TSTP Solution File: GRP715-11 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP715-11 : TPTP v8.1.0. Released v8.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n018.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 : Fri Sep 16 22:29:37 EDT 2022
% Result : Unsatisfiable 0.20s 0.48s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 34
% Syntax : Number of formulae : 99 ( 71 unt; 5 typ; 0 def)
% Number of atoms : 129 ( 122 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 43 ( 14 ~; 10 |; 0 &)
% ( 19 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 6 ( 6 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 2 ( 1 >; 1 *; 0 +; 0 <<)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 4 con; 0-2 aty)
% Number of variables : 88 ( 79 !; 0 ?; 88 :)
% Comments :
%------------------------------------------------------------------------------
tff(x0_type,type,
x0: $i ).
tff(mult_type,type,
mult: ( $i * $i ) > $i ).
tff(op_a_type,type,
op_a: $i ).
tff(op_b_type,type,
op_b: $i ).
tff(unit_type,type,
unit: $i ).
tff(1,plain,
^ [A: $i] :
refl(
( ( mult(A,mult(op_a,op_b)) = A )
<=> ( mult(A,mult(op_a,op_b)) = A ) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [A: $i] : ( mult(A,mult(op_a,op_b)) = A )
<=> ! [A: $i] : ( mult(A,mult(op_a,op_b)) = A ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
^ [A: $i] :
rewrite(
( ( mult(A,unit) = A )
<=> ( mult(A,mult(op_a,op_b)) = A ) )),
inference(bind,[status(th)],]) ).
tff(4,plain,
( ! [A: $i] : ( mult(A,unit) = A )
<=> ! [A: $i] : ( mult(A,mult(op_a,op_b)) = A ) ),
inference(quant_intro,[status(thm)],[3]) ).
tff(5,plain,
( ! [A: $i] : ( mult(A,unit) = A )
<=> ! [A: $i] : ( mult(A,unit) = A ) ),
inference(rewrite,[status(thm)],]) ).
tff(6,axiom,
! [A: $i] : ( mult(A,unit) = A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f08) ).
tff(7,plain,
! [A: $i] : ( mult(A,unit) = A ),
inference(modus_ponens,[status(thm)],[6,5]) ).
tff(8,plain,
! [A: $i] : ( mult(A,mult(op_a,op_b)) = A ),
inference(modus_ponens,[status(thm)],[7,4]) ).
tff(9,plain,
! [A: $i] : ( mult(A,mult(op_a,op_b)) = A ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [A: $i] : ( mult(A,mult(op_a,op_b)) = A ),
inference(modus_ponens,[status(thm)],[9,2]) ).
tff(11,plain,
( ~ ! [A: $i] : ( mult(A,mult(op_a,op_b)) = A )
| ( mult(x0,mult(op_a,op_b)) = x0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(12,plain,
mult(x0,mult(op_a,op_b)) = x0,
inference(unit_resolution,[status(thm)],[11,10]) ).
tff(13,plain,
^ [A: $i] :
refl(
( ( mult(mult(op_a,op_b),A) = A )
<=> ( mult(mult(op_a,op_b),A) = A ) )),
inference(bind,[status(th)],]) ).
tff(14,plain,
( ! [A: $i] : ( mult(mult(op_a,op_b),A) = A )
<=> ! [A: $i] : ( mult(mult(op_a,op_b),A) = A ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
^ [A: $i] :
rewrite(
( ( mult(unit,A) = A )
<=> ( mult(mult(op_a,op_b),A) = A ) )),
inference(bind,[status(th)],]) ).
tff(16,plain,
( ! [A: $i] : ( mult(unit,A) = A )
<=> ! [A: $i] : ( mult(mult(op_a,op_b),A) = A ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [A: $i] : ( mult(unit,A) = A )
<=> ! [A: $i] : ( mult(unit,A) = A ) ),
inference(rewrite,[status(thm)],]) ).
tff(18,axiom,
! [A: $i] : ( mult(unit,A) = A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f09) ).
tff(19,plain,
! [A: $i] : ( mult(unit,A) = A ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
! [A: $i] : ( mult(mult(op_a,op_b),A) = A ),
inference(modus_ponens,[status(thm)],[19,16]) ).
tff(21,plain,
! [A: $i] : ( mult(mult(op_a,op_b),A) = A ),
inference(skolemize,[status(sab)],[20]) ).
tff(22,plain,
! [A: $i] : ( mult(mult(op_a,op_b),A) = A ),
inference(modus_ponens,[status(thm)],[21,14]) ).
tff(23,plain,
( ~ ! [A: $i] : ( mult(mult(op_a,op_b),A) = A )
| ( mult(mult(op_a,op_b),mult(op_a,op_b)) = mult(op_a,op_b) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(24,plain,
mult(mult(op_a,op_b),mult(op_a,op_b)) = mult(op_a,op_b),
inference(unit_resolution,[status(thm)],[23,22]) ).
tff(25,plain,
mult(op_a,op_b) = mult(mult(op_a,op_b),mult(op_a,op_b)),
inference(symmetry,[status(thm)],[24]) ).
tff(26,plain,
mult(x0,mult(op_a,op_b)) = mult(x0,mult(mult(op_a,op_b),mult(op_a,op_b))),
inference(monotonicity,[status(thm)],[25]) ).
tff(27,plain,
mult(x0,mult(mult(op_a,op_b),mult(op_a,op_b))) = mult(x0,mult(op_a,op_b)),
inference(symmetry,[status(thm)],[26]) ).
tff(28,plain,
( ~ ! [A: $i] : ( mult(mult(op_a,op_b),A) = A )
| ( mult(mult(op_a,op_b),op_b) = op_b ) ),
inference(quant_inst,[status(thm)],]) ).
tff(29,plain,
mult(mult(op_a,op_b),op_b) = op_b,
inference(unit_resolution,[status(thm)],[28,22]) ).
tff(30,plain,
^ [B: $i,A: $i] :
refl(
( ( mult(A,mult(B,B)) = mult(mult(A,B),B) )
<=> ( mult(A,mult(B,B)) = mult(mult(A,B),B) ) )),
inference(bind,[status(th)],]) ).
tff(31,plain,
( ! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) )
<=> ! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) )
<=> ! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) ) ),
inference(rewrite,[status(thm)],]) ).
tff(33,axiom,
! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f07) ).
tff(34,plain,
! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ~ ! [B: $i,A: $i] : ( mult(A,mult(B,B)) = mult(mult(A,B),B) )
| ( mult(op_a,mult(op_b,op_b)) = mult(mult(op_a,op_b),op_b) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(38,plain,
mult(op_a,mult(op_b,op_b)) = mult(mult(op_a,op_b),op_b),
inference(unit_resolution,[status(thm)],[37,36]) ).
tff(39,plain,
mult(op_a,mult(op_b,op_b)) = op_b,
inference(transitivity,[status(thm)],[38,29]) ).
tff(40,plain,
mult(mult(op_a,mult(op_b,op_b)),op_a) = mult(op_b,op_a),
inference(monotonicity,[status(thm)],[39]) ).
tff(41,plain,
mult(op_b,op_a) = mult(mult(op_a,mult(op_b,op_b)),op_a),
inference(symmetry,[status(thm)],[40]) ).
tff(42,plain,
( ( mult(op_b,op_a) = unit )
<=> ( mult(op_b,op_a) = mult(op_a,op_b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(43,plain,
( ( mult(op_b,op_a) = unit )
<=> ( mult(op_b,op_a) = unit ) ),
inference(rewrite,[status(thm)],]) ).
tff(44,axiom,
mult(op_b,op_a) = unit,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f11) ).
tff(45,plain,
mult(op_b,op_a) = unit,
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
mult(op_b,op_a) = mult(op_a,op_b),
inference(modus_ponens,[status(thm)],[45,42]) ).
tff(47,plain,
mult(op_a,op_b) = mult(op_b,op_a),
inference(symmetry,[status(thm)],[46]) ).
tff(48,plain,
mult(mult(op_a,op_b),mult(op_a,op_b)) = mult(mult(op_a,mult(op_b,op_b)),op_a),
inference(transitivity,[status(thm)],[24,47,41]) ).
tff(49,plain,
mult(x0,mult(mult(op_a,op_b),mult(op_a,op_b))) = mult(x0,mult(mult(op_a,mult(op_b,op_b)),op_a)),
inference(monotonicity,[status(thm)],[48]) ).
tff(50,plain,
mult(x0,mult(mult(op_a,mult(op_b,op_b)),op_a)) = mult(x0,mult(mult(op_a,op_b),mult(op_a,op_b))),
inference(symmetry,[status(thm)],[49]) ).
tff(51,plain,
^ [B: $i,A: $i,C: $i] :
refl(
( ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) )
<=> ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ) )),
inference(bind,[status(th)],]) ).
tff(52,plain,
( ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) )
<=> ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ) ),
inference(quant_intro,[status(thm)],[51]) ).
tff(53,plain,
( ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) )
<=> ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(54,axiom,
! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f06) ).
tff(55,plain,
! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ),
inference(skolemize,[status(sab)],[55]) ).
tff(57,plain,
! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) ),
inference(modus_ponens,[status(thm)],[56,52]) ).
tff(58,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) )
| ( mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a) = mult(x0,mult(mult(op_a,mult(op_b,op_b)),op_a)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(59,plain,
mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a) = mult(x0,mult(mult(op_a,mult(op_b,op_b)),op_a)),
inference(unit_resolution,[status(thm)],[58,57]) ).
tff(60,plain,
mult(mult(mult(op_a,op_b),mult(op_a,op_b)),mult(mult(op_a,op_b),mult(op_a,op_b))) = mult(mult(op_a,op_b),mult(op_a,op_b)),
inference(monotonicity,[status(thm)],[24,24]) ).
tff(61,plain,
( ~ ! [A: $i] : ( mult(A,mult(op_a,op_b)) = A )
| ( mult(op_a,mult(op_a,op_b)) = op_a ) ),
inference(quant_inst,[status(thm)],]) ).
tff(62,plain,
mult(op_a,mult(op_a,op_b)) = op_a,
inference(unit_resolution,[status(thm)],[61,10]) ).
tff(63,plain,
mult(mult(op_a,mult(op_a,op_b)),op_b) = mult(op_a,op_b),
inference(monotonicity,[status(thm)],[62]) ).
tff(64,plain,
mult(mult(op_a,mult(op_a,op_b)),op_b) = mult(mult(op_a,op_b),mult(op_a,op_b)),
inference(transitivity,[status(thm)],[63,25]) ).
tff(65,plain,
mult(mult(mult(op_a,mult(op_a,op_b)),op_b),mult(op_a,op_b)) = mult(mult(mult(op_a,op_b),mult(op_a,op_b)),mult(mult(op_a,op_b),mult(op_a,op_b))),
inference(monotonicity,[status(thm)],[64,25]) ).
tff(66,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) )
| ( mult(mult(mult(op_a,mult(op_a,op_b)),op_b),mult(op_a,op_b)) = mult(op_a,mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(67,plain,
mult(mult(mult(op_a,mult(op_a,op_b)),op_b),mult(op_a,op_b)) = mult(op_a,mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b))),
inference(unit_resolution,[status(thm)],[66,57]) ).
tff(68,plain,
mult(op_a,mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b))) = mult(mult(mult(op_a,mult(op_a,op_b)),op_b),mult(op_a,op_b)),
inference(symmetry,[status(thm)],[67]) ).
tff(69,plain,
( ~ ! [A: $i] : ( mult(A,mult(op_a,op_b)) = A )
| ( mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b)) = mult(mult(op_a,op_b),op_b) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(70,plain,
mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b)) = mult(mult(op_a,op_b),op_b),
inference(unit_resolution,[status(thm)],[69,10]) ).
tff(71,plain,
mult(mult(op_a,op_b),op_b) = mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b)),
inference(symmetry,[status(thm)],[70]) ).
tff(72,plain,
mult(op_a,mult(mult(op_a,op_b),op_b)) = mult(op_a,mult(mult(mult(op_a,op_b),op_b),mult(op_a,op_b))),
inference(monotonicity,[status(thm)],[71]) ).
tff(73,plain,
mult(op_a,mult(mult(op_a,op_b),op_b)) = mult(mult(op_a,op_b),mult(op_a,op_b)),
inference(transitivity,[status(thm)],[72,68,65,60]) ).
tff(74,plain,
mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a) = mult(mult(mult(op_a,op_b),mult(op_a,op_b)),op_a),
inference(monotonicity,[status(thm)],[73]) ).
tff(75,plain,
mult(mult(mult(op_a,op_b),mult(op_a,op_b)),op_a) = mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a),
inference(symmetry,[status(thm)],[74]) ).
tff(76,plain,
mult(mult(mult(op_a,op_b),mult(op_a,op_b)),op_a) = mult(mult(op_a,op_b),op_a),
inference(monotonicity,[status(thm)],[24]) ).
tff(77,plain,
mult(mult(op_a,op_b),op_a) = mult(mult(mult(op_a,op_b),mult(op_a,op_b)),op_a),
inference(symmetry,[status(thm)],[76]) ).
tff(78,plain,
( ~ ! [A: $i] : ( mult(mult(op_a,op_b),A) = A )
| ( mult(mult(op_a,op_b),op_a) = op_a ) ),
inference(quant_inst,[status(thm)],]) ).
tff(79,plain,
mult(mult(op_a,op_b),op_a) = op_a,
inference(unit_resolution,[status(thm)],[78,22]) ).
tff(80,plain,
op_a = mult(mult(op_a,op_b),op_a),
inference(symmetry,[status(thm)],[79]) ).
tff(81,plain,
op_a = mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a),
inference(transitivity,[status(thm)],[80,77,75]) ).
tff(82,plain,
mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a) = mult(mult(mult(x0,op_a),mult(op_b,op_b)),mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a)),
inference(monotonicity,[status(thm)],[81]) ).
tff(83,plain,
mult(mult(mult(x0,op_a),mult(op_b,op_b)),mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a)) = mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a),
inference(symmetry,[status(thm)],[82]) ).
tff(84,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( mult(mult(mult(A,B),C),B) = mult(A,mult(mult(B,C),B)) )
| ( mult(mult(mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a),mult(mult(op_a,op_b),op_b)),op_a) = mult(mult(mult(x0,op_a),mult(op_b,op_b)),mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(85,plain,
mult(mult(mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a),mult(mult(op_a,op_b),op_b)),op_a) = mult(mult(mult(x0,op_a),mult(op_b,op_b)),mult(mult(op_a,mult(mult(op_a,op_b),op_b)),op_a)),
inference(unit_resolution,[status(thm)],[84,57]) ).
tff(86,plain,
mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a) = x0,
inference(transitivity,[status(thm)],[59,50,27,12]) ).
tff(87,plain,
mult(mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a),mult(mult(op_a,op_b),op_b)) = mult(x0,op_b),
inference(monotonicity,[status(thm)],[86,29]) ).
tff(88,plain,
mult(mult(mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a),mult(mult(op_a,op_b),op_b)),op_a) = mult(mult(x0,op_b),op_a),
inference(monotonicity,[status(thm)],[87]) ).
tff(89,plain,
mult(mult(x0,op_b),op_a) = mult(mult(mult(mult(mult(x0,op_a),mult(op_b,op_b)),op_a),mult(mult(op_a,op_b),op_b)),op_a),
inference(symmetry,[status(thm)],[88]) ).
tff(90,plain,
mult(mult(x0,op_b),op_a) = x0,
inference(transitivity,[status(thm)],[89,85,83,59,50,27,12]) ).
tff(91,plain,
( ( mult(mult(x0,op_b),op_a) != x0 )
<=> ( mult(mult(x0,op_b),op_a) != x0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(92,axiom,
mult(mult(x0,op_b),op_a) != x0,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',goal) ).
tff(93,plain,
mult(mult(x0,op_b),op_a) != x0,
inference(modus_ponens,[status(thm)],[92,91]) ).
tff(94,plain,
$false,
inference(unit_resolution,[status(thm)],[93,90]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GRP715-11 : TPTP v8.1.0. Released v8.1.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n018.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 31 20:27:41 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.35 Usage: tptp [options] [-file:]file
% 0.12/0.35 -h, -? prints this message.
% 0.12/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.35 -m, -model generate model.
% 0.12/0.35 -p, -proof generate proof.
% 0.12/0.35 -c, -core generate unsat core of named formulas.
% 0.12/0.35 -st, -statistics display statistics.
% 0.12/0.35 -t:timeout set timeout (in second).
% 0.12/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.35 -<param>:<value> configuration parameter and value.
% 0.12/0.35 -o:<output-file> file to place output in.
% 0.20/0.48 % SZS status Unsatisfiable
% 0.20/0.48 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------