%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET009-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:45 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.36s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 44
% Syntax : Number of formulae : 97 ( 24 unt; 0 typ; 0 def)
% Number of atoms : 528 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 749 ( 346 ~; 352 |; 0 &)
% ( 51 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 28 ( 28 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 6 ( 6 usr; 5 con; 0-2 aty)
% Number of variables : 335 ( 301 !; 0 ?; 335 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(b_type,type,
b: $i ).
tff(member_of_1_not_of_2_type,type,
member_of_1_not_of_2: ( $i * $i ) > $i ).
tff(bDd_type,type,
bDd: $i ).
tff(bDa_type,type,
bDa: $i ).
tff(d_type,type,
d: $i ).
tff(a_type,type,
a: $i ).
tff(subset_type,type,
subset: ( $i * $i ) > $o ).
tff(difference_type,type,
difference: ( $i * $i * $i ) > $o ).
tff(1,plain,
( ~ subset(bDa,bDd)
<=> ~ subset(bDa,bDd) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
~ subset(bDa,bDd),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_bDa_is_a_subset_of_bDd) ).
tff(3,plain,
~ subset(bDa,bDd),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Subset: $i,Superset: $i] :
refl(( ( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
<=> ( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
<=> ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
<=> ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET001-0.ax',subsets_axiom1) ).
tff(8,plain,
! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
| subset(bDa,bDd)
| member(member_of_1_not_of_2(bDa,bDd),bDa) )
<=> ( ~ ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
| subset(bDa,bDd)
| member(member_of_1_not_of_2(bDa,bDd),bDa) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
| subset(bDa,bDd)
| member(member_of_1_not_of_2(bDa,bDd),bDa) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [Subset: $i,Superset: $i] :
( subset(Subset,Superset)
| member(member_of_1_not_of_2(Subset,Superset),Subset) )
| subset(bDa,bDd)
| member(member_of_1_not_of_2(bDa,bDd),bDa) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
member(member_of_1_not_of_2(bDa,bDd),bDa),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
( difference(b,a,bDa)
<=> difference(b,a,bDa) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,axiom,
difference(b,a,bDa),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_minus_a) ).
tff(17,plain,
difference(b,a,bDa),
inference(modus_ponens,[status(thm)],[16,15]) ).
tff(18,plain,
^ [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
refl(( ( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
<=> ( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ) )),
inference(bind,[status(th)],[]) ).
tff(19,plain,
( ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
<=> ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,plain,
( ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
<=> ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(21,plain,
^ [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
trans(monotonicity(rewrite(( ( ~ member(Element,Set1)
| ~ member(Element,Set2) )
<=> ( ~ member(Element,Set2)
| ~ member(Element,Set1) ) )),
( ( ~ member(Element,Set1)
| ~ member(Element,Set2)
| ~ difference(A_set,Set1,Set2) )
<=> ( ~ member(Element,Set2)
| ~ member(Element,Set1)
| ~ difference(A_set,Set1,Set2) ) )),
rewrite(( ( ~ member(Element,Set2)
| ~ member(Element,Set1)
| ~ difference(A_set,Set1,Set2) )
<=> ( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ) )),
( ( ~ member(Element,Set1)
| ~ member(Element,Set2)
| ~ difference(A_set,Set1,Set2) )
<=> ( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ member(Element,Set1)
| ~ member(Element,Set2)
| ~ difference(A_set,Set1,Set2) )
<=> ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,axiom,
! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ member(Element,Set1)
| ~ member(Element,Set2)
| ~ difference(A_set,Set1,Set2) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET001-3.ax',not_member_of_difference) ).
tff(24,plain,
! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) ),
inference(modus_ponens,[status(thm)],[26,19]) ).
tff(28,plain,
( ( ~ ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),a) )
<=> ( ~ ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
( ~ ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(30,plain,
( ~ ! [Set1: $i,Set2: $i,A_set: $i,Element: $i] :
( ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set2)
| ~ member(Element,Set1) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),a) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
~ member(member_of_1_not_of_2(bDa,bDd),a),
inference(unit_resolution,[status(thm)],[30,27,17,14]) ).
tff(32,plain,
( subset(d,a)
<=> subset(d,a) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
subset(d,a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d_is_a_subset_of_a) ).
tff(34,plain,
subset(d,a),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
^ [Subset: $i,Element: $i,Superset: $i] :
refl(( ( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
<=> ( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ) )),
inference(bind,[status(th)],[]) ).
tff(36,plain,
( ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
<=> ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ) ),
inference(quant_intro,[status(thm)],[35]) ).
tff(37,plain,
( ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
<=> ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(38,plain,
^ [Subset: $i,Element: $i,Superset: $i] :
rewrite(( ( ~ member(Element,Subset)
| ~ subset(Subset,Superset)
| member(Element,Superset) )
<=> ( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [Subset: $i,Element: $i,Superset: $i] :
( ~ member(Element,Subset)
| ~ subset(Subset,Superset)
| member(Element,Superset) )
<=> ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,axiom,
! [Subset: $i,Element: $i,Superset: $i] :
( ~ member(Element,Subset)
| ~ subset(Subset,Superset)
| member(Element,Superset) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET001-0.ax',membership_in_subsets) ).
tff(41,plain,
! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ),
inference(modus_ponens,[status(thm)],[41,37]) ).
tff(43,plain,
! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) ),
inference(modus_ponens,[status(thm)],[43,36]) ).
tff(45,plain,
( ( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| ~ subset(d,a)
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) )
<=> ( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| ~ subset(d,a)
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(46,plain,
( ( member(member_of_1_not_of_2(bDa,bDd),a)
| ~ subset(d,a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) )
<=> ( ~ subset(d,a)
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,plain,
( ( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ subset(d,a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) )
<=> ( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| ~ subset(d,a)
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) ) ),
inference(monotonicity,[status(thm)],[46]) ).
tff(48,plain,
( ( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ subset(d,a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) )
<=> ( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| ~ subset(d,a)
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) ) ),
inference(transitivity,[status(thm)],[47,45]) ).
tff(49,plain,
( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ subset(d,a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [Subset: $i,Element: $i,Superset: $i] :
( member(Element,Superset)
| ~ subset(Subset,Superset)
| ~ member(Element,Subset) )
| ~ subset(d,a)
| member(member_of_1_not_of_2(bDa,bDd),a)
| ~ member(member_of_1_not_of_2(bDa,bDd),d) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
~ member(member_of_1_not_of_2(bDa,bDd),d),
inference(unit_resolution,[status(thm)],[50,44,34,31]) ).
tff(52,plain,
^ [Subset: $i,Superset: $i] :
refl(( ( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
<=> ( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ) )),
inference(bind,[status(th)],[]) ).
tff(53,plain,
( ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
<=> ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ) ),
inference(quant_intro,[status(thm)],[52]) ).
tff(54,plain,
( ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
<=> ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,axiom,
! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET001-0.ax',subsets_axiom2) ).
tff(56,plain,
! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ),
inference(skolemize,[status(sab)],[56]) ).
tff(58,plain,
! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) ),
inference(modus_ponens,[status(thm)],[57,53]) ).
tff(59,plain,
( ( ~ ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
| ~ member(member_of_1_not_of_2(bDa,bDd),bDd)
| subset(bDa,bDd) )
<=> ( ~ ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
| ~ member(member_of_1_not_of_2(bDa,bDd),bDd)
| subset(bDa,bDd) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(60,plain,
( ~ ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
| ~ member(member_of_1_not_of_2(bDa,bDd),bDd)
| subset(bDa,bDd) ),
inference(quant_inst,[status(thm)],[]) ).
tff(61,plain,
( ~ ! [Subset: $i,Superset: $i] :
( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
| subset(Subset,Superset) )
| ~ member(member_of_1_not_of_2(bDa,bDd),bDd)
| subset(bDa,bDd) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
~ member(member_of_1_not_of_2(bDa,bDd),bDd),
inference(unit_resolution,[status(thm)],[61,58,3]) ).
tff(63,plain,
( difference(b,d,bDd)
<=> difference(b,d,bDd) ),
inference(rewrite,[status(thm)],[]) ).
tff(64,axiom,
difference(b,d,bDd),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_minus_d) ).
tff(65,plain,
difference(b,d,bDd),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
^ [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
refl(( ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
<=> ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ) )),
inference(bind,[status(th)],[]) ).
tff(67,plain,
( ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
<=> ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
<=> ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,plain,
^ [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
trans(monotonicity(rewrite(( ( ~ member(Element,Set1)
| ~ difference(Set1,Set2,Difference)
| member(Element,Difference) )
<=> ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| ~ member(Element,Set1) ) )),
( ( ~ member(Element,Set1)
| ~ difference(Set1,Set2,Difference)
| member(Element,Difference)
| member(Element,Set2) )
<=> ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| ~ member(Element,Set1)
| member(Element,Set2) ) )),
rewrite(( ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| ~ member(Element,Set1)
| member(Element,Set2) )
<=> ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ) )),
( ( ~ member(Element,Set1)
| ~ difference(Set1,Set2,Difference)
| member(Element,Difference)
| member(Element,Set2) )
<=> ( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ) )),
inference(bind,[status(th)],[]) ).
tff(70,plain,
( ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( ~ member(Element,Set1)
| ~ difference(Set1,Set2,Difference)
| member(Element,Difference)
| member(Element,Set2) )
<=> ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,axiom,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( ~ member(Element,Set1)
| ~ difference(Set1,Set2,Difference)
| member(Element,Difference)
| member(Element,Set2) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET001-3.ax',member_of_difference_or_set2) ).
tff(72,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ),
inference(modus_ponens,[status(thm)],[71,70]) ).
tff(73,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ),
inference(skolemize,[status(sab)],[73]) ).
tff(75,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) ),
inference(modus_ponens,[status(thm)],[74,67]) ).
tff(76,plain,
( ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
| member(member_of_1_not_of_2(bDa,bDd),bDd)
| ~ difference(b,d,bDd)
| member(member_of_1_not_of_2(bDa,bDd),d)
| ~ member(member_of_1_not_of_2(bDa,bDd),b) )
<=> ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
| member(member_of_1_not_of_2(bDa,bDd),bDd)
| ~ difference(b,d,bDd)
| member(member_of_1_not_of_2(bDa,bDd),d)
| ~ member(member_of_1_not_of_2(bDa,bDd),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(77,plain,
( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
| member(member_of_1_not_of_2(bDa,bDd),bDd)
| ~ difference(b,d,bDd)
| member(member_of_1_not_of_2(bDa,bDd),d)
| ~ member(member_of_1_not_of_2(bDa,bDd),b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(78,plain,
( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set2)
| ~ member(Element,Set1) )
| member(member_of_1_not_of_2(bDa,bDd),bDd)
| ~ difference(b,d,bDd)
| member(member_of_1_not_of_2(bDa,bDd),d)
| ~ member(member_of_1_not_of_2(bDa,bDd),b) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
( member(member_of_1_not_of_2(bDa,bDd),d)
| ~ member(member_of_1_not_of_2(bDa,bDd),b) ),
inference(unit_resolution,[status(thm)],[78,75,65,62]) ).
tff(80,plain,
~ member(member_of_1_not_of_2(bDa,bDd),b),
inference(unit_resolution,[status(thm)],[79,51]) ).
tff(81,plain,
^ [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
refl(( ( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
<=> ( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) )),
inference(bind,[status(th)],[]) ).
tff(82,plain,
( ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
<=> ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) ),
inference(quant_intro,[status(thm)],[81]) ).
tff(83,plain,
( ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
<=> ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,plain,
^ [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
trans(monotonicity(rewrite(( ( ~ difference(Set1,Set2,Difference)
| ~ member(Element,Difference) )
<=> ( ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) )),
( ( ~ difference(Set1,Set2,Difference)
| ~ member(Element,Difference)
| member(Element,Set1) )
<=> ( ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set1) ) )),
rewrite(( ( ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference)
| member(Element,Set1) )
<=> ( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) )),
( ( ~ difference(Set1,Set2,Difference)
| ~ member(Element,Difference)
| member(Element,Set1) )
<=> ( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) )),
inference(bind,[status(th)],[]) ).
tff(85,plain,
( ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( ~ difference(Set1,Set2,Difference)
| ~ member(Element,Difference)
| member(Element,Set1) )
<=> ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ) ),
inference(quant_intro,[status(thm)],[84]) ).
tff(86,axiom,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( ~ difference(Set1,Set2,Difference)
| ~ member(Element,Difference)
| member(Element,Set1) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET001-3.ax',member_of_difference) ).
tff(87,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ),
inference(modus_ponens,[status(thm)],[87,83]) ).
tff(89,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ),
inference(skolemize,[status(sab)],[88]) ).
tff(90,plain,
! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) ),
inference(modus_ponens,[status(thm)],[89,82]) ).
tff(91,plain,
( ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| member(member_of_1_not_of_2(bDa,bDd),b) )
<=> ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| member(member_of_1_not_of_2(bDa,bDd),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(92,plain,
( ( member(member_of_1_not_of_2(bDa,bDd),b)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ difference(b,a,bDa) )
<=> ( ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| member(member_of_1_not_of_2(bDa,bDd),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,plain,
( ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| member(member_of_1_not_of_2(bDa,bDd),b)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ difference(b,a,bDa) )
<=> ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| member(member_of_1_not_of_2(bDa,bDd),b) ) ),
inference(monotonicity,[status(thm)],[92]) ).
tff(94,plain,
( ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| member(member_of_1_not_of_2(bDa,bDd),b)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ difference(b,a,bDa) )
<=> ( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| member(member_of_1_not_of_2(bDa,bDd),b) ) ),
inference(transitivity,[status(thm)],[93,91]) ).
tff(95,plain,
( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| member(member_of_1_not_of_2(bDa,bDd),b)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| ~ difference(b,a,bDa) ),
inference(quant_inst,[status(thm)],[]) ).
tff(96,plain,
( ~ ! [Set1: $i,Set2: $i,Difference: $i,Element: $i] :
( member(Element,Set1)
| ~ member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )
| ~ difference(b,a,bDa)
| ~ member(member_of_1_not_of_2(bDa,bDd),bDa)
| member(member_of_1_not_of_2(bDa,bDd),b) ),
inference(modus_ponens,[status(thm)],[95,94]) ).
tff(97,plain,
$false,
inference(unit_resolution,[status(thm)],[96,90,17,14,80]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.01/0.12 % Problem : SET009-1 : TPTP v9.0.0. Released v1.0.0.
% 0.01/0.12 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n005.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 04:00:50 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.36 % SZS status Unsatisfiable
% 0.19/0.36 % SZS output start Proof
% See solution above
% 0.19/0.38 % E exiting
%------------------------------------------------------------------------------