%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWX228-1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n024.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 : Tue May 5 07:08:19 PM UTC 2026
% Result : Unsatisfiable 0.06s 0.30s
% Output : Proof 0.06s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 57
% Syntax : Number of formulae : 154 ( 105 unt; 0 typ; 0 def)
% Number of atoms : 245 ( 231 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 129 ( 51 ~; 38 |; 0 &)
% ( 40 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 3 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of FOOLs : 13 ( 13 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 15 ( 15 usr; 4 con; 0-4 aty)
% Number of variables : 262 ( 238 !; 0 ?; 262 :)
% Comments :
%------------------------------------------------------------------------------
tff(btrue_type,type,
btrue: $i ).
tff(eq3_type,type,
eq3: ( $i * $i ) > $i ).
tff(bfalse_type,type,
bfalse: $i ).
tff(prop_union_comm_type,type,
prop_union_comm: ( $i * $i ) > $i ).
tff(cons_type,type,
cons: ( $i * $i ) > $i ).
tff(nil_type,type,
nil: $i ).
tff(s_type,type,
s: $i > $i ).
tff(z_type,type,
z: $i ).
tff(eq_type,type,
eq: ( $i * $i ) > $i ).
tff(union_type,type,
union: ( $i * $i ) > $i ).
tff(eq2_type,type,
eq2: ( $i * $i ) > $i ).
tff(aux_type,type,
aux: ( $i * $i * $i * $i ) > $i ).
tff(elem_type,type,
elem: ( $i * $i ) > $i ).
tff(barbar_type,type,
barbar: ( $i * $i ) > $i ).
tff(eqNat_type,type,
eqNat: ( $i * $i ) > $i ).
tff(1,plain,
^ [X: $i] :
refl(( ( eq3(X,X) = btrue )
<=> ( eq3(X,X) = btrue ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( eq3(X,X) = btrue )
<=> ! [X: $i] : ( eq3(X,X) = btrue ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( eq3(X,X) = btrue )
<=> ! [X: $i] : ( eq3(X,X) = btrue ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( eq3(X,X) = btrue ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_020) ).
tff(5,plain,
! [X: $i] : ( eq3(X,X) = btrue ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( eq3(X,X) = btrue ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( eq3(X,X) = btrue ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( eq3(X,X) = btrue )
| ( eq3(bfalse,bfalse) = btrue ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
eq3(bfalse,bfalse) = btrue,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( ( eq2(z,s(X)) = bfalse )
<=> ( eq2(z,s(X)) = bfalse ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : ( eq2(z,s(X)) = bfalse )
<=> ! [X: $i] : ( eq2(z,s(X)) = bfalse ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : ( eq2(z,s(X)) = bfalse )
<=> ! [X: $i] : ( eq2(z,s(X)) = bfalse ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : ( eq2(z,s(X)) = bfalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_017) ).
tff(14,plain,
! [X: $i] : ( eq2(z,s(X)) = bfalse ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : ( eq2(z,s(X)) = bfalse ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : ( eq2(z,s(X)) = bfalse ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : ( eq2(z,s(X)) = bfalse )
| ( eq2(z,s(z)) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
eq2(z,s(z)) = bfalse,
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [Z: $i,Y: $i,X: $i,X2: $i] :
refl(( ( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
<=> ( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ) )),
inference(bind,[status(th)],[]) ).
tff(20,plain,
( ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
<=> ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
<=> ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_021) ).
tff(23,plain,
! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
| ( eq2(z,s(z)) != bfalse )
| ( eq(cons(z,cons(s(z),nil)),cons(s(z),union(nil,cons(z,nil)))) = bfalse ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
| ( eq2(z,s(z)) != bfalse )
| ( eq(cons(z,cons(s(z),nil)),cons(s(z),union(nil,cons(z,nil)))) = bfalse ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
| ( eq2(z,s(z)) != bfalse )
| ( eq(cons(z,cons(s(z),nil)),cons(s(z),union(nil,cons(z,nil)))) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
( ~ ! [Z: $i,Y: $i,X: $i,X2: $i] :
( ( eq2(X,Z) != bfalse )
| ( eq(cons(X,Y),cons(Z,X2)) = bfalse ) )
| ( eq2(z,s(z)) != bfalse )
| ( eq(cons(z,cons(s(z),nil)),cons(s(z),union(nil,cons(z,nil)))) = bfalse ) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
eq(cons(z,cons(s(z),nil)),cons(s(z),union(nil,cons(z,nil)))) = bfalse,
inference(unit_resolution,[status(thm)],[28,25,18]) ).
tff(30,plain,
^ [Xs: $i,Z: $i,Y: $i] :
refl(( ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) )
<=> ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) )
<=> ! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) )
<=> ! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_001) ).
tff(34,plain,
! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ~ ! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) )
| ( aux(cons(z,nil),s(z),nil,bfalse) = cons(s(z),union(nil,cons(z,nil))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,plain,
aux(cons(z,nil),s(z),nil,bfalse) = cons(s(z),union(nil,cons(z,nil))),
inference(unit_resolution,[status(thm)],[37,36]) ).
tff(39,plain,
^ [Xs: $i,Z: $i,X: $i] :
refl(( ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) )
<=> ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) )
<=> ! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) )
<=> ! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_009) ).
tff(43,plain,
! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
( ~ ! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) )
| ( elem(s(z),cons(z,nil)) = barbar(eqNat(s(z),z),elem(s(z),nil)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
elem(s(z),cons(z,nil)) = barbar(eqNat(s(z),z),elem(s(z),nil)),
inference(unit_resolution,[status(thm)],[46,45]) ).
tff(48,plain,
barbar(eqNat(s(z),z),elem(s(z),nil)) = elem(s(z),cons(z,nil)),
inference(symmetry,[status(thm)],[47]) ).
tff(49,plain,
^ [X: $i] :
refl(( ( elem(X,nil) = bfalse )
<=> ( elem(X,nil) = bfalse ) )),
inference(bind,[status(th)],[]) ).
tff(50,plain,
( ! [X: $i] : ( elem(X,nil) = bfalse )
<=> ! [X: $i] : ( elem(X,nil) = bfalse ) ),
inference(quant_intro,[status(thm)],[49]) ).
tff(51,plain,
( ! [X: $i] : ( elem(X,nil) = bfalse )
<=> ! [X: $i] : ( elem(X,nil) = bfalse ) ),
inference(rewrite,[status(thm)],[]) ).
tff(52,axiom,
! [X: $i] : ( elem(X,nil) = bfalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_008) ).
tff(53,plain,
! [X: $i] : ( elem(X,nil) = bfalse ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
! [X: $i] : ( elem(X,nil) = bfalse ),
inference(skolemize,[status(sab)],[53]) ).
tff(55,plain,
! [X: $i] : ( elem(X,nil) = bfalse ),
inference(modus_ponens,[status(thm)],[54,50]) ).
tff(56,plain,
( ~ ! [X: $i] : ( elem(X,nil) = bfalse )
| ( elem(z,nil) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(57,plain,
elem(z,nil) = bfalse,
inference(unit_resolution,[status(thm)],[56,55]) ).
tff(58,plain,
bfalse = elem(z,nil),
inference(symmetry,[status(thm)],[57]) ).
tff(59,plain,
( ~ ! [X: $i] : ( elem(X,nil) = bfalse )
| ( elem(s(z),nil) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(60,plain,
elem(s(z),nil) = bfalse,
inference(unit_resolution,[status(thm)],[59,55]) ).
tff(61,plain,
elem(s(z),nil) = elem(z,nil),
inference(transitivity,[status(thm)],[60,58]) ).
tff(62,plain,
^ [X2: $i] :
refl(( ( eqNat(z,s(X2)) = bfalse )
<=> ( eqNat(z,s(X2)) = bfalse ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [X2: $i] : ( eqNat(z,s(X2)) = bfalse )
<=> ! [X2: $i] : ( eqNat(z,s(X2)) = bfalse ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
( ! [X2: $i] : ( eqNat(z,s(X2)) = bfalse )
<=> ! [X2: $i] : ( eqNat(z,s(X2)) = bfalse ) ),
inference(rewrite,[status(thm)],[]) ).
tff(65,axiom,
! [X2: $i] : ( eqNat(z,s(X2)) = bfalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_006) ).
tff(66,plain,
! [X2: $i] : ( eqNat(z,s(X2)) = bfalse ),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,plain,
! [X2: $i] : ( eqNat(z,s(X2)) = bfalse ),
inference(skolemize,[status(sab)],[66]) ).
tff(68,plain,
! [X2: $i] : ( eqNat(z,s(X2)) = bfalse ),
inference(modus_ponens,[status(thm)],[67,63]) ).
tff(69,plain,
( ~ ! [X2: $i] : ( eqNat(z,s(X2)) = bfalse )
| ( eqNat(z,s(z)) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(70,plain,
eqNat(z,s(z)) = bfalse,
inference(unit_resolution,[status(thm)],[69,68]) ).
tff(71,plain,
bfalse = eqNat(z,s(z)),
inference(symmetry,[status(thm)],[70]) ).
tff(72,plain,
^ [Z: $i] :
refl(( ( eqNat(s(Z),z) = bfalse )
<=> ( eqNat(s(Z),z) = bfalse ) )),
inference(bind,[status(th)],[]) ).
tff(73,plain,
( ! [Z: $i] : ( eqNat(s(Z),z) = bfalse )
<=> ! [Z: $i] : ( eqNat(s(Z),z) = bfalse ) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,plain,
( ! [Z: $i] : ( eqNat(s(Z),z) = bfalse )
<=> ! [Z: $i] : ( eqNat(s(Z),z) = bfalse ) ),
inference(rewrite,[status(thm)],[]) ).
tff(75,axiom,
! [Z: $i] : ( eqNat(s(Z),z) = bfalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_005) ).
tff(76,plain,
! [Z: $i] : ( eqNat(s(Z),z) = bfalse ),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [Z: $i] : ( eqNat(s(Z),z) = bfalse ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [Z: $i] : ( eqNat(s(Z),z) = bfalse ),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,plain,
( ~ ! [Z: $i] : ( eqNat(s(Z),z) = bfalse )
| ( eqNat(s(z),z) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(80,plain,
eqNat(s(z),z) = bfalse,
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
eqNat(s(z),z) = eqNat(z,s(z)),
inference(transitivity,[status(thm)],[80,71]) ).
tff(82,plain,
barbar(eqNat(s(z),z),elem(s(z),nil)) = barbar(eqNat(z,s(z)),elem(z,nil)),
inference(monotonicity,[status(thm)],[81,61]) ).
tff(83,plain,
barbar(eqNat(z,s(z)),elem(z,nil)) = barbar(eqNat(s(z),z),elem(s(z),nil)),
inference(symmetry,[status(thm)],[82]) ).
tff(84,plain,
barbar(eqNat(z,s(z)),elem(z,nil)) = barbar(bfalse,bfalse),
inference(monotonicity,[status(thm)],[70,57]) ).
tff(85,plain,
barbar(bfalse,bfalse) = barbar(eqNat(z,s(z)),elem(z,nil)),
inference(symmetry,[status(thm)],[84]) ).
tff(86,plain,
^ [Y: $i] :
refl(( ( barbar(bfalse,Y) = Y )
<=> ( barbar(bfalse,Y) = Y ) )),
inference(bind,[status(th)],[]) ).
tff(87,plain,
( ! [Y: $i] : ( barbar(bfalse,Y) = Y )
<=> ! [Y: $i] : ( barbar(bfalse,Y) = Y ) ),
inference(quant_intro,[status(thm)],[86]) ).
tff(88,plain,
( ! [Y: $i] : ( barbar(bfalse,Y) = Y )
<=> ! [Y: $i] : ( barbar(bfalse,Y) = Y ) ),
inference(rewrite,[status(thm)],[]) ).
tff(89,axiom,
! [Y: $i] : ( barbar(bfalse,Y) = Y ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_003) ).
tff(90,plain,
! [Y: $i] : ( barbar(bfalse,Y) = Y ),
inference(modus_ponens,[status(thm)],[89,88]) ).
tff(91,plain,
! [Y: $i] : ( barbar(bfalse,Y) = Y ),
inference(skolemize,[status(sab)],[90]) ).
tff(92,plain,
! [Y: $i] : ( barbar(bfalse,Y) = Y ),
inference(modus_ponens,[status(thm)],[91,87]) ).
tff(93,plain,
( ~ ! [Y: $i] : ( barbar(bfalse,Y) = Y )
| ( barbar(bfalse,bfalse) = bfalse ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(94,plain,
barbar(bfalse,bfalse) = bfalse,
inference(unit_resolution,[status(thm)],[93,92]) ).
tff(95,plain,
bfalse = barbar(bfalse,bfalse),
inference(symmetry,[status(thm)],[94]) ).
tff(96,plain,
bfalse = elem(s(z),cons(z,nil)),
inference(transitivity,[status(thm)],[95,85,83,48]) ).
tff(97,plain,
aux(cons(z,nil),s(z),nil,bfalse) = aux(cons(z,nil),s(z),nil,elem(s(z),cons(z,nil))),
inference(monotonicity,[status(thm)],[96]) ).
tff(98,plain,
aux(cons(z,nil),s(z),nil,elem(s(z),cons(z,nil))) = aux(cons(z,nil),s(z),nil,bfalse),
inference(symmetry,[status(thm)],[97]) ).
tff(99,plain,
^ [Xs: $i,Z: $i,Y: $i] :
refl(( ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) )
<=> ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(100,plain,
( ! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) )
<=> ! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ) ),
inference(quant_intro,[status(thm)],[99]) ).
tff(101,plain,
( ! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) )
<=> ! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(102,axiom,
! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_011) ).
tff(103,plain,
! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ),
inference(modus_ponens,[status(thm)],[102,101]) ).
tff(104,plain,
! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ),
inference(skolemize,[status(sab)],[103]) ).
tff(105,plain,
! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) ),
inference(modus_ponens,[status(thm)],[104,100]) ).
tff(106,plain,
( ~ ! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) )
| ( union(cons(s(z),nil),cons(z,nil)) = aux(cons(z,nil),s(z),nil,elem(s(z),cons(z,nil))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
union(cons(s(z),nil),cons(z,nil)) = aux(cons(z,nil),s(z),nil,elem(s(z),cons(z,nil))),
inference(unit_resolution,[status(thm)],[106,105]) ).
tff(108,plain,
union(cons(s(z),nil),cons(z,nil)) = cons(s(z),union(nil,cons(z,nil))),
inference(transitivity,[status(thm)],[107,98,38]) ).
tff(109,plain,
^ [Y: $i] :
refl(( ( union(nil,Y) = Y )
<=> ( union(nil,Y) = Y ) )),
inference(bind,[status(th)],[]) ).
tff(110,plain,
( ! [Y: $i] : ( union(nil,Y) = Y )
<=> ! [Y: $i] : ( union(nil,Y) = Y ) ),
inference(quant_intro,[status(thm)],[109]) ).
tff(111,plain,
( ! [Y: $i] : ( union(nil,Y) = Y )
<=> ! [Y: $i] : ( union(nil,Y) = Y ) ),
inference(rewrite,[status(thm)],[]) ).
tff(112,axiom,
! [Y: $i] : ( union(nil,Y) = Y ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_010) ).
tff(113,plain,
! [Y: $i] : ( union(nil,Y) = Y ),
inference(modus_ponens,[status(thm)],[112,111]) ).
tff(114,plain,
! [Y: $i] : ( union(nil,Y) = Y ),
inference(skolemize,[status(sab)],[113]) ).
tff(115,plain,
! [Y: $i] : ( union(nil,Y) = Y ),
inference(modus_ponens,[status(thm)],[114,110]) ).
tff(116,plain,
( ~ ! [Y: $i] : ( union(nil,Y) = Y )
| ( union(nil,cons(s(z),nil)) = cons(s(z),nil) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(117,plain,
union(nil,cons(s(z),nil)) = cons(s(z),nil),
inference(unit_resolution,[status(thm)],[116,115]) ).
tff(118,plain,
cons(s(z),nil) = union(nil,cons(s(z),nil)),
inference(symmetry,[status(thm)],[117]) ).
tff(119,plain,
cons(z,cons(s(z),nil)) = cons(z,union(nil,cons(s(z),nil))),
inference(monotonicity,[status(thm)],[118]) ).
tff(120,plain,
cons(z,union(nil,cons(s(z),nil))) = cons(z,cons(s(z),nil)),
inference(symmetry,[status(thm)],[119]) ).
tff(121,plain,
( ~ ! [Xs: $i,Z: $i,Y: $i] : ( aux(Y,Z,Xs,bfalse) = cons(Z,union(Xs,Y)) )
| ( aux(cons(s(z),nil),z,nil,bfalse) = cons(z,union(nil,cons(s(z),nil))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(122,plain,
aux(cons(s(z),nil),z,nil,bfalse) = cons(z,union(nil,cons(s(z),nil))),
inference(unit_resolution,[status(thm)],[121,36]) ).
tff(123,plain,
( ~ ! [Xs: $i,Z: $i,X: $i] : ( elem(X,cons(Z,Xs)) = barbar(eqNat(X,Z),elem(X,Xs)) )
| ( elem(z,cons(s(z),nil)) = barbar(eqNat(z,s(z)),elem(z,nil)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(124,plain,
elem(z,cons(s(z),nil)) = barbar(eqNat(z,s(z)),elem(z,nil)),
inference(unit_resolution,[status(thm)],[123,45]) ).
tff(125,plain,
barbar(eqNat(z,s(z)),elem(z,nil)) = elem(z,cons(s(z),nil)),
inference(symmetry,[status(thm)],[124]) ).
tff(126,plain,
bfalse = elem(z,cons(s(z),nil)),
inference(transitivity,[status(thm)],[95,85,125]) ).
tff(127,plain,
aux(cons(s(z),nil),z,nil,bfalse) = aux(cons(s(z),nil),z,nil,elem(z,cons(s(z),nil))),
inference(monotonicity,[status(thm)],[126]) ).
tff(128,plain,
aux(cons(s(z),nil),z,nil,elem(z,cons(s(z),nil))) = aux(cons(s(z),nil),z,nil,bfalse),
inference(symmetry,[status(thm)],[127]) ).
tff(129,plain,
( ~ ! [Xs: $i,Z: $i,Y: $i] : ( union(cons(Z,Xs),Y) = aux(Y,Z,Xs,elem(Z,Y)) )
| ( union(cons(z,nil),cons(s(z),nil)) = aux(cons(s(z),nil),z,nil,elem(z,cons(s(z),nil))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(130,plain,
union(cons(z,nil),cons(s(z),nil)) = aux(cons(s(z),nil),z,nil,elem(z,cons(s(z),nil))),
inference(unit_resolution,[status(thm)],[129,105]) ).
tff(131,plain,
union(cons(z,nil),cons(s(z),nil)) = cons(z,cons(s(z),nil)),
inference(transitivity,[status(thm)],[130,128,122,120]) ).
tff(132,plain,
eq(union(cons(z,nil),cons(s(z),nil)),union(cons(s(z),nil),cons(z,nil))) = eq(cons(z,cons(s(z),nil)),cons(s(z),union(nil,cons(z,nil)))),
inference(monotonicity,[status(thm)],[131,108]) ).
tff(133,plain,
^ [Y: $i,X: $i] :
refl(( ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) )
<=> ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(134,plain,
( ! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) )
<=> ! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ) ),
inference(quant_intro,[status(thm)],[133]) ).
tff(135,plain,
( ! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) )
<=> ! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(136,axiom,
! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_012) ).
tff(137,plain,
! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ),
inference(modus_ponens,[status(thm)],[136,135]) ).
tff(138,plain,
! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ),
inference(skolemize,[status(sab)],[137]) ).
tff(139,plain,
! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) ),
inference(modus_ponens,[status(thm)],[138,134]) ).
tff(140,plain,
( ~ ! [Y: $i,X: $i] : ( prop_union_comm(X,Y) = eq(union(X,Y),union(Y,X)) )
| ( prop_union_comm(cons(z,nil),cons(s(z),nil)) = eq(union(cons(z,nil),cons(s(z),nil)),union(cons(s(z),nil),cons(z,nil))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(141,plain,
prop_union_comm(cons(z,nil),cons(s(z),nil)) = eq(union(cons(z,nil),cons(s(z),nil)),union(cons(s(z),nil),cons(z,nil))),
inference(unit_resolution,[status(thm)],[140,139]) ).
tff(142,plain,
prop_union_comm(cons(z,nil),cons(s(z),nil)) = bfalse,
inference(transitivity,[status(thm)],[141,132,29]) ).
tff(143,plain,
eq3(prop_union_comm(cons(z,nil),cons(s(z),nil)),bfalse) = eq3(bfalse,bfalse),
inference(monotonicity,[status(thm)],[142]) ).
tff(144,plain,
eq3(prop_union_comm(cons(z,nil),cons(s(z),nil)),bfalse) = btrue,
inference(transitivity,[status(thm)],[143,9]) ).
tff(145,plain,
^ [Y: $i,X: $i] :
refl(( ( eq3(prop_union_comm(X,Y),bfalse) != btrue )
<=> ( eq3(prop_union_comm(X,Y),bfalse) != btrue ) )),
inference(bind,[status(th)],[]) ).
tff(146,plain,
( ! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue )
<=> ! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue ) ),
inference(quant_intro,[status(thm)],[145]) ).
tff(147,plain,
( ! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue )
<=> ! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue ) ),
inference(rewrite,[status(thm)],[]) ).
tff(148,axiom,
! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',goal) ).
tff(149,plain,
! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue ),
inference(modus_ponens,[status(thm)],[148,147]) ).
tff(150,plain,
! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue ),
inference(skolemize,[status(sab)],[149]) ).
tff(151,plain,
! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue ),
inference(modus_ponens,[status(thm)],[150,146]) ).
tff(152,plain,
( ~ ! [Y: $i,X: $i] : ( eq3(prop_union_comm(X,Y),bfalse) != btrue )
| ( eq3(prop_union_comm(cons(z,nil),cons(s(z),nil)),bfalse) != btrue ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(153,plain,
eq3(prop_union_comm(cons(z,nil),cons(s(z),nil)),bfalse) != btrue,
inference(unit_resolution,[status(thm)],[152,151]) ).
tff(154,plain,
$false,
inference(unit_resolution,[status(thm)],[153,144]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SWX228-1 : TPTP v9.3.0. Released v9.3.0.
% 0.00/0.06 % Command : run_E %s %d THM
% 0.06/0.24 % Computer : n024.cluster.edu
% 0.06/0.24 % Model : x86_64 x86_64
% 0.06/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.24 % Memory : 8042.1875MB
% 0.06/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.24 % CPULimit : 300
% 0.06/0.24 % WCLimit : 300
% 0.06/0.24 % DateTime : Tue May 5 07:56:13 EDT 2026
% 0.06/0.24 % CPUTime :
% 0.06/0.30 % SZS status Unsatisfiable
% 0.06/0.30 % SZS output start Proof
% See solution above
% 0.06/0.31 % E exiting
%------------------------------------------------------------------------------