%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : FLD067-3 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:05:07 AM UTC 2025
% Result : Unsatisfiable 0.11s 0.38s
% Output : Proof 0.11s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 38
% Syntax : Number of formulae : 82 ( 22 unt; 0 typ; 0 def)
% Number of atoms : 422 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 586 ( 273 ~; 269 |; 0 &)
% ( 44 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 27 ( 27 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 : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 220 ( 197 !; 0 ?; 220 :)
% Comments :
%------------------------------------------------------------------------------
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(additive_inverse_type,type,
additive_inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(defined_type,type,
defined: $i > $o ).
tff(add_type,type,
add: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(less_or_equal_type,type,
less_or_equal: ( $i * $i ) > $o ).
tff(1,plain,
( defined(a)
<=> defined(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
defined(a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_is_defined) ).
tff(3,plain,
defined(a),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,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(5,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)],[4]) ).
tff(6,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(7,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(8,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)],[7]) ).
tff(9,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(10,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,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(15,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(16,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)],[15,14]) ).
tff(17,plain,
sum(additive_inverse(a),a,additive_identity),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,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(19,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)],[18]) ).
tff(20,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(21,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(22,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)],[21]) ).
tff(23,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(24,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[26,19]) ).
tff(28,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(a,additive_inverse(a),additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(a,additive_inverse(a),additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(a,additive_inverse(a),additive_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(30,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(a),a,additive_identity)
| sum(a,additive_inverse(a),additive_identity) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
sum(a,additive_inverse(a),additive_identity),
inference(unit_resolution,[status(thm)],[30,27,17]) ).
tff(32,plain,
^ [X: $i] :
refl(( ( defined(additive_inverse(X))
| ~ defined(X) )
<=> ( defined(additive_inverse(X))
| ~ defined(X) ) )),
inference(bind,[status(th)],[]) ).
tff(33,plain,
( ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
<=> ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ) ),
inference(quant_intro,[status(thm)],[32]) ).
tff(34,plain,
( ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
<=> ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(35,axiom,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',well_definedness_of_additive_inverse) ).
tff(36,plain,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
inference(skolemize,[status(sab)],[36]) ).
tff(38,plain,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
inference(modus_ponens,[status(thm)],[37,33]) ).
tff(39,plain,
( ( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(a))
| ~ defined(a) )
<=> ( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(a))
| ~ defined(a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(40,plain,
( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(a))
| ~ defined(a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(41,plain,
( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(a))
| ~ defined(a) ),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
defined(additive_inverse(a)),
inference(unit_resolution,[status(thm)],[41,38,3]) ).
tff(43,plain,
( defined(b)
<=> defined(b) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,axiom,
defined(b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_is_defined) ).
tff(45,plain,
defined(b),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
<=> ( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(47,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ) ),
inference(quant_intro,[status(thm)],[46]) ).
tff(48,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,plain,
^ [Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( sum(X,Y,add(X,Y))
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(X,Y,add(X,Y)) ) )),
( ( sum(X,Y,add(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(X)
| sum(X,Y,add(X,Y))
| ~ defined(Y) ) )),
rewrite(( ( ~ defined(X)
| sum(X,Y,add(X,Y))
| ~ defined(Y) )
<=> ( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ) )),
( ( sum(X,Y,add(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(50,plain,
( ! [Y: $i,X: $i] :
( sum(X,Y,add(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ) ),
inference(quant_intro,[status(thm)],[49]) ).
tff(51,axiom,
! [Y: $i,X: $i] :
( sum(X,Y,add(X,Y))
| ~ defined(X)
| ~ defined(Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',totality_of_addition) ).
tff(52,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ),
inference(modus_ponens,[status(thm)],[51,50]) ).
tff(53,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ),
inference(modus_ponens,[status(thm)],[52,48]) ).
tff(54,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ),
inference(skolemize,[status(sab)],[53]) ).
tff(55,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) ),
inference(modus_ponens,[status(thm)],[54,47]) ).
tff(56,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
| ~ defined(b)
| ~ defined(additive_inverse(a))
| sum(b,additive_inverse(a),add(b,additive_inverse(a))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
| ~ defined(b)
| ~ defined(additive_inverse(a))
| sum(b,additive_inverse(a),add(b,additive_inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(57,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
| ~ defined(b)
| ~ defined(additive_inverse(a))
| sum(b,additive_inverse(a),add(b,additive_inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(58,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| sum(X,Y,add(X,Y)) )
| ~ defined(b)
| ~ defined(additive_inverse(a))
| sum(b,additive_inverse(a),add(b,additive_inverse(a))) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
sum(b,additive_inverse(a),add(b,additive_inverse(a))),
inference(unit_resolution,[status(thm)],[58,55,45,42]) ).
tff(60,plain,
( ~ less_or_equal(additive_identity,add(b,additive_inverse(a)))
<=> ~ less_or_equal(additive_identity,add(b,additive_inverse(a))) ),
inference(rewrite,[status(thm)],[]) ).
tff(61,axiom,
~ less_or_equal(additive_identity,add(b,additive_inverse(a))),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_less_or_equal_4) ).
tff(62,plain,
~ less_or_equal(additive_identity,add(b,additive_inverse(a))),
inference(modus_ponens,[status(thm)],[61,60]) ).
tff(63,plain,
( less_or_equal(a,b)
<=> less_or_equal(a,b) ),
inference(rewrite,[status(thm)],[]) ).
tff(64,axiom,
less_or_equal(a,b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',less_or_equal_3) ).
tff(65,plain,
less_or_equal(a,b),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
^ [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(( ( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) )),
inference(bind,[status(th)],[]) ).
tff(67,plain,
( ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
<=> ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
<=> ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,plain,
^ [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( less_or_equal(U,V)
| ~ less_or_equal(X,Y) )
<=> ( ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) )),
( ( less_or_equal(U,V)
| ~ less_or_equal(X,Y)
| ~ sum(X,Z,U) )
<=> ( ~ less_or_equal(X,Y)
| less_or_equal(U,V)
| ~ sum(X,Z,U) ) )),
rewrite(( ( ~ less_or_equal(X,Y)
| less_or_equal(U,V)
| ~ sum(X,Z,U) )
<=> ( ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) )),
( ( less_or_equal(U,V)
| ~ less_or_equal(X,Y)
| ~ sum(X,Z,U) )
<=> ( ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) )),
( ( less_or_equal(U,V)
| ~ less_or_equal(X,Y)
| ~ sum(X,Z,U)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V)
| ~ sum(Y,Z,V) ) )),
rewrite(( ( ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) )),
( ( less_or_equal(U,V)
| ~ less_or_equal(X,Y)
| ~ sum(X,Z,U)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) )),
inference(bind,[status(th)],[]) ).
tff(70,plain,
( ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( less_or_equal(U,V)
| ~ less_or_equal(X,Y)
| ~ sum(X,Z,U)
| ~ sum(Y,Z,V) )
<=> ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,axiom,
! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( less_or_equal(U,V)
| ~ less_or_equal(X,Y)
| ~ sum(X,Z,U)
| ~ sum(Y,Z,V) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',compatibility_of_order_relation_and_addition) ).
tff(72,plain,
! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ),
inference(modus_ponens,[status(thm)],[71,70]) ).
tff(73,plain,
! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ),
inference(skolemize,[status(sab)],[73]) ).
tff(75,plain,
! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) ),
inference(modus_ponens,[status(thm)],[74,67]) ).
tff(76,plain,
( ( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| less_or_equal(additive_identity,add(b,additive_inverse(a)))
| ~ less_or_equal(a,b)
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity) )
<=> ( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| less_or_equal(additive_identity,add(b,additive_inverse(a)))
| ~ less_or_equal(a,b)
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(77,plain,
( ( ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ less_or_equal(a,b)
| less_or_equal(additive_identity,add(b,additive_inverse(a))) )
<=> ( less_or_equal(additive_identity,add(b,additive_inverse(a)))
| ~ less_or_equal(a,b)
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(78,plain,
( ( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ less_or_equal(a,b)
| less_or_equal(additive_identity,add(b,additive_inverse(a))) )
<=> ( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| less_or_equal(additive_identity,add(b,additive_inverse(a)))
| ~ less_or_equal(a,b)
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity) ) ),
inference(monotonicity,[status(thm)],[77]) ).
tff(79,plain,
( ( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ less_or_equal(a,b)
| less_or_equal(additive_identity,add(b,additive_inverse(a))) )
<=> ( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| less_or_equal(additive_identity,add(b,additive_inverse(a)))
| ~ less_or_equal(a,b)
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity) ) ),
inference(transitivity,[status(thm)],[78,76]) ).
tff(80,plain,
( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ less_or_equal(a,b)
| less_or_equal(additive_identity,add(b,additive_inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(81,plain,
( ~ ! [V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(Y,Z,V)
| ~ sum(X,Z,U)
| ~ less_or_equal(X,Y)
| less_or_equal(U,V) )
| less_or_equal(additive_identity,add(b,additive_inverse(a)))
| ~ less_or_equal(a,b)
| ~ sum(b,additive_inverse(a),add(b,additive_inverse(a)))
| ~ sum(a,additive_inverse(a),additive_identity) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
$false,
inference(unit_resolution,[status(thm)],[81,75,65,62,59,31]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : FLD067-3 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.11/0.11 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n015.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % 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 : Thu Jun 19 13:41:41 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.11/0.38 % SZS status Unsatisfiable
% 0.11/0.38 % SZS output start Proof
% See solution above
% 0.11/0.39 % E exiting
%------------------------------------------------------------------------------