f43,plain,
( mforall_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] : ( X0 @ X2 @ X1 ) ) ),
inference(rectify,[],[f11]) ).
thf(f44,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f43]) ).
thf(f45,plain,
( ( ^ [X0: ( mu > $i > $o ) > $i > $o,X1: $i] :
! [X2: mu > $i > $o] : ( X0 @ X2 @ X1 ) )
= mforall_indset ),
inference(rectify,[],[f12]) ).
thf(f46,plain,
( mforall_indset
= ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f45]) ).
thf(f48,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
& ( X0 @ X2 ) ) )
= mand ),
inference(rectify,[],[f6]) ).
thf(f49,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f48]) ).
thf(f50,plain,
( ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) )
= mnot ),
inference(rectify,[],[f4]) ).
thf(f51,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(fool_elimination,[],[f50]) ).
thf(f54,plain,
( mbox_generic
= ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( X1 @ X3 )
| ~ ( X0 @ X2 @ X3 ) ) ) ),
inference(rectify,[],[f17]) ).
thf(f55,plain,
( mbox_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( Y0 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) ) ) ),
inference(fool_elimination,[],[f54]) ).
thf(f58,plain,
( mdia_generic
= ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
? [X3: $i] :
( ( X1 @ X3 )
& ( X0 @ X2 @ X3 ) ) ) ),
inference(rectify,[],[f18]) ).
thf(f59,plain,
( mdia_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( Y1 @ Y3 ) ) ) ) ),
inference(fool_elimination,[],[f58]) ).
thf(f60,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
=> ( X1 @ X2 ) ) )
= mimplies ),
inference(rectify,[],[f7]) ).
thf(f61,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f60]) ).
thf(f64,plain,
~ ( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mimplies @ ( positive @ X0 )
@ ( mdia
@ ( mexists_ind
@ ^ [X1: mu] : ( X0 @ X1 ) ) ) ) ) ),
inference(rectify,[],[f32]) ).
thf(f65,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mimplies @ ( positive @ Y0 )
@ ( mdia
@ ( mexists_ind
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(fool_elimination,[],[f64]) ).
thf(f73,plain,
( mvalid
= ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) ) ),
inference(rectify,[],[f21]) ).
thf(f74,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f73]) ).
thf(f77,plain,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) )
@ ( mnot @ ( positive @ X0 ) ) ) ) ),
inference(rectify,[],[f23]) ).
thf(f78,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) )
@ ( mnot @ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(fool_elimination,[],[f77]) ).
thf(f81,plain,
( ( ^ [X0: mu > $i > $o,X1: $i] :
? [X2: mu] : ( X0 @ X2 @ X1 ) )
= mexists_ind ),
inference(rectify,[],[f14]) ).
thf(f82,plain,
( mexists_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( ?? @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f81]) ).
thf(f88,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mimplies @ ( positive @ Y0 )
@ ( mdia
@ ( mexists_ind
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(flattening,[],[f65]) ).
thf(f89,plain,
( mforall_indset
= ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f46]) ).
thf(f90,plain,
( mbox_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( Y0 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) ) ) ),
inference(cnf_transformation,[],[f55]) ).
thf(f92,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) )
@ ( mnot @ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(cnf_transformation,[],[f78]) ).
thf(f97,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f61]) ).
thf(f98,plain,
( mdia_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( Y1 @ Y3 ) ) ) ) ),
inference(cnf_transformation,[],[f59]) ).
thf(f102,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f49]) ).
thf(f104,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f44]) ).
thf(f105,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
= ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f38]) ).
thf(f106,plain,
( mbox
= ( mbox_generic @ rel ) ),
inference(cnf_transformation,[],[f19]) ).
thf(f110,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(cnf_transformation,[],[f51]) ).
thf(f111,plain,
( mexists_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( ?? @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f82]) ).
thf(f112,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f74]) ).
thf(f115,plain,
( mdia
= ( mdia_generic @ rel ) ),
inference(cnf_transformation,[],[f20]) ).
thf(f116,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mimplies @ ( positive @ Y0 )
@ ( mdia
@ ( mexists_ind
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
!= $true ),
inference(cnf_transformation,[],[f88]) ).
thf(f117,plain,
( $true
= ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mforall_indset
@ ^ [Y1: mu > $i > $o] :
( mimplies
@ ( mand @ ( positive @ Y0 )
@ ( mbox
@ ( mforall_ind
@ ^ [Y2: mu] : ( mimplies @ ( Y0 @ Y2 ) @ ( Y1 @ Y2 ) ) ) ) )
@ ( positive @ Y1 ) ) ) ) ) ),
inference(cnf_transformation,[],[f42]) ).
thf(f122,plain,
( mbox
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( Y0 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ rel ) ),
inference(definition_unfolding,[],[f106,f90]) ).
thf(f123,plain,
( mdia
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( Y1 @ Y3 ) ) )
@ rel ) ),
inference(definition_unfolding,[],[f115,f98]) ).
thf(f127,plain,
( $true
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
= ( Y2 @ Y3 ) )
@ ( positive
@ ^ [Y1: mu] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( Y0 @ Y1 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( positive @ Y0 ) ) ) ) ) ),
inference(definition_unfolding,[],[f92,f112,f89,f105,f110,f110]) ).
thf(f130,plain,
( $true
!= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 @ Y3 ) )
@ ( positive @ Y0 )
@ ( ^ [Y1: $i > $i > $o,Y2: $i > $o,Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1 @ Y3 @ Y4 )
& ( Y2 @ Y4 ) ) )
@ rel
@ ( ^ [Y1: mu > $i > $o,Y2: $i] :
( ?? @ mu
@ ^ [Y3: mu] : ( Y1 @ Y3 @ Y2 ) )
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) ) ),
inference(definition_unfolding,[],[f116,f112,f89,f97,f123,f111]) ).
thf(f131,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: ( mu > $i > $o ) > $i > $o,Y2: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] : ( Y1 @ Y3 @ Y2 ) )
@ ^ [Y1: mu > $i > $o] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
=> ( Y3 @ Y4 ) )
@ ( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 @ Y4 ) )
@ ( positive @ Y0 )
@ ( ^ [Y2: $i > $i > $o,Y3: $i > $o,Y4: $i] :
( !! @ $i
@ ^ [Y5: $i] :
( ~ ( Y2 @ Y4 @ Y5 )
| ( Y3 @ Y5 ) ) )
@ rel
@ ( ^ [Y2: mu > $i > $o,Y3: $i] :
( !! @ mu
@ ^ [Y4: mu] : ( Y2 @ Y4 @ Y3 ) )
@ ^ [Y2: mu] :
( ^ [Y3: $i > $o,Y4: $i > $o,Y5: $i] :
( ( Y3 @ Y5 )
=> ( Y4 @ Y5 ) )
@ ( Y0 @ Y2 )
@ ( Y1 @ Y2 ) ) ) ) )
@ ( positive @ Y1 ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f117,f112,f89,f89,f97,f102,f122,f104,f97]) ).
thf(f133,definition,
( sF0
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 @ Y3 ) )
@ ( positive @ Y0 )
@ ( ^ [Y1: $i > $i > $o,Y2: $i > $o,Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1 @ Y3 @ Y4 )
& ( Y2 @ Y4 ) ) )
@ rel
@ ( ^ [Y1: mu > $i > $o,Y2: $i] :
( ?? @ mu
@ ^ [Y3: mu] : ( Y1 @ Y3 @ Y2 ) )
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sF0])],[function_definition]) ).
thf(f134,plain,
sF0 != $true,
inference(definition_folding,[],[f130,f133]) ).
thf(f135,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive
@ ^ [Y2: mu,Y3: $i] :
~ ( Y1 @ Y2 @ Y3 )
@ Y0 )
= ( ~ ( positive @ Y1 @ Y0 ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f127]) ).
thf(f136,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive
@ ^ [Y2: mu,Y3: $i] :
~ ( Y1 @ Y2 @ Y3 )
@ Y0 )
= ( ~ ( positive @ Y1 @ Y0 ) ) ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f135]) ).
thf(f137,plain,
! [X1: $i] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive
@ ^ [Y1: mu,Y2: $i] :
~ ( Y0 @ Y1 @ Y2 )
@ X1 )
= ( ~ ( positive @ Y0 @ X1 ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f136]) ).
thf(f138,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ^ [Y0: mu > $i > $o] :
( ( positive
@ ^ [Y1: mu,Y2: $i] :
~ ( Y0 @ Y1 @ Y2 )
@ X1 )
= ( ~ ( positive @ Y0 @ X1 ) ) )
@ X2 )
= $true ),
inference(pi_proxy_clausification,[],[f137]) ).
thf(f139,plain,
! [X2: mu > $i > $o,X1: $i] :
( $true
= ( ( positive
@ ^ [Y0: mu,Y1: $i] :
~ ( X2 @ Y0 @ Y1 )
@ X1 )
= ( ~ ( positive @ X2 @ X1 ) ) ) ),
inference(beta-eta_normalization,[],[f138]) ).
thf(f140,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ~ ( positive @ X2 @ X1 ) )
= ( positive
@ ^ [Y0: mu,Y1: $i] :
~ ( X2 @ Y0 @ Y1 )
@ X1 ) ),
inference(iff_proxy_clausification,[],[f139]) ).
thf(f141,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ( ~ ( positive @ X2 @ X1 ) )
= $true )
| ( ( positive
@ ^ [Y0: mu,Y1: $i] :
~ ( X2 @ Y0 @ Y1 )
@ X1 )
= $false ) ),
inference(iff_proxy_clausification,[],[f140]) ).
thf(f144,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ( positive
@ ^ [Y0: mu,Y1: $i] :
~ ( X2 @ Y0 @ Y1 )
@ X1 )
= $false )
| ( ( positive @ X2 @ X1 )
= $false ) ),
inference(not_proxy_clausification,[],[f141]) ).
thf(f147,plain,
( ( sF0 = $true )
| ( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 @ Y3 ) )
@ ( positive @ Y0 )
@ ( ^ [Y1: $i > $i > $o,Y2: $i > $o,Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1 @ Y3 @ Y4 )
& ( Y2 @ Y4 ) ) )
@ rel
@ ( ^ [Y1: mu > $i > $o,Y2: $i] :
( ?? @ mu
@ ^ [Y3: mu] : ( Y1 @ Y3 @ Y2 ) )
@ ^ [Y1: mu] : ( Y0 @ Y1 ) ) ) ) ) )
= $false ) ),
inference(iff_proxy_clausification,[],[f133]) ).
thf(f161,plain,
( ( sF0 = $true )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ Y0 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( rel @ Y0 @ Y2 )
& ( ?? @ mu
@ ^ [Y3: mu] : ( Y1 @ Y3 @ Y2 ) ) ) ) ) ) )
= $false ) ),
inference(beta-eta_normalization,[],[f147]) ).
thf(f162,plain,
( ( ( ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ Y0 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( rel @ Y0 @ Y2 )
& ( ?? @ mu
@ ^ [Y3: mu] : ( Y1 @ Y3 @ Y2 ) ) ) ) ) )
@ sK3 )
= $false )
| ( sF0 = $true ) ),
inference(sigma_proxy_clausification,[],[f161]) ).
thf(f163,plain,
( ( sF0 = $true )
| ( $false
= ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK3 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( rel @ sK3 @ Y1 )
& ( ?? @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f162]) ).
thf(f164,plain,
( ( sF0 = $true )
| ( ( ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK3 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( rel @ sK3 @ Y1 )
& ( ?? @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ) )
@ sK4 )
= $false ) ),
inference(sigma_proxy_clausification,[],[f163]) ).
thf(f165,plain,
( ( ( ( positive @ sK4 @ sK3 )
=> ( ?? @ $i
@ ^ [Y0: $i] :
( ( rel @ sK3 @ Y0 )
& ( ?? @ mu
@ ^ [Y1: mu] : ( sK4 @ Y1 @ Y0 ) ) ) ) )
= $false )
| ( sF0 = $true ) ),
inference(beta-eta_normalization,[],[f164]) ).
thf(f166,plain,
( ( sF0 = $true )
| ( ( ?? @ $i
@ ^ [Y0: $i] :
( ( rel @ sK3 @ Y0 )
& ( ?? @ mu
@ ^ [Y1: mu] : ( sK4 @ Y1 @ Y0 ) ) ) )
= $false ) ),
inference(imp_proxy_clausification,[],[f165]) ).
thf(f167,plain,
( ( sF0 = $true )
| ( ( positive @ sK4 @ sK3 )
= $true ) ),
inference(imp_proxy_clausification,[],[f165]) ).
thf(f168,plain,
! [X1: $i] :
( ( ( ^ [Y0: $i] :
( ( rel @ sK3 @ Y0 )
& ( ?? @ mu
@ ^ [Y1: mu] : ( sK4 @ Y1 @ Y0 ) ) )
@ X1 )
= $false )
| ( sF0 = $true ) ),
inference(pi_proxy_clausification,[],[f166]) ).
thf(f169,plain,
! [X1: $i] :
( ( sF0 = $true )
| ( ( ( rel @ sK3 @ X1 )
& ( ?? @ mu
@ ^ [Y0: mu] : ( sK4 @ Y0 @ X1 ) ) )
= $false ) ),
inference(beta-eta_normalization,[],[f168]) ).
thf(f170,plain,
! [X1: $i] :
( ( ( rel @ sK3 @ X1 )
= $false )
| ( sF0 = $true )
| ( ( ?? @ mu
@ ^ [Y0: mu] : ( sK4 @ Y0 @ X1 ) )
= $false ) ),
inference(and_proxy_clausification,[],[f169]) ).
thf(f171,plain,
! [X2: mu,X1: $i] :
( ( ( rel @ sK3 @ X1 )
= $false )
| ( sF0 = $true )
| ( ( ^ [Y0: mu] : ( sK4 @ Y0 @ X1 )
@ X2 )
= $false ) ),
inference(pi_proxy_clausification,[],[f170]) ).
thf(f172,plain,
! [X2: mu,X1: $i] :
( ( sF0 = $true )
| ( ( sK4 @ X2 @ X1 )
= $false )
| ( ( rel @ sK3 @ X1 )
= $false ) ),
inference(beta-eta_normalization,[],[f171]) ).
thf(f173,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( ( positive @ Y1 @ Y0 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel @ Y0 @ Y3 )
| ( !! @ mu
@ ^ [Y4: mu] :
( ( Y1 @ Y4 @ Y3 )
=> ( Y2 @ Y4 @ Y3 ) ) ) ) ) )
=> ( positive @ Y2 @ Y0 ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f131]) ).
thf(f174,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( ( positive @ Y1 @ Y0 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel @ Y0 @ Y3 )
| ( !! @ mu
@ ^ [Y4: mu] :
( ( Y1 @ Y4 @ Y3 )
=> ( Y2 @ Y4 @ Y3 ) ) ) ) ) )
=> ( positive @ Y2 @ Y0 ) ) ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f173]) ).
thf(f175,plain,
! [X1: $i] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( ( positive @ Y0 @ X1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel @ X1 @ Y2 )
| ( !! @ mu
@ ^ [Y3: mu] :
( ( Y0 @ Y3 @ Y2 )
=> ( Y1 @ Y3 @ Y2 ) ) ) ) ) )
=> ( positive @ Y1 @ X1 ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f174]) ).
thf(f176,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ^ [Y0: mu > $i > $o] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( ( positive @ Y0 @ X1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel @ X1 @ Y2 )
| ( !! @ mu
@ ^ [Y3: mu] :
( ( Y0 @ Y3 @ Y2 )
=> ( Y1 @ Y3 @ Y2 ) ) ) ) ) )
=> ( positive @ Y1 @ X1 ) ) )
@ X2 )
= $true ),
inference(pi_proxy_clausification,[],[f175]) ).
thf(f177,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( ( positive @ X2 @ X1 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ X1 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( X2 @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) )
=> ( positive @ Y0 @ X1 ) ) )
= $true ),
inference(beta-eta_normalization,[],[f176]) ).
thf(f178,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ^ [Y0: mu > $i > $o] :
( ( ( positive @ X2 @ X1 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ X1 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( X2 @ Y2 @ Y1 )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) )
=> ( positive @ Y0 @ X1 ) )
@ X3 )
= $true ),
inference(pi_proxy_clausification,[],[f177]) ).
thf(f179,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( ( positive @ X2 @ X1 )
& ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ X1 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( X2 @ Y1 @ Y0 )
=> ( X3 @ Y1 @ Y0 ) ) ) ) ) )
=> ( positive @ X3 @ X1 ) )
= $true ),
inference(beta-eta_normalization,[],[f178]) ).
thf(f180,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( positive @ X3 @ X1 )
= $true )
| ( ( ( positive @ X2 @ X1 )
& ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ X1 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( X2 @ Y1 @ Y0 )
=> ( X3 @ Y1 @ Y0 ) ) ) ) ) )
= $false ) ),
inference(imp_proxy_clausification,[],[f179]) ).
thf(f181,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( positive @ X3 @ X1 )
= $true )
| ( ( positive @ X2 @ X1 )
= $false )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ X1 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( X2 @ Y1 @ Y0 )
=> ( X3 @ Y1 @ Y0 ) ) ) ) )
= $false ) ),
inference(and_proxy_clausification,[],[f180]) ).
thf(f182,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( ^ [Y0: $i] :
( ~ ( rel @ X1 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( X2 @ Y1 @ Y0 )
=> ( X3 @ Y1 @ Y0 ) ) ) )
@ ( sK5 @ X3 @ X1 @ X2 ) )
= $false )
| ( ( positive @ X2 @ X1 )
= $false )
| ( ( positive @ X3 @ X1 )
= $true ) ),
inference(sigma_proxy_clausification,[],[f181]) ).
thf(f183,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( positive @ X2 @ X1 )
= $false )
| ( ( ~ ( rel @ X1 @ ( sK5 @ X3 @ X1 @ X2 ) )
| ( !! @ mu
@ ^ [Y0: mu] :
( ( X2 @ Y0 @ ( sK5 @ X3 @ X1 @ X2 ) )
=> ( X3 @ Y0 @ ( sK5 @ X3 @ X1 @ X2 ) ) ) ) )
= $false )
| ( ( positive @ X3 @ X1 )
= $true ) ),
inference(beta-eta_normalization,[],[f182]) ).
thf(f184,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( !! @ mu
@ ^ [Y0: mu] :
( ( X2 @ Y0 @ ( sK5 @ X3 @ X1 @ X2 ) )
=> ( X3 @ Y0 @ ( sK5 @ X3 @ X1 @ X2 ) ) ) )
= $false )
| ( ( positive @ X3 @ X1 )
= $true )
| ( ( positive @ X2 @ X1 )
= $false ) ),
inference(or_proxy_clausification,[],[f183]) ).
thf(f185,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( positive @ X3 @ X1 )
= $true )
| ( ( positive @ X2 @ X1 )
= $false )
| ( ( ~ ( rel @ X1 @ ( sK5 @ X3 @ X1 @ X2 ) ) )
= $false ) ),
inference(or_proxy_clausification,[],[f183]) ).
thf(f186,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( rel @ X1 @ ( sK5 @ X3 @ X1 @ X2 ) )
= $true )
| ( ( positive @ X2 @ X1 )
= $false )
| ( ( positive @ X3 @ X1 )
= $true ) ),
inference(not_proxy_clausification,[],[f185]) ).
thf(f187,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( positive @ X3 @ X1 )
= $true )
| ( $false
= ( ^ [Y0: mu] :
( ( X2 @ Y0 @ ( sK5 @ X3 @ X1 @ X2 ) )
=> ( X3 @ Y0 @ ( sK5 @ X3 @ X1 @ X2 ) ) )
@ ( sK6 @ X3 @ X1 @ X2 ) ) )
| ( ( positive @ X2 @ X1 )
= $false ) ),
inference(sigma_proxy_clausification,[],[f184]) ).
thf(f188,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( positive @ X3 @ X1 )
= $true )
| ( ( ( X2 @ ( sK6 @ X3 @ X1 @ X2 ) @ ( sK5 @ X3 @ X1 @ X2 ) )
=> ( X3 @ ( sK6 @ X3 @ X1 @ X2 ) @ ( sK5 @ X3 @ X1 @ X2 ) ) )
= $false )
| ( ( positive @ X2 @ X1 )
= $false ) ),
inference(beta-eta_normalization,[],[f187]) ).
thf(f190,plain,
! [X2: mu > $i > $o,X3: mu > $i > $o,X1: $i] :
( ( ( X2 @ ( sK6 @ X3 @ X1 @ X2 ) @ ( sK5 @ X3 @ X1 @ X2 ) )
= $true )
| ( ( positive @ X3 @ X1 )
= $true )
| ( ( positive @ X2 @ X1 )
= $false ) ),
inference(imp_proxy_clausification,[],[f188]) ).
thf(f216,definition,
( spl7_4
<=> ( ( positive @ sK4 @ sK3 )
= $true ) ),
introduced(definition,[new_symbols(definition,[spl7_4])],[avatar_definition]) ).
thf(f218,plain,
( ( ( positive @ sK4 @ sK3 )
= $true )
| ~ spl7_4 ),
inference(avatar_component_clause,[],[f216]) ).
thf(f220,definition,
( spl7_5
<=> ( sF0 = $true ) ),
introduced(definition,[new_symbols(definition,[spl7_5])],[avatar_definition]) ).
thf(f223,plain,
( spl7_4
| spl7_5 ),
inference(avatar_split_clause,[],[f167,f220,f216]) ).
thf(f225,definition,
( spl7_6
<=> ! [X2: mu,X1: $i] :
( ( ( sK4 @ X2 @ X1 )
= $false )
| ( ( rel @ sK3 @ X1 )
= $false ) ) ),
introduced(definition,[new_symbols(definition,[spl7_6])],[avatar_definition]) ).
thf(f226,plain,
( ! [X2: mu,X1: $i] :
( ( ( sK4 @ X2 @ X1 )
= $false )
| ( ( rel @ sK3 @ X1 )
= $false ) )
| ~ spl7_6 ),
inference(avatar_component_clause,[],[f225]) ).
thf(f227,plain,
( spl7_6
| spl7_5 ),
inference(avatar_split_clause,[],[f172,f220,f225]) ).
thf(f228,plain,
~ spl7_5,
inference(avatar_split_clause,[],[f134,f220]) ).
thf(f295,plain,
( ! [X0: mu > $i > $o,X1: $i] :
( ( ( positive @ sK4 @ X1 )
= $false )
| ( ( positive @ X0 @ X1 )
= $true )
| ( ( rel @ sK3 @ ( sK5 @ X0 @ X1 @ sK4 ) )
= $false )
| ( $true = $false ) )
| ~ spl7_6 ),
inference(constrained_superposition,[],[f190,f226]) ).
thf(f299,plain,
( ! [X0: mu > $i > $o,X1: $i] :
( ( ( rel @ sK3 @ ( sK5 @ X0 @ X1 @ sK4 ) )
= $false )
| ( ( positive @ sK4 @ X1 )
= $false )
| ( ( positive @ X0 @ X1 )
= $true ) )
| ~ spl7_6 ),
inference(trivial_inequality_removal,[],[f295]) ).
thf(f307,plain,
( ! [X0: mu > $i > $o] :
( ( ( positive @ X0 @ sK3 )
= $true )
| ( $true = $false )
| ( ( positive @ sK4 @ sK3 )
= $false )
| ( ( positive @ X0 @ sK3 )
= $true )
| ( ( positive @ sK4 @ sK3 )
= $false ) )
| ~ spl7_6 ),
inference(constrained_superposition,[],[f186,f299]) ).
thf(f308,plain,
( ! [X0: mu > $i > $o] :
( ( ( positive @ sK4 @ sK3 )
= $false )
| ( ( positive @ X0 @ sK3 )
= $true )
| ( $true = $false ) )
| ~ spl7_6 ),
inference(duplicate_literal_removal,[],[f307]) ).
thf(f309,plain,
( ! [X0: mu > $i > $o] :
( ( ( positive @ X0 @ sK3 )
= $true )
| ( ( positive @ sK4 @ sK3 )
= $false ) )
| ~ spl7_6 ),
inference(trivial_inequality_removal,[],[f308]) ).
thf(f313,plain,
( ! [X0: mu > $i > $o] :
( ( $true = $false )
| ( ( positive @ X0 @ sK3 )
= $true ) )
| ~ spl7_4
| ~ spl7_6 ),
inference(forward_demodulation,[],[f309,f218]) ).
thf(f314,plain,
( ! [X0: mu > $i > $o] :
( ( positive @ X0 @ sK3 )
= $true )
| ~ spl7_4
| ~ spl7_6 ),
inference(trivial_inequality_removal,[],[f313]) ).
thf(f326,plain,
( ! [X0: mu > $i > $o] :
( ( ( positive @ X0 @ sK3 )
= $false )
| ( $true = $false ) )
| ~ spl7_4
| ~ spl7_6 ),
inference(constrained_superposition,[],[f144,f314]) ).
thf(f330,plain,
( ! [X0: mu > $i > $o] :
( ( positive @ X0 @ sK3 )
= $false )
| ~ spl7_4
| ~ spl7_6 ),
inference(trivial_inequality_removal,[],[f326]) ).
thf(f340,plain,
( ( $true = $false )
| ~ spl7_4
| ~ spl7_6 ),
inference(forward_demodulation,[],[f330,f314]) ).
thf(f341,plain,
( $false
| ~ spl7_4
| ~ spl7_6 ),
inference(trivial_inequality_removal,[],[f340]) ).
thf(f342,plain,
( ~ spl7_4
| ~ spl7_6 ),
inference(avatar_contradiction_clause,[],[f341]) ).
cnf(s3,plain,
( spl7_4
| spl7_5 ),
inference(sat_conversion,[],[f223]) ).
cnf(s4,plain,
( spl7_5
| spl7_6 ),
inference(sat_conversion,[],[f227]) ).
cnf(s5,plain,
~ spl7_5,
inference(sat_conversion,[],[f228]) ).
cnf(s10,plain,
( ~ spl7_4
| ~ spl7_6 ),
inference(sat_conversion,[],[f342]) ).
cnf(s12,plain,
spl7_6,
inference(rat,[],[s4,s5]) ).
cnf(s13,plain,
~ spl7_4,
inference(rat,[],[s10,s12]) ).
cnf(s14,plain,
$false,
inference(rat,[],[s3,s5,s13]) ).
thf(f343,plain,
$false,
inference(avatar_sat_refutation,[],[s14]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : PHI002^1 : TPTP v9.3.1. Released v6.1.0.
% 0.00/0.05 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.07/0.19 % Computer : n015.cluster.edu
% 0.07/0.19 % Model : x86_64 x86_64
% 0.07/0.19 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.19 % Memory : 8046.5625MB
% 0.07/0.19 % OS : Linux 6.8.0-71-generic
% 0.07/0.19 % CPULimit : 300
% 0.07/0.19 % WCLimit : 300
% 0.07/0.19 % DateTime : Tue Sep 29 13:19:58 UTC 2026
% 0.07/0.19 % CPUTime :
% 0.07/0.19 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.07/0.23 Running higher-order theorem proving
% 0.07/0.25 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
% 0.22/0.31 % (3496067)Detected a higher-order problem, will run a greedy HOL sequence.
% 0.22/0.31 % (3496077)lrs+10_1_to=lpo:sil=128000:e2e=on:si=on:random_seed=2877825653:s2a=on:i=75:s2at=3:aac=none:bd=preordered:rtra=on:fe=abstraction_2999 on theBenchmark for (2999ds/75Mi)
% 0.22/0.31 % (3496078)WARNING Broken Constraint: if ho_split_queue_ratios(1,8) has been set then ho_split_queue(off) is equal to on
% 0.22/0.31 % (3496078)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
% 0.22/0.31 % (3496076)dis+21_4_fde=none:e2e=on:si=on:uwa=off:foolp=on:random_seed=2380853825:i=24:av=off:rtra=on_2999 on theBenchmark for (2999ds/24Mi)
% 0.22/0.31 % (3496073)lrs+10_16_si=on:nwc=1.5:random_seed=72567284:i=18:kws=arity_squared:rtra=on:fe=abstraction:ntd=on_2999 on theBenchmark for (2999ds/18Mi)
% 0.22/0.31 % (3496072)lrs+10_40_drc=off:e2e=on:si=on:uwa=one_side_interpreted:random_seed=1461998584:s2a=on:i=87:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/87Mi)
% 0.22/0.31 % (3496074)lrs+10_1_cnfonf=off:si=on:uwa=one_side_interpreted:random_seed=3801300277:i=3:rtra=on:inj=on:ntd=on_2999 on theBenchmark for (2999ds/3Mi)
% 0.22/0.31 % (3496075)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=3925500348: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)
% 0.22/0.31 % (3496078)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=3867901452:hsqr=1,8:i=157:s2at=5:add=on:nm=2:rtra=on_2999 on theBenchmark for (2999ds/157Mi)
% 0.22/0.31 % (3496074)Instruction limit reached!
% 0.22/0.31 % (3496074)------------------------------
% 0.22/0.31 % (3496074)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.31 % (3496074)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.31 % (3496074)CaDiCaL version: 2.1.3
% 0.22/0.31 % (3496074)Termination reason: Instruction limit
% 0.22/0.31 % (3496074)Termination phase: Property scanning
% 0.22/0.31 % (3496074)Time elapsed: 0.002 s
% 0.22/0.31 % (3496074)Peak memory usage: 10 MB
% 0.22/0.31 % (3496074)Instructions burned: 3 (million)
% 0.22/0.31 % (3496073)Instruction limit reached!
% 0.22/0.31 % (3496073)------------------------------
% 0.22/0.31 % (3496073)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.31 % (3496073)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.31 % (3496073)CaDiCaL version: 2.1.3
% 0.22/0.31 % (3496073)Termination reason: Instruction limit
% 0.22/0.31 % (3496073)Termination phase: Saturation
% 0.22/0.31 % (3496073)Time elapsed: 0.010 s
% 0.22/0.31 % (3496073)Peak memory usage: 12 MB
% 0.22/0.31 % (3496073)Instructions burned: 19 (million)
% 0.22/0.31 % (3496077)Instruction limit reached!
% 0.22/0.31 % (3496077)------------------------------
% 0.22/0.31 % (3496077)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.31 % (3496077)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.31 % (3496077)CaDiCaL version: 2.1.3
% 0.22/0.31 % (3496077)Termination reason: Instruction limit
% 0.22/0.31 % (3496077)Termination phase: Saturation
% 0.22/0.31 % (3496077)Time elapsed: 0.021 s
% 0.22/0.31 % (3496077)Peak memory usage: 12 MB
% 0.22/0.31 % (3496077)Instructions burned: 78 (million)
% 0.22/0.31 % (3496076)Instruction limit reached!
% 0.22/0.31 % (3496076)------------------------------
% 0.22/0.31 % (3496076)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.31 % (3496076)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.31 % (3496076)CaDiCaL version: 2.1.3
% 0.22/0.31 % (3496076)Termination reason: Instruction limit
% 0.22/0.31 % (3496076)Termination phase: Saturation
% 0.22/0.31 % (3496076)Time elapsed: 0.014 s
% 0.22/0.31 % (3496076)Peak memory usage: 11 MB
% 0.22/0.31 % (3496076)Instructions burned: 25 (million)
% 0.22/0.31 % (3496078) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-3496067-3496078"...
% 0.22/0.31 % (3496078)...printing done.
% 0.22/0.31 % (3496078)Refutation found. Thanks to Tanya!
% 0.22/0.31 % SZS status Theorem for theBenchmark
% 0.22/0.31 % SZS output start Proof for theBenchmark
% See solution above
% 0.22/0.32 % (3496078)------------------------------
% 0.22/0.32 % (3496078)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.22/0.32 % (3496078)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.22/0.32 % (3496078)CaDiCaL version: 2.1.3
% 0.22/0.32 % (3496078)Termination reason: Refutation
% 0.22/0.32 % (3496078)Time elapsed: 0.019 s
% 0.22/0.32 % (3496078)Peak memory usage: 13 MB
% 0.22/0.32 % (3496078)Instructions burned: 34 (million)
% 0.22/0.32 % (3496067)Success in time 0.06 s
% 0.22/0.32 % Vampire exiting
%------------------------------------------------------------------------------