%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : FLD071-3 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 05:05:09 AM UTC 2025
% Result : Unsatisfiable 0.12s 0.38s
% Output : Proof 0.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 23
% Syntax : Number of formulae : 46 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 290 ( 0 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 419 ( 197 ~; 193 |; 0 &)
% ( 29 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 22 ( 22 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 : 4 ( 4 usr; 3 con; 0-2 aty)
% Number of variables : 97 ( 87 !; 0 ?; 97 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(defined_type,type,
defined: $i > $o ).
tff(less_or_equal_type,type,
less_or_equal: ( $i * $i ) > $o ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(1,plain,
( defined(b)
<=> defined(b) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
defined(b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_is_defined) ).
tff(3,plain,
defined(b),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
( defined(a)
<=> defined(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(5,axiom,
defined(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_is_defined) ).
tff(6,plain,
defined(a),
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
<=> ( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(10,plain,
^ [Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( product(X,Y,multiply(X,Y))
| ~ defined(X) )
<=> ( ~ defined(X)
| product(X,Y,multiply(X,Y)) ) )),
( ( product(X,Y,multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(X)
| product(X,Y,multiply(X,Y))
| ~ defined(Y) ) )),
rewrite(( ( ~ defined(X)
| product(X,Y,multiply(X,Y))
| ~ defined(Y) )
<=> ( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ) )),
( ( product(X,Y,multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [Y: $i,X: $i] :
( product(X,Y,multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,axiom,
! [Y: $i,X: $i] :
( product(X,Y,multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD002-0.ax',totality_of_multiplication) ).
tff(13,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ),
inference(modus_ponens,[status(thm)],[13,9]) ).
tff(15,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) ),
inference(modus_ponens,[status(thm)],[15,8]) ).
tff(17,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
| ~ defined(a)
| ~ defined(b)
| product(a,b,multiply(a,b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
| ~ defined(a)
| ~ defined(b)
| product(a,b,multiply(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
| ~ defined(a)
| ~ defined(b)
| product(a,b,multiply(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X)
| ~ defined(Y)
| product(X,Y,multiply(X,Y)) )
| ~ defined(a)
| ~ defined(b)
| product(a,b,multiply(a,b)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
product(a,b,multiply(a,b)),
inference(unit_resolution,[status(thm)],[19,16,6,3]) ).
tff(21,plain,
( ~ less_or_equal(additive_identity,multiply(a,b))
<=> ~ less_or_equal(additive_identity,multiply(a,b)) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
~ less_or_equal(additive_identity,multiply(a,b)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_less_or_equal_5) ).
tff(23,plain,
~ less_or_equal(additive_identity,multiply(a,b)),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
( less_or_equal(additive_identity,b)
<=> less_or_equal(additive_identity,b) ),
inference(rewrite,[status(thm)],[]) ).
tff(25,axiom,
less_or_equal(additive_identity,b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',less_or_equal_4) ).
tff(26,plain,
less_or_equal(additive_identity,b),
inference(modus_ponens,[status(thm)],[25,24]) ).
tff(27,plain,
( less_or_equal(additive_identity,a)
<=> less_or_equal(additive_identity,a) ),
inference(rewrite,[status(thm)],[]) ).
tff(28,axiom,
less_or_equal(additive_identity,a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',less_or_equal_3) ).
tff(29,plain,
less_or_equal(additive_identity,a),
inference(modus_ponens,[status(thm)],[28,27]) ).
tff(30,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
<=> ( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X) )
<=> ( ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) )),
( ( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X)
| ~ less_or_equal(additive_identity,Y) )
<=> ( ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,Y) ) )),
rewrite(( ( ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,Y) )
<=> ( ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) )),
( ( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X)
| ~ less_or_equal(additive_identity,Y) )
<=> ( ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) )),
( ( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X)
| ~ less_or_equal(additive_identity,Y)
| ~ product(X,Y,Z) )
<=> ( ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z)
| ~ product(X,Y,Z) ) )),
rewrite(( ( ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z)
| ~ product(X,Y,Z) )
<=> ( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) )),
( ( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X)
| ~ less_or_equal(additive_identity,Y)
| ~ product(X,Y,Z) )
<=> ( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(34,plain,
( ! [Z: $i,Y: $i,X: $i] :
( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X)
| ~ less_or_equal(additive_identity,Y)
| ~ product(X,Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,axiom,
! [Z: $i,Y: $i,X: $i] :
( less_or_equal(additive_identity,Z)
| ~ less_or_equal(additive_identity,X)
| ~ less_or_equal(additive_identity,Y)
| ~ product(X,Y,Z) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD002-0.ax',compatibility_of_order_relation_and_multiplication) ).
tff(36,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) ),
inference(modus_ponens,[status(thm)],[38,31]) ).
tff(40,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| less_or_equal(additive_identity,multiply(a,b))
| ~ less_or_equal(additive_identity,a)
| ~ less_or_equal(additive_identity,b)
| ~ product(a,b,multiply(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| less_or_equal(additive_identity,multiply(a,b))
| ~ less_or_equal(additive_identity,a)
| ~ less_or_equal(additive_identity,b)
| ~ product(a,b,multiply(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
( ( ~ product(a,b,multiply(a,b))
| ~ less_or_equal(additive_identity,b)
| ~ less_or_equal(additive_identity,a)
| less_or_equal(additive_identity,multiply(a,b)) )
<=> ( less_or_equal(additive_identity,multiply(a,b))
| ~ less_or_equal(additive_identity,a)
| ~ less_or_equal(additive_identity,b)
| ~ product(a,b,multiply(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| ~ product(a,b,multiply(a,b))
| ~ less_or_equal(additive_identity,b)
| ~ less_or_equal(additive_identity,a)
| less_or_equal(additive_identity,multiply(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| less_or_equal(additive_identity,multiply(a,b))
| ~ less_or_equal(additive_identity,a)
| ~ less_or_equal(additive_identity,b)
| ~ product(a,b,multiply(a,b)) ) ),
inference(monotonicity,[status(thm)],[41]) ).
tff(43,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| ~ product(a,b,multiply(a,b))
| ~ less_or_equal(additive_identity,b)
| ~ less_or_equal(additive_identity,a)
| less_or_equal(additive_identity,multiply(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| less_or_equal(additive_identity,multiply(a,b))
| ~ less_or_equal(additive_identity,a)
| ~ less_or_equal(additive_identity,b)
| ~ product(a,b,multiply(a,b)) ) ),
inference(transitivity,[status(thm)],[42,40]) ).
tff(44,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| ~ product(a,b,multiply(a,b))
| ~ less_or_equal(additive_identity,b)
| ~ less_or_equal(additive_identity,a)
| less_or_equal(additive_identity,multiply(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(45,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ less_or_equal(additive_identity,Y)
| ~ less_or_equal(additive_identity,X)
| less_or_equal(additive_identity,Z) )
| less_or_equal(additive_identity,multiply(a,b))
| ~ less_or_equal(additive_identity,a)
| ~ less_or_equal(additive_identity,b)
| ~ product(a,b,multiply(a,b)) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
$false,
inference(unit_resolution,[status(thm)],[45,39,29,26,23,20]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : FLD071-3 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n004.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 13:45:08 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.38 % SZS status Unsatisfiable
% 0.12/0.38 % SZS output start Proof
% See solution above
% 0.12/0.39 % E exiting
%------------------------------------------------------------------------------