%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ALG002-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 04:57:45 AM UTC 2025
% Result : Unsatisfiable 109.66s 109.96s
% Output : Proof 109.66s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 59
% Syntax : Number of formulae : 132 ( 33 unt; 0 typ; 0 def)
% Number of atoms : 674 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 906 ( 395 ~; 446 |; 0 &)
% ( 65 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 31 ( 31 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; 0-1 aty)
% Number of variables : 316 ( 289 !; 0 ?; 316 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(additive_inverse_type,type,
additive_inverse: $i > $i ).
tff(multiplicative_identity_type,type,
multiplicative_identity: $i ).
tff(multiplicative_inverse_type,type,
multiplicative_inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(greater_than_0_type,type,
greater_than_0: $i > $o ).
tff(1,plain,
( greater_than_0(a)
<=> greater_than_0(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
greater_than_0(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_greater_than_0) ).
tff(3,plain,
greater_than_0(a),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [X: $i] :
refl(( ( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
<=> ( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
<=> ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
<=> ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',greater_than_0_square) ).
tff(8,plain,
! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
| ~ greater_than_0(a)
| ~ product(a,a,additive_identity) )
<=> ( ~ ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
| ~ greater_than_0(a)
| ~ product(a,a,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
| ~ greater_than_0(a)
| ~ product(a,a,additive_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [X: $i] :
( ~ greater_than_0(X)
| ~ product(X,X,additive_identity) )
| ~ greater_than_0(a)
| ~ product(a,a,additive_identity) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
~ product(a,a,additive_identity),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
^ [X: $i] :
refl(( ( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
<=> ( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(16,plain,
( ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
<=> ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
<=> ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
^ [X: $i] :
rewrite(( ( product(X,multiplicative_inverse(X),multiplicative_identity)
| product(X,X,additive_identity) )
<=> ( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ) )),
inference(bind,[status(th)],[]) ).
tff(19,plain,
( ! [X: $i] :
( product(X,multiplicative_inverse(X),multiplicative_identity)
| product(X,X,additive_identity) )
<=> ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,axiom,
! [X: $i] :
( product(X,multiplicative_inverse(X),multiplicative_identity)
| product(X,X,additive_identity) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse_and_identity) ).
tff(21,plain,
! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ),
inference(skolemize,[status(sab)],[22]) ).
tff(24,plain,
! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[23,16]) ).
tff(25,plain,
( ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
| product(a,a,additive_identity)
| product(a,multiplicative_inverse(a),multiplicative_identity) )
<=> ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
| product(a,a,additive_identity)
| product(a,multiplicative_inverse(a),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(26,plain,
( ~ ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
| product(a,a,additive_identity)
| product(a,multiplicative_inverse(a),multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [X: $i] :
( product(X,X,additive_identity)
| product(X,multiplicative_inverse(X),multiplicative_identity) )
| product(a,a,additive_identity)
| product(a,multiplicative_inverse(a),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
( product(a,a,additive_identity)
| product(a,multiplicative_inverse(a),multiplicative_identity) ),
inference(unit_resolution,[status(thm)],[27,24]) ).
tff(29,plain,
product(a,multiplicative_inverse(a),multiplicative_identity),
inference(unit_resolution,[status(thm)],[28,14]) ).
tff(30,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
<=> ( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',product_to_inverse) ).
tff(34,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(38,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(X,additive_inverse(Y),additive_inverse(Z)) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)),
inference(unit_resolution,[status(thm)],[39,36,29]) ).
tff(41,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(X,Y,Z)
| product(Y,X,Z) )
<=> ( ~ product(X,Y,Z)
| product(Y,X,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(42,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_product) ).
tff(45,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[46,42]) ).
tff(48,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(multiplicative_inverse(a),a,multiplicative_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(multiplicative_inverse(a),a,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(multiplicative_inverse(a),a,multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,multiplicative_inverse(a),multiplicative_identity)
| product(multiplicative_inverse(a),a,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
product(multiplicative_inverse(a),a,multiplicative_identity),
inference(unit_resolution,[status(thm)],[50,47,29]) ).
tff(52,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
<=> ( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) )),
inference(bind,[status(th)],[]) ).
tff(53,plain,
( ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) ),
inference(quant_intro,[status(thm)],[52]) ).
tff(54,plain,
( ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ product(Y,Z,X)
| ~ product(Y,Y,additive_identity) )
<=> ( ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) )),
( ( ~ product(Y,Z,X)
| ~ product(Y,Y,additive_identity)
| product(X,X,additive_identity) )
<=> ( ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X)
| product(X,X,additive_identity) ) )),
rewrite(( ( ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X)
| product(X,X,additive_identity) )
<=> ( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) )),
( ( ~ product(Y,Z,X)
| ~ product(Y,Y,additive_identity)
| product(X,X,additive_identity) )
<=> ( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) )),
inference(bind,[status(th)],[]) ).
tff(56,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(Y,Z,X)
| ~ product(Y,Y,additive_identity)
| product(X,X,additive_identity) )
<=> ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ) ),
inference(quant_intro,[status(thm)],[55]) ).
tff(57,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ product(Y,Z,X)
| ~ product(Y,Y,additive_identity)
| product(X,X,additive_identity) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',square_to_0) ).
tff(58,plain,
! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ),
inference(modus_ponens,[status(thm)],[58,54]) ).
tff(60,plain,
! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ),
inference(skolemize,[status(sab)],[59]) ).
tff(61,plain,
! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) ),
inference(modus_ponens,[status(thm)],[60,53]) ).
tff(62,plain,
( ~ product(multiplicative_identity,multiplicative_identity,additive_identity)
<=> ~ product(multiplicative_identity,multiplicative_identity,additive_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(63,axiom,
~ product(multiplicative_identity,multiplicative_identity,additive_identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_abelian) ).
tff(64,plain,
~ product(multiplicative_identity,multiplicative_identity,additive_identity),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
| product(multiplicative_identity,multiplicative_identity,additive_identity)
| ~ product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| ~ product(multiplicative_inverse(a),a,multiplicative_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
| product(multiplicative_identity,multiplicative_identity,additive_identity)
| ~ product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| ~ product(multiplicative_inverse(a),a,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(66,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
| product(multiplicative_identity,multiplicative_identity,additive_identity)
| ~ product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| ~ product(multiplicative_inverse(a),a,multiplicative_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(67,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( product(X,X,additive_identity)
| ~ product(Y,Y,additive_identity)
| ~ product(Y,Z,X) )
| product(multiplicative_identity,multiplicative_identity,additive_identity)
| ~ product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| ~ product(multiplicative_inverse(a),a,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[66,65]) ).
tff(68,plain,
~ product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity),
inference(unit_resolution,[status(thm)],[67,64,61,51]) ).
tff(69,plain,
( ~ greater_than_0(multiplicative_inverse(a))
<=> ~ greater_than_0(multiplicative_inverse(a)) ),
inference(rewrite,[status(thm)],[]) ).
tff(70,axiom,
~ greater_than_0(multiplicative_inverse(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_a_inverse_greater_than_0) ).
tff(71,plain,
~ greater_than_0(multiplicative_inverse(a)),
inference(modus_ponens,[status(thm)],[70,69]) ).
tff(72,plain,
^ [X: $i] :
refl(( ( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
<=> ( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ) )),
inference(bind,[status(th)],[]) ).
tff(73,plain,
( ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
<=> ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,plain,
( ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
<=> ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(75,plain,
^ [X: $i] :
rewrite(( ( greater_than_0(X)
| product(X,X,additive_identity)
| greater_than_0(additive_inverse(X)) )
<=> ( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ) )),
inference(bind,[status(th)],[]) ).
tff(76,plain,
( ! [X: $i] :
( greater_than_0(X)
| product(X,X,additive_identity)
| greater_than_0(additive_inverse(X)) )
<=> ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ) ),
inference(quant_intro,[status(thm)],[75]) ).
tff(77,axiom,
! [X: $i] :
( greater_than_0(X)
| product(X,X,additive_identity)
| greater_than_0(additive_inverse(X)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',product_and_inverse) ).
tff(78,plain,
! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ),
inference(modus_ponens,[status(thm)],[78,74]) ).
tff(80,plain,
! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ),
inference(skolemize,[status(sab)],[79]) ).
tff(81,plain,
! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) ),
inference(modus_ponens,[status(thm)],[80,73]) ).
tff(82,plain,
( ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| greater_than_0(multiplicative_inverse(a))
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a))) )
<=> ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| greater_than_0(multiplicative_inverse(a))
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(83,plain,
( ( product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a)))
| greater_than_0(multiplicative_inverse(a)) )
<=> ( greater_than_0(multiplicative_inverse(a))
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,plain,
( ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a)))
| greater_than_0(multiplicative_inverse(a)) )
<=> ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| greater_than_0(multiplicative_inverse(a))
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a))) ) ),
inference(monotonicity,[status(thm)],[83]) ).
tff(85,plain,
( ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a)))
| greater_than_0(multiplicative_inverse(a)) )
<=> ( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| greater_than_0(multiplicative_inverse(a))
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a))) ) ),
inference(transitivity,[status(thm)],[84,82]) ).
tff(86,plain,
( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a)))
| greater_than_0(multiplicative_inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(87,plain,
( ~ ! [X: $i] :
( product(X,X,additive_identity)
| greater_than_0(additive_inverse(X))
| greater_than_0(X) )
| greater_than_0(multiplicative_inverse(a))
| product(multiplicative_inverse(a),multiplicative_inverse(a),additive_identity)
| greater_than_0(additive_inverse(multiplicative_inverse(a))) ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
greater_than_0(additive_inverse(multiplicative_inverse(a))),
inference(unit_resolution,[status(thm)],[87,81,71,68]) ).
tff(89,plain,
^ [X: $i] :
refl(( product(X,multiplicative_identity,X)
<=> product(X,multiplicative_identity,X) )),
inference(bind,[status(th)],[]) ).
tff(90,plain,
( ! [X: $i] : product(X,multiplicative_identity,X)
<=> ! [X: $i] : product(X,multiplicative_identity,X) ),
inference(quant_intro,[status(thm)],[89]) ).
tff(91,plain,
( ! [X: $i] : product(X,multiplicative_identity,X)
<=> ! [X: $i] : product(X,multiplicative_identity,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(92,axiom,
! [X: $i] : product(X,multiplicative_identity,X),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',right_identity) ).
tff(93,plain,
! [X: $i] : product(X,multiplicative_identity,X),
inference(modus_ponens,[status(thm)],[92,91]) ).
tff(94,plain,
! [X: $i] : product(X,multiplicative_identity,X),
inference(skolemize,[status(sab)],[93]) ).
tff(95,plain,
! [X: $i] : product(X,multiplicative_identity,X),
inference(modus_ponens,[status(thm)],[94,90]) ).
tff(96,plain,
( ~ ! [X: $i] : product(X,multiplicative_identity,X)
| product(multiplicative_inverse(a),multiplicative_identity,multiplicative_inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(97,plain,
product(multiplicative_inverse(a),multiplicative_identity,multiplicative_inverse(a)),
inference(unit_resolution,[status(thm)],[96,95]) ).
tff(98,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
<=> ( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ) )),
inference(bind,[status(th)],[]) ).
tff(99,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ) ),
inference(quant_intro,[status(thm)],[98]) ).
tff(100,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(101,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',product_of_inverses1) ).
tff(102,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ),
inference(skolemize,[status(sab)],[102]) ).
tff(104,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) ),
inference(modus_ponens,[status(thm)],[103,99]) ).
tff(105,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
| ~ product(multiplicative_inverse(a),multiplicative_identity,multiplicative_inverse(a))
| product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
| ~ product(multiplicative_inverse(a),multiplicative_identity,multiplicative_inverse(a))
| product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(106,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
| ~ product(multiplicative_inverse(a),multiplicative_identity,multiplicative_inverse(a))
| product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(additive_inverse(X),additive_inverse(Y),Z) )
| ~ product(multiplicative_inverse(a),multiplicative_identity,multiplicative_inverse(a))
| product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ),
inference(modus_ponens,[status(thm)],[106,105]) ).
tff(108,plain,
product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)),
inference(unit_resolution,[status(thm)],[107,104,97]) ).
tff(109,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
<=> ( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) )),
inference(bind,[status(th)],[]) ).
tff(110,plain,
( ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) ),
inference(quant_intro,[status(thm)],[109]) ).
tff(111,plain,
( ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(112,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ~ product(Y,Z,X)
| ~ greater_than_0(Y) )
<=> ( ~ product(Y,Z,X)
| ~ greater_than_0(Y) ) )),
( ( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z) )
<=> ( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z) ) )),
rewrite(( ( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z) )
<=> ( ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) )),
( ( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z) )
<=> ( ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) )),
( ( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z)
| greater_than_0(X) )
<=> ( ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y)
| greater_than_0(X) ) )),
rewrite(( ( ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y)
| greater_than_0(X) )
<=> ( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) )),
( ( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z)
| greater_than_0(X) )
<=> ( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) )),
inference(bind,[status(th)],[]) ).
tff(113,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z)
| greater_than_0(X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ) ),
inference(quant_intro,[status(thm)],[112]) ).
tff(114,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ product(Y,Z,X)
| ~ greater_than_0(Y)
| ~ greater_than_0(Z)
| greater_than_0(X) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',product_and_greater_than_0) ).
tff(115,plain,
! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ),
inference(modus_ponens,[status(thm)],[115,111]) ).
tff(117,plain,
! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ),
inference(skolemize,[status(sab)],[116]) ).
tff(118,plain,
! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) ),
inference(modus_ponens,[status(thm)],[117,110]) ).
tff(119,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(120,plain,
( ( greater_than_0(multiplicative_inverse(a))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a))) )
<=> ( greater_than_0(multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(121,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ) ),
inference(monotonicity,[status(thm)],[120]) ).
tff(122,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ) ),
inference(transitivity,[status(thm)],[121,119]) ).
tff(123,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(124,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(multiplicative_inverse(a))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity),multiplicative_inverse(a)) ),
inference(modus_ponens,[status(thm)],[123,122]) ).
tff(125,plain,
~ greater_than_0(additive_inverse(multiplicative_identity)),
inference(unit_resolution,[status(thm)],[124,118,71,88,108]) ).
tff(126,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(a)
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(a)
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(127,plain,
( ( greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(a) )
<=> ( greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(a)
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(128,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(a)
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ) ),
inference(monotonicity,[status(thm)],[127]) ).
tff(129,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(a)
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ) ),
inference(transitivity,[status(thm)],[128,126]) ).
tff(130,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity))
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ greater_than_0(a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(131,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( greater_than_0(X)
| ~ product(Y,Z,X)
| ~ greater_than_0(Z)
| ~ greater_than_0(Y) )
| greater_than_0(additive_inverse(multiplicative_identity))
| ~ greater_than_0(a)
| ~ greater_than_0(additive_inverse(multiplicative_inverse(a)))
| ~ product(a,additive_inverse(multiplicative_inverse(a)),additive_inverse(multiplicative_identity)) ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
$false,
inference(unit_resolution,[status(thm)],[131,118,3,125,88,40]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ALG002-1 : TPTP v9.0.0. Released v1.0.0.
% 0.00/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n013.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 10:41:53 EDT 2025
% 0.11/0.32 % CPUTime :
% 109.66/109.96 % SZS status Unsatisfiable
% 109.66/109.96 % SZS output start Proof
% See solution above
% 109.66/109.98 % E exiting
%------------------------------------------------------------------------------