%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : FLD032-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% 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 05:04:59 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.50s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 61
% Syntax : Number of formulae : 136 ( 30 unt; 0 typ; 0 def)
% Number of atoms : 769 ( 0 equ)
% Maximal formula atoms : 12 ( 5 avg)
% Number of connectives : 1067 ( 470 ~; 521 |; 0 &)
% ( 76 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 36 ( 36 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 : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 298 ( 277 !; 0 ?; 298 :)
% Comments :
%------------------------------------------------------------------------------
tff(equalish_type,type,
equalish: ( $i * $i ) > $o ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(multiplicative_identity_type,type,
multiplicative_identity: $i ).
tff(multiplicative_inverse_type,type,
multiplicative_inverse: $i > $i ).
tff(defined_type,type,
defined: $i > $o ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(1,plain,
( equalish(multiplicative_inverse(a),multiplicative_identity)
<=> equalish(multiplicative_inverse(a),multiplicative_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
equalish(multiplicative_inverse(a),multiplicative_identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',multiplicative_inverses_equal) ).
tff(3,plain,
equalish(multiplicative_inverse(a),multiplicative_identity),
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,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
<=> ( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(10,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z) )
<=> ( ~ defined(Z)
| equalish(multiply(X,Z),multiply(Y,Z)) ) )),
( ( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z)
| ~ equalish(X,Y) )
<=> ( ~ defined(Z)
| equalish(multiply(X,Z),multiply(Y,Z))
| ~ equalish(X,Y) ) )),
rewrite(( ( ~ defined(Z)
| equalish(multiply(X,Z),multiply(Y,Z))
| ~ equalish(X,Y) )
<=> ( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ) )),
( ( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z)
| ~ equalish(X,Y) )
<=> ( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [Z: $i,Y: $i,X: $i] :
( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z)
| ~ equalish(X,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,axiom,
! [Z: $i,Y: $i,X: $i] :
( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z)
| ~ equalish(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',compatibility_of_equality_and_multiplication) ).
tff(13,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[13,9]) ).
tff(15,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[15,8]) ).
tff(17,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_inverse(a),multiplicative_identity)
| equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_inverse(a),multiplicative_identity)
| equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_inverse(a),multiplicative_identity)
| equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_inverse(a),multiplicative_identity)
| equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a)),
inference(unit_resolution,[status(thm)],[19,16,6,3]) ).
tff(21,plain,
( ~ equalish(a,additive_identity)
<=> ~ equalish(a,additive_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
~ equalish(a,additive_identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_not_equal_to_additive_identity_2) ).
tff(23,plain,
~ equalish(a,additive_identity),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
^ [X: $i] :
refl(( ( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
<=> ( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(25,plain,
( ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,plain,
( ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,plain,
^ [X: $i] :
rewrite(( ( defined(multiplicative_inverse(X))
| ~ defined(X)
| equalish(X,additive_identity) )
<=> ( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(28,plain,
( ! [X: $i] :
( defined(multiplicative_inverse(X))
| ~ defined(X)
| equalish(X,additive_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ) ),
inference(quant_intro,[status(thm)],[27]) ).
tff(29,axiom,
! [X: $i] :
( defined(multiplicative_inverse(X))
| ~ defined(X)
| equalish(X,additive_identity) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',well_definedness_of_multiplicative_inverse) ).
tff(30,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(modus_ponens,[status(thm)],[30,26]) ).
tff(32,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(skolemize,[status(sab)],[31]) ).
tff(33,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(modus_ponens,[status(thm)],[32,25]) ).
tff(34,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(a,additive_identity)
| ~ defined(a)
| defined(multiplicative_inverse(a)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(a,additive_identity)
| ~ defined(a)
| defined(multiplicative_inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(35,plain,
( ( ~ defined(a)
| equalish(a,additive_identity)
| defined(multiplicative_inverse(a)) )
<=> ( equalish(a,additive_identity)
| ~ defined(a)
| defined(multiplicative_inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| ~ defined(a)
| equalish(a,additive_identity)
| defined(multiplicative_inverse(a)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(a,additive_identity)
| ~ defined(a)
| defined(multiplicative_inverse(a)) ) ),
inference(monotonicity,[status(thm)],[35]) ).
tff(37,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| ~ defined(a)
| equalish(a,additive_identity)
| defined(multiplicative_inverse(a)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(a,additive_identity)
| ~ defined(a)
| defined(multiplicative_inverse(a)) ) ),
inference(transitivity,[status(thm)],[36,34]) ).
tff(38,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| ~ defined(a)
| equalish(a,additive_identity)
| defined(multiplicative_inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(a,additive_identity)
| ~ defined(a)
| defined(multiplicative_inverse(a)) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
defined(multiplicative_inverse(a)),
inference(unit_resolution,[status(thm)],[39,33,6,23]) ).
tff(41,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
<=> ( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(42,plain,
( ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,plain,
( ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,plain,
^ [Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X) )
<=> ( ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) )),
( ( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(Y) ) )),
rewrite(( ( ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(Y) )
<=> ( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) )),
( ( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(45,plain,
( ! [Y: $i,X: $i] :
( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X)
| ~ defined(Y) )
<=> ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ) ),
inference(quant_intro,[status(thm)],[44]) ).
tff(46,axiom,
! [Y: $i,X: $i] :
( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X)
| ~ defined(Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',commutativity_multiplication) ).
tff(47,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(modus_ponens,[status(thm)],[46,45]) ).
tff(48,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(modus_ponens,[status(thm)],[47,43]) ).
tff(49,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(skolemize,[status(sab)],[48]) ).
tff(50,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(modus_ponens,[status(thm)],[49,42]) ).
tff(51,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiplicative_inverse(a))
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiplicative_inverse(a))
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(52,plain,
( ( ~ defined(multiplicative_inverse(a))
| ~ defined(a)
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) )
<=> ( ~ defined(a)
| ~ defined(multiplicative_inverse(a))
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(53,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(multiplicative_inverse(a))
| ~ defined(a)
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiplicative_inverse(a))
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) ) ),
inference(monotonicity,[status(thm)],[52]) ).
tff(54,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(multiplicative_inverse(a))
| ~ defined(a)
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiplicative_inverse(a))
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) ) ),
inference(transitivity,[status(thm)],[53,51]) ).
tff(55,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(multiplicative_inverse(a))
| ~ defined(a)
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(56,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiplicative_inverse(a))
| equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)) ),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a)),
inference(unit_resolution,[status(thm)],[56,50,6,40]) ).
tff(58,plain,
^ [X: $i] :
refl(( ( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
<=> ( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(59,plain,
( ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) ),
inference(quant_intro,[status(thm)],[58]) ).
tff(60,plain,
( ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(61,plain,
^ [X: $i] :
trans(monotonicity(rewrite(( ( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X) )
<=> ( ~ defined(X)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) )),
( ( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X)
| equalish(X,additive_identity) )
<=> ( ~ defined(X)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| equalish(X,additive_identity) ) )),
rewrite(( ( ~ defined(X)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| equalish(X,additive_identity) )
<=> ( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) )),
( ( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X)
| equalish(X,additive_identity) )
<=> ( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(62,plain,
( ! [X: $i] :
( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X)
| equalish(X,additive_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ) ),
inference(quant_intro,[status(thm)],[61]) ).
tff(63,axiom,
! [X: $i] :
( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X)
| equalish(X,additive_identity) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',existence_of_inverse_multiplication) ).
tff(64,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[64,60]) ).
tff(66,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(skolemize,[status(sab)],[65]) ).
tff(67,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[66,59]) ).
tff(68,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(a)
| equalish(a,additive_identity)
| equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(a)
| equalish(a,additive_identity)
| equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(a)
| equalish(a,additive_identity)
| equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(70,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(a)
| equalish(a,additive_identity)
| equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity),
inference(unit_resolution,[status(thm)],[70,67,6,23]) ).
tff(72,plain,
^ [Y: $i,X: $i] :
refl(( ( equalish(X,Y)
| ~ equalish(Y,X) )
<=> ( equalish(X,Y)
| ~ equalish(Y,X) ) )),
inference(bind,[status(th)],[]) ).
tff(73,plain,
( ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
<=> ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,plain,
( ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
<=> ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(75,axiom,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',symmetry_of_equality) ).
tff(76,plain,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,plain,
( ( equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) )
<=> ( ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))) ) ),
inference(monotonicity,[status(thm)],[80]) ).
tff(82,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))) ) ),
inference(transitivity,[status(thm)],[81,79]) ).
tff(83,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(84,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))) ),
inference(modus_ponens,[status(thm)],[83,82]) ).
tff(85,plain,
equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a))),
inference(unit_resolution,[status(thm)],[84,78,71]) ).
tff(86,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( equalish(X,Z)
| ~ equalish(X,Y) )
<=> ( ~ equalish(X,Y)
| equalish(X,Z) ) )),
( ( equalish(X,Z)
| ~ equalish(X,Y)
| ~ equalish(Y,Z) )
<=> ( ~ equalish(X,Y)
| equalish(X,Z)
| ~ equalish(Y,Z) ) )),
rewrite(( ( ~ equalish(X,Y)
| equalish(X,Z)
| ~ equalish(Y,Z) )
<=> ( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ) )),
( ( equalish(X,Z)
| ~ equalish(X,Y)
| ~ equalish(Y,Z) )
<=> ( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(89,plain,
( ! [Z: $i,Y: $i,X: $i] :
( equalish(X,Z)
| ~ equalish(X,Y)
| ~ equalish(Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ) ),
inference(quant_intro,[status(thm)],[88]) ).
tff(90,axiom,
! [Z: $i,Y: $i,X: $i] :
( equalish(X,Z)
| ~ equalish(X,Y)
| ~ equalish(Y,Z) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',transitivity_of_equality) ).
tff(91,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(modus_ponens,[status(thm)],[91,87]) ).
tff(93,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(skolemize,[status(sab)],[92]) ).
tff(94,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(modus_ponens,[status(thm)],[93,86]) ).
tff(95,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(96,plain,
( ( ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) )
<=> ( ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(97,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) ) ),
inference(monotonicity,[status(thm)],[96]) ).
tff(98,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) ) ),
inference(transitivity,[status(thm)],[97,95]) ).
tff(99,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(100,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(a,multiplicative_inverse(a)))
| ~ equalish(multiply(a,multiplicative_inverse(a)),multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)) ),
inference(modus_ponens,[status(thm)],[99,98]) ).
tff(101,plain,
equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a)),
inference(unit_resolution,[status(thm)],[100,94,85,57]) ).
tff(102,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
( ( ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) )
<=> ( ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(104,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(monotonicity,[status(thm)],[103]) ).
tff(105,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(transitivity,[status(thm)],[104,102]) ).
tff(106,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiplicative_identity,multiply(multiplicative_inverse(a),a))
| ~ equalish(multiply(multiplicative_inverse(a),a),multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ),
inference(modus_ponens,[status(thm)],[106,105]) ).
tff(108,plain,
equalish(multiplicative_identity,multiply(multiplicative_identity,a)),
inference(unit_resolution,[status(thm)],[107,94,101,20]) ).
tff(109,plain,
( ~ equalish(a,multiplicative_identity)
<=> ~ equalish(a,multiplicative_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(110,axiom,
~ equalish(a,multiplicative_identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_not_equal_to_multiplicative_identity_4) ).
tff(111,plain,
~ equalish(a,multiplicative_identity),
inference(modus_ponens,[status(thm)],[110,109]) ).
tff(112,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiplicative_identity)
| ~ equalish(multiplicative_identity,a) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiplicative_identity)
| ~ equalish(multiplicative_identity,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(113,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiplicative_identity)
| ~ equalish(multiplicative_identity,a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(114,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiplicative_identity)
| ~ equalish(multiplicative_identity,a) ),
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
~ equalish(multiplicative_identity,a),
inference(unit_resolution,[status(thm)],[114,78,111]) ).
tff(116,plain,
^ [X: $i] :
refl(( ( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
<=> ( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ) )),
inference(bind,[status(th)],[]) ).
tff(117,plain,
( ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ) ),
inference(quant_intro,[status(thm)],[116]) ).
tff(118,plain,
( ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(119,plain,
^ [X: $i] :
rewrite(( ( equalish(multiply(multiplicative_identity,X),X)
| ~ defined(X) )
<=> ( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ) )),
inference(bind,[status(th)],[]) ).
tff(120,plain,
( ! [X: $i] :
( equalish(multiply(multiplicative_identity,X),X)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ) ),
inference(quant_intro,[status(thm)],[119]) ).
tff(121,axiom,
! [X: $i] :
( equalish(multiply(multiplicative_identity,X),X)
| ~ defined(X) ),
file('/export/starexec/sandbox/benchmark/Axioms/FLD001-0.ax',existence_of_identity_multiplication) ).
tff(122,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(modus_ponens,[status(thm)],[122,118]) ).
tff(124,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(skolemize,[status(sab)],[123]) ).
tff(125,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(modus_ponens,[status(thm)],[124,117]) ).
tff(126,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
| ~ defined(a)
| equalish(multiply(multiplicative_identity,a),a) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
| ~ defined(a)
| equalish(multiply(multiplicative_identity,a),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(127,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
| ~ defined(a)
| equalish(multiply(multiplicative_identity,a),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(128,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
| ~ defined(a)
| equalish(multiply(multiplicative_identity,a),a) ),
inference(modus_ponens,[status(thm)],[127,126]) ).
tff(129,plain,
equalish(multiply(multiplicative_identity,a),a),
inference(unit_resolution,[status(thm)],[128,125,6]) ).
tff(130,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(multiplicative_identity,a)
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(multiplicative_identity,a)
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(131,plain,
( ( ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,a) )
<=> ( equalish(multiplicative_identity,a)
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(132,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(multiplicative_identity,a)
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(monotonicity,[status(thm)],[131]) ).
tff(133,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(multiplicative_identity,a)
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ) ),
inference(transitivity,[status(thm)],[132,130]) ).
tff(134,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a))
| equalish(multiplicative_identity,a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(135,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(multiplicative_identity,a)
| ~ equalish(multiply(multiplicative_identity,a),a)
| ~ equalish(multiplicative_identity,multiply(multiplicative_identity,a)) ),
inference(modus_ponens,[status(thm)],[134,133]) ).
tff(136,plain,
$false,
inference(unit_resolution,[status(thm)],[135,94,129,115,108]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : FLD032-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.13/0.33 % Computer : n005.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Thu Jun 19 13:35:20 EDT 2025
% 0.13/0.33 % CPUTime :
% 0.20/0.50 % SZS status Unsatisfiable
% 0.20/0.50 % SZS output start Proof
% See solution above
% 0.20/0.51 % E exiting
%------------------------------------------------------------------------------