TSTP Solution File: ARI742_1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : ARI742_1 : TPTP v8.1.0. Released v7.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n009.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 : Tue Sep 6 17:02:49 EDT 2022
% Result : Theorem 0.12s 0.37s
% Output : Proof 0.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 46
% Syntax : Number of formulae : 117 ( 30 unt; 2 typ; 0 def)
% Number of atoms : 445 ( 158 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 497 ( 198 ~; 183 |; 30 &)
% ( 74 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of FOOLs : 31 ( 31 fml; 0 var)
% Number arithmetic : 1506 ( 251 atm; 400 fun; 731 num; 124 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 2 ( 2 >; 0 *; 0 +; 0 <<)
% Number of predicates : 9 ( 4 usr; 2 prp; 0-3 aty)
% Number of functors : 6 ( 2 usr; 2 con; 0-2 aty)
% Number of variables : 124 ( 106 !; 0 ?; 124 :)
% Comments :
%------------------------------------------------------------------------------
tff(sqrt_type,type,
sqrt: $real > $real ).
tff(sqr_type,type,
sqr: $real > $real ).
tff(1,plain,
( ( 0 = sqrt(0) )
<=> ( sqrt(0) = 0 ) ),
inference(commutativity,[status(thm)],]) ).
tff(2,plain,
( ( sqrt(0) = 0 )
<=> ( 0 = sqrt(0) ) ),
inference(symmetry,[status(thm)],[1]) ).
tff(3,plain,
( ( sqrt(0) != 0 )
<=> ( 0 != sqrt(0) ) ),
inference(monotonicity,[status(thm)],[2]) ).
tff(4,plain,
( ( sqrt(0) != 0 )
<=> ( sqrt(0) != 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(5,axiom,
sqrt(0) != 0,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sqrt_zero) ).
tff(6,plain,
sqrt(0) != 0,
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,plain,
0 != sqrt(0),
inference(modus_ponens,[status(thm)],[6,3]) ).
tff(8,plain,
( ( 0 = $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) )
<=> ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(commutativity,[status(thm)],]) ).
tff(9,plain,
( ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 )
<=> ( 0 = $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) ) ),
inference(symmetry,[status(thm)],[8]) ).
tff(10,plain,
^ [X: $real,Y: $real] :
refl(
( ( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
<=> ( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
<=> ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
^ [X: $real,Y: $real] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( $greatereq(X,0)
& $greatereq(Y,0) )
<=> ~ ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0) ) )),
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
<=> ~ ~ ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0) ) )),
rewrite(
( ~ ~ ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0) )
<=> ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0) ) )),
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
<=> ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0) ) )),
( ( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) )
<=> ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) ) )),
rewrite(
( ( ~ $greatereq(Y,0)
| ~ $greatereq(X,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) )
<=> ( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ) )),
( ( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) )
<=> ( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ) )),
inference(bind,[status(th)],]) ).
tff(13,plain,
( ! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) )
<=> ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ) ),
inference(quant_intro,[status(thm)],[12]) ).
tff(14,plain,
^ [X: $real,Y: $real] :
rewrite(
( ( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum(sqrt($product(Y,X)),$product(-1,$product(sqrt(Y),sqrt(X)))) = 0 ) )
<=> ( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(15,plain,
( ! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum(sqrt($product(Y,X)),$product(-1,$product(sqrt(Y),sqrt(X)))) = 0 ) )
<=> ! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[14]) ).
tff(16,plain,
^ [X: $real,Y: $real] :
rewrite(
( ( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) )
<=> ( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum(sqrt($product(Y,X)),$product(-1,$product(sqrt(Y),sqrt(X)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(17,plain,
( ! [X: $real,Y: $real] :
( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) )
<=> ! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum(sqrt($product(Y,X)),$product(-1,$product(sqrt(Y),sqrt(X)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,plain,
( ! [X: $real,Y: $real] :
( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) )
<=> ! [X: $real,Y: $real] :
( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(19,plain,
^ [X: $real,Y: $real] :
trans(
monotonicity(
rewrite(
( ( sqrt($product(X,Y)) = $product(sqrt(X),sqrt(Y)) )
<=> ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) )),
( ( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ( sqrt($product(X,Y)) = $product(sqrt(X),sqrt(Y)) ) )
<=> ( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ) )),
rewrite(
( ( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) )
<=> ( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ) )),
( ( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ( sqrt($product(X,Y)) = $product(sqrt(X),sqrt(Y)) ) )
<=> ( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ! [X: $real,Y: $real] :
( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ( sqrt($product(X,Y)) = $product(sqrt(X),sqrt(Y)) ) )
<=> ! [X: $real,Y: $real] :
( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,axiom,
! [X: $real,Y: $real] :
( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ( sqrt($product(X,Y)) = $product(sqrt(X),sqrt(Y)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sqrt_mul) ).
tff(22,plain,
! [X: $real,Y: $real] :
( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
! [X: $real,Y: $real] :
( ~ ( $lesseq(0,X)
& $lesseq(0,Y) )
| ( sqrt($product(Y,X)) = $product(sqrt(Y),sqrt(X)) ) ),
inference(modus_ponens,[status(thm)],[22,18]) ).
tff(24,plain,
! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum(sqrt($product(Y,X)),$product(-1,$product(sqrt(Y),sqrt(X)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[23,17]) ).
tff(25,plain,
! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[24,15]) ).
tff(26,plain,
! [X: $real,Y: $real] :
( ~ ( $greatereq(X,0)
& $greatereq(Y,0) )
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 ) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ),
inference(modus_ponens,[status(thm)],[26,13]) ).
tff(28,plain,
! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) ),
inference(modus_ponens,[status(thm)],[27,11]) ).
tff(29,plain,
( ( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) )
<=> ( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(30,plain,
( ( $false
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 )
| $false )
<=> ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(31,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(32,plain,
( $greatereq(0,0)
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(33,plain,
( ~ $greatereq(0,0)
<=> ~ $true ),
inference(monotonicity,[status(thm)],[32]) ).
tff(34,plain,
( ~ $greatereq(0,0)
<=> $false ),
inference(transitivity,[status(thm)],[33,31]) ).
tff(35,plain,
( ( $sum($product(-1,sqrt(0)),$product(sqrt(0),sqrt(0))) = 0 )
<=> ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(36,plain,
$sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt(0))) = $sum($product(-1,sqrt(0)),$product(sqrt(0),sqrt(0))),
inference(rewrite,[status(thm)],]) ).
tff(37,plain,
$product(-1,sqrt($product(0,0))) = $product(-1,sqrt(0)),
inference(rewrite,[status(thm)],]) ).
tff(38,plain,
$sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt(0))),
inference(monotonicity,[status(thm)],[37]) ).
tff(39,plain,
$sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = $sum($product(-1,sqrt(0)),$product(sqrt(0),sqrt(0))),
inference(transitivity,[status(thm)],[38,36]) ).
tff(40,plain,
( ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
<=> ( $sum($product(-1,sqrt(0)),$product(sqrt(0),sqrt(0))) = 0 ) ),
inference(monotonicity,[status(thm)],[39]) ).
tff(41,plain,
( ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
<=> ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(transitivity,[status(thm)],[40,35]) ).
tff(42,plain,
( ( ~ $greatereq(0,0)
| ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
| ~ $greatereq(0,0) )
<=> ( $false
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 )
| $false ) ),
inference(monotonicity,[status(thm)],[34,41,34]) ).
tff(43,plain,
( ( ~ $greatereq(0,0)
| ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
| ~ $greatereq(0,0) )
<=> ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(transitivity,[status(thm)],[42,30]) ).
tff(44,plain,
( ( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ~ $greatereq(0,0)
| ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
| ~ $greatereq(0,0) )
<=> ( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ) ),
inference(monotonicity,[status(thm)],[43]) ).
tff(45,plain,
( ( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ~ $greatereq(0,0)
| ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
| ~ $greatereq(0,0) )
<=> ( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ) ),
inference(transitivity,[status(thm)],[44,29]) ).
tff(46,plain,
( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ~ $greatereq(0,0)
| ( $sum($product(sqrt(0),sqrt(0)),$product(-1,sqrt($product(0,0)))) = 0 )
| ~ $greatereq(0,0) ),
inference(quant_inst,[status(thm)],]) ).
tff(47,plain,
( ~ ! [X: $real,Y: $real] :
( ~ $greatereq(Y,0)
| ( $sum($product(sqrt(Y),sqrt(X)),$product(-1,sqrt($product(Y,X)))) = 0 )
| ~ $greatereq(X,0) )
| ( $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[46,45]) ).
tff(48,plain,
$sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) = 0,
inference(unit_resolution,[status(thm)],[47,28]) ).
tff(49,plain,
0 = $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))),
inference(modus_ponens,[status(thm)],[48,9]) ).
tff(50,plain,
( ( 0 != $sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))) )
| $lesseq($sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(51,plain,
$lesseq($sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))),0),
inference(unit_resolution,[status(thm)],[50,49]) ).
tff(52,plain,
( ( 0 = $product(sqrt(0),sqrt(0)) )
<=> ( $product(sqrt(0),sqrt(0)) = 0 ) ),
inference(commutativity,[status(thm)],]) ).
tff(53,plain,
( ( $product(sqrt(0),sqrt(0)) = 0 )
<=> ( 0 = $product(sqrt(0),sqrt(0)) ) ),
inference(symmetry,[status(thm)],[52]) ).
tff(54,plain,
^ [X: $real] :
refl(
( ( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
<=> ( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(55,plain,
( ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
<=> ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[54]) ).
tff(56,plain,
^ [X: $real] :
rewrite(
( ( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,sqr(sqrt(X)))) = 0 ) )
<=> ( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(57,plain,
( ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,sqr(sqrt(X)))) = 0 ) )
<=> ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[56]) ).
tff(58,plain,
^ [X: $real] :
rewrite(
( ( ~ $greatereq(X,0)
| ( $sum(sqr(sqrt(X)),$product(-1,X)) = 0 ) )
<=> ( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,sqr(sqrt(X)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(59,plain,
( ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(sqr(sqrt(X)),$product(-1,X)) = 0 ) )
<=> ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,sqr(sqrt(X)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[58]) ).
tff(60,plain,
^ [X: $real] :
rewrite(
( ( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) )
<=> ( ~ $greatereq(X,0)
| ( $sum(sqr(sqrt(X)),$product(-1,X)) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(61,plain,
( ! [X: $real] :
( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) )
<=> ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(sqr(sqrt(X)),$product(-1,X)) = 0 ) ) ),
inference(quant_intro,[status(thm)],[60]) ).
tff(62,plain,
( ! [X: $real] :
( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) )
<=> ! [X: $real] :
( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(63,plain,
^ [X: $real] :
rewrite(
( ( $lesseq(0,X)
=> ( sqr(sqrt(X)) = X ) )
<=> ( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) ) )),
inference(bind,[status(th)],]) ).
tff(64,plain,
( ! [X: $real] :
( $lesseq(0,X)
=> ( sqr(sqrt(X)) = X ) )
<=> ! [X: $real] :
( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) ) ),
inference(quant_intro,[status(thm)],[63]) ).
tff(65,axiom,
! [X: $real] :
( $lesseq(0,X)
=> ( sqr(sqrt(X)) = X ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sqrt_square) ).
tff(66,plain,
! [X: $real] :
( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) ),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,plain,
! [X: $real] :
( ~ $lesseq(0,X)
| ( sqr(sqrt(X)) = X ) ),
inference(modus_ponens,[status(thm)],[66,62]) ).
tff(68,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(sqr(sqrt(X)),$product(-1,X)) = 0 ) ),
inference(modus_ponens,[status(thm)],[67,61]) ).
tff(69,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,sqr(sqrt(X)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[68,59]) ).
tff(70,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[69,57]) ).
tff(71,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ),
inference(skolemize,[status(sab)],[70]) ).
tff(72,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[71,55]) ).
tff(73,plain,
( ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ( $product(sqrt(0),sqrt(0)) = 0 ) )
<=> ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ( $product(sqrt(0),sqrt(0)) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(74,plain,
( ( $false
| ( $product(sqrt(0),sqrt(0)) = 0 ) )
<=> ( $product(sqrt(0),sqrt(0)) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(75,plain,
( ( $product(-1,$product(sqrt(0),sqrt(0))) = 0 )
<=> ( $product(sqrt(0),sqrt(0)) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(76,plain,
$sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = $product(-1,$product(sqrt(0),sqrt(0))),
inference(rewrite,[status(thm)],]) ).
tff(77,plain,
( ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 )
<=> ( $product(-1,$product(sqrt(0),sqrt(0))) = 0 ) ),
inference(monotonicity,[status(thm)],[76]) ).
tff(78,plain,
( ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 )
<=> ( $product(sqrt(0),sqrt(0)) = 0 ) ),
inference(transitivity,[status(thm)],[77,75]) ).
tff(79,plain,
( ( ~ $greatereq(0,0)
| ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) )
<=> ( $false
| ( $product(sqrt(0),sqrt(0)) = 0 ) ) ),
inference(monotonicity,[status(thm)],[34,78]) ).
tff(80,plain,
( ( ~ $greatereq(0,0)
| ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) )
<=> ( $product(sqrt(0),sqrt(0)) = 0 ) ),
inference(transitivity,[status(thm)],[79,74]) ).
tff(81,plain,
( ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ~ $greatereq(0,0)
| ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) )
<=> ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ( $product(sqrt(0),sqrt(0)) = 0 ) ) ),
inference(monotonicity,[status(thm)],[80]) ).
tff(82,plain,
( ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ~ $greatereq(0,0)
| ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) )
<=> ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ( $product(sqrt(0),sqrt(0)) = 0 ) ) ),
inference(transitivity,[status(thm)],[81,73]) ).
tff(83,plain,
( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ~ $greatereq(0,0)
| ( $sum(0,$product(-1,$product(sqrt(0),sqrt(0)))) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(84,plain,
( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| ( $sum(X,$product(-1,$product(sqrt(X),sqrt(X)))) = 0 ) )
| ( $product(sqrt(0),sqrt(0)) = 0 ) ),
inference(modus_ponens,[status(thm)],[83,82]) ).
tff(85,plain,
$product(sqrt(0),sqrt(0)) = 0,
inference(unit_resolution,[status(thm)],[84,72]) ).
tff(86,plain,
0 = $product(sqrt(0),sqrt(0)),
inference(modus_ponens,[status(thm)],[85,53]) ).
tff(87,plain,
( ( 0 != $product(sqrt(0),sqrt(0)) )
| $lesseq($product(sqrt(0),sqrt(0)),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(88,plain,
$lesseq($product(sqrt(0),sqrt(0)),0),
inference(unit_resolution,[status(thm)],[87,86]) ).
tff(89,plain,
( ~ $lesseq($product(sqrt(0),sqrt(0)),0)
| ~ $lesseq($sum(sqrt(0),$product(-1,$product(sqrt(0),sqrt(0)))),0)
| $lesseq(sqrt(0),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(90,plain,
$lesseq(sqrt(0),0),
inference(unit_resolution,[status(thm)],[89,88,51]) ).
tff(91,plain,
^ [X: $real] :
refl(
( ( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
<=> ( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ) )),
inference(bind,[status(th)],]) ).
tff(92,plain,
( ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
<=> ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ) ),
inference(quant_intro,[status(thm)],[91]) ).
tff(93,plain,
^ [X: $real] :
rewrite(
( ( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) )
<=> ( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ) )),
inference(bind,[status(th)],]) ).
tff(94,plain,
( ! [X: $real] :
( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) )
<=> ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ) ),
inference(quant_intro,[status(thm)],[93]) ).
tff(95,plain,
( ! [X: $real] :
( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) )
<=> ! [X: $real] :
( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(96,plain,
^ [X: $real] :
rewrite(
( ( $lesseq(0,X)
=> $lesseq(0,sqrt(X)) )
<=> ( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) ) )),
inference(bind,[status(th)],]) ).
tff(97,plain,
( ! [X: $real] :
( $lesseq(0,X)
=> $lesseq(0,sqrt(X)) )
<=> ! [X: $real] :
( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) ) ),
inference(quant_intro,[status(thm)],[96]) ).
tff(98,axiom,
! [X: $real] :
( $lesseq(0,X)
=> $lesseq(0,sqrt(X)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sqrt_positive) ).
tff(99,plain,
! [X: $real] :
( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) ),
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
! [X: $real] :
( ~ $lesseq(0,X)
| $lesseq(0,sqrt(X)) ),
inference(modus_ponens,[status(thm)],[99,95]) ).
tff(101,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ),
inference(modus_ponens,[status(thm)],[100,94]) ).
tff(102,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ),
inference(skolemize,[status(sab)],[101]) ).
tff(103,plain,
! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) ),
inference(modus_ponens,[status(thm)],[102,92]) ).
tff(104,plain,
( ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| $greatereq(sqrt(0),0) )
<=> ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| $greatereq(sqrt(0),0) ) ),
inference(rewrite,[status(thm)],]) ).
tff(105,plain,
( ( $false
| $greatereq(sqrt(0),0) )
<=> $greatereq(sqrt(0),0) ),
inference(rewrite,[status(thm)],]) ).
tff(106,plain,
( ( ~ $greatereq(0,0)
| $greatereq(sqrt(0),0) )
<=> ( $false
| $greatereq(sqrt(0),0) ) ),
inference(monotonicity,[status(thm)],[34]) ).
tff(107,plain,
( ( ~ $greatereq(0,0)
| $greatereq(sqrt(0),0) )
<=> $greatereq(sqrt(0),0) ),
inference(transitivity,[status(thm)],[106,105]) ).
tff(108,plain,
( ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| ~ $greatereq(0,0)
| $greatereq(sqrt(0),0) )
<=> ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| $greatereq(sqrt(0),0) ) ),
inference(monotonicity,[status(thm)],[107]) ).
tff(109,plain,
( ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| ~ $greatereq(0,0)
| $greatereq(sqrt(0),0) )
<=> ( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| $greatereq(sqrt(0),0) ) ),
inference(transitivity,[status(thm)],[108,104]) ).
tff(110,plain,
( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| ~ $greatereq(0,0)
| $greatereq(sqrt(0),0) ),
inference(quant_inst,[status(thm)],]) ).
tff(111,plain,
( ~ ! [X: $real] :
( ~ $greatereq(X,0)
| $greatereq(sqrt(X),0) )
| $greatereq(sqrt(0),0) ),
inference(modus_ponens,[status(thm)],[110,109]) ).
tff(112,plain,
$greatereq(sqrt(0),0),
inference(unit_resolution,[status(thm)],[111,103]) ).
tff(113,plain,
( ( 0 = sqrt(0) )
| ~ $lesseq(sqrt(0),0)
| ~ $greatereq(sqrt(0),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(114,plain,
( ( 0 = sqrt(0) )
| ~ $lesseq(sqrt(0),0) ),
inference(unit_resolution,[status(thm)],[113,112]) ).
tff(115,plain,
$false,
inference(unit_resolution,[status(thm)],[114,90,7]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : ARI742_1 : TPTP v8.1.0. Released v7.0.0.
% 0.10/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.32 % Computer : n009.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Tue Aug 30 02:05:50 EDT 2022
% 0.12/0.32 % CPUTime :
% 0.12/0.33 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.33 Usage: tptp [options] [-file:]file
% 0.12/0.33 -h, -? prints this message.
% 0.12/0.33 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.33 -m, -model generate model.
% 0.12/0.33 -p, -proof generate proof.
% 0.12/0.33 -c, -core generate unsat core of named formulas.
% 0.12/0.33 -st, -statistics display statistics.
% 0.12/0.33 -t:timeout set timeout (in second).
% 0.12/0.33 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.33 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.33 -<param>:<value> configuration parameter and value.
% 0.12/0.33 -o:<output-file> file to place output in.
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------