%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : FLD016-5 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:04:54 AM UTC 2025
% Result : Unsatisfiable 0.13s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 52
% Syntax : Number of formulae : 109 ( 32 unt; 0 typ; 0 def)
% Number of atoms : 600 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 861 ( 404 ~; 399 |; 0 &)
% ( 58 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 34 ( 34 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 6 con; 0-1 aty)
% Number of variables : 377 ( 343 !; 0 ?; 377 :)
% Comments :
%------------------------------------------------------------------------------
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(v_type,type,
v: $i ).
tff(r_type,type,
r: $i ).
tff(additive_inverse_type,type,
additive_inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(defined_type,type,
defined: $i > $o ).
tff(u_type,type,
u: $i ).
tff(q_type,type,
q: $i ).
tff(1,plain,
( defined(v)
<=> defined(v) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
defined(v),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',v_is_defined) ).
tff(3,plain,
defined(v),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [X: $i] :
refl(( ( ~ defined(X)
| sum(additive_identity,X,X) )
<=> ( ~ defined(X)
| sum(additive_identity,X,X) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,plain,
^ [X: $i] :
rewrite(( ( sum(additive_identity,X,X)
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(additive_identity,X,X) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [X: $i] :
( sum(additive_identity,X,X)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,axiom,
! [X: $i] :
( sum(additive_identity,X,X)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',existence_of_identity_addition) ).
tff(10,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(v)
| sum(additive_identity,v,v) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(v)
| sum(additive_identity,v,v) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(v)
| sum(additive_identity,v,v) ),
inference(quant_inst,[status(thm)],[]) ).
tff(16,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(v)
| sum(additive_identity,v,v) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
sum(additive_identity,v,v),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
( defined(a)
<=> defined(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
defined(a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_is_defined) ).
tff(20,plain,
defined(a),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
^ [X: $i] :
refl(( ( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
<=> ( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
^ [X: $i] :
rewrite(( ( sum(additive_inverse(X),X,additive_identity)
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(25,plain,
( ! [X: $i] :
( sum(additive_inverse(X),X,additive_identity)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,axiom,
! [X: $i] :
( sum(additive_inverse(X),X,additive_identity)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',existence_of_inverse_addition) ).
tff(27,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[27,23]) ).
tff(29,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(skolemize,[status(sab)],[28]) ).
tff(30,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[29,22]) ).
tff(31,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(a)
| sum(additive_inverse(a),a,additive_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(a)
| sum(additive_inverse(a),a,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(a)
| sum(additive_inverse(a),a,additive_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(33,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(a)
| sum(additive_inverse(a),a,additive_identity) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
sum(additive_inverse(a),a,additive_identity),
inference(unit_resolution,[status(thm)],[33,30,20]) ).
tff(35,plain,
( sum(a,v,r)
<=> sum(a,v,r) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,axiom,
sum(a,v,r),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_7) ).
tff(37,plain,
sum(a,v,r),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(( ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( sum(X,V,W)
| ~ sum(X,Y,U) )
<=> ( ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(Y,Z,V) ) )),
rewrite(( ( ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(U,Z,W) ) )),
rewrite(( ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(U,Z,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
inference(bind,[status(th)],[]) ).
tff(42,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',associativity_addition_1) ).
tff(44,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[46,39]) ).
tff(48,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,v,v)
| sum(additive_inverse(a),r,v) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,v,v)
| sum(additive_inverse(a),r,v) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,plain,
( ( ~ sum(additive_identity,v,v)
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),r,v) )
<=> ( ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,v,v)
| sum(additive_inverse(a),r,v) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,v,v)
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),r,v) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,v,v)
| sum(additive_inverse(a),r,v) ) ),
inference(monotonicity,[status(thm)],[49]) ).
tff(51,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,v,v)
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),r,v) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,v,v)
| sum(additive_inverse(a),r,v) ) ),
inference(transitivity,[status(thm)],[50,48]) ).
tff(52,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,v,v)
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),r,v) ),
inference(quant_inst,[status(thm)],[]) ).
tff(53,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,v,r)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,v,v)
| sum(additive_inverse(a),r,v) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
sum(additive_inverse(a),r,v),
inference(unit_resolution,[status(thm)],[53,47,37,34,17]) ).
tff(55,plain,
( defined(u)
<=> defined(u) ),
inference(rewrite,[status(thm)],[]) ).
tff(56,axiom,
defined(u),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',u_is_defined) ).
tff(57,plain,
defined(u),
inference(modus_ponens,[status(thm)],[56,55]) ).
tff(58,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(u)
| sum(additive_identity,u,u) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(u)
| sum(additive_identity,u,u) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(59,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(u)
| sum(additive_identity,u,u) ),
inference(quant_inst,[status(thm)],[]) ).
tff(60,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(u)
| sum(additive_identity,u,u) ),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
sum(additive_identity,u,u),
inference(unit_resolution,[status(thm)],[60,13,57]) ).
tff(62,plain,
( sum(a,u,q)
<=> sum(a,u,q) ),
inference(rewrite,[status(thm)],[]) ).
tff(63,axiom,
sum(a,u,q),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_6) ).
tff(64,plain,
sum(a,u,q),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,u,u)
| sum(additive_inverse(a),q,u) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,u,u)
| sum(additive_inverse(a),q,u) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(66,plain,
( ( ~ sum(additive_identity,u,u)
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),q,u) )
<=> ( ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,u,u)
| sum(additive_inverse(a),q,u) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(67,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,u,u)
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),q,u) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,u,u)
| sum(additive_inverse(a),q,u) ) ),
inference(monotonicity,[status(thm)],[66]) ).
tff(68,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,u,u)
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),q,u) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,u,u)
| sum(additive_inverse(a),q,u) ) ),
inference(transitivity,[status(thm)],[67,65]) ).
tff(69,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,u,u)
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(additive_inverse(a),q,u) ),
inference(quant_inst,[status(thm)],[]) ).
tff(70,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(a,u,q)
| ~ sum(additive_inverse(a),a,additive_identity)
| ~ sum(additive_identity,u,u)
| sum(additive_inverse(a),q,u) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
sum(additive_inverse(a),q,u),
inference(unit_resolution,[status(thm)],[70,47,64,34,61]) ).
tff(72,plain,
( ~ sum(additive_identity,u,v)
<=> ~ sum(additive_identity,u,v) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,axiom,
~ sum(additive_identity,u,v),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_sum_9) ).
tff(74,plain,
~ sum(additive_identity,u,v),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(76,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[75]) ).
tff(77,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(78,plain,
^ [Z: $i,Y: $i,X: $i] :
rewrite(( ( sum(Y,X,Z)
| ~ sum(X,Y,Z) )
<=> ( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(79,plain,
( ! [Z: $i,Y: $i,X: $i] :
( sum(Y,X,Z)
| ~ sum(X,Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[78]) ).
tff(80,axiom,
! [Z: $i,Y: $i,X: $i] :
( sum(Y,X,Z)
| ~ sum(X,Y,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',commutativity_addition) ).
tff(81,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[81,77]) ).
tff(83,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(skolemize,[status(sab)],[82]) ).
tff(84,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[83,76]) ).
tff(85,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(u,additive_identity,v)
| sum(additive_identity,u,v) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(u,additive_identity,v)
| sum(additive_identity,u,v) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(86,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(u,additive_identity,v)
| sum(additive_identity,u,v) ),
inference(quant_inst,[status(thm)],[]) ).
tff(87,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(u,additive_identity,v)
| sum(additive_identity,u,v) ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
~ sum(u,additive_identity,v),
inference(unit_resolution,[status(thm)],[87,84,74]) ).
tff(89,plain,
( sum(additive_identity,q,r)
<=> sum(additive_identity,q,r) ),
inference(rewrite,[status(thm)],[]) ).
tff(90,axiom,
sum(additive_identity,q,r),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_8) ).
tff(91,plain,
sum(additive_identity,q,r),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,q,r)
| sum(q,additive_identity,r) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,q,r)
| sum(q,additive_identity,r) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,q,r)
| sum(q,additive_identity,r) ),
inference(quant_inst,[status(thm)],[]) ).
tff(94,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,q,r)
| sum(q,additive_identity,r) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
sum(q,additive_identity,r),
inference(unit_resolution,[status(thm)],[94,84,91]) ).
tff(96,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(( ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
inference(bind,[status(th)],[]) ).
tff(97,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[96]) ).
tff(98,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(99,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( sum(U,Z,W)
| ~ sum(X,Y,U) )
<=> ( sum(U,Z,W)
| ~ sum(X,Y,U) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) ) )),
rewrite(( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
rewrite(( ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
inference(bind,[status(th)],[]) ).
tff(100,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[99]) ).
tff(101,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',associativity_addition_2) ).
tff(102,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[102,98]) ).
tff(104,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(skolemize,[status(sab)],[103]) ).
tff(105,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[104,97]) ).
tff(106,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(u,additive_identity,v)
| ~ sum(q,additive_identity,r)
| ~ sum(additive_inverse(a),q,u)
| ~ sum(additive_inverse(a),r,v) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(u,additive_identity,v)
| ~ sum(q,additive_identity,r)
| ~ sum(additive_inverse(a),q,u)
| ~ sum(additive_inverse(a),r,v) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(107,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(u,additive_identity,v)
| ~ sum(q,additive_identity,r)
| ~ sum(additive_inverse(a),q,u)
| ~ sum(additive_inverse(a),r,v) ),
inference(quant_inst,[status(thm)],[]) ).
tff(108,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(u,additive_identity,v)
| ~ sum(q,additive_identity,r)
| ~ sum(additive_inverse(a),q,u)
| ~ sum(additive_inverse(a),r,v) ),
inference(modus_ponens,[status(thm)],[107,106]) ).
tff(109,plain,
$false,
inference(unit_resolution,[status(thm)],[108,105,95,88,71,54]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : FLD016-5 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n011.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 13:32:12 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.13/0.40 % SZS status Unsatisfiable
% 0.13/0.40 % SZS output start Proof
% See solution above
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------