%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : FLD004-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:04:51 AM UTC 2025
% Result : Unsatisfiable 0.21s 0.44s
% Output : Proof 0.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 66
% Syntax : Number of formulae : 149 ( 33 unt; 0 typ; 0 def)
% Number of atoms : 807 ( 0 equ)
% Maximal formula atoms : 12 ( 5 avg)
% Number of connectives : 1110 ( 490 ~; 541 |; 0 &)
% ( 79 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of FOOLs : 38 ( 38 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 : 6 ( 6 usr; 4 con; 0-2 aty)
% Number of variables : 316 ( 291 !; 0 ?; 316 :)
% 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_inverse_type,type,
multiplicative_inverse: $i > $i ).
tff(b_type,type,
b: $i ).
tff(multiplicative_identity_type,type,
multiplicative_identity: $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(defined_type,type,
defined: $i > $o ).
tff(1,plain,
( ~ equalish(b,additive_identity)
<=> ~ equalish(b,additive_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
~ equalish(b,additive_identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_not_equal_to_additive_identity_3) ).
tff(3,plain,
~ equalish(b,additive_identity),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
( defined(b)
<=> defined(b) ),
inference(rewrite,[status(thm)],[]) ).
tff(5,axiom,
defined(b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_is_defined) ).
tff(6,plain,
defined(b),
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,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(8,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)],[7]) ).
tff(9,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(10,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(11,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)],[10]) ).
tff(12,axiom,
! [X: $i] :
( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X)
| equalish(X,additive_identity) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',existence_of_inverse_multiplication) ).
tff(13,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[13,9]) ).
tff(15,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[15,8]) ).
tff(17,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(b)
| equalish(b,additive_identity)
| equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(b)
| equalish(b,additive_identity)
| equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(b)
| equalish(b,additive_identity)
| equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity) )
| ~ defined(b)
| equalish(b,additive_identity)
| equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity),
inference(unit_resolution,[status(thm)],[19,16,6,3]) ).
tff(21,plain,
^ [Y: $i,X: $i] :
refl(( ( equalish(X,Y)
| ~ equalish(Y,X) )
<=> ( equalish(X,Y)
| ~ equalish(Y,X) ) )),
inference(bind,[status(th)],[]) ).
tff(22,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)],[21]) ).
tff(23,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(24,axiom,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',symmetry_of_equality) ).
tff(25,plain,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) ),
inference(modus_ponens,[status(thm)],[26,22]) ).
tff(28,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
( ( equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) )
<=> ( ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))) ) ),
inference(monotonicity,[status(thm)],[29]) ).
tff(31,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))) ) ),
inference(transitivity,[status(thm)],[30,28]) ).
tff(32,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(33,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| ~ equalish(multiply(b,multiplicative_inverse(b)),multiplicative_identity)
| equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b))),
inference(unit_resolution,[status(thm)],[33,27,20]) ).
tff(35,plain,
( defined(a)
<=> defined(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,axiom,
defined(a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_is_defined) ).
tff(37,plain,
defined(a),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,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(39,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)],[38]) ).
tff(40,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(41,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(42,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)],[41]) ).
tff(43,axiom,
! [Z: $i,Y: $i,X: $i] :
( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z)
| ~ equalish(X,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',compatibility_of_equality_and_multiplication) ).
tff(44,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[46,39]) ).
tff(48,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(Z)
| ~ equalish(X,Y)
| equalish(multiply(X,Z),multiply(Y,Z)) )
| ~ defined(a)
| ~ equalish(multiplicative_identity,multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a)) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a)),
inference(unit_resolution,[status(thm)],[50,47,37,34]) ).
tff(52,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(53,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)],[52]) ).
tff(54,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(55,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(56,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)],[55]) ).
tff(57,axiom,
! [X: $i] :
( equalish(multiply(multiplicative_identity,X),X)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',existence_of_identity_multiplication) ).
tff(58,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(modus_ponens,[status(thm)],[58,54]) ).
tff(60,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(skolemize,[status(sab)],[59]) ).
tff(61,plain,
! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) ),
inference(modus_ponens,[status(thm)],[60,53]) ).
tff(62,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(63,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(64,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(multiply(multiplicative_identity,X),X) )
| ~ defined(a)
| equalish(multiply(multiplicative_identity,a),a) ),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
equalish(multiply(multiplicative_identity,a),a),
inference(unit_resolution,[status(thm)],[64,61,37]) ).
tff(66,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),a) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(67,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(68,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),a) ),
inference(modus_ponens,[status(thm)],[67,66]) ).
tff(69,plain,
equalish(a,multiply(multiplicative_identity,a)),
inference(unit_resolution,[status(thm)],[68,27,65]) ).
tff(70,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(71,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(72,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(73,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)],[72]) ).
tff(74,axiom,
! [Z: $i,Y: $i,X: $i] :
( equalish(X,Z)
| ~ equalish(X,Y)
| ~ equalish(Y,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',transitivity_of_equality) ).
tff(75,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(modus_ponens,[status(thm)],[74,73]) ).
tff(76,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(modus_ponens,[status(thm)],[75,71]) ).
tff(77,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) ),
inference(modus_ponens,[status(thm)],[77,70]) ).
tff(79,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,plain,
( ( ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| ~ equalish(a,multiply(multiplicative_identity,a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) )
<=> ( ~ equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| ~ equalish(a,multiply(multiplicative_identity,a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(monotonicity,[status(thm)],[80]) ).
tff(82,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| ~ equalish(a,multiply(multiplicative_identity,a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(transitivity,[status(thm)],[81,79]) ).
tff(83,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| ~ equalish(a,multiply(multiplicative_identity,a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(84,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(a,multiply(multiplicative_identity,a))
| ~ equalish(multiply(multiplicative_identity,a),multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ),
inference(modus_ponens,[status(thm)],[83,82]) ).
tff(85,plain,
equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)),
inference(unit_resolution,[status(thm)],[84,78,69,51]) ).
tff(86,plain,
( ~ equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a)
<=> ~ equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,axiom,
~ equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',multiply_not_equal_to_a_4) ).
tff(88,plain,
~ equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a),
inference(modus_ponens,[status(thm)],[87,86]) ).
tff(89,plain,
( ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a)
| ~ equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) )
<=> ( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a)
| ~ equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(90,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a)
| ~ equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(91,plain,
( ~ ! [Y: $i,X: $i] :
( equalish(X,Y)
| ~ equalish(Y,X) )
| equalish(multiply(a,multiply(b,multiplicative_inverse(b))),a)
| ~ equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
~ equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))),
inference(unit_resolution,[status(thm)],[91,27,88]) ).
tff(93,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(94,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)],[93]) ).
tff(95,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(96,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(97,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)],[96]) ).
tff(98,axiom,
! [X: $i] :
( defined(multiplicative_inverse(X))
| ~ defined(X)
| equalish(X,additive_identity) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',well_definedness_of_multiplicative_inverse) ).
tff(99,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(modus_ponens,[status(thm)],[99,95]) ).
tff(101,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(skolemize,[status(sab)],[100]) ).
tff(102,plain,
! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) ),
inference(modus_ponens,[status(thm)],[101,94]) ).
tff(103,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(b,additive_identity)
| ~ defined(b)
| defined(multiplicative_inverse(b)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(b,additive_identity)
| ~ defined(b)
| defined(multiplicative_inverse(b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(104,plain,
( ( ~ defined(b)
| equalish(b,additive_identity)
| defined(multiplicative_inverse(b)) )
<=> ( equalish(b,additive_identity)
| ~ defined(b)
| defined(multiplicative_inverse(b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(105,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| ~ defined(b)
| equalish(b,additive_identity)
| defined(multiplicative_inverse(b)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(b,additive_identity)
| ~ defined(b)
| defined(multiplicative_inverse(b)) ) ),
inference(monotonicity,[status(thm)],[104]) ).
tff(106,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| ~ defined(b)
| equalish(b,additive_identity)
| defined(multiplicative_inverse(b)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(b,additive_identity)
| ~ defined(b)
| defined(multiplicative_inverse(b)) ) ),
inference(transitivity,[status(thm)],[105,103]) ).
tff(107,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| ~ defined(b)
| equalish(b,additive_identity)
| defined(multiplicative_inverse(b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(108,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| equalish(X,additive_identity)
| defined(multiplicative_inverse(X)) )
| equalish(b,additive_identity)
| ~ defined(b)
| defined(multiplicative_inverse(b)) ),
inference(modus_ponens,[status(thm)],[107,106]) ).
tff(109,plain,
defined(multiplicative_inverse(b)),
inference(unit_resolution,[status(thm)],[108,102,6,3]) ).
tff(110,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
<=> ( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(111,plain,
( ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,plain,
( ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(113,plain,
^ [Y: $i,X: $i] :
rewrite(( ( defined(multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(114,plain,
( ! [Y: $i,X: $i] :
( defined(multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) )
<=> ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ) ),
inference(quant_intro,[status(thm)],[113]) ).
tff(115,axiom,
! [Y: $i,X: $i] :
( defined(multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',well_definedness_of_multiplication) ).
tff(116,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ),
inference(modus_ponens,[status(thm)],[116,112]) ).
tff(118,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ),
inference(skolemize,[status(sab)],[117]) ).
tff(119,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) ),
inference(modus_ponens,[status(thm)],[118,111]) ).
tff(120,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(b)
| ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(b)
| ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(121,plain,
( ( ~ defined(multiplicative_inverse(b))
| ~ defined(b)
| defined(multiply(b,multiplicative_inverse(b))) )
<=> ( ~ defined(b)
| ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(122,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(multiplicative_inverse(b))
| ~ defined(b)
| defined(multiply(b,multiplicative_inverse(b))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(b)
| ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) ) ),
inference(monotonicity,[status(thm)],[121]) ).
tff(123,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(multiplicative_inverse(b))
| ~ defined(b)
| defined(multiply(b,multiplicative_inverse(b))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(b)
| ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) ) ),
inference(transitivity,[status(thm)],[122,120]) ).
tff(124,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(multiplicative_inverse(b))
| ~ defined(b)
| defined(multiply(b,multiplicative_inverse(b))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(125,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| defined(multiply(X,Y)) )
| ~ defined(b)
| ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) ),
inference(modus_ponens,[status(thm)],[124,123]) ).
tff(126,plain,
( ~ defined(multiplicative_inverse(b))
| defined(multiply(b,multiplicative_inverse(b))) ),
inference(unit_resolution,[status(thm)],[125,119,6]) ).
tff(127,plain,
defined(multiply(b,multiplicative_inverse(b))),
inference(unit_resolution,[status(thm)],[126,109]) ).
tff(128,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(129,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)],[128]) ).
tff(130,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(131,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(132,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)],[131]) ).
tff(133,axiom,
! [Y: $i,X: $i] :
( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X)
| ~ defined(Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD001-0.ax',commutativity_multiplication) ).
tff(134,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(modus_ponens,[status(thm)],[133,132]) ).
tff(135,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(modus_ponens,[status(thm)],[134,130]) ).
tff(136,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(skolemize,[status(sab)],[135]) ).
tff(137,plain,
! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) ),
inference(modus_ponens,[status(thm)],[136,129]) ).
tff(138,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b)))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(139,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(140,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(Y)
| ~ defined(X)
| equalish(multiply(X,Y),multiply(Y,X)) )
| ~ defined(a)
| ~ defined(multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b)))) ),
inference(modus_ponens,[status(thm)],[139,138]) ).
tff(141,plain,
( ~ defined(multiply(b,multiplicative_inverse(b)))
| equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b)))) ),
inference(unit_resolution,[status(thm)],[140,137,37]) ).
tff(142,plain,
equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b)))),
inference(unit_resolution,[status(thm)],[141,127]) ).
tff(143,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(144,plain,
( ( ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) )
<=> ( equalish(a,multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(145,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(monotonicity,[status(thm)],[144]) ).
tff(146,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ) ),
inference(transitivity,[status(thm)],[145,143]) ).
tff(147,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a))
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(148,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ equalish(Y,Z)
| ~ equalish(X,Y)
| equalish(X,Z) )
| equalish(a,multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(multiply(multiply(b,multiplicative_inverse(b)),a),multiply(a,multiply(b,multiplicative_inverse(b))))
| ~ equalish(a,multiply(multiply(b,multiplicative_inverse(b)),a)) ),
inference(modus_ponens,[status(thm)],[147,146]) ).
tff(149,plain,
$false,
inference(unit_resolution,[status(thm)],[148,78,142,92,85]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : FLD004-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.04/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n003.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Jun 19 13:32:24 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.21/0.44 % SZS status Unsatisfiable
% 0.21/0.44 % SZS output start Proof
% See solution above
% 0.21/0.46 % E exiting
%------------------------------------------------------------------------------