%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : PHI004^2 : 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 : n026.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 08:20:17 AM UTC 2026
% Result : Theorem 0.21s 0.35s
% Output : Refutation 0.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 32
% Number of leaves : 14
% Syntax : Number of formulae : 117 ( 108 unt; 0 typ; 0 def)
% Number of atoms : 700 ( 119 equ; 0 cnn)
% Maximal formula atoms : 7 ( 5 avg)
% Number of connectives : 1305 ( 67 ~; 35 |; 13 &; 971 @)
% ( 2 <=>; 98 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 2 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 473 ( 473 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 46 usr; 7 con; 0-4 aty)
% ( 119 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 385 ( 355 ^; 30 !; 0 ?; 385 :)
% 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,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_28,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_29,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_30,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_31,type,
vNOT: $o > $o ).
thf(func_def_34,type,
vIFF: $o > $o > $o ).
thf(func_def_35,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_36,type,
vOR: $o > $o > $o ).
thf(func_def_37,type,
vAND: $o > $o > $o ).
thf(func_def_38,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_39,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_40,type,
vIMP: $o > $o > $o ).
thf(func_def_41,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_42,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_44,type,
sK1: mu ).
thf(func_def_45,type,
sK2: mu > $i > $o ).
thf(func_def_47,type,
sK4: mu ).
thf(f4,axiom,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mnot) ).
thf(f6,axiom,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
& ( X1 @ 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(f17,axiom,
( ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( X1 @ X3 )
| ~ ( X0 @ X2 @ X3 ) ) )
= mbox_generic ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mbox_generic) ).
thf(f19,axiom,
( mbox
= ( mbox_generic @ rel ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/LCL016^0.ax',mbox) ).
thf(f21,axiom,
( ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) )
= mvalid ),
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/theBenchmark.p',axA1) ).
thf(f24,axiom,
( ( ^ [X0: mu] :
( mforall_indset
@ ^ [X1: mu > $i > $o] : ( mimplies @ ( positive @ X1 ) @ ( X1 @ X0 ) ) ) )
= god ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',defD1) ).
thf(f25,axiom,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] : ( mimplies @ ( positive @ X0 ) @ ( mbox @ ( positive @ X0 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axA4) ).
thf(f26,axiom,
( ( ^ [X0: mu > $i > $o,X1: mu] :
( mand @ ( X0 @ X1 )
@ ( mforall_indset
@ ^ [X2: mu > $i > $o] :
( mimplies @ ( X2 @ X1 )
@ ( mbox
@ ( mforall_ind
@ ^ [X3: mu] : ( mimplies @ ( X0 @ X3 ) @ ( X2 @ X3 ) ) ) ) ) ) ) )
= essence ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',defD2) ).
thf(f27,conjecture,
( mvalid
@ ( mforall_ind
@ ^ [X0: mu] : ( mimplies @ ( god @ X0 ) @ ( essence @ god @ X0 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',thmT2) ).
thf(f28,negated_conjecture,
~ ( mvalid
@ ( mforall_ind
@ ^ [X0: mu] : ( mimplies @ ( god @ X0 ) @ ( essence @ god @ X0 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f27]) ).
thf(f29,plain,
( ( ^ [X0: $i > $o] :
! [X1: $i] : ( X0 @ X1 ) )
= mvalid ),
inference(rectify,[],[f21]) ).
thf(f30,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f29]) ).
thf(f31,plain,
~ ( mvalid
@ ( mforall_ind
@ ^ [X0: mu] : ( mimplies @ ( god @ X0 ) @ ( essence @ god @ X0 ) ) ) ),
inference(rectify,[],[f28]) ).
thf(f32,plain,
( ( ~ ( mvalid
@ ( mforall_ind
@ ^ [Y0: mu] : ( mimplies @ ( god @ Y0 ) @ ( essence @ god @ Y0 ) ) ) ) )
= $true ),
inference(fool_elimination,[],[f31]) ).
thf(f33,plain,
( mequiv
= ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
<=> ( X1 @ X2 ) ) ) ),
inference(rectify,[],[f9]) ).
thf(f34,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
= ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f33]) ).
thf(f35,plain,
( mnot
= ( ^ [X0: $i > $o,X1: $i] :
~ ( X0 @ X1 ) ) ),
inference(rectify,[],[f4]) ).
thf(f36,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(fool_elimination,[],[f35]) ).
thf(f38,plain,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] : ( mimplies @ ( positive @ X0 ) @ ( mbox @ ( positive @ X0 ) ) ) ) ),
inference(rectify,[],[f25]) ).
thf(f39,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] : ( mimplies @ ( positive @ Y0 ) @ ( mbox @ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(fool_elimination,[],[f38]) ).
thf(f42,plain,
( ( ^ [X0: mu] :
( mforall_indset
@ ^ [X1: mu > $i > $o] : ( mimplies @ ( positive @ X1 ) @ ( X1 @ X0 ) ) ) )
= god ),
inference(rectify,[],[f24]) ).
thf(f43,plain,
( god
= ( ^ [Y0: mu] :
( mforall_indset
@ ^ [Y1: mu > $i > $o] : ( mimplies @ ( positive @ Y1 ) @ ( Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[],[f42]) ).
thf(f47,plain,
( ( ^ [X0: mu > $i > $o,X1: mu] :
( mand @ ( X0 @ X1 )
@ ( mforall_indset
@ ^ [X2: mu > $i > $o] :
( mimplies @ ( X2 @ X1 )
@ ( mbox
@ ( mforall_ind
@ ^ [X3: mu] : ( mimplies @ ( X0 @ X3 ) @ ( X2 @ X3 ) ) ) ) ) ) ) )
= essence ),
inference(rectify,[],[f26]) ).
thf(f48,plain,
( essence
= ( ^ [Y0: mu > $i > $o,Y1: mu] :
( mand @ ( Y0 @ Y1 )
@ ( mforall_indset
@ ^ [Y2: mu > $i > $o] :
( mimplies @ ( Y2 @ Y1 )
@ ( mbox
@ ( mforall_ind
@ ^ [Y3: mu] : ( mimplies @ ( Y0 @ Y3 ) @ ( Y2 @ Y3 ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[],[f47]) ).
thf(f49,plain,
( mforall_ind
= ( ^ [X0: mu > $i > $o,X1: $i] :
! [X2: mu] : ( X0 @ X2 @ X1 ) ) ),
inference(rectify,[],[f11]) ).
thf(f50,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f49]) ).
thf(f51,plain,
( ( ^ [X0: $i > $i > $o,X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( X1 @ X3 )
| ~ ( X0 @ X2 @ X3 ) ) )
= mbox_generic ),
inference(rectify,[],[f17]) ).
thf(f52,plain,
( mbox_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( Y0 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) ) ) ),
inference(fool_elimination,[],[f51]) ).
thf(f53,plain,
( mbox
= ( mbox_generic @ rel ) ),
inference(fool_elimination,[],[f19]) ).
thf(f63,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
=> ( X1 @ X2 ) ) )
= mimplies ),
inference(rectify,[],[f7]) ).
thf(f64,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f63]) ).
thf(f67,plain,
( ( ^ [X0: $i > $o,X1: $i > $o,X2: $i] :
( ( X0 @ X2 )
& ( X1 @ X2 ) ) )
= mand ),
inference(rectify,[],[f6]) ).
thf(f68,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0 @ Y2 ) ) ) ),
inference(fool_elimination,[],[f67]) ).
thf(f74,plain,
( ( ^ [X0: ( mu > $i > $o ) > $i > $o,X1: $i] :
! [X2: mu > $i > $o] : ( X0 @ X2 @ X1 ) )
= mforall_indset ),
inference(rectify,[],[f12]) ).
thf(f75,plain,
( mforall_indset
= ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[],[f74]) ).
thf(f76,plain,
( mvalid
@ ( mforall_indset
@ ^ [X0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [X1: mu] : ( mnot @ ( X0 @ X1 ) ) )
@ ( mnot @ ( positive @ X0 ) ) ) ) ),
inference(rectify,[],[f23]) ).
thf(f77,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) )
@ ( mnot @ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(fool_elimination,[],[f76]) ).
thf(f78,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] : ( mimplies @ ( positive @ Y0 ) @ ( mbox @ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(cnf_transformation,[],[f39]) ).
thf(f80,plain,
( mforall_ind
= ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f50]) ).
thf(f83,plain,
( mforall_indset
= ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f75]) ).
thf(f85,plain,
( mequiv
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
= ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f34]) ).
thf(f87,plain,
( mvalid
= ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[],[f30]) ).
thf(f89,plain,
( mbox_generic
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( Y0 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) ) ) ),
inference(cnf_transformation,[],[f52]) ).
thf(f91,plain,
( essence
= ( ^ [Y0: mu > $i > $o,Y1: mu] :
( mand @ ( Y0 @ Y1 )
@ ( mforall_indset
@ ^ [Y2: mu > $i > $o] :
( mimplies @ ( Y2 @ Y1 )
@ ( mbox
@ ( mforall_ind
@ ^ [Y3: mu] : ( mimplies @ ( Y0 @ Y3 ) @ ( Y2 @ Y3 ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[],[f48]) ).
thf(f94,plain,
( mand
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f68]) ).
thf(f96,plain,
( mnot
= ( ^ [Y0: $i > $o,Y1: $i] :
~ ( Y0 @ Y1 ) ) ),
inference(cnf_transformation,[],[f36]) ).
thf(f100,plain,
( mbox
= ( mbox_generic @ rel ) ),
inference(cnf_transformation,[],[f53]) ).
thf(f101,plain,
( ( mvalid
@ ( mforall_indset
@ ^ [Y0: mu > $i > $o] :
( mequiv
@ ( positive
@ ^ [Y1: mu] : ( mnot @ ( Y0 @ Y1 ) ) )
@ ( mnot @ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(cnf_transformation,[],[f77]) ).
thf(f102,plain,
( mimplies
= ( ^ [Y0: $i > $o,Y1: $i > $o,Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[],[f64]) ).
thf(f103,plain,
( god
= ( ^ [Y0: mu] :
( mforall_indset
@ ^ [Y1: mu > $i > $o] : ( mimplies @ ( positive @ Y1 ) @ ( Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[],[f43]) ).
thf(f104,plain,
( ( ~ ( mvalid
@ ( mforall_ind
@ ^ [Y0: mu] : ( mimplies @ ( god @ Y0 ) @ ( essence @ god @ Y0 ) ) ) ) )
= $true ),
inference(cnf_transformation,[],[f32]) ).
thf(f107,plain,
( mbox
= ( ^ [Y0: $i > $i > $o,Y1: $i > $o,Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ~ ( Y0 @ Y2 @ Y3 )
| ( Y1 @ Y3 ) ) )
@ rel ) ),
inference(definition_unfolding,[],[f100,f89]) ).
thf(f109,plain,
( god
= ( ^ [Y0: mu] :
( ^ [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 ) )
@ ( positive @ Y1 )
@ ( Y1 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[],[f103,f83,f102]) ).
thf(f110,plain,
( essence
= ( ^ [Y0: mu > $i > $o,Y1: mu] :
( ^ [Y2: $i > $o,Y3: $i > $o,Y4: $i] :
( ( Y3 @ Y4 )
& ( Y2 @ Y4 ) )
@ ( Y0 @ Y1 )
@ ( ^ [Y2: ( mu > $i > $o ) > $i > $o,Y3: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y4: mu > $i > $o] : ( Y2 @ Y4 @ Y3 ) )
@ ^ [Y2: mu > $i > $o] :
( ^ [Y3: $i > $o,Y4: $i > $o,Y5: $i] :
( ( Y3 @ Y5 )
=> ( Y4 @ Y5 ) )
@ ( Y2 @ Y1 )
@ ( ^ [Y3: $i > $i > $o,Y4: $i > $o,Y5: $i] :
( !! @ $i
@ ^ [Y6: $i] :
( ~ ( Y3 @ Y5 @ Y6 )
| ( Y4 @ Y6 ) ) )
@ rel
@ ( ^ [Y3: mu > $i > $o,Y4: $i] :
( !! @ mu
@ ^ [Y5: mu] : ( Y3 @ Y5 @ Y4 ) )
@ ^ [Y3: mu] :
( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
=> ( Y5 @ Y6 ) )
@ ( Y0 @ Y3 )
@ ( Y2 @ Y3 ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[],[f91,f94,f83,f102,f107,f80,f102]) ).
thf(f111,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 @ Y3 ) )
@ ( positive @ Y0 )
@ ( ^ [Y1: $i > $i > $o,Y2: $i > $o,Y3: $i] :
( !! @ $i
@ ^ [Y4: $i] :
( ~ ( Y1 @ Y3 @ Y4 )
| ( Y2 @ Y4 ) ) )
@ rel
@ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f78,f87,f83,f102,f107]) ).
thf(f112,plain,
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: ( mu > $i > $o ) > $i > $o,Y1: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu > $i > $o] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
= ( Y2 @ Y3 ) )
@ ( positive
@ ^ [Y1: mu] :
( ^ [Y2: $i > $o,Y3: $i] :
~ ( Y2 @ Y3 )
@ ( Y0 @ Y1 ) ) )
@ ( ^ [Y1: $i > $o,Y2: $i] :
~ ( Y1 @ Y2 )
@ ( positive @ Y0 ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f101,f87,f83,f85,f96,f96]) ).
thf(f113,plain,
( ( ~ ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
@ ( ^ [Y0: mu > $i > $o,Y1: $i] :
( !! @ mu
@ ^ [Y2: mu] : ( Y0 @ Y2 @ Y1 ) )
@ ^ [Y0: mu] :
( ^ [Y1: $i > $o,Y2: $i > $o,Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 @ Y3 ) )
@ ( ^ [Y1: mu] :
( ^ [Y2: ( mu > $i > $o ) > $i > $o,Y3: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y4: mu > $i > $o] : ( Y2 @ Y4 @ Y3 ) )
@ ^ [Y2: mu > $i > $o] :
( ^ [Y3: $i > $o,Y4: $i > $o,Y5: $i] :
( ( Y3 @ Y5 )
=> ( Y4 @ Y5 ) )
@ ( positive @ Y2 )
@ ( Y2 @ Y1 ) ) )
@ Y0 )
@ ( ^ [Y1: mu > $i > $o,Y2: mu] :
( ^ [Y3: $i > $o,Y4: $i > $o,Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3 @ Y5 ) )
@ ( Y1 @ Y2 )
@ ( ^ [Y3: ( mu > $i > $o ) > $i > $o,Y4: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y5: mu > $i > $o] : ( Y3 @ Y5 @ Y4 ) )
@ ^ [Y3: mu > $i > $o] :
( ^ [Y4: $i > $o,Y5: $i > $o,Y6: $i] :
( ( Y4 @ Y6 )
=> ( Y5 @ Y6 ) )
@ ( Y3 @ Y2 )
@ ( ^ [Y4: $i > $i > $o,Y5: $i > $o,Y6: $i] :
( !! @ $i
@ ^ [Y7: $i] :
( ~ ( Y4 @ Y6 @ Y7 )
| ( Y5 @ Y7 ) ) )
@ rel
@ ( ^ [Y4: mu > $i > $o,Y5: $i] :
( !! @ mu
@ ^ [Y6: mu] : ( Y4 @ Y6 @ Y5 ) )
@ ^ [Y4: mu] :
( ^ [Y5: $i > $o,Y6: $i > $o,Y7: $i] :
( ( Y5 @ Y7 )
=> ( Y6 @ Y7 ) )
@ ( Y1 @ Y4 )
@ ( Y3 @ Y4 ) ) ) ) ) ) )
@ ^ [Y1: mu] :
( ^ [Y2: ( mu > $i > $o ) > $i > $o,Y3: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y4: mu > $i > $o] : ( Y2 @ Y4 @ Y3 ) )
@ ^ [Y2: mu > $i > $o] :
( ^ [Y3: $i > $o,Y4: $i > $o,Y5: $i] :
( ( Y3 @ Y5 )
=> ( Y4 @ Y5 ) )
@ ( positive @ Y2 )
@ ( Y2 @ Y1 ) ) )
@ Y0 ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f104,f87,f80,f102,f109,f110,f109]) ).
thf(f114,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,[],[f112]) ).
thf(f115,plain,
( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( !! @ mu
@ ^ [Y1: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) )
=> ( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( Y2 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel @ Y0 @ Y3 )
| ( !! @ mu
@ ^ [Y4: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y5: mu > $i > $o] :
( ( positive @ Y5 @ Y3 )
=> ( Y5 @ Y4 @ Y3 ) ) )
=> ( Y2 @ Y4 @ Y3 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) ) ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f113]) ).
thf(f116,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel @ Y0 @ Y2 )
| ( positive @ Y1 @ Y2 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f111]) ).
thf(f117,plain,
( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ mu
@ ^ [Y1: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) )
=> ( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( Y2 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel @ Y0 @ Y3 )
| ( !! @ mu
@ ^ [Y4: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y5: mu > $i > $o] :
( ( positive @ Y5 @ Y3 )
=> ( Y5 @ Y4 @ Y3 ) ) )
=> ( Y2 @ Y4 @ Y3 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) ) ) ) ) ) ),
inference(not_proxy_clausification,[],[f115]) ).
thf(f118,plain,
( $false
= ( ^ [Y0: $i] :
( !! @ mu
@ ^ [Y1: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) )
=> ( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( Y2 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ~ ( rel @ Y0 @ Y3 )
| ( !! @ mu
@ ^ [Y4: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y5: mu > $i > $o] :
( ( positive @ Y5 @ Y3 )
=> ( Y5 @ Y4 @ Y3 ) ) )
=> ( Y2 @ Y4 @ Y3 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) ) ) ) )
@ sK0 ) ),
inference(sigma_proxy_clausification,[],[f117]) ).
thf(f120,plain,
( $false
= ( !! @ mu
@ ^ [Y0: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK0 )
=> ( Y1 @ Y0 @ sK0 ) ) )
=> ( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( Y1 @ Y0 @ sK0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel @ sK0 @ Y2 )
| ( !! @ mu
@ ^ [Y3: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y4: mu > $i > $o] :
( ( positive @ Y4 @ Y2 )
=> ( Y4 @ Y3 @ Y2 ) ) )
=> ( Y1 @ Y3 @ Y2 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK0 )
=> ( Y1 @ Y0 @ sK0 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f118]) ).
thf(f122,plain,
( ( ^ [Y0: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK0 )
=> ( Y1 @ Y0 @ sK0 ) ) )
=> ( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( Y1 @ Y0 @ sK0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel @ sK0 @ Y2 )
| ( !! @ mu
@ ^ [Y3: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y4: mu > $i > $o] :
( ( positive @ Y4 @ Y2 )
=> ( Y4 @ Y3 @ Y2 ) ) )
=> ( Y1 @ Y3 @ Y2 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK0 )
=> ( Y1 @ Y0 @ sK0 ) ) ) ) )
@ sK1 )
= $false ),
inference(sigma_proxy_clausification,[],[f120]) ).
thf(f123,plain,
( ( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK0 )
=> ( Y0 @ sK1 @ sK0 ) ) )
=> ( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( Y0 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ sK0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] :
( ( positive @ Y3 @ Y1 )
=> ( Y3 @ Y2 @ Y1 ) ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK0 )
=> ( Y0 @ sK1 @ sK0 ) ) ) ) )
= $false ),
inference(beta-eta_normalization,[],[f122]) ).
thf(f124,plain,
( $false
= ( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( Y0 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ sK0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] :
( ( positive @ Y3 @ Y1 )
=> ( Y3 @ Y2 @ Y1 ) ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) )
& ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK0 )
=> ( Y0 @ sK1 @ sK0 ) ) ) ) ),
inference(imp_proxy_clausification,[],[f123]) ).
thf(f125,plain,
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK0 )
=> ( Y0 @ sK1 @ sK0 ) ) )
= $true ),
inference(imp_proxy_clausification,[],[f123]) ).
thf(f126,plain,
! [X1: mu > $i > $o] :
( ( ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK0 )
=> ( Y0 @ sK1 @ sK0 ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f125]) ).
thf(f127,plain,
! [X1: mu > $i > $o] :
( ( ( positive @ X1 @ sK0 )
=> ( X1 @ sK1 @ sK0 ) )
= $true ),
inference(beta-eta_normalization,[],[f126]) ).
thf(f129,plain,
( ( $false
= ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( Y0 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ sK0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] :
( ( positive @ Y3 @ Y1 )
=> ( Y3 @ Y2 @ Y1 ) ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ) )
| ( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK0 )
=> ( Y0 @ sK1 @ sK0 ) ) )
= $false ) ),
inference(and_proxy_clausification,[],[f124]) ).
thf(f130,plain,
( ( $false = $true )
| ( $false
= ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( Y0 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ sK0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] :
( ( positive @ Y3 @ Y1 )
=> ( Y3 @ Y2 @ Y1 ) ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ) ) ),
inference(forward_demodulation,[],[f129,f125]) ).
thf(f131,plain,
( $false
= ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( Y0 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ sK0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] :
( ( positive @ Y3 @ Y1 )
=> ( Y3 @ Y2 @ Y1 ) ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) ) ) ),
inference(trivial_inequality_removal,[],[f130]) ).
thf(f132,plain,
( $false
= ( ^ [Y0: mu > $i > $o] :
( ( Y0 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ sK0 @ Y1 )
| ( !! @ mu
@ ^ [Y2: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y3: mu > $i > $o] :
( ( positive @ Y3 @ Y1 )
=> ( Y3 @ Y2 @ Y1 ) ) )
=> ( Y0 @ Y2 @ Y1 ) ) ) ) ) )
@ sK2 ) ),
inference(sigma_proxy_clausification,[],[f131]) ).
thf(f133,plain,
( $false
= ( ( sK2 @ sK1 @ sK0 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ sK0 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) )
=> ( sK2 @ Y1 @ Y0 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f132]) ).
thf(f134,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ sK0 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) )
=> ( sK2 @ Y1 @ Y0 ) ) ) ) )
= $false ),
inference(imp_proxy_clausification,[],[f133]) ).
thf(f135,plain,
( ( sK2 @ sK1 @ sK0 )
= $true ),
inference(imp_proxy_clausification,[],[f133]) ).
thf(f140,plain,
( $false
= ( ^ [Y0: $i] :
( ~ ( rel @ sK0 @ Y0 )
| ( !! @ mu
@ ^ [Y1: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y2: mu > $i > $o] :
( ( positive @ Y2 @ Y0 )
=> ( Y2 @ Y1 @ Y0 ) ) )
=> ( sK2 @ Y1 @ Y0 ) ) ) )
@ sK3 ) ),
inference(sigma_proxy_clausification,[],[f134]) ).
thf(f141,plain,
( ( ~ ( rel @ sK0 @ sK3 )
| ( !! @ mu
@ ^ [Y0: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK3 )
=> ( Y1 @ Y0 @ sK3 ) ) )
=> ( sK2 @ Y0 @ sK3 ) ) ) )
= $false ),
inference(beta-eta_normalization,[],[f140]) ).
thf(f142,plain,
( $false
= ( !! @ mu
@ ^ [Y0: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK3 )
=> ( Y1 @ Y0 @ sK3 ) ) )
=> ( sK2 @ Y0 @ sK3 ) ) ) ),
inference(or_proxy_clausification,[],[f141]) ).
thf(f143,plain,
( $false
= ( ~ ( rel @ sK0 @ sK3 ) ) ),
inference(or_proxy_clausification,[],[f141]) ).
thf(f152,plain,
( ( ^ [Y0: mu] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ sK3 )
=> ( Y1 @ Y0 @ sK3 ) ) )
=> ( sK2 @ Y0 @ sK3 ) )
@ sK4 )
= $false ),
inference(sigma_proxy_clausification,[],[f142]) ).
thf(f153,plain,
( $false
= ( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK3 )
=> ( Y0 @ sK4 @ sK3 ) ) )
=> ( sK2 @ sK4 @ sK3 ) ) ),
inference(beta-eta_normalization,[],[f152]) ).
thf(f154,plain,
( $false
= ( sK2 @ sK4 @ sK3 ) ),
inference(imp_proxy_clausification,[],[f153]) ).
thf(f155,plain,
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK3 )
=> ( Y0 @ sK4 @ sK3 ) ) )
= $true ),
inference(imp_proxy_clausification,[],[f153]) ).
thf(f160,plain,
! [X1: mu > $i > $o] :
( ( ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ sK3 )
=> ( Y0 @ sK4 @ sK3 ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f155]) ).
thf(f161,plain,
! [X1: mu > $i > $o] :
( ( ( positive @ X1 @ sK3 )
=> ( X1 @ sK4 @ sK3 ) )
= $true ),
inference(beta-eta_normalization,[],[f160]) ).
thf(f163,plain,
( ( ( positive @ sK2 @ sK3 )
=> $false )
= $true ),
inference(constrained_superposition,[],[f161,f154]) ).
thf(f164,plain,
( ( ~ ( positive @ sK2 @ sK3 ) )
= $true ),
inference(boolean_simplification,[],[f163]) ).
thf(f165,plain,
( $false
= ( positive @ sK2 @ sK3 ) ),
inference(not_proxy_clausification,[],[f164]) ).
thf(f170,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,[],[f114]) ).
thf(f171,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,[],[f170]) ).
thf(f172,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,[],[f171]) ).
thf(f173,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ( positive
@ ^ [Y0: mu,Y1: $i] :
~ ( X2 @ Y0 @ Y1 )
@ X1 )
= ( ~ ( positive @ X2 @ X1 ) ) )
= $true ),
inference(beta-eta_normalization,[],[f172]) ).
thf(f174,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( positive
@ ^ [Y0: mu,Y1: $i] :
~ ( X2 @ Y0 @ Y1 )
@ X1 )
= ( ~ ( positive @ X2 @ X1 ) ) ),
inference(iff_proxy_clausification,[],[f173]) ).
thf(f177,plain,
! [X0: mu > $i > $o] :
( ( ~ ( positive @ X0 @ sK0 )
=> ( ^ [Y0: mu,Y1: $i] :
~ ( X0 @ Y0 @ Y1 )
@ sK1
@ sK0 ) )
= $true ),
inference(constrained_superposition,[],[f127,f174]) ).
thf(f180,plain,
! [X0: mu > $i > $o] :
( ( ~ ( positive @ X0 @ sK0 )
=> ~ ( X0 @ sK1 @ sK0 ) )
= $true ),
inference(beta-eta_normalization,[],[f177]) ).
thf(f184,plain,
( ( ~ ( positive @ sK2 @ sK0 )
=> ~ $true )
= $true ),
inference(constrained_superposition,[],[f180,f135]) ).
thf(f189,plain,
( ( ~ ( positive @ sK2 @ sK0 )
=> $false )
= $true ),
inference(boolean_simplification,[],[f184]) ).
thf(f190,plain,
( ( ~ ~ ( positive @ sK2 @ sK0 ) )
= $true ),
inference(boolean_simplification,[],[f189]) ).
thf(f191,plain,
( ( positive @ sK2 @ sK0 )
= $true ),
inference(boolean_simplification,[],[f190]) ).
thf(f214,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ ( mu > $i > $o )
@ ^ [Y1: mu > $i > $o] :
( ( positive @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ~ ( rel @ Y0 @ Y2 )
| ( positive @ Y1 @ Y2 ) ) ) ) )
@ X1 )
= $true ),
inference(pi_proxy_clausification,[],[f116]) ).
thf(f215,plain,
! [X1: $i] :
( ( !! @ ( mu > $i > $o )
@ ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ X1 @ Y1 )
| ( positive @ Y0 @ Y1 ) ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f214]) ).
thf(f216,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ^ [Y0: mu > $i > $o] :
( ( positive @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ~ ( rel @ X1 @ Y1 )
| ( positive @ Y0 @ Y1 ) ) ) )
@ X2 )
= $true ),
inference(pi_proxy_clausification,[],[f215]) ).
thf(f217,plain,
! [X2: mu > $i > $o,X1: $i] :
( ( ( positive @ X2 @ X1 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ X1 @ Y0 )
| ( positive @ X2 @ Y0 ) ) ) )
= $true ),
inference(beta-eta_normalization,[],[f216]) ).
thf(f219,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ sK0 @ Y0 )
| ( positive @ sK2 @ Y0 ) ) ) )
= $true ),
inference(constrained_superposition,[],[f217,f191]) ).
thf(f223,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ~ ( rel @ sK0 @ Y0 )
| ( positive @ sK2 @ Y0 ) ) )
= $true ),
inference(boolean_simplification,[],[f219]) ).
thf(f224,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ~ ( rel @ sK0 @ Y0 )
| ( positive @ sK2 @ Y0 ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f223]) ).
thf(f225,plain,
! [X1: $i] :
( ( ~ ( rel @ sK0 @ X1 )
| ( positive @ sK2 @ X1 ) )
= $true ),
inference(beta-eta_normalization,[],[f224]) ).
thf(f230,plain,
( ( $false
| ( positive @ sK2 @ sK3 ) )
= $true ),
inference(constrained_superposition,[],[f225,f143]) ).
thf(f231,plain,
( ( positive @ sK2 @ sK3 )
= $true ),
inference(boolean_simplification,[],[f230]) ).
thf(f236,plain,
$false = $true,
inference(forward_demodulation,[],[f231,f165]) ).
thf(f237,plain,
$false,
inference(trivial_inequality_removal,[],[f236]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : PHI004^2 : TPTP v9.3.1. Released v6.1.0.
% 0.00/0.06 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.08/0.18 % Computer : n026.cluster.edu
% 0.08/0.18 % Model : x86_64 x86_64
% 0.08/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.18 % Memory : 8046.5625MB
% 0.08/0.18 % OS : Linux 6.8.0-71-generic
% 0.08/0.18 % CPULimit : 300
% 0.08/0.18 % WCLimit : 300
% 0.08/0.18 % DateTime : Tue Sep 29 13:20:12 UTC 2026
% 0.08/0.18 % CPUTime :
% 0.08/0.18 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.08/0.21 Running higher-order theorem proving
% 0.08/0.23 Running: /export/starexec/sandbox2/solver/bin/vampire-ho --input_syntax tptp --output_axiom_names on --mode casc -m 16384 --cores 7 -t 300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.21/0.35 % (567479)Detected a higher-order problem, will run a greedy HOL sequence.
% 0.21/0.35 % (567484)lrs+10_40_drc=off:e2e=on:si=on:uwa=one_side_interpreted:random_seed=1246831496:s2a=on:i=87:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/87Mi)
% 0.21/0.35 % (567486)lrs+10_1_cnfonf=off:si=on:uwa=one_side_interpreted:random_seed=252107204:i=3:rtra=on:inj=on:ntd=on_2999 on theBenchmark for (2999ds/3Mi)
% 0.21/0.35 % (567485)lrs+10_16_si=on:nwc=1.5:random_seed=3182025761:i=18:kws=arity_squared:rtra=on:fe=abstraction:ntd=on_2999 on theBenchmark for (2999ds/18Mi)
% 0.21/0.35 % (567486)Instruction limit reached!
% 0.21/0.35 % (567486)------------------------------
% 0.21/0.35 % (567486)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567486)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567486)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567486)Termination reason: Instruction limit
% 0.21/0.35 % (567486)Termination phase: Function definition elimination
% 0.21/0.35 % (567486)Time elapsed: 0.002 s
% 0.21/0.35 % (567486)Peak memory usage: 10 MB
% 0.21/0.35 % (567486)Instructions burned: 4 (million)
% 0.21/0.35 % (567485)Instruction limit reached!
% 0.21/0.35 % (567485)------------------------------
% 0.21/0.35 % (567485)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567485)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567485)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567485)Termination reason: Instruction limit
% 0.21/0.35 % (567485)Termination phase: Saturation
% 0.21/0.35 % (567485)Time elapsed: 0.010 s
% 0.21/0.35 % (567485)Peak memory usage: 12 MB
% 0.21/0.35 % (567485)Instructions burned: 19 (million)
% 0.21/0.35 % (567487)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=3449304893: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)
% 0.21/0.35 % (567484)Instruction limit reached!
% 0.21/0.35 % (567484)------------------------------
% 0.21/0.35 % (567484)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567484)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567484)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567484)Termination reason: Instruction limit
% 0.21/0.35 % (567484)Termination phase: Saturation
% 0.21/0.35 % (567484)Time elapsed: 0.021 s
% 0.21/0.35 % (567484)Peak memory usage: 12 MB
% 0.21/0.35 % (567484)Instructions burned: 87 (million)
% 0.21/0.35 % (567494)dis+10_1024_sil=128000:si=on:sp=unary_first:urr=on:uwa=all:fd=off:random_seed=3734159828:i=2:hud=10:rtra=on_2999 on theBenchmark for (2999ds/2Mi)
% 0.21/0.35 % (567497)WARNING Broken Constraint: if forward_subsumption_demodulation_max_matches(5) has been set then forward_subsumption_demodulation(off) is equal to on
% 0.21/0.35 % (567494)Instruction limit reached!
% 0.21/0.35 % (567494)------------------------------
% 0.21/0.35 % (567494)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567494)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567494)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567494)Termination reason: Instruction limit
% 0.21/0.35 % (567494)Termination phase: Function definition elimination
% 0.21/0.35 % (567494)Time elapsed: 0.002 s
% 0.21/0.35 % (567494)Peak memory usage: 10 MB
% 0.21/0.35 % (567494)Instructions burned: 4 (million)
% 0.21/0.35 % (567497)dis+21_1_to=kbo:sil=128000:plsq=on:plsqc=1:cnfonf=lazy_gen:si=on:plsqr=64,1:uwa=hol:random_seed=113783733:uwa_fpi=on:i=7:fgj=on:hud=10:fsr=off:rtra=on:rawr=on:fsdmm=5_2999 on theBenchmark for (2999ds/7Mi)
% 0.21/0.35 % (567497)Instruction limit reached!
% 0.21/0.35 % (567497)------------------------------
% 0.21/0.35 % (567497)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567497)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567497)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567497)Termination reason: Instruction limit
% 0.21/0.35 % (567497)Termination phase: Saturation
% 0.21/0.35 % (567497)Time elapsed: 0.002 s
% 0.21/0.35 % (567497)Peak memory usage: 11 MB
% 0.21/0.35 % (567497)Instructions burned: 8 (million)
% 0.21/0.35 % (567495)lrs+1010_2:3_cha=on:si=on:uwa=off:nwc=1:random_seed=3903495880:i=5:fgj=on:av=off:rtra=on:fe=axiom:ntd=on_2999 on theBenchmark for (2999ds/5Mi)
% 0.21/0.35 % (567490)WARNING Broken Constraint: if ho_split_queue_ratios(1,8) has been set then ho_split_queue(off) is equal to on
% 0.21/0.35 % (567490)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
% 0.21/0.35 % (567495)Instruction limit reached!
% 0.21/0.35 % (567495)------------------------------
% 0.21/0.35 % (567495)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567495)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567495)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567495)Termination reason: Instruction limit
% 0.21/0.35 % (567495)Termination phase: Property scanning
% 0.21/0.35 % (567495)Time elapsed: 0.003 s
% 0.21/0.35 % (567495)Peak memory usage: 10 MB
% 0.21/0.35 % (567495)Instructions burned: 5 (million)
% 0.21/0.35 % (567501)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
% 0.21/0.35 % (567501)WARNING Broken Constraint: if forward_subsumption_demodulation_max_matches(1) has been set then forward_subsumption_demodulation(off) is equal to on
% 0.21/0.35 % (567501)ott+1010_64_tgt=ground:cnfonf=lazy_simp:si=on:lma=off:spb=goal:lcm=predicate:random_seed=454046323:i=28:s2at=5:piset=not:hud=10:bd=all:av=off:rtra=on:ixr=off:fsdmm=1_2999 on theBenchmark for (2999ds/28Mi)
% 0.21/0.35 % (567489)lrs+10_1_to=lpo:sil=128000:e2e=on:si=on:random_seed=2832998170:s2a=on:i=75:s2at=3:aac=none:bd=preordered:rtra=on:fe=abstraction_2999 on theBenchmark for (2999ds/75Mi)
% 0.21/0.35 % (567490)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=4050747687:hsqr=1,8:i=157:s2at=5:add=on:nm=2:rtra=on_2999 on theBenchmark for (2999ds/157Mi)
% 0.21/0.35 % (567499)lrs+10_1_sil=128000:si=on:urr=on:slsqc=1:slsq=on:random_seed=920407176:i=12:s2at=2:kws=inv_frequency:bd=all:rtra=on_2999 on theBenchmark for (2999ds/12Mi)
% 0.21/0.35 % (567488)dis+21_4_fde=none:e2e=on:si=on:uwa=off:foolp=on:random_seed=1014189935:i=24:av=off:rtra=on_2999 on theBenchmark for (2999ds/24Mi)
% 0.21/0.35 % (567501)Instruction limit reached!
% 0.21/0.35 % (567501)------------------------------
% 0.21/0.35 % (567501)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567501)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567501)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567501)Termination reason: Instruction limit
% 0.21/0.35 % (567501)Termination phase: Saturation
% 0.21/0.35 % (567501)Time elapsed: 0.008 s
% 0.21/0.35 % (567501)Peak memory usage: 12 MB
% 0.21/0.35 % (567501)Instructions burned: 31 (million)
% 0.21/0.35 % (567499)Instruction limit reached!
% 0.21/0.35 % (567499)------------------------------
% 0.21/0.35 % (567499)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567499)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567499)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567499)Termination reason: Instruction limit
% 0.21/0.35 % (567499)Termination phase: Saturation
% 0.21/0.35 % (567499)Time elapsed: 0.006 s
% 0.21/0.35 % (567499)Peak memory usage: 12 MB
% 0.21/0.35 % (567499)Instructions burned: 12 (million)
% 0.21/0.35 % (567503)lrs+1002_64_sil=128000:plsq=on:plsqc=1:cnfonf=lazy_gen:si=on:sp=occurrence:lma=off:plsqr=32,1:uwa=interpreted_only:random_seed=3836884258:i=86:piset=equals:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/86Mi)
% 0.21/0.35 % (567509)lrs+10_1_si=on:cs=on:random_seed=1185634664:i=8:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/8Mi)
% 0.21/0.35 % (567509)Instruction limit reached!
% 0.21/0.35 % (567509)------------------------------
% 0.21/0.35 % (567509)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567509)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567509)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567509)Termination reason: Instruction limit
% 0.21/0.35 % (567509)Termination phase: Saturation
% 0.21/0.35 % (567509)Time elapsed: 0.002 s
% 0.21/0.35 % (567509)Peak memory usage: 11 MB
% 0.21/0.35 % (567509)Instructions burned: 9 (million)
% 0.21/0.35 % (567510)WARNING Broken Constraint: if positive_literal_split_queue_ratios(1,32) has been set then positive_literal_split_queue(off) is equal to on
% 0.21/0.35 % (567503) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-567479-567503"...
% 0.21/0.35 % (567510)ott+1002_20_sil=128000:cnfonf=lazy_not_gen_be_off:si=on:sp=unary_frequency:plsqr=1,32:bce=on:uwa=interpreted_only:foolp=on:random_seed=1890323494:i=2:add=on:rtra=on_2999 on theBenchmark for (2999ds/2Mi)
% 0.21/0.35 % (567513)lrs+1002_3:1_sil=128000:e2e=on:si=on:urr=on:uwa=one_side_constant:nwc=1.5:random_seed=1724833611:i=38:bd=all:rtra=on:amm=off:ss=axioms:ntd=on_2999 on theBenchmark for (2999ds/38Mi)
% 0.21/0.35 % (567503)...printing done.
% 0.21/0.35 % (567513)Refutation not found, incomplete strategy
% 0.21/0.35 % (567513)------------------------------
% 0.21/0.35 % (567513)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567513)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567513)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567513)Termination reason: Refutation not found, incomplete strategy
% 0.21/0.35 % (567513)Time elapsed: 0.002 s
% 0.21/0.35 % (567513)Peak memory usage: 12 MB
% 0.21/0.35 % (567513)Instructions burned: 6 (million)
% 0.21/0.35 % (567510)Instruction limit reached!
% 0.21/0.35 % (567510)------------------------------
% 0.21/0.35 % (567510)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567510)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567510)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567510)Termination reason: Instruction limit
% 0.21/0.35 % (567510)Termination phase: Saturation
% 0.21/0.35 % (567510)Time elapsed: 0.003 s
% 0.21/0.35 % (567510)Peak memory usage: 11 MB
% 0.21/0.35 % (567510)Instructions burned: 6 (million)
% 0.21/0.35 % (567513)------------------------------
% 0.21/0.35 % (567513)------------------------------
% 0.21/0.35 % (567503)Refutation found. Thanks to Tanya!
% 0.21/0.35 % SZS status Theorem for theBenchmark
% 0.21/0.35 % SZS output start Proof for theBenchmark
% See solution above
% 0.21/0.35 % (567503)------------------------------
% 0.21/0.35 % (567503)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.21/0.35 % (567503)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.21/0.35 % (567503)CaDiCaL version: 2.1.3
% 0.21/0.35 % (567503)Termination reason: Refutation
% 0.21/0.35 % (567503)Time elapsed: 0.016 s
% 0.21/0.35 % (567503)Peak memory usage: 12 MB
% 0.21/0.35 % (567503)Instructions burned: 30 (million)
% 0.21/0.35 % (567479)Success in time 0.112 s
% 0.21/0.35 % Vampire exiting
%------------------------------------------------------------------------------